Introduction
This chapter continues the study of formal algebraic geometry and in particular the question of whether a formal object is the completion of an algebraic one. A fundamental reference is [SGA2]. Here is a list of results we have already discussed in the Stacks project:
The theorem on formal functions, see Cohomology of Schemes, Section 02O7.
Coherent formal modules, see Cohomology of Schemes, Section 0EHN.
Grothendieck’s existence theorem, see Cohomology of Schemes, Sections 087V, 0886, and 0CYW.
Grothendieck’s algebraization theorem, see Cohomology of Schemes, Section 0898.
Grothendieck’s existence theorem more generally, see More on Flatness, Sections 0CTB and 0DIA.
Let us give an overview of the contents of this chapter.
Let \(X\) be a scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a finite type quasi-coherent sheaf of ideals. Many questions in this chapter have to do with inverse systems \((\mathcal{F}_n)\) of quasi-coherent \(\mathcal{O}_X\)-modules such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/\mathcal{I}^n\mathcal{F}_{n + 1}\). An important special case is where \(X\) is a scheme over a Noetherian ring \(A\) and \(\mathcal{I} = I \mathcal{O}_X\) for some ideal \(I \subset A\). In Cohomology, Sections 0GYJ, 0H38, and 0H3B we have some general results. In this chapter, Sections 0EH3 and 0BLA contain results specific to schemes and quasi-coherent modules. In Section 0EI9 we prove that the limit topology on \(\lim H^p(X, \mathcal{F}_n)\) is \(I\)-adic in case \(\text{cd}(A, I) = 1\). One of the themes of this chapter will be to show that results proven in the principal ideal case \(I = (f)\) also hold when we only assume \(\text{cd}(A, I) = 1\).
In Section 0995 we discuss derived completion of modules on a ringed site \((\mathcal{C}, \mathcal{O})\) with respect to a finite type sheaf of ideals \(\mathcal{I}\). This section is the natural continuation of the theory of derived completion in commutative algebra as described in More on Algebra, Section 091N. The first main result is that derived completion exists. The second main result is that for a morphism \(f\) of ringed sites derived completion commutes with derived pushforward: \[(Rf_*K)^\wedge = Rf_*(K^\wedge)\] if the ideal sheaf upstairs is locally generated by sections coming from the ideal downstairs, see Lemma 0A0G. We stress that both main results are very elementary in case the ideals in question are globally finitely generated which will be true for all applications of this theory in this chapter. The displayed equality is the “correct” version of the theorem on formal functions, see discussion in Section 0A0H.
Let \(A\) be a Noetherian ring and let \(I, J\) be two ideals of \(A\). Let \(M\) be a finite \(A\)-module. The next topic in this chapter is the map \[R\Gamma_J(M) \longrightarrow R\Gamma_J(M)^\wedge\] from local cohomology of \(M\) into the derived \(I\)-adic completion of the same. It turns out that if we impose suitable depth conditions this map becomes an isomorphism on cohomology in a range of degrees. In Section 0EFF we work essentially in the generality just mentioned. In Section 0EFP we assume \(A\) is a local ring and \(J = \mathfrak m\) is a maximal ideal. We encourage the reader to read this section before the other two in this part of the chapter. Finally, in Section 0EFT we bootstrap the local case to obtain stronger results back in the general case.
In the next part of this chapter we use the results on completion of local cohomology to get a nonexhaustive list of results on cohomology of the completion of coherent modules. More precisely, let \(A\) be a Noetherian ring, let \(I \subset A\) be an ideal, and let \(U \subset \Spec(A)\) be an open subscheme. If \(\mathcal{F}\) is a coherent \(\mathcal{O}_U\)-module, then we may consider the maps \[H^i(U, \mathcal{F}) \longrightarrow \lim H^i(U, \mathcal{F}/I^n\mathcal{F})\] and ask if we get an isomorphism in a certain range of degrees. In Section 0DXH we work out some examples where \(U\) is the punctured spectrum of a local ring. In Section 0EG1 we discuss the general case. In Section 0ECQ we apply some of the results obtained to questions of connectedness in algebraic geometry.
The remaining sections of this chapter are devoted to a discussion of algebraization of coherent formal modules. In other words, given an inverse system of coherent modules \((\mathcal{F}_n)\) on \(U\) as above with \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\) we ask whether there exists a coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\) such that \(\mathcal{F}_n = \mathcal{F}/I^n\mathcal{F}\) for all \(n\). We encourage the reader to read Section 0DXS for a precise statement of the question, a useful general result (Lemma 0EHH), and a nontrivial application (Lemma 0DXW). To prove a result going essentially beyond this case quite a bit more theory has to be developed. Please see Section 0EJH for the strongest results of this type obtained in this chapter.
Formal sections, I
We suggest looking at Cohomology, Section 0GYJ first.
Lemma
Let \(X\) be a scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of quasi-coherent \(\mathcal{O}_X\)-modules such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/\mathcal{I}^n\mathcal{F}_{n + 1}\). Set \(\mathcal{F} = \lim \mathcal{F}_n\). Then
\(\mathcal{F} = R\lim \mathcal{F}_n\),
for any affine open \(U \subset X\) we have \(H^p(U, \mathcal{F}) = 0\) for \(p > 0\), and
for each \(p\) there is a short exact sequence \(0 \to R^1\lim H^{p - 1}(X, \mathcal{F}_n) \to H^p(X, \mathcal{F}) \to \lim H^p(X, \mathcal{F}_n) \to 0\).
If moreover \(\mathcal{I}\) is of finite type, then
\(\mathcal{F}_n = \mathcal{F}/\mathcal{I}^n\mathcal{F}\), and
\(\mathcal{I}^n \mathcal{F} = \lim_{m \geq n} \mathcal{I}^n\mathcal{F}_m\).
Proof
Parts (1), (2), and (3) are general facts about inverse systems of quasi-coherent modules with surjective transition maps, see Derived Categories of Schemes, Lemma 0A0J and Cohomology, Lemma 0D60. Next, assume \(\mathcal{I}\) is of finite type. Let \(U \subset X\) be affine open. Say \(U = \Spec(A)\) and \(\mathcal{I}|_U\) corresponds to \(I \subset A\). Observe that \(I\) is a finitely generated ideal. By the equivalence of categories between quasi-coherent \(\mathcal{O}_U\)-modules and \(A\)-modules (Schemes, Lemma 01IB) we find that \(M_n = \mathcal{F}_n(U)\) is an inverse system of \(A\)-modules with \(M_n = M_{n + 1}/I^nM_{n + 1}\). Thus \[M = \mathcal{F}(U) = \lim \mathcal{F}_n(U) = \lim M_n\] is an \(I\)-adically complete module with \(M/I^nM = M_n\) by Algebra, Lemma 09B8. This proves (4). Part (5) translates into the statement that \(\lim_{m \geq n} I^nM/I^mM = I^nM\). Since \(I^mM = I^{m - n} \cdot I^nM\) this is just the statement that \(I^mM\) is \(I\)-adically complete. This follows from Algebra, Lemma 05GG and the fact that \(M\) is complete.
Formal sections, II
We suggest looking at Cohomology, Sections 0H38 and 0H3B first.
Lemma
Let \(X\) be a scheme. Let \(f \in \Gamma(X, \mathcal{O}_X)\). Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of quasi-coherent \(\mathcal{O}_X\)-modules. The following are equivalent
for all \(n \geq 1\) the map \(f : \mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1}\) factors through \(\mathcal{F}_{n + 1} \to \mathcal{F}_n\) to give a short exact sequence \(0 \to \mathcal{F}_n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0\),
for all \(n \geq 1\) the map \(f^n : \mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1}\) factors through \(\mathcal{F}_{n + 1} \to \mathcal{F}_1\) to give a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_{n + 1} \to \mathcal{F}_n \to 0\)
there exists an \(\mathcal{O}_X\)-module \(\mathcal{G}\) which is \(f\)-divisible such that \(\mathcal{F}_n = \mathcal{G}[f^n]\).
there exists an \(\mathcal{O}_X\)-module \(\mathcal{F}\) which is \(f\)-torsion free such that \(\mathcal{F}_n = \mathcal{F}/f^n\mathcal{F}\).
Proof
The equivalence of (1), (2), (3) and the implication (4) \(\Rightarrow\) (1) are proven in Cohomology, Lemma 0H39. Assume (1) holds. Set \(\mathcal{F} = \lim \mathcal{F}_n\). By Lemma 0EI8 part (4) we have \(\mathcal{F}_n = \mathcal{F}/f^n\mathcal{F}\). Let \(U \subset X\) be open and \(s = (s_n) \in \mathcal{F}(U) = \lim \mathcal{F}_n(U)\). Choose \(n \geq 1\). If \(fs = 0\), then \(s_{n + 1}\) is in the kernel of \(\mathcal{F}_{n + 1} \to \mathcal{F}_n\) by condition (1). Hence \(s_n = 0\). Since \(n\) was arbitrary, we see \(s = 0\). Thus \(\mathcal{F}\) is \(f\)-torsion free.
Lemma
Let \(A\) be a ring and \(f \in A\). Let \(X\) be a scheme over \(A\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume that \(\mathcal{F}[f^n] = \Ker(f^n : \mathcal{F} \to \mathcal{F})\) stabilizes. Then \[R\Gamma(X, \lim \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, \mathcal{F})^\wedge\] where the right hand side indicates the derived completion with respect to the ideal \((f) \subset A\). Consequently, for \(p \in \mathbf{Z}\) we obtain a commutative diagram \[\xymatrix{ & 0 & 0 \\ 0 \ar[r] & \widehat{H^p(X, \mathcal{F})} \ar[r] \ar[u] & \lim H^p(X, \mathcal{F}/f^n\mathcal{F}) \ar[r] \ar[u] & T_f(H^{p + 1}(X, \mathcal{F})) \ar[r] & 0 \\ 0 \ar[r] & H^0(H^p(X, \mathcal{F})^\wedge) \ar[r] \ar[u] & H^p(X, \lim \mathcal{F}/f^n\mathcal{F}) \ar[r] \ar[u] & T_f(H^{p + 1}(X, \mathcal{F})) \ar[r] \ar@{=}[u] & 0 \\ & R^1\lim H^p(X, \mathcal{F})[f^n] \ar[u] \ar[r]^\cong & R^1\lim H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \ar[u] \\ & 0 \ar[u] & 0 \ar[u] }\] with exact rows and columns where \(\widehat{H^p(X, \mathcal{F})} = \lim H^p(X, \mathcal{F})/f^n H^p(X, \mathcal{F})\) is the usual \(f\)-adic completion and \(T_f(-)\) denotes the \(f\)-adic Tate module as in More on Algebra, Example 0BKG.
Proof
By Lemma 0EI8 we have \(\lim \mathcal{F}/f^n\mathcal{F} = R\lim \mathcal{F}/f^n \mathcal{F}\). Everything else follows from Cohomology, Example 0H3E.
Formal sections, III
In this section we prove Lemma 0EH7 which (in the setting of Noetherian schemes and coherent modules) is the analogue of Cohomology, Lemma 0EHA in case the ideal \(I\) is not assumed principal but has the property that \(\text{cd}(A, I) = 1\).
Lemma
Let \(I = (f_1, \ldots, f_r)\) be an ideal of a Noetherian ring \(A\). If \(\text{cd}(A, I) = 1\), then there exist \(c \geq 1\) and maps \(\varphi_j : I^c \to A\) such that \(\sum f_j \varphi_j : I^c \to I\) is the inclusion map.
Proof
Since \(\text{cd}(A, I) = 1\) the complement \(U = \Spec(A) \setminus V(I)\) is affine (Local Cohomology, Lemma 0DXD). Say \(U = \Spec(B)\). Then \(IB = B\) and we can write \(1 = \sum_{j = 1, \ldots, r} f_j b_j\) for some \(b_j \in B\). By Cohomology of Schemes, Lemma 01YB we can represent \(b_j\) by maps \(\varphi_j : I^c \to A\) for some \(c \geq 0\). Then \(\sum f_j \varphi_j : I^c \to I \subset A\) is the canonical embedding, after possibly replacing \(c\) by a larger integer, by the same lemma.
Lemma
Let \(I = (f_1, \ldots, f_r)\) be an ideal of a Noetherian ring \(A\) with \(\text{cd}(A, I) = 1\). Let \(c \geq 1\) and \(\varphi_j : I^c \to A\), \(j = 1, \ldots, r\) be as in Lemma 0EIA. Then there is a unique graded \(A\)-algebra map \[\Phi : \bigoplus\nolimits_{n \geq 0} I^{nc} \to A[T_1, \ldots, T_r]\] with \(\Phi(g) = \sum \varphi_j(g) T_j\) for \(g \in I^c\). Moreover, the composition of \(\Phi\) with the map \(A[T_1, \ldots, T_r] \to \bigoplus_{n \geq 0} I^n\), \(T_j \mapsto f_j\) is the inclusion map \(\bigoplus_{n \geq 0} I^{nc} \to \bigoplus_{n \geq 0} I^n\).
Proof
For each \(j\) and \(m \geq c\) the restriction of \(\varphi_j\) to \(I^m\) is a map \(\varphi_j : I^m \to I^{m - c}\). Given \(j_1, \ldots, j_n \in \{1, \ldots, r\}\) we claim that the composition \[\varphi_{j_1} \ldots \varphi_{j_n} : I^{nc} \to I^{(n - 1)c} \to \ldots \to I^c \to A\] is independent of the order of the indices \(j_1, \ldots, j_n\). Namely, if \(g = g_1 \ldots g_n\) with \(g_i \in I^c\), then we see that \[(\varphi_{j_1} \ldots \varphi_{j_n})(g) = \varphi_{j_1}(g_1) \ldots \varphi_{j_n}(g_n)\] is independent of the ordering as multiplication in \(A\) is commutative. Thus we can define \(\Phi\) by sending \(g \in I^{nc}\) to \[\Phi(g) = \sum\nolimits_{e_1 + \ldots + e_r = n} (\varphi_1^{e_1} \circ \ldots \circ \varphi_r^{e_r})(g) T_1^{e_1} \ldots T_r^{e_r}\] It is straightforward to prove that this is a graded \(A\)-algebra homomorphism with the desired property. Uniqueness is immediate as is the final property. This proves the lemma.
Lemma
Let \(I = (f_1, \ldots, f_r)\) be an ideal of a Noetherian ring \(A\) with \(\text{cd}(A, I) = 1\). Let \(c \geq 1\) and \(\varphi_j : I^c \to A\), \(j = 1, \ldots, r\) be as in Lemma 0EIA. Let \(A \to B\) be a ring map with \(B\) Noetherian and let \(N\) be a finite \(B\)-module. Then, after possibly increasing \(c\) and adjusting \(\varphi_j\) accordingly, there is a unique unique graded \(B\)-module map \[\Phi_N : \bigoplus\nolimits_{n \geq 0} I^{nc}N \to N[T_1, \ldots, T_r]\] with \(\Phi_N(g x) = \Phi(g) x\) for \(g \in I^{nc}\) and \(x \in N\) where \(\Phi\) is as in Lemma 0EIB. The composition of \(\Phi_N\) with the map \(N[T_1, \ldots, T_r] \to \bigoplus_{n \geq 0} I^nN\), \(T_j \mapsto f_j\) is the inclusion map \(\bigoplus_{n \geq 0} I^{nc}N \to \bigoplus_{n \geq 0} I^nN\).
Proof
The uniqueness is clear from the formula and the uniqueness of \(\Phi\) in Lemma 0EIB. Consider the Noetherian \(A\)-algebra \(B' = B \oplus N\) where \(N\) is an ideal of square zero. To show the existence of \(\Phi_N\) it is enough (via Lemma 0EIA) to show that \(\varphi_j\) extends to a map \(\varphi'_j : I^cB' \to B'\) after possibly increasing \(c\) to some \(c'\) (and replacing \(\varphi_j\) by the composition of the inclusion \(I^{c'} \to I^c\) with \(\varphi_j\)). Recall that \(\varphi_j\) corresponds to a section \[h_j \in \Gamma(\Spec(A) \setminus V(I), \mathcal{O}_{\Spec(A)})\] see Cohomology of Schemes, Lemma 01YB. (This is in fact how we chose our \(\varphi_j\) in the proof of Lemma 0EIA.) Let us use the same lemma to represent the pullback \[h'_j \in \Gamma(\Spec(B') \setminus V(IB'), \mathcal{O}_{\Spec(B')})\] of \(h_j\) by a \(B'\)-linear map \(\varphi'_j : I^{c'}B' \to B'\) for some \(c' \geq c\). The agreement with \(\varphi_j\) will hold for \(c'\) sufficiently large by a further application of the lemma: namely we can test agreement on a finite list of generators of \(I^{c'}\). Small detail omitted.
Lemma
Let \(I = (f_1, \ldots, f_r)\) be an ideal of a Noetherian ring \(A\) with \(\text{cd}(A, I) = 1\). Let \(c \geq 1\) and \(\varphi_j : I^c \to A\), \(j = 1, \ldots, r\) be as in Lemma 0EIA. Let \(X\) be a Noetherian scheme over \(\Spec(A)\). Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of coherent \(\mathcal{O}_X\)-modules such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\). Set \(\mathcal{F} = \lim \mathcal{F}_n\). Then, after possibly increasing \(c\) and adjusting \(\varphi_j\) accordingly, there exists a unique graded \(\mathcal{O}_X\)-module map \[\Phi_\mathcal{F} : \bigoplus\nolimits_{n \geq 0} I^{nc}\mathcal{F} \longrightarrow \mathcal{F}[T_1, \ldots, T_r]\] with \(\Phi_\mathcal{F}(g s) = \Phi(g) s\) for \(g \in I^{nc}\) and \(s\) a local section of \(\mathcal{F}\) where \(\Phi\) is as in Lemma 0EIB. The composition of \(\Phi_\mathcal{F}\) with the map \(\mathcal{F}[T_1, \ldots, T_r] \to \bigoplus_{n \geq 0} I^n\mathcal{F}\), \(T_j \mapsto f_j\) is the canonical inclusion \(\bigoplus_{n \geq 0} I^{nc}\mathcal{F} \to \bigoplus_{n \geq 0} I^n\mathcal{F}\).
Proof
The uniqueness is immediate from the \(\mathcal{O}_X\)-linearity and the requirement that \(\Phi_\mathcal{F}(g s) = \Phi(g) s\) for \(g \in I^{nc}\) and \(s\) a local section of \(\mathcal{F}\). Thus we may assume \(X = \Spec(B)\) is affine. Observe that \((\mathcal{F}_n)\) is an object of the category \(\textit{Coh}(X, I\mathcal{O}_X)\) introduced in Cohomology of Schemes, Section 0EHN. Let \(B' = B^\wedge\) be the \(I\)-adic completion of \(B\). By Cohomology of Schemes, Lemma 087W the object \((\mathcal{F}_n)\) corresponds to a finite \(B'\)-module \(N\) in the sense that \(\mathcal{F}_n\) is the coherent module associated to the finite \(B\)-module \(N/I^n N\). Applying Lemma 0EIC to \(I \subset A \to B'\) and \(N\) we see that, after possibly increasing \(c\) and adjusting \(\varphi_j\) accordingly, we get unique maps \[\Phi_N : \bigoplus\nolimits_{n \geq 0} I^{nc}N \to N[T_1, \ldots, T_r]\] with the corresponding properties. Note that in degree \(n\) we obtain an inverse system of maps \(N/I^mN \to \bigoplus_{e_1 + \ldots + e_r = n} N/I^{m - nc}N \cdot T_1^{e_1} \ldots T_r^{e_r}\) for \(m \geq nc\). Translating back into coherent sheaves we see that \(\Phi_N\) corresponds to a system of maps \[\Phi^n_m : I^{nc}\mathcal{F}_m \longrightarrow \bigoplus\nolimits_{e_1 + \ldots + e_r = n} \mathcal{F}_{m - nc} \cdot T_1^{e_1} \ldots T_r^{e_r}\] for varying \(m \geq nc\) and \(n \geq 1\). Taking the inverse limit of these maps over \(m\) we obtain \(\Phi_\mathcal{F} = \bigoplus_n \lim_m \Phi^n_m\). Note that \(\lim_m I^t\mathcal{F}_m = I^t \mathcal{F}\) as can be seen by evaluating on affines for example, but in fact we don’t need this because it is clear there is a map \(I^t\mathcal{F} \to \lim_m I^t\mathcal{F}_m\).
Lemma
Let \(I\) be an ideal of a Noetherian ring \(A\). Let \(X\) be a Noetherian scheme over \(\Spec(A)\). Let \[\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1\] be an inverse system of coherent \(\mathcal{O}_X\)-modules such that \(\mathcal{F}_n = \mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1}\). If \(\text{cd}(A, I) = 1\), then for all \(p \in \mathbf{Z}\) the limit topology on \(\lim H^p(X, \mathcal{F}_n)\) is \(I\)-adic.
Proof
First it is clear that \(I^t \lim H^p(X, \mathcal{F}_n)\) maps to zero in \(H^p(X, \mathcal{F}_t)\). Thus the \(I\)-adic topology is finer than the limit topology. For the converse we set \(\mathcal{F} = \lim \mathcal{F}_n\), we pick generators \(f_1, \ldots, f_r\) of \(I\), we pick \(c \geq 1\), and we choose \(\Phi_\mathcal{F}\) as in Lemma 0EH6. We will use the results of Lemma 0EI8 without further mention. In particular we have a short exact sequence \[0 \to R^1\lim H^{p - 1}(X, \mathcal{F}_n) \to H^p(X, \mathcal{F}) \to \lim H^p(X, \mathcal{F}_n) \to 0\] Thus we can lift any element \(\xi\) of \(\lim H^p(X, \mathcal{F}_n)\) to an element \(\xi' \in H^p(X, \mathcal{F})\). Suppose \(\xi\) maps to zero in \(H^p(X, \mathcal{F}_{nc})\) for some \(n\), in other words, suppose \(\xi\) is “small” in the limit topology. We have a short exact sequence \[0 \to I^{nc}\mathcal{F} \to \mathcal{F} \to \mathcal{F}_{nc} \to 0\] and hence the assumption means we can lift \(\xi'\) to an element \(\xi'' \in H^p(X, I^{nc}\mathcal{F})\). Applying \(\Phi_\mathcal{F}\) we get \[\Phi_\mathcal{F}(\xi'') = \sum\nolimits_{e_1 + \ldots + e_r = n} \xi'_{e_1, \ldots, e_r} \cdot T_1^{e_1} \ldots T_r^{e_r}\] for some \(\xi'_{e_1, \ldots, e_r} \in H^p(X, \mathcal{F})\). Letting \(\xi_{e_1, \ldots, e_r} \in \lim H^p(X, \mathcal{F}_n)\) be the images and using the final assertion of Lemma 0EH6 we conclude that \[\xi = \sum f_1^{e_1} \ldots f_r^{e_r} \xi_{e_1, \ldots, e_r}\] is in \(I^n \lim H^p(X, \mathcal{F}_n)\) as desired.
Example
Let \(k\) be a field. Let \(A = k[x, y][[s, t]]/(xs - yt)\). Let \(I = (s, t)\) and \(\mathfrak a = (x, y, s, t)\). Let \(X = \Spec(A) - V(\mathfrak a)\) and \(\mathcal{F}_n = \mathcal{O}_X/I^n\mathcal{O}_X\). Observe that the rational function \[g = \frac{t}{x} = \frac{s}{y}\] is regular in an open neighbourhood \(V \subset X\) of \(V(I\mathcal{O}_X)\). Hence every power \(g^e\) determines a section \(g^e \in M = \lim H^0(X, \mathcal{F}_n)\). Observe that \(g^e \to 0\) as \(e \to \infty\) in the limit topology on \(M\) since \(g^e\) maps to zero in \(\mathcal{F}_e\). On the other hand, \(g^e \not \in IM\) for any \(e\) as the reader can see by computing \(H^0(U, \mathcal{F}_n)\); computation omitted. Observe that \(\text{cd}(A, I) = 2\). Thus the result of Lemma 0EH7 is sharp.
Mittag-Leffler conditions
When taking local cohomology with respect to the maximal ideal of a local Noetherian ring, we often get the Mittag-Leffler condition for free. This implies the same thing is true for higher cohomology groups of an inverse system of coherent sheaves with surjective transition maps on the puncture spectrum.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring.
Let \(M\) be a finite \(A\)-module. Then the \(A\)-module \(H^i_\mathfrak m(M)\) satisfies the descending chain condition for any \(i\).
Let \(U = \Spec(A) \setminus \{\mathfrak m\}\) be the punctured spectrum of \(A\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module. Then the \(A\)-module \(H^i(U, \mathcal{F})\) satisfies the descending chain condition for \(i > 0\).
Proof
We will prove part (1) by induction on the dimension of the support of \(M\). The statement holds if \(M = 0\), thus we may and do assume \(M\) is not zero.
Base case of the induction. If \(\dim(\text{Supp}(M)) = 0\), then the support of \(M\) is \(\{\mathfrak m\}\) and we see that \(H^0_\mathfrak m(M) = M\) and \(H^i_\mathfrak m(M) = 0\) for \(i > 0\) as is clear from the construction of local cohomology, see Dualizing Complexes, Section 0952. Since \(M\) has finite length (Algebra, Lemma 00J0) it has the descending chain condition.
Induction step. Assume \(\dim(\text{Supp}(M)) > 0\). By the base case the finite module \(H^0_\mathfrak m(M) \subset M\) has the descending chain condition. By Dualizing Complexes, Lemma 0AW0 we may replace \(M\) by \(M/H^0_\mathfrak m(M)\). Then \(H^0_\mathfrak m(M) = 0\), i.e., \(M\) has depth \(\geq 1\), see Dualizing Complexes, Lemma 0AVZ. Choose \(x \in \mathfrak m\) such that \(x : M \to M\) is injective. By Algebra, Lemma 0B52 we have \(\dim(\text{Supp}(M/xM)) = \dim(\text{Supp}(M)) - 1\) and the induction hypothesis applies. Pick an index \(i\) and consider the exact sequence \[H^{i - 1}_\mathfrak m(M/xM) \to H^i_\mathfrak m(M) \xrightarrow{x} H^i_\mathfrak m(M)\] coming from the short exact sequence \(0 \to M \xrightarrow{x} M \to M/xM \to 0\). It follows that the \(x\)-torsion \(H^i_\mathfrak m(M)[x]\) is a quotient of a module with the descending chain condition, and hence has the descending chain condition itself. Hence the \(\mathfrak m\)-torsion submodule \(H^i_\mathfrak m(M)[\mathfrak m]\) has the descending chain condition (and hence is finite dimensional over \(A/\mathfrak m\)). Thus we conclude that the \(\mathfrak m\)-power torsion module \(H^i_\mathfrak m(M)\) has the descending chain condition by Dualizing Complexes, Lemma 08Z8.
Part (2) follows from (1) via Local Cohomology, Lemma 0BK0.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring.
Let \((M_n)\) be an inverse system of finite \(A\)-modules. Then the inverse system \(H^i_\mathfrak m(M_n)\) satisfies the Mittag-Leffler condition for any \(i\).
Let \(U = \Spec(A) \setminus \{\mathfrak m\}\) be the punctured spectrum of \(A\). Let \(\mathcal{F}_n\) be an inverse system of coherent \(\mathcal{O}_U\)-modules. Then the inverse system \(H^i(U, \mathcal{F}_n)\) satisfies the Mittag-Leffler condition for \(i > 0\).
Proof
Follows immediately from Lemma 0DX0.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \((M_n)\) be an inverse system of finite \(A\)-modules. Let \(M \to \lim M_n\) be a map where \(M\) is a finite \(A\)-module such that for some \(i\) the map \(H^i_\mathfrak m(M) \to \lim H^i_\mathfrak m(M_n)\) is an isomorphism. Then the inverse system \(H^i_\mathfrak m(M_n)\) is essentially constant with value \(H^i_\mathfrak m(M)\).
Proof
By Lemma 0DX1 the inverse system \(H^i_\mathfrak m(M_n)\) satisfies the Mittag-Leffler condition. Let \(E_n \subset H^i_\mathfrak m(M_n)\) be the image of \(H^i_\mathfrak m(M_{n'})\) for \(n' \gg n\). Then \((E_n)\) is an inverse system with surjective transition maps and \(H^i_\mathfrak m(M) = \lim E_n\). Since \(H^i_\mathfrak m(M)\) has the descending chain condition by Lemma 0DX0 we find there can only be a finite number of nontrivial kernels of the surjections \(H^i_\mathfrak m(M) \to E_n\). Thus \(E_n \to E_{n - 1}\) is an isomorphism for all \(n \gg 0\) as desired.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(I \subset A\) be an ideal. Let \(M\) be a finite \(A\)-module. Then \[H^i(R\Gamma_\mathfrak m(M)^\wedge) = \lim H^i_\mathfrak m(M/I^nM)\] for all \(i\) where \(R\Gamma_\mathfrak m(M)^\wedge\) denotes the derived \(I\)-adic completion.
Proof
Apply Dualizing Complexes, Lemma 0EEW and Lemma 0DX1 to see the vanishing of the \(R^1\lim\) terms.
Derived completion on a ringed site
We urge the reader to skip this section on a first reading.
The algebra version of this material can be found in More on Algebra, Section 091N. Let \(\mathcal{O}\) be a sheaf of rings on a site \(\mathcal{C}\). Let \(f\) be a global section of \(\mathcal{O}\). We denote \(\mathcal{O}_f\) the sheaf associated to the presheaf of localizations \(U \mapsto \mathcal{O}(U)_f\).
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(f\) be a global section of \(\mathcal{O}\).
For \(L, N \in D(\mathcal{O}_f)\) we have \(R\SheafHom_\mathcal{O}(L, N) = R\SheafHom_{\mathcal{O}_f}(L, N)\). In particular the two \(\mathcal{O}_f\)-structures on \(R\SheafHom_\mathcal{O}(L, N)\) agree.
For \(K \in D(\mathcal{O})\) and \(L \in D(\mathcal{O}_f)\) we have \[R\SheafHom_\mathcal{O}(L, K) = R\SheafHom_{\mathcal{O}_f}(L, R\SheafHom_\mathcal{O}(\mathcal{O}_f, K))\] In particular \(R\SheafHom_\mathcal{O}(\mathcal{O}_f, R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)) = R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)\).
If \(g\) is a second global section of \(\mathcal{O}\), then \[R\SheafHom_\mathcal{O}(\mathcal{O}_f, R\SheafHom_\mathcal{O}(\mathcal{O}_g, K)) = R\SheafHom_\mathcal{O}(\mathcal{O}_{gf}, K).\]
Proof
Proof of (1). Let \(\mathcal{J}^\bullet\) be a K-injective complex of \(\mathcal{O}_f\)-modules representing \(N\). By Cohomology on Sites, Lemma 093Y it follows that \(\mathcal{J}^\bullet\) is a K-injective complex of \(\mathcal{O}\)-modules as well. Let \(\mathcal{F}^\bullet\) be a complex of \(\mathcal{O}_f\)-modules representing \(L\). Then \[R\SheafHom_\mathcal{O}(L, N) = R\SheafHom_\mathcal{O}(\mathcal{F}^\bullet, \mathcal{J}^\bullet) = R\SheafHom_{\mathcal{O}_f}(\mathcal{F}^\bullet, \mathcal{J}^\bullet)\] by Modules on Sites, Lemma 0930 because \(\mathcal{J}^\bullet\) is a K-injective complex of \(\mathcal{O}\) and of \(\mathcal{O}_f\)-modules.
Proof of (2). Let \(\mathcal{I}^\bullet\) be a K-injective complex of \(\mathcal{O}\)-modules representing \(K\). Then \(R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)\) is represented by \(\SheafHom_\mathcal{O}(\mathcal{O}_f, \mathcal{I}^\bullet)\) which is a K-injective complex of \(\mathcal{O}_f\)-modules and of \(\mathcal{O}\)-modules by Cohomology on Sites, Lemmas 093Z and 093Y. Let \(\mathcal{F}^\bullet\) be a complex of \(\mathcal{O}_f\)-modules representing \(L\). Then \[R\SheafHom_\mathcal{O}(L, K) = R\SheafHom_\mathcal{O}(\mathcal{F}^\bullet, \mathcal{I}^\bullet) = R\SheafHom_{\mathcal{O}_f}(\mathcal{F}^\bullet, \SheafHom_\mathcal{O}(\mathcal{O}_f, \mathcal{I}^\bullet))\] by Modules on Sites, Lemma 0932 and because \(\SheafHom_\mathcal{O}(\mathcal{O}_f, \mathcal{I}^\bullet)\) is a K-injective complex of \(\mathcal{O}_f\)-modules.
Proof of (3). This follows from the fact that \(R\SheafHom_\mathcal{O}(\mathcal{O}_g, \mathcal{I}^\bullet)\) is K-injective as a complex of \(\mathcal{O}\)-modules and the fact that \(\SheafHom_\mathcal{O}(\mathcal{O}_f, \SheafHom_\mathcal{O}(\mathcal{O}_g, \mathcal{H})) = \SheafHom_\mathcal{O}(\mathcal{O}_{gf}, \mathcal{H})\) for all sheaves of \(\mathcal{O}\)-modules \(\mathcal{H}\).
Let \(K \in D(\mathcal{O})\). We denote \(T(K, f)\) a derived limit (Derived Categories, Definition 08TC) of the inverse system \[\ldots \to K \xrightarrow{f} K \xrightarrow{f} K\] in \(D(\mathcal{O})\).
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(f\) be a global section of \(\mathcal{O}\). Let \(K \in D(\mathcal{O})\). The following are equivalent
\(R\SheafHom_\mathcal{O}(\mathcal{O}_f, K) = 0\),
\(R\SheafHom_\mathcal{O}(L, K) = 0\) for all \(L\) in \(D(\mathcal{O}_f)\),
\(T(K, f) = 0\).
Proof
It is clear that (2) implies (1). The implication (1) \(\Rightarrow\) (2) follows from Lemma 0996. A free resolution of the \(\mathcal{O}\)-module \(\mathcal{O}_f\) is given by \[0 \to \bigoplus\nolimits_{n \in \mathbf{N}} \mathcal{O} \to \bigoplus\nolimits_{n \in \mathbf{N}} \mathcal{O} \to \mathcal{O}_f \to 0\] where the first map sends a local section \((x_0, x_1, \ldots)\) to \((x_0, x_1 - fx_0, x_2 - fx_1, \ldots)\) and the second map sends \((x_0, x_1, \ldots)\) to \(x_0 + x_1/f + x_2/f^2 + \ldots\). Applying \(\SheafHom_\mathcal{O}(-, \mathcal{I}^\bullet)\) where \(\mathcal{I}^\bullet\) is a K-injective complex of \(\mathcal{O}\)-modules representing \(K\) we get a short exact sequence of complexes \[0 \to \SheafHom_\mathcal{O}(\mathcal{O}_f, \mathcal{I}^\bullet) \to \prod \mathcal{I}^\bullet \to \prod \mathcal{I}^\bullet \to 0\] because \(\mathcal{I}^n\) is an injective \(\mathcal{O}\)-module. The products are products in \(D(\mathcal{O})\), see Injectives, Lemma 07D9. This means that the object \(T(K, f)\) is a representative of \(R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)\) in \(D(\mathcal{O})\). Thus the equivalence of (1) and (3).
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(K \in D(\mathcal{O})\). The rule which associates to \(U\) the set \(\mathcal{I}(U)\) of sections \(f \in \mathcal{O}(U)\) such that \(T(K|_U, f) = 0\) is a sheaf of ideals in \(\mathcal{O}\).
Proof
We will use the results of Lemma 0997 without further mention. If \(f \in \mathcal{I}(U)\), and \(g \in \mathcal{O}(U)\), then \(\mathcal{O}_{U, gf}\) is an \(\mathcal{O}_{U, f}\)-module hence \(R\SheafHom_\mathcal{O}(\mathcal{O}_{U, gf}, K|_U) = 0\), hence \(gf \in \mathcal{I}(U)\). Suppose \(f, g \in \mathcal{O}(U)\). Then there is a short exact sequence \[0 \to \mathcal{O}_{U, f + g} \to \mathcal{O}_{U, f(f + g)} \oplus \mathcal{O}_{U, g(f + g)} \to \mathcal{O}_{U, gf(f + g)} \to 0\] because \(f, g\) generate the unit ideal in \(\mathcal{O}(U)_{f + g}\). This follows from Algebra, Lemma 00EJ and the easy fact that the last arrow is surjective. Because \(R\SheafHom_\mathcal{O}( - , K|_U)\) is an exact functor of triangulated categories the vanishing of \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, f(f + g)}, K|_U)\), \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, g(f + g)}, K|_U)\), and \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, gf(f + g)}, K|_U)\), implies the vanishing of \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, f + g}, K|_U)\). We omit the verification of the sheaf condition.
We can make the following definition for any ringed site.
Definition
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a sheaf of ideals. Let \(K \in D(\mathcal{O})\). We say that \(K\) is derived complete with respect to \(\mathcal{I}\) if for every object \(U\) of \(\mathcal{C}\) and \(f \in \mathcal{I}(U)\) the object \(T(K|_U, f)\) of \(D(\mathcal{O}_U)\) is zero.
It is clear that the full subcategory \(D_{comp}(\mathcal{O}) = D_{comp}(\mathcal{O}, \mathcal{I}) \subset D(\mathcal{O})\) consisting of derived complete objects is a saturated triangulated subcategory, see Derived Categories, Definitions 05QM and 05RB. This subcategory is preserved under products and homotopy limits in \(D(\mathcal{O})\). But it is not preserved under countable direct sums in general.
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a sheaf of ideals. If \(K \in D(\mathcal{O})\) and \(L \in D_{comp}(\mathcal{O})\), then \(R\SheafHom_\mathcal{O}(K, L) \in D_{comp}(\mathcal{O})\).
Proof
Let \(U\) be an object of \(\mathcal{C}\) and let \(f \in \mathcal{I}(U)\). Recall that \[\Hom_{D(\mathcal{O}_U)}(\mathcal{O}_{U, f}, R\SheafHom_\mathcal{O}(K, L)|_U) = \Hom_{D(\mathcal{O}_U)}( K|_U \otimes_{\mathcal{O}_U}^\mathbf{L} \mathcal{O}_{U, f}, L|_U)\] by Cohomology on Sites, Lemma 08J9. The right hand side is zero by Lemma 0997 and the relationship between internal hom and actual hom, see Cohomology on Sites, Lemma 08JA. The same vanishing holds for all \(U'/U\). Thus the object \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, f}, R\SheafHom_\mathcal{O}(K, L)|_U)\) of \(D(\mathcal{O}_U)\) has vanishing \(0\)th cohomology sheaf (by locus citatus). Similarly for the other cohomology sheaves, i.e., \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, f}, R\SheafHom_\mathcal{O}(K, L)|_U)\) is zero in \(D(\mathcal{O}_U)\). By Lemma 0997 we conclude.
Lemma
Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}'\) be a homomorphism of sheaves of rings. Let \(\mathcal{I} \subset \mathcal{O}\) be a sheaf of ideals. The inverse image of \(D_{comp}(\mathcal{O}, \mathcal{I})\) under the restriction functor \(D(\mathcal{O}') \to D(\mathcal{O})\) is \(D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')\).
Proof
Using Lemma 0998 we see that \(K' \in D(\mathcal{O}')\) is in \(D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')\) if and only if \(T(K'|_U, f)\) is zero for every local section \(f \in \mathcal{I}(U)\). Observe that the cohomology sheaves of \(T(K'|_U, f)\) are computed in the category of abelian sheaves, so it doesn’t matter whether we think of \(f\) as a section of \(\mathcal{O}\) or take the image of \(f\) as a section of \(\mathcal{O}'\). The lemma follows immediately from this and the definition of derived complete objects.
Lemma
Let \(f : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) be a morphism of ringed topoi. Let \(\mathcal{I} \subset \mathcal{O}\) and \(\mathcal{I}' \subset \mathcal{O}'\) be sheaves of ideals such that \(f^\sharp\) sends \(f^{-1}\mathcal{I}\) into \(\mathcal{I}'\). Then \(Rf_*\) sends \(D_{comp}(\mathcal{O}', \mathcal{I}')\) into \(D_{comp}(\mathcal{O}, \mathcal{I})\).
Proof
We may assume \(f\) is given by a morphism of ringed sites corresponding to a continuous functor \(\mathcal{C} \to \mathcal{D}\) (Modules on Sites, Lemma 03CR ). Let \(U\) be an object of \(\mathcal{C}\) and let \(g\) be a section of \(\mathcal{I}\) over \(U\). We have to show that \(\Hom_{D(\mathcal{O}_U)}(\mathcal{O}_{U, g}, Rf_*K|_U) = 0\) whenever \(K\) is derived complete with respect to \(\mathcal{I}'\). Namely, by Cohomology on Sites, Lemma 08JA this, applied to all objects over \(U\) and all shifts of \(K\), will imply that \(R\SheafHom_{\mathcal{O}_U}(\mathcal{O}_{U, g}, Rf_*K|_U)\) is zero, which implies that \(T(Rf_*K|_U, g)\) is zero (Lemma 0997) which is what we have to show (Definition 0999). Let \(V\) in \(\mathcal{D}\) be the image of \(U\). Then \[\Hom_{D(\mathcal{O}_U)}(\mathcal{O}_{U, g}, Rf_*K|_U) = \Hom_{D(\mathcal{O}'_V)}(\mathcal{O}'_{V, g'}, K|_V) = 0\] where \(g' = f^\sharp(g) \in \mathcal{I}'(V)\). The second equality because \(K\) is derived complete and the first equality because the derived pullback of \(\mathcal{O}_{U, g}\) is \(\mathcal{O}'_{V, g'}\) and Cohomology on Sites, Lemma 07A6.
The following lemma is the simplest case where one has derived completion.
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed on a site. Let \(f_1, \ldots, f_r\) be global sections of \(\mathcal{O}\). Let \(\mathcal{I} \subset \mathcal{O}\) be the ideal sheaf generated by \(f_1, \ldots, f_r\). Then the inclusion functor \(D_{comp}(\mathcal{O}) \to D(\mathcal{O})\) has a left adjoint, i.e., given any object \(K\) of \(D(\mathcal{O})\) there exists a map \(K \to K^\wedge\) with \(K^\wedge\) in \(D_{comp}(\mathcal{O})\) such that the map \[\Hom_{D(\mathcal{O})}(K^\wedge, E) \longrightarrow \Hom_{D(\mathcal{O})}(K, E)\] is bijective whenever \(E\) is in \(D_{comp}(\mathcal{O})\). In fact we have \[K^\wedge = R\SheafHom_\mathcal{O} (\mathcal{O} \to \prod\nolimits_{i_0} \mathcal{O}_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_r}, K)\] functorially in \(K\).
Proof
Define \(K^\wedge\) by the last displayed formula of the lemma. There is a map of complexes \[(\mathcal{O} \to \prod\nolimits_{i_0} \mathcal{O}_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_r}) \longrightarrow \mathcal{O}\] which induces a map \(K \to K^\wedge\). It suffices to prove that \(K^\wedge\) is derived complete and that \(K \to K^\wedge\) is an isomorphism if \(K\) is derived complete.
Let \(f\) be a global section of \(\mathcal{O}\). By Lemma 0996 the object \(R\SheafHom_\mathcal{O}(\mathcal{O}_f, K^\wedge)\) is equal to \[R\SheafHom_\mathcal{O}( (\mathcal{O}_f \to \prod\nolimits_{i_0} \mathcal{O}_{ff_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{ff_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{ff_1\ldots f_r}), K)\] If \(f = f_i\) for some \(i\), then \(f_1, \ldots, f_r\) generate the unit ideal in \(\mathcal{O}_f\), hence the extended alternating Čech complex \[\mathcal{O}_f \to \prod\nolimits_{i_0} \mathcal{O}_{ff_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{ff_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{ff_1\ldots f_r}\] is zero (even homotopic to zero). In this way we see that \(K^\wedge\) is derived complete.
If \(K\) is derived complete, then \(R\SheafHom_\mathcal{O}(\mathcal{O}_f, K)\) is zero for all \(f = f_{i_0} \ldots f_{i_p}\), \(p \geq 0\). Thus \(K \to K^\wedge\) is an isomorphism in \(D(\mathcal{O})\).
Next we explain why derived completion is a completion.
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed on a site. Let \(f_1, \ldots, f_r\) be global sections of \(\mathcal{O}\). Let \(\mathcal{I} \subset \mathcal{O}\) be the ideal sheaf generated by \(f_1, \ldots, f_r\). Let \(K \in D(\mathcal{O})\). The derived completion \(K^\wedge\) of Lemma 099B is given by the formula \[K^\wedge = R\lim K \otimes^\mathbf{L}_\mathcal{O} K_n\] where \(K_n = K(\mathcal{O}, f_1^n, \ldots, f_r^n)\) is the Koszul complex on \(f_1^n, \ldots, f_r^n\) over \(\mathcal{O}\).
Proof
In More on Algebra, Lemma 0913 we have seen that the extended alternating Čech complex \[\mathcal{O} \to \prod\nolimits_{i_0} \mathcal{O}_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_r}\] is a colimit of the Koszul complexes \(K^n = K(\mathcal{O}, f_1^n, \ldots, f_r^n)\) sitting in degrees \(0, \ldots, r\). Note that \(K^n\) is a finite chain complex of finite free \(\mathcal{O}\)-modules with dual \(\SheafHom_\mathcal{O}(K^n, \mathcal{O}) = K_n\) where \(K_n\) is the Koszul cochain complex sitting in degrees \(-r, \ldots, 0\) (as usual). By Lemma 099B the functor \(E \mapsto E^\wedge\) is gotten by taking \(R\SheafHom\) from the extended alternating Čech complex into \(E\): \[E^\wedge = R\SheafHom(\colim K^n, E)\] This is equal to \(R\lim (E \otimes_\mathcal{O}^\mathbf{L} K_n)\) by Cohomology on Sites, Lemma 0A0A.
Lemma
There exist a way to construct
for every pair \((A, I)\) consisting of a ring \(A\) and a finitely generated ideal \(I \subset A\) a complex \(K(A, I)\) of \(A\)-modules,
a map \(K(A, I) \to A\) of complexes of \(A\)-modules,
for every ring map \(A \to B\) and finitely generated ideal \(I \subset A\) a map of complexes \(K(A, I) \to K(B, IB)\),
such that
for \(A \to B\) and \(I \subset A\) finitely generated the diagram \[\xymatrix{ K(A, I) \ar[r] \ar[d] & A \ar[d] \\ K(B, IB) \ar[r] & B }\] commutes,
for \(A \to B \to C\) and \(I \subset A\) finitely generated the composition of the maps \(K(A, I) \to K(B, IB) \to K(C, IC)\) is the map \(K(A, I) \to K(C, IC)\).
for \(A \to B\) and a finitely generated ideal \(I \subset A\) the induced map \(K(A, I) \otimes_A^\mathbf{L} B \to K(B, IB)\) is an isomorphism in \(D(B)\), and
if \(I = (f_1, \ldots, f_r) \subset A\) then there is a commutative diagram \[\xymatrix{ (A \to \prod\nolimits_{i_0} A_{f_{i_0}} \to \prod\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_r}) \ar[r] \ar[d] & K(A, I) \ar[d] \\ A \ar[r]^1 & A }\] in \(D(A)\) whose horizontal arrows are isomorphisms.
Proof
Let \(S\) be the set of rings \(A_0\) of the form \(A_0 = \mathbf{Z}[x_1, \ldots, x_n]/J\). Every finite type \(\mathbf{Z}\)-algebra is isomorphic to an element of \(S\). Let \(\mathcal{A}_0\) be the category whose objects are pairs \((A_0, I_0)\) where \(A_0 \in S\) and \(I_0 \subset A_0\) is an ideal and whose morphisms \((A_0, I_0) \to (B_0, J_0)\) are ring maps \(\varphi : A_0 \to B_0\) such that \(J_0 = \varphi(I_0)B_0\).
Suppose we can construct \(K(A_0, I_0) \to A_0\) functorially for objects of \(\mathcal{A}_0\) having properties (a), (b), (c), and (d). Then we take \[K(A, I) = \colim_{\varphi : (A_0, I_0) \to (A, I)} K(A_0, I_0)\] where the colimit is over ring maps \(\varphi : A_0 \to A\) such that \(\varphi(I_0)A = I\) with \((A_0, I_0)\) in \(\mathcal{A}_0\). A morphism between \((A_0, I_0) \to (A, I)\) and \((A_0', I_0') \to (A, I)\) are given by maps \((A_0, I_0) \to (A_0', I_0')\) in \(\mathcal{A}_0\) commuting with maps to \(A\). The category of these \((A_0, I_0) \to (A, I)\) is filtered (details omitted). Moreover, \(\colim_{\varphi : (A_0, I_0) \to (A, I)} A_0 = A\) so that \(K(A, I)\) is a complex of \(A\)-modules. Finally, given \(\varphi : A \to B\) and \(I \subset A\) for every \((A_0, I_0) \to (A, I)\) in the colimit, the composition \((A_0, I_0) \to (B, IB)\) lives in the colimit for \((B, IB)\). In this way we get a map on colimits. Properties (a), (b), (c), and (d) follow readily from this and the corresponding properties of the complexes \(K(A_0, I_0)\).
Endow \(\mathcal{C}_0 = \mathcal{A}_0^{opp}\) with the chaotic topology. We equip \(\mathcal{C}_0\) with the sheaf of rings \(\mathcal{O} : (A, I) \mapsto A\). The ideals \(I\) fit together to give a sheaf of ideals \(\mathcal{I} \subset \mathcal{O}\). Choose an injective resolution \(\mathcal{O} \to \mathcal{J}^\bullet\). Consider the object \[\mathcal{F}^\bullet = \bigcup\nolimits_n \mathcal{J}^\bullet[\mathcal{I}^n]\] Let \(U = (A, I) \in \Ob(\mathcal{C}_0)\). Since the topology in \(\mathcal{C}_0\) is chaotic, the value \(\mathcal{J}^\bullet(U)\) is a resolution of \(A\) by injective \(A\)-modules. Hence the value \(\mathcal{F}^\bullet(U)\) is an object of \(D(A)\) representing the image of \(R\Gamma_I(A)\) in \(D(A)\), see Dualizing Complexes, Section 0952. Choose a complex of \(\mathcal{O}\)-modules \(\mathcal{K}^\bullet\) and a commutative diagram \[\xymatrix{ \mathcal{O} \ar[r] & \mathcal{J}^\bullet \\ \mathcal{K}^\bullet \ar[r] \ar[u] & \mathcal{F}^\bullet \ar[u] }\] where the horizontal arrows are quasi-isomorphisms. This is possible by the construction of the derived category \(D(\mathcal{O})\). Set \(K(A, I) = \mathcal{K}^\bullet(U)\) where \(U = (A, I)\). Properties (a) and (b) are clear and properties (c) and (d) follow from Dualizing Complexes, Lemmas 0956 and 0957.
Lemma
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. There exists a map \(K \to \mathcal{O}\) in \(D(\mathcal{O})\) such that for every \(U \in \Ob(\mathcal{C})\) such that \(\mathcal{I}|_U\) is generated by \(f_1, \ldots, f_r \in \mathcal{I}(U)\) there is an isomorphism \[(\mathcal{O}_U \to \prod\nolimits_{i_0} \mathcal{O}_{U, f_{i_0}} \to \prod\nolimits_{i_0 < i_1} \mathcal{O}_{U, f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{U, f_1\ldots f_r}) \longrightarrow K|_U\] compatible with maps to \(\mathcal{O}_U\).
Proof
Let \(\mathcal{C}' \subset \mathcal{C}\) be the full subcategory of objects \(U\) such that \(\mathcal{I}|_U\) is generated by finitely many sections. Then \(\mathcal{C}' \to \mathcal{C}\) is a special cocontinuous functor (Sites, Definition 03CG). Hence it suffices to work with \(\mathcal{C}'\), see Sites, Lemma 03A0. In other words we may assume that for every object \(U\) of \(\mathcal{C}\) there exists a finitely generated ideal \(I \subset \mathcal{I}(U)\) such that \(\mathcal{I}|_U = \Im(I \otimes \mathcal{O}_U \to \mathcal{O}_U)\). We will say that \(I\) generates \(\mathcal{I}|_U\). Warning: We do not know that \(\mathcal{I}(U)\) is a finitely generated ideal in \(\mathcal{O}(U)\).
Let \(U\) be an object and \(I \subset \mathcal{O}(U)\) a finitely generated ideal which generates \(\mathcal{I}|_U\). On the category \(\mathcal{C}/U\) consider the complex of presheaves \[K_{U, I}^\bullet : U'/U \longmapsto K(\mathcal{O}(U'), I\mathcal{O}(U'))\] with \(K(-, -)\) as in Lemma 099D. We claim that the sheafification of this is independent of the choice of \(I\). Indeed, if \(I' \subset \mathcal{O}(U)\) is a finitely generated ideal which also generates \(\mathcal{I}|_U\), then there exists a covering \(\{U_j \to U\}\) such that \(I\mathcal{O}(U_j) = I'\mathcal{O}(U_j)\). (Hint: this works because both \(I\) and \(I'\) are finitely generated and generate \(\mathcal{I}|_U\).) Hence \(K_{U, I}^\bullet\) and \(K_{U, I'}^\bullet\) are the same for any object lying over one of the \(U_j\). The statement on sheafifications follows. Denote \(K_U^\bullet\) the common value.
The independence of choice of \(I\) also shows that \(K_U^\bullet|_{\mathcal{C}/U'} = K_{U'}^\bullet\) whenever we are given a morphism \(U' \to U\) and hence a localization morphism \(\mathcal{C}/U' \to \mathcal{C}/U\). Thus the complexes \(K_U^\bullet\) glue to give a single well defined complex \(K^\bullet\) of \(\mathcal{O}\)-modules. The existence of the map \(K^\bullet \to \mathcal{O}\) and the quasi-isomorphism of the lemma follow immediately from the corresponding properties of the complexes \(K(-, -)\) in Lemma 099D.
Proposition
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. There exists a left adjoint to the inclusion functor \(D_{comp}(\mathcal{O}) \to D(\mathcal{O})\).
Proof
Let \(K \to \mathcal{O}\) in \(D(\mathcal{O})\) be as constructed in Lemma 099E. Let \(E \in D(\mathcal{O})\). Then \(E^\wedge = R\SheafHom(K, E)\) together with the map \(E \to E^\wedge\) will do the job. Namely, locally on the site \(\mathcal{C}\) we recover the adjoint of Lemma 099B. This shows that \(E^\wedge\) is always derived complete and that \(E \to E^\wedge\) is an isomorphism if \(E\) is derived complete.
Remark
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. Let \(K \mapsto K^\wedge\) be the derived completion functor of Proposition 099F. For any \(n \geq 1\) the object \(K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n\) is derived complete as it is annihilated by powers of local sections of \(\mathcal{I}\). Hence there is a canonical factorization \[K \to K^\wedge \to K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n\] of the canonical map \(K \to K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n\). These maps are compatible for varying \(n\) and we obtain a comparison map \[K^\wedge \longrightarrow R\lim \left(K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}/\mathcal{I}^n\right)\] The right hand side is more recognizable as a kind of completion. In general this comparison map is not an isomorphism.
Remark
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. Let \(K \mapsto K^\wedge\) be the derived completion functor of Proposition 099F. It follows from the construction in the proof of the proposition that \(K^\wedge|_U\) is the derived completion of \(K|_U\) for any \(U \in \Ob(\mathcal{C})\). But we can also prove this as follows. From the definition of derived complete objects it follows that \(K^\wedge|_U\) is derived complete. Thus we obtain a canonical map \(a : (K|_U)^\wedge \to K^\wedge|_U\). On the other hand, if \(E\) is a derived complete object of \(D(\mathcal{O}_U)\), then \(Rj_*E\) is a derived complete object of \(D(\mathcal{O})\) by Lemma 099J. Here \(j\) is the localization morphism (Modules on Sites, Section 03DH). Hence we also obtain a canonical map \(b : K^\wedge \to Rj_*((K|_U)^\wedge)\). We omit the (formal) verification that the adjoint of \(b\) is the inverse of \(a\).
Remark
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. Denote \(K \mapsto K^\wedge\) the adjoint of Proposition 099F. Then we set \[K \otimes^\wedge_\mathcal{O} L = (K \otimes_\mathcal{O}^\mathbf{L} L)^\wedge\] This completed tensor product defines a functor \(D_{comp}(\mathcal{O}) \times D_{comp}(\mathcal{O}) \to D_{comp}(\mathcal{O})\) such that we have \[\Hom_{D_{comp}(\mathcal{O})}(K, R\SheafHom_\mathcal{O}(L, M)) = \Hom_{D_{comp}(\mathcal{O})}(K \otimes_\mathcal{O}^\wedge L, M)\] for \(K, L, M \in D_{comp}(\mathcal{O})\). Note that \(R\SheafHom_\mathcal{O}(L, M) \in D_{comp}(\mathcal{O})\) by Lemma 099A.
Lemma
Let \(\mathcal{C}\) be a site. Assume \(\varphi : \mathcal{O} \to \mathcal{O}'\) is a flat homomorphism of sheaves of rings. Let \(f_1, \ldots, f_r\) be global sections of \(\mathcal{O}\) such that \(\mathcal{O}/(f_1, \ldots, f_r) \cong \mathcal{O}'/(f_1, \ldots, f_r)\mathcal{O}'\). Then the map of extended alternating Čech complexes \[\xymatrix{ \mathcal{O} \to \prod_{i_0} \mathcal{O}_{f_{i_0}} \to \prod_{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_r} \ar[d] \\ \mathcal{O}' \to \prod_{i_0} \mathcal{O}'_{f_{i_0}} \to \prod_{i_0 < i_1} \mathcal{O}'_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}'_{f_1\ldots f_r} }\] is a quasi-isomorphism.
Proof
Observe that the second complex is the tensor product of the first complex with \(\mathcal{O}'\). We can write the first extended alternating Čech complex as a colimit of the Koszul complexes \(K_n = K(\mathcal{O}, f_1^n, \ldots, f_r^n)\), see More on Algebra, Lemma 0913. Hence it suffices to prove \(K_n \to K_n \otimes_\mathcal{O} \mathcal{O}'\) is a quasi-isomorphism. Since \(\mathcal{O} \to \mathcal{O}'\) is flat it suffices to show that \(H^i \to H^i \otimes_\mathcal{O} \mathcal{O}'\) is an isomorphism where \(H^i\) is the \(i\)th cohomology sheaf \(H^i = H^i(K_n)\). These sheaves are annihilated by \(f_1^n, \ldots, f_r^n\), see More on Algebra, Lemma 0663. Hence these sheaves are annihilated by \((f_1, \ldots, f_r)^m\) for some \(m \gg 0\). Thus \(H^i \to H^i \otimes_\mathcal{O} \mathcal{O}'\) is an isomorphism by Modules on Sites, Lemma 0GLY.
Lemma
Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}'\) be a homomorphism of sheaves of rings. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. If \(\mathcal{O} \to \mathcal{O}'\) is flat and \(\mathcal{O}/\mathcal{I} \cong \mathcal{O}'/\mathcal{I}\mathcal{O}'\), then the restriction functor \(D(\mathcal{O}') \to D(\mathcal{O})\) induces an equivalence \(D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}') \to D_{comp}(\mathcal{O}, \mathcal{I})\).
Proof
Lemma 099J implies restriction \(r : D(\mathcal{O}') \to D(\mathcal{O})\) sends \(D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}')\) into \(D_{comp}(\mathcal{O}, \mathcal{I})\). We will construct a quasi-inverse \(E \mapsto E'\).
Let \(K \to \mathcal{O}\) be the morphism of \(D(\mathcal{O})\) constructed in Lemma 099E. Set \(K' = K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}'\) in \(D(\mathcal{O}')\). Then \(K' \to \mathcal{O}'\) is a map in \(D(\mathcal{O}')\) which satisfies the conclusions of Lemma 099E with respect to \(\mathcal{I}' = \mathcal{I}\mathcal{O}'\). The map \(K \to r(K')\) is a quasi-isomorphism by Lemma 099H. Now, for \(E \in D_{comp}(\mathcal{O}, \mathcal{I})\) we set \[E' = R\SheafHom_\mathcal{O}(r(K'), E)\] viewed as an object in \(D(\mathcal{O}')\) using the \(\mathcal{O}'\)-module structure on \(K'\). Since \(E\) is derived complete we have \(E = R\SheafHom_\mathcal{O}(K, E)\), see proof of Proposition 099F. On the other hand, since \(K \to r(K')\) is an isomorphism in we see that there is an isomorphism \(E \to r(E')\) in \(D(\mathcal{O})\). To finish the proof we have to show that, if \(E = r(M')\) for an object \(M'\) of \(D_{comp}(\mathcal{O}', \mathcal{I}')\), then \(E' \cong M'\). To get a map we use \[M' = R\SheafHom_{\mathcal{O}'}(\mathcal{O}', M') \to R\SheafHom_\mathcal{O}(r(\mathcal{O}'), r(M')) \to R\SheafHom_\mathcal{O}(r(K'), r(M')) = E'\] where the second arrow uses the map \(K' \to \mathcal{O}'\). To see that this is an isomorphism, one shows that \(r\) applied to this arrow is the same as the isomorphism \(E \to r(E')\) above. Details omitted.
Lemma
Let \(f : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) be a morphism of ringed topoi. Let \(\mathcal{I} \subset \mathcal{O}\) and \(\mathcal{I}' \subset \mathcal{O}'\) be finite type sheaves of ideals such that \(f^\sharp\) sends \(f^{-1}\mathcal{I}\) into \(\mathcal{I}'\). Then \(Rf_*\) sends \(D_{comp}(\mathcal{O}', \mathcal{I}')\) into \(D_{comp}(\mathcal{O}, \mathcal{I})\) and has a left adjoint \(Lf_{comp}^*\) which is \(Lf^*\) followed by derived completion.
Proof
The first statement we have seen in Lemma 099J. Note that the second statement makes sense as we have a derived completion functor \(D(\mathcal{O}') \to D_{comp}(\mathcal{O}', \mathcal{I}')\) by Proposition 099F. OK, so now let \(K \in D_{comp}(\mathcal{O}, \mathcal{I})\) and \(M \in D_{comp}(\mathcal{O}', \mathcal{I}')\). Then we have \[\Hom(K, Rf_*M) = \Hom(Lf^*K, M) = \Hom(Lf_{comp}^*K, M)\] by the universal property of derived completion.
Lemma
Let \(f : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) be a morphism of ringed topoi. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. Let \(\mathcal{I}' \subset \mathcal{O}'\) be the ideal generated by \(f^\sharp(f^{-1}\mathcal{I})\). Then \(Rf_*\) commutes with derived completion, i.e., \(Rf_*(K^\wedge) = (Rf_*K)^\wedge\).
Proof
By Proposition 099F the derived completion functors exist. By Lemma 099J the object \(Rf_*(K^\wedge)\) is derived complete, and hence we obtain a canonical map \((Rf_*K)^\wedge \to Rf_*(K^\wedge)\) by the universal property of derived completion. We may check this map is an isomorphism locally on \(\mathcal{C}\). Thus, since derived completion commutes with localization (Remark 0A0F) we may assume that \(\mathcal{I}\) is generated by global sections \(f_1, \ldots, f_r\). Then \(\mathcal{I}'\) is generated by \(g_i = f^\sharp(f_i)\). By Lemma 0A0E we have to prove that \[R\lim \left( Rf_*K \otimes^\mathbf{L}_\mathcal{O} K(\mathcal{O}, f_1^n, \ldots, f_r^n) \right) = Rf_*\left( R\lim K \otimes^\mathbf{L}_{\mathcal{O}'} K(\mathcal{O}', g_1^n, \ldots, g_r^n) \right)\] Because \(Rf_*\) commutes with \(R\lim\) (Cohomology on Sites, Lemma 0A07) it suffices to prove that \[Rf_*K \otimes^\mathbf{L}_\mathcal{O} K(\mathcal{O}, f_1^n, \ldots, f_r^n) = Rf_*\left( K \otimes^\mathbf{L}_{\mathcal{O}'} K(\mathcal{O}', g_1^n, \ldots, g_r^n) \right)\] This follows from the projection formula (Cohomology on Sites, Lemma 0944) and the fact that \(Lf^*K(\mathcal{O}, f_1^n, \ldots, f_r^n) = K(\mathcal{O}', g_1^n, \ldots, g_r^n)\).
Lemma
Let \(A\) be a ring and let \(I \subset A\) be a finitely generated ideal. Let \(\mathcal{C}\) be a site and let \(\mathcal{O}\) be a sheaf of \(A\)-algebras. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Then we have \[R\Gamma(\mathcal{C}, \mathcal{F})^\wedge = R\Gamma(\mathcal{C}, \mathcal{F}^\wedge)\] in \(D(A)\) where \(\mathcal{F}^\wedge\) is the derived completion of \(\mathcal{F}\) with respect to \(I\mathcal{O}\) and on the left hand wide we have the derived completion with respect to \(I\). This produces two spectral sequences \[E_2^{i, j} = H^i(H^j(\mathcal{C}, \mathcal{F})^\wedge) \quad\text{and}\quad E_2^{p, q} = H^p(\mathcal{C}, H^q(\mathcal{F}^\wedge))\] both converging to \(H^*(R\Gamma(\mathcal{C}, \mathcal{F})^\wedge) = H^*(\mathcal{C}, \mathcal{F}^\wedge)\)
Proof
Apply Lemma 0A0G to the morphism of ringed topoi \((\mathcal{C}, \mathcal{O}) \to (pt, A)\) and take cohomology to get the first statement. The second spectral sequence is the second spectral sequence of Derived Categories, Lemma 015J. The first spectral sequence is the spectral sequence of More on Algebra, Example 0BKE applied to \(R\Gamma(\mathcal{C}, \mathcal{F})^\wedge\).
Remark
Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\mathcal{I} \subset \mathcal{O}\) be a finite type sheaf of ideals. Let \(K \mapsto K^\wedge\) be the derived completion of Proposition 099F. Let \(U \in \Ob(\mathcal{C})\) be an object such that \(\mathcal{I}\) is generated as an ideal sheaf by \(f_1, \ldots, f_r \in \mathcal{I}(U)\). Set \(A = \mathcal{O}(U)\) and \(I = (f_1, \ldots, f_r) \subset A\). Warning: it may not be the case that \(I = \mathcal{I}(U)\). Then we have \[R\Gamma(U, K^\wedge) = R\Gamma(U, K)^\wedge\] where the right hand side is the derived completion of the object \(R\Gamma(U, K)\) of \(D(A)\) with respect to \(I\). This is true because derived completion commutes with localization (Remark 0A0F) and Lemma 0BLX.
The theorem on formal functions
We interrupt the flow of the exposition to talk a little bit about derived completion in the setting of quasi-coherent modules on schemes and to use this to give a somewhat different proof of the theorem on formal functions. We give some pointers to the literature in Remark 0AKL.
Lemma 0A0G is a (very formal) derived version of the theorem on formal functions (Cohomology of Schemes, Theorem 02OC). To make this more explicit, suppose \(f : X \to S\) is a morphism of schemes, \(\mathcal{I} \subset \mathcal{O}_S\) is a quasi-coherent sheaf of ideals of finite type, and \(\mathcal{F}\) is a quasi-coherent sheaf on \(X\). Then the lemma says that [0A0I]\[\begin{equation} Rf_*(\mathcal{F}^\wedge) = (Rf_*\mathcal{F})^\wedge \end{equation}\] where \(\mathcal{F}^\wedge\) is the derived completion of \(\mathcal{F}\) with respect to \(f^{-1}\mathcal{I} \cdot \mathcal{O}_X\) and the right hand side is the derived completion of \(Rf_*\mathcal{F}\) with respect to \(\mathcal{I}\). To see that this gives back the theorem on formal functions we have to do a bit of work.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(K\) be a pseudo-coherent object of \(D(\mathcal{O}_X)\) with derived completion \(K^\wedge\). Then \[H^p(U, K^\wedge) = \lim H^p(U, K)/I^nH^p(U, K) = H^p(U, K)^\wedge\] for any affine open \(U \subset X\) where \(I = \mathcal{I}(U)\) and where on the right we have the derived completion with respect to \(I\).
Proof
Write \(U = \Spec(A)\). The ring \(A\) is Noetherian and hence \(I \subset A\) is finitely generated. Then we have \[R\Gamma(U, K^\wedge) = R\Gamma(U, K)^\wedge\] by Remark 0CQI. Now \(R\Gamma(U, K)\) is a pseudo-coherent complex of \(A\)-modules (Derived Categories of Schemes, Lemma 08E7). By More on Algebra, Lemma 0A06 we conclude that the \(p\)th cohomology module of \(R\Gamma(U, K^\wedge)\) is equal to the \(I\)-adic completion of \(H^p(U, K)\). This proves the first equality. The second (less important) equality follows immediately from a second application of the lemma just used.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(K\) be an object of \(D(\mathcal{O}_X)\). Then
the derived completion \(K^\wedge\) is equal to \(R\lim (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X/\mathcal{I}^n)\).
Let \(K\) is a pseudo-coherent object of \(D(\mathcal{O}_X)\). Then
the cohomology sheaf \(H^q(K^\wedge)\) is equal to \(\lim H^q(K)/\mathcal{I}^nH^q(K)\).
Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module1. Then
the derived completion \(\mathcal{F}^\wedge\) is equal to \(\lim \mathcal{F}/\mathcal{I}^n\mathcal{F}\),
\(\lim \mathcal{F}/\mathcal{I}^n \mathcal{F} = R\lim \mathcal{F}/\mathcal{I}^n \mathcal{F}\),
\(H^p(U, \mathcal{F}^\wedge) = 0\) for \(p \not = 0\) for all affine opens \(U \subset X\).
Proof
Proof of (1). There is a canonical map \[K \longrightarrow R\lim (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X/\mathcal{I}^n),\] see Remark 0CQH. Derived completion commutes with passing to open subschemes (Remark 0A0F). Formation of \(R\lim\) commutes with passing to open subschemes. It follows that to check our map is an isomorphism, we may work locally. Thus we may assume \(X = U = \Spec(A)\). Say \(I = (f_1, \ldots, f_r)\). Let \(K_n = K(A, f_1^n, \ldots, f_r^n)\) be the Koszul complex. By More on Algebra, Lemma 0921 we have seen that the pro-systems \(\{K_n\}\) and \(\{A/I^n\}\) of \(D(A)\) are isomorphic. Using the equivalence \(D(A) = D_{\QCoh}(\mathcal{O}_X)\) of Derived Categories of Schemes, Lemma 06Z0 we see that the pro-systems \(\{K(\mathcal{O}_X, f_1^n, \ldots, f_r^n)\}\) and \(\{\mathcal{O}_X/\mathcal{I}^n\}\) are isomorphic in \(D(\mathcal{O}_X)\). This proves the second equality in \[K^\wedge = R\lim \left( K \otimes_{\mathcal{O}_X}^\mathbf{L} K(\mathcal{O}_X, f_1^n, \ldots, f_r^n) \right) = R\lim (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X/\mathcal{I}^n)\] The first equality is Lemma 0A0E.
Assume \(K\) is pseudo-coherent. For \(U \subset X\) affine open we have \(H^q(U, K^\wedge) = \lim H^q(U, K)/\mathcal{I}^n(U)H^q(U, K)\) by Lemma 0A0L. As this is true for every \(U\) we see that \(H^q(K^\wedge) = \lim H^q(K)/\mathcal{I}^nH^q(K)\) as sheaves. This proves (2).
Part (3) is a special case of (2). Parts (4) and (5) follow from Derived Categories of Schemes, Lemma 0A0J.
Lemma
Let \(A\) be a Noetherian ring and let \(I \subset A\) be an ideal. Let \(X\) be a Noetherian scheme over \(A\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Assume that \(H^p(X, \mathcal{F})\) is a finite \(A\)-module for all \(p\). Then there are short exact sequences \[0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/I^n\mathcal{F}) \to H^p(X, \mathcal{F})^\wedge \to \lim H^p(X, \mathcal{F}/I^n\mathcal{F}) \to 0\] of \(A\)-modules where \(H^p(X, \mathcal{F})^\wedge\) is the usual \(I\)-adic completion. If \(f\) is proper, then the \(R^1\lim\) term is zero.
Proof
Consider the two spectral sequences of Lemma 0BLX. The first degenerates by More on Algebra, Lemma 0A06. We obtain \(H^p(X, \mathcal{F})^\wedge\) in degree \(p\). This is where we use the assumption that \(H^p(X, \mathcal{F})\) is a finite \(A\)-module. The second degenerates because \[\mathcal{F}^\wedge = \lim \mathcal{F}/I^n\mathcal{F} = R\lim \mathcal{F}/I^n\mathcal{F}\] is a sheaf by Lemma 0A0K. We obtain \(H^p(X, \lim \mathcal{F}/I^n\mathcal{F})\) in degree \(p\). Since \(R\Gamma(X, -)\) commutes with derived limits (Injectives, Lemma 08U1) we also get \[R\Gamma(X, \lim \mathcal{F}/I^n\mathcal{F}) = R\Gamma(X, R\lim \mathcal{F}/I^n\mathcal{F}) = R\lim R\Gamma(X, \mathcal{F}/I^n\mathcal{F})\] By More on Algebra, Remark 07KZ we obtain exact sequences \[0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/I^n\mathcal{F}) \to H^p(X, \lim \mathcal{F}/I^n\mathcal{F}) \to \lim H^p(X, \mathcal{F}/I^n\mathcal{F}) \to 0\] of \(A\)-modules. Combining the above we get the first statement of the lemma. The vanishing of the \(R^1\lim\) term follows from Cohomology of Schemes, Lemma 02OB.
Remark
Here are some references to discussions of related material the literature. It seems that a “derived formal functions theorem” for proper maps goes back to [lurie-thesis, Theorem 6.3.1]. There is the discussion in [dag12], especially Chapter 4 which discusses the affine story, see More on Algebra, Section 091N. In [G-R, Section 2.9] one finds a discussion of proper base change and derived completion using (ind) coherent modules. An analogue of (0A0I) for complexes of quasi-coherent modules can be found as [HL-P, Theorem 6.5]
Algebraization of local cohomology, I
Let \(A\) be a Noetherian ring and let \(I\) and \(J\) be two ideals of \(A\). Let \(M\) be a finite \(A\)-module. In this section we study the cohomology groups of the object \[R\Gamma_J(M)^\wedge \quad\text{of}\quad D(A)\] where \({}^\wedge\) denotes derived \(I\)-adic completion. Observe that in Dualizing Complexes, Lemma 0EEX we have shown, if \(A\) is complete with respect to \(I\), that there is an isomorphism \[\colim H^0_Z(M) \longrightarrow H^0(R\Gamma_J(M)^\wedge)\] where the (directed) colimit is over the closed subsets \(Z = V(J')\) with \(J' \subset J\) and \(V(J') \cap V(I) = V(J) \cap V(I)\). The union of these closed subsets is [0EFG]\[\begin{equation} T = \{\mathfrak p \in \Spec(A) : V(\mathfrak p) \cap V(I) \subset V(J) \cap V(I)\} \end{equation}\] This is a subset of \(\Spec(A)\) stable under specialization. The result above becomes the statement that \[H^0_T(M) \longrightarrow H^0(R\Gamma_J(M)^\wedge)\] is an isomorphism provided \(A\) is complete with respect to \(I\), see Local Cohomology, Lemma 0EF1 and Remark 0EF4. Our method to extend this isomorphism to higher cohomology groups rests on the following lemma.
Lemma
Let \(I, J\) be ideals of a Noetherian ring \(A\). Let \(M\) be a finite \(A\)-module. Let \(\mathfrak p \subset A\) be a prime. Let \(s\) and \(d\) be integers. Assume
\(A\) has a dualizing complex,
\(\mathfrak p \not \in V(J) \cap V(I)\),
\(\text{cd}(A, I) \leq d\), and
for all primes \(\mathfrak p' \subset \mathfrak p\) we have \(\text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_\mathfrak q) > d + s\) for all \(\mathfrak q \in V(\mathfrak p') \cap V(J) \cap V(I)\).
Then there exists an \(f \in A\), \(f \not \in \mathfrak p\) which annihilates \(H^i(R\Gamma_J(M)^\wedge)\) for \(i \leq s\) where \({}^\wedge\) indicates \(I\)-adic completion.
Proof
We will use that \(R\Gamma_J = R\Gamma_{V(J)}\) and similarly for \(I + J\), see Dualizing Complexes, Lemma 0955. Observe that \(R\Gamma_J(M)^\wedge = R\Gamma_I(R\Gamma_J(M))^\wedge = R\Gamma_{I + J}(M)^\wedge\), see Dualizing Complexes, Lemmas 0A6W and 0BJC. Thus we may replace \(J\) by \(I + J\) and assume \(I \subset J\) and \(\mathfrak p \not \in V(J)\). Recall that \[R\Gamma_J(M)^\wedge = R\Hom_A(R\Gamma_I(A), R\Gamma_J(M))\] by the description of derived completion in More on Algebra, Lemma 091V combined with the description of local cohomology in Dualizing Complexes, Lemma 0956. Assumption (3) means that \(R\Gamma_I(A)\) has nonzero cohomology only in degrees \(\leq d\). Using the canonical truncations of \(R\Gamma_I(A)\) we find it suffices to show that \[\text{Ext}^i(N, R\Gamma_J(M))\] is annihilated by an \(f \in A\), \(f \not \in \mathfrak p\) for \(i \leq s + d\) and any \(A\)-module \(N\). In turn using the canonical truncations for \(R\Gamma_J(M)\) we see that it suffices to show \(H^i_J(M)\) is annihilated by an \(f \in A\), \(f \not \in \mathfrak p\) for \(i \leq s + d\). This follows from Local Cohomology, Lemma 0EFE.
Lemma
Let \(I, J\) be ideals of a Noetherian ring. Let \(M\) be a finite \(A\)-module. Let \(s\) and \(d\) be integers. With \(T\) as in (0EFG) assume
\(A\) has a dualizing complex,
if \(\mathfrak p \in V(I)\), then no condition,
if \(\mathfrak p \not \in V(I)\), \(\mathfrak p \in T\), then \(\dim((A/\mathfrak p)_\mathfrak q) \leq d\) for some \(\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)\),
if \(\mathfrak p \not \in V(I)\), \(\mathfrak p \not \in T\), then \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s\] for all \(\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)\).
Then there exists an ideal \(J_0 \subset J\) with \(V(J_0) \cap V(I) = V(J) \cap V(I)\) such that for any \(J' \subset J_0\) with \(V(J') \cap V(I) = V(J) \cap V(I)\) the map \[R\Gamma_{J'}(M) \longrightarrow R\Gamma_{J_0}(M)\] induces an isomorphism in cohomology in degrees \(\leq s\) and moreover these modules are annihilated by a power of \(J_0I\).
Proof
Let us consider the set \[B = \{\mathfrak p \not \in V(I),\ \mathfrak p \in T,\text{ and } \text{depth}(M_\mathfrak p) \leq s\}\] Choose \(J_0 \subset J\) such that \(V(J_0)\) is the closure of \(B \cup V(J)\).
Claim I: \(V(J_0) \cap V(I) = V(J) \cap V(I)\).
Proof of Claim I. The inclusion \(\supset\) holds by construction. Let \(\mathfrak p\) be a minimal prime of \(V(J_0)\). If \(\mathfrak p \in B \cup V(J)\), then either \(\mathfrak p \in T\) or \(\mathfrak p \in V(J)\) and in both cases \(V(\mathfrak p) \cap V(I) \subset V(J) \cap V(I)\) as desired. If \(\mathfrak p \not \in B \cup V(J)\), then \(V(\mathfrak p) \cap B\) is dense, hence infinite, and we conclude that \(\text{depth}(M_\mathfrak p) < s\) by Local Cohomology, Lemma 0DWY. In fact, let \(V(\mathfrak p) \cap B = \{\mathfrak p_\lambda\}_{\lambda \in \Lambda}\). Pick \(\mathfrak q_\lambda \in V(\mathfrak p_\lambda) \cap V(J) \cap V(I)\) as in (3). Let \(\delta : \Spec(A) \to \mathbf{Z}\) be the dimension function associated to a dualizing complex \(\omega_A^\bullet\) for \(A\). Since \(\Lambda\) is infinite and \(\delta\) is bounded, there exists an infinite subset \(\Lambda' \subset \Lambda\) on which \(\delta(\mathfrak q_\lambda)\) is constant. For \(\lambda \in \Lambda'\) we have \[\text{depth}(M_{\mathfrak p_\lambda}) + \delta(\mathfrak p_\lambda) - \delta(\mathfrak q_\lambda) = \text{depth}(M_{\mathfrak p_\lambda}) + \dim((A/\mathfrak p_\lambda)_{\mathfrak q_\lambda}) \leq d + s\] by (3) and the definition of \(B\). By the semi-continuity of the function \(\text{depth} + \delta\) proved in Duality for Schemes, Lemma 0ECM we conclude that \[\text{depth}(M_\mathfrak p) + \dim((A/\mathfrak p)_{\mathfrak q_\lambda}) = \text{depth}(M_\mathfrak p) + \delta(\mathfrak p) - \delta(\mathfrak q_\lambda) \leq d + s\] Since also \(\mathfrak p \not \in V(I)\) we read off from (4) that \(\mathfrak p \in T\), i.e., \(V(\mathfrak p) \cap V(I) \subset V(J) \cap V(I)\). This finishes the proof of Claim I.
Claim II: \(H^i_{J_0}(M) \to H^i_J(M)\) is an isomorphism for \(i \leq s\) and \(J' \subset J_0\) with \(V(J') \cap V(I) = V(J) \cap V(I)\).
Proof of claim II. Choose \(\mathfrak p \in V(J')\) not in \(V(J_0)\). It suffices to show that \(H^i_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = 0\) for \(i \leq s\), see Local Cohomology, Lemma 0DWV. Observe that \(\mathfrak p \in T\). Hence since \(\mathfrak p\) is not in \(B\) we see that \(\text{depth}(M_\mathfrak p) > s\) and the groups vanish by Dualizing Complexes, Lemma 0AVZ.
Claim III. The final statement of the lemma is true.
By Claim II for \(i \leq s\) we have \[H^i_T(M) = H^i_{J_0}(M) = H^i_{J'}(M)\] for all ideals \(J' \subset J_0\) with \(V(J') \cap V(I) = V(J) \cap V(I)\). See Local Cohomology, Lemma 0EF1. Let us check the hypotheses of Local Cohomology, Proposition 0EFC for the subsets \(T \subset T \cup V(I)\), the module \(M\), and the integer \(s\). We have to show that given \(\mathfrak p \subset \mathfrak q\) with \(\mathfrak p \not \in T \cup V(I)\) and \(\mathfrak q \in T\) we have \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s\] If \(\text{depth}(M_\mathfrak p) \geq s\), then this is true because the dimension of \((A/\mathfrak p)_\mathfrak q\) is at least \(1\). Thus we may assume \(\text{depth}(M_\mathfrak p) < s\). If \(\mathfrak q \in V(I)\), then \(\mathfrak q \in V(J) \cap V(I)\) and the inequality holds by (4). If \(\mathfrak q \not \in V(I)\), then we can use (3) to pick \(\mathfrak q' \in V(\mathfrak q) \cap V(J) \cap V(I)\) with \(\dim((A/\mathfrak q)_{\mathfrak q'}) \leq d\). Then assumption (4) gives \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_{\mathfrak q'}) > s + d\] Since \(A\) is catenary this implies the inequality we want. Applying Local Cohomology, Proposition 0EFC we find \(J'' \subset A\) with \(V(J'') \subset T \cup V(I)\) such that \(J''\) annihilates \(H^i_T(M)\) for \(i \leq s\). Then we can write \(V(J'') \cup V(J_0) \cup V(I) = V(J'I)\) for some \(J' \subset J_0\) with \(V(J') \cap V(I) = V(J) \cap V(I)\). Replacing \(J_0\) by \(J'\) the proof is complete.
Lemma
In Lemma 0EFI if instead of the empty condition (2) we assume
if \(\mathfrak p \in V(I)\), \(\mathfrak p \not \in V(J) \cap V(I)\), then \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s\) for all \(\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)\),
then the conditions also imply that \(H^i_{J_0}(M)\) is a finite \(A\)-module for \(i \leq s\).
Proof
Recall that \(H^i_{J_0}(M) = H^i_T(M)\), see proof of Lemma 0EFI. Thus it suffices to check that for \(\mathfrak p \not \in T\) and \(\mathfrak q \in T\) with \(\mathfrak p \subset \mathfrak q\) we have \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > s\), see Local Cohomology, Proposition 0EFD. Condition (2’) tells us this is true for \(\mathfrak p \in V(I)\). Since we know \(H^i_T(M)\) is annihilated by a power of \(IJ_0\) we know the condition holds if \(\mathfrak p \not \in V(IJ_0)\) by Local Cohomology, Proposition 0EFC. This covers all cases and the proof is complete.
Lemma
If in Lemma 0EFI we additionally assume
if \(\mathfrak p \not \in V(I)\), \(\mathfrak p \in T\), then \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s\),
then \(H^i_{J_0}(M) = H^i_J(M) = H^i_{J + I}(M)\) for \(i \leq s\) and these modules are annihilated by a power of \(I\).
Proof
Choose \(\mathfrak p \in V(J)\) or \(\mathfrak p \in V(J_0)\) but \(\mathfrak p \not \in V(J + I) = V(J_0 + I)\). It suffices to show that \(H^i_{\mathfrak pA_\mathfrak p}(M_\mathfrak p) = 0\) for \(i \leq s\), see Local Cohomology, Lemma 0DWV. These groups vanish by condition (6) and Dualizing Complexes, Lemma 0AVZ. The final statement follows from Local Cohomology, Proposition 0EFC.
Lemma
Let \(I, J\) be ideals of a Noetherian ring \(A\). Let \(M\) be a finite \(A\)-module. Let \(s\) and \(d\) be integers. With \(T\) as in (0EFG) assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(\mathfrak p \in V(I)\) no condition,
\(\text{cd}(A, I) \leq d\),
if \(\mathfrak p \not \in V(I)\), \(\mathfrak p \not \in T\) then \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s\] for all \(\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)\),
if \(\mathfrak p \not \in V(I)\), \(\mathfrak p \not \in T\), \(V(\mathfrak p) \cap V(J) \cap V(I) \not = \emptyset\), and \(\text{depth}(M_\mathfrak p) < s\), then one of the following holds2:
\(\dim(\text{Supp}(M_\mathfrak p)) < s + 2\)3, or
\(\delta(\mathfrak p) > d + \delta_{max} - 1\) where \(\delta\) is a dimension function and \(\delta_{max}\) is the maximum of \(\delta\) on \(V(J) \cap V(I)\), or
\(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s + \delta_{max} - \delta_{min} - 2\) for all \(\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)\).
Then there exists an ideal \(J_0 \subset J\) with \(V(J_0) \cap V(I) = V(J) \cap V(I)\) such that for any \(J' \subset J_0\) with \(V(J') \cap V(I) = V(J) \cap V(I)\) the map \[R\Gamma_{J'}(M) \longrightarrow R\Gamma_J(M)^\wedge\] induces an isomorphism on cohomology in degrees \(\leq s\). Here \({}^\wedge\) denotes derived \(I\)-adic completion.
We encourage the reader to read the proof in the local case first (Lemma 0DXP) as it explains the structure of the proof without having to deal with all the inequalities.
Proof
For an ideal \(\mathfrak a \subset A\) we have \(R\Gamma_\mathfrak a = R\Gamma_{V(\mathfrak a)}\), see Dualizing Complexes, Lemma 0955. Next, we observe that \[R\Gamma_J(M)^\wedge = R\Gamma_I(R\Gamma_J(M))^\wedge = R\Gamma_{I + J}(M)^\wedge = R\Gamma_{I + J'}(M)^\wedge = R\Gamma_I(R\Gamma_{J'}(M))^\wedge = R\Gamma_{J'}(M)^\wedge\] by Dualizing Complexes, Lemmas 0BJC and 0A6W. This explains how we define the arrow in the statement of the lemma.
We claim that the hypotheses of Lemma 0EFI are implied by our current hypotheses on \(M\). The only thing to verify is hypothesis (3). Thus let \(\mathfrak p \not \in V(I)\), \(\mathfrak p \in T\). Then \(V(\mathfrak p) \cap V(I)\) is nonempty as \(I\) is contained in the Jacobson radical of \(A\) (Algebra, Lemma 05GI). Since \(\mathfrak p \in T\) we have \(V(\mathfrak p) \cap V(I) = V(\mathfrak p) \cap V(J) \cap V(I)\). Let \(\mathfrak q \in V(\mathfrak p) \cap V(I)\) be the generic point of an irreducible component. We have \(\text{cd}(A_\mathfrak q, I_\mathfrak q) \leq d\) by Local Cohomology, Lemma 0DXB. We have \(V(\mathfrak pA_\mathfrak q) \cap V(I_\mathfrak q) = \{\mathfrak qA_\mathfrak q\}\) by our choice of \(\mathfrak q\) and we conclude \(\dim((A/\mathfrak p)_\mathfrak q) \leq d\) by Local Cohomology, Lemma 0DXF.
Observe that the lemma holds for \(s < 0\). This is not a trivial case because it is not a priori clear that \(H^i(R\Gamma_J(M)^\wedge)\) is zero for \(i < 0\). However, this vanishing was established in Dualizing Complexes, Lemma 0EEW. We will prove the lemma by induction for \(s \geq 0\).
The lemma for \(s = 0\) follows immediately from the conclusion of Lemma 0EFI and Dualizing Complexes, Lemma 0EEX.
Assume \(s > 0\) and the lemma has been shown for smaller values of \(s\). Let \(M' \subset M\) be the maximal submodule whose support is contained in \(V(I) \cup T\). Then \(M'\) is a finite \(A\)-module whose support is contained in \(V(J') \cup V(I)\) for some ideal \(J' \subset J\) with \(V(J') \cap V(I) = V(J) \cap V(I)\). We claim that \[R\Gamma_{J'}(M') \to R\Gamma_J(M')^\wedge\] is an isomorphism for any choice of \(J'\). Namely, we can choose a short exact sequence \(0 \to M_1 \oplus M_2 \to M' \to N \to 0\) with \(M_1\) annihilated by a power of \(J'\), with \(M_2\) annihilated by a power of \(I\), and with \(N\) annihilated by a power of \(I + J'\). Thus it suffices to show that the claim holds for \(M_1\), \(M_2\), and \(N\). In the case of \(M_1\) we see that \(R\Gamma_{J'}(M_1) = M_1\) and since \(M_1\) is a finite \(A\)-module and \(I\)-adically complete we have \(M_1^\wedge = M_1\). This proves the claim for \(M_1\) by the initial remarks of the proof. In the case of \(M_2\) we see that \(H^i_J(M_2) = H^i_{I + J}(M) = H^i_{I + J'}(M) = H^i_{J'}(M_2)\) are annihilated by a power of \(I\) and hence derived complete. Thus the claim in this case also. For \(N\) we can use either of the arguments just given. Considering the short exact sequence \(0 \to M' \to M \to M/M' \to 0\) we see that it suffices to prove the lemma for \(M/M'\). Thus we may assume \(\text{Ass}(M) \cap (V(I) \cup T) = \emptyset\).
Let \(\mathfrak p \in \text{Ass}(M)\) be such that \(V(\mathfrak p) \cap V(J) \cap V(I) = \emptyset\). Since \(I\) is contained in the Jacobson radical of \(A\) this implies that \(V(\mathfrak p) \cap V(J') = \emptyset\) for any \(J' \subset J\) with \(V(J') \cap V(I) = V(J) \cap V(I)\). Thus setting \(N = H^0_\mathfrak p(M)\) we see that \(R\Gamma_J(N) = R\Gamma_{J'}(N) = 0\) for all \(J' \subset J\) with \(V(J') \cap V(I) = V(J) \cap V(I)\). In particular \(R\Gamma_J(N)^\wedge = 0\). Thus we may replace \(M\) by \(M/N\) as this changes the structure of \(M\) only in primes which do not play a role in conditions (4) or (5). Repeating we may assume that \(V(\mathfrak p) \cap V(J) \cap V(I) \not = \emptyset\) for all \(\mathfrak p \in \text{Ass}(M)\).
Assume \(\text{Ass}(M) \cap (V(I) \cup T) = \emptyset\) and that \(V(\mathfrak p) \cap V(J) \cap V(I) \not = \emptyset\) for all \(\mathfrak p \in \text{Ass}(M)\). Let \(\mathfrak p \in \text{Ass}(M)\). We want to show that we may apply Lemma 0EFH. It is in the verification of this that we will use the supplemental condition (5). Choose \(\mathfrak p' \subset \mathfrak p\) and \(\mathfrak q' \subset V(\mathfrak p) \cap V(J) \cap V(I)\).
If \(M_{\mathfrak p'} = 0\), then \(\text{depth}(M_{\mathfrak p'}) = \infty\) and \(\text{depth}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_{\mathfrak q'}) > d + s\).
If \(\text{depth}(M_{\mathfrak p'}) < s\), then \(\text{depth}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_{\mathfrak q'}) > d + s\) by (4).
In the remaining cases we have \(M_{\mathfrak p'} \not = 0\) and \(\text{depth}(M_{\mathfrak p'}) \geq s\). In particular, we see that \(\mathfrak p'\) is in the support of \(M\) and we can choose \(\mathfrak p'' \subset \mathfrak p'\) with \(\mathfrak p'' \in \text{Ass}(M)\).
Observe that \(\dim((A/\mathfrak p'')_{\mathfrak p'}) \geq \text{depth}(M_{\mathfrak p'})\) by Algebra, Lemma 0BK4. If equality holds, then we have \[\text{depth}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_{\mathfrak q'}) = \text{depth}(M_{\mathfrak p''}) + \dim((A/\mathfrak p'')_{\mathfrak q'}) > s + d\] by (4) applied to \(\mathfrak p''\) and we are done. This means we are only in trouble if \(\dim((A/\mathfrak p'')_{\mathfrak p'}) > \text{depth}(M_{\mathfrak p'})\). This implies that \(\dim(M_\mathfrak p) \geq s + 2\). Thus if (5)(a) holds, then this does not occur.
If (5)(b) holds, then we get \[\text{depth}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_{\mathfrak q'}) \geq s + \delta(\mathfrak p') - \delta(\mathfrak q') \geq s + 1 + \delta(\mathfrak p) - \delta_{max} > s + d\] as desired.
If (5)(c) holds, then we get \[\begin{align*} \text{depth}(M_{\mathfrak p'}) + \dim((A/\mathfrak p')_{\mathfrak q'}) & \geq s + \delta(\mathfrak p') - \delta(\mathfrak q') \\ & \geq s + 1 + \delta(\mathfrak p) - \delta(\mathfrak q') \\ & = s + 1 + \delta(\mathfrak p) - \delta(\mathfrak q) + \delta(\mathfrak q) - \delta(\mathfrak q') \\ & > s + 1 + (s + d + \delta_{max} - \delta_{min} - 2) + \delta(\mathfrak q) - \delta(\mathfrak q') \\ & \geq 2s + d - 1 \geq s + d \end{align*}\] as desired. Observe that this argument works because we know that a prime \(\mathfrak q \in V(\mathfrak p) \cap V(J) \cap V(I)\) exists.
Now we are ready to do the induction step.
Choose an ideal \(J_0\) as in Lemma 0EFI and an integer \(t > 0\) such that \((J_0I)^t\) annihilates \(H^s_J(M)\). The assumptions of Lemma 0EFH are satisfied for every \(\mathfrak p \in \text{Ass}(M)\) (see previous paragraph). Thus the annihilator \(\mathfrak a \subset A\) of \(H^s(R\Gamma_J(M)^\wedge)\) is not contained in \(\mathfrak p\) for \(\mathfrak p \in \text{Ass}(M)\). Thus we can find an \(f \in \mathfrak a(J_0I)^t\) not in any associated prime of \(M\) which is an annihilator of both \(H^s(R\Gamma_J(M)^\wedge)\) and \(H^s_J(M)\). Then \(f\) is a nonzerodivisor on \(M\) and we can consider the short exact sequence \[0 \to M \xrightarrow{f} M \to M/fM \to 0\] Our choice of \(f\) shows that we obtain \[\xymatrix{ H^{s - 1}_{J'}(M) \ar[d] \ar[r] & H^{s - 1}_{J'}(M/fM) \ar[d] \ar[r] & H^s_{J'}(M) \ar[d] \ar[r] & 0 \\ H^{s - 1}(R\Gamma_J(M)^\wedge) \ar[r] & H^{s - 1}(R\Gamma_J(M/fM)^\wedge) \ar[r] & H^s(R\Gamma_J(M)^\wedge) \ar[r] & 0 }\] for any \(J' \subset J_0\) with \(V(J') \cap V(I) = V(J) \cap V(I)\). Thus if we choose \(J'\) such that it works for \(M\) and \(M/fM\) and \(s - 1\) (possible by induction hypothesis – see next paragraph), then we conclude that the lemma is true.
To finish the proof we have to show that the module \(M/fM\) satisfies the hypotheses (4) and (5) for \(s - 1\). Thus we let \(\mathfrak p\) be a prime in the support of \(M/fM\) with \(\text{depth}((M/fM)_\mathfrak p) < s - 1\) and with \(V(\mathfrak p) \cap V(J) \cap V(I)\) nonempty. Then \(\dim(M_\mathfrak p) = \dim((M/fM)_\mathfrak p) + 1\) and \(\text{depth}(M_\mathfrak p) = \text{depth}((M/fM)_\mathfrak p) + 1\). In particular, we know (4) and (5) hold for \(\mathfrak p\) and \(M\) with the original value \(s\). The desired inequalities then follow by inspection.
Example
In Lemma 0EFL we do not know that the inverse systems \(H^i_J(M/I^nM)\) satisfy the Mittag-Leffler condition. For example, suppose that \(A = \mathbf{Z}_p[[x, y]]\), \(I = (p)\), \(J = (p, x)\), and \(M = A/(xy - p)\). Then the image of \(H^0_J(M/p^nM) \to H^0_J(M/pM)\) is the ideal generated by \(y^n\) in \(M/pM = A/(p, xy)\).
Algebraization of local cohomology, II
In this section we redo the arguments of Section 0EFF when \((A, \mathfrak m)\) is a local ring and we take local cohomology \(R\Gamma_\mathfrak m\) with respect to \(\mathfrak m\). As before our main tool is the following lemma.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(I \subset A\) be an ideal. Let \(M\) be a finite \(A\)-module and let \(\mathfrak p \subset A\) be a prime. Let \(s\) and \(d\) be integers. Assume
\(A\) has a dualizing complex,
\(\text{cd}(A, I) \leq d\), and
\(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s\).
Then there exists an \(f \in A \setminus \mathfrak p\) which annihilates \(H^i(R\Gamma_\mathfrak m(M)^\wedge)\) for \(i \leq s\) where \({}^\wedge\) indicates \(I\)-adic completion.
Proof
According to Local Cohomology, Lemma 0DWZ the function \[\mathfrak p' \longmapsto \text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim(A/\mathfrak p')\] is lower semi-continuous on \(\Spec(A)\). Thus the value of this function on \(\mathfrak p' \subset \mathfrak p\) is \(> s + d\). Thus our lemma is a special case of Lemma 0EFH provided that \(\mathfrak p \not = \mathfrak m\). If \(\mathfrak p = \mathfrak m\), then we have \(H^i_\mathfrak m(M) = 0\) for \(i \leq s + d\) by the relationship between depth and local cohomology (Dualizing Complexes, Lemma 0AVZ). Thus the argument given in the proof of Lemma 0EFH shows that \(H^i(R\Gamma_\mathfrak m(M)^\wedge) = 0\) for \(i \leq s\) in this (degenerate) case.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(I \subset A\) be an ideal. Let \(M\) be a finite \(A\)-module. Let \(s\) and \(d\) be integers. Assume
\(A\) has a dualizing complex,
if \(\mathfrak p \in V(I)\), then no condition,
if \(\mathfrak p \not \in V(I)\) and \(V(\mathfrak p) \cap V(I) = \{\mathfrak m\}\), then \(\dim(A/\mathfrak p) \leq d\),
if \(\mathfrak p \not \in V(I)\) and \(V(\mathfrak p) \cap V(I) \not = \{\mathfrak m\}\), then \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s\]
Then there exists an ideal \(J_0 \subset A\) with \(V(J_0) \cap V(I) = \{\mathfrak m\}\) such that for any \(J \subset J_0\) with \(V(J) \cap V(I) = \{\mathfrak m\}\) the map \[R\Gamma_J(M) \longrightarrow R\Gamma_{J_0}(M)\] induces an isomorphism in cohomology in degrees \(\leq s\) and moreover these modules are annihilated by a power of \(J_0I\).
Proof
This is a special case of Lemma 0EFI.
Lemma
In Lemma 0DXM if instead of the empty condition (2) we assume
if \(\mathfrak p \in V(I)\) and \(\mathfrak p \not = \mathfrak m\), then \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > s\),
then the conditions also imply that \(H^i_{J_0}(M)\) is a finite \(A\)-module for \(i \leq s\).
Proof
This is a special case of Lemma 0EFJ.
Lemma
If in Lemma 0DXM we additionally assume
if \(\mathfrak p \not \in V(I)\) and \(V(\mathfrak p) \cap V(I) = \{\mathfrak m\}\), then \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s\),
then \(H^i_{J_0}(M) = H^i_J(M) = H^i_\mathfrak m(M)\) for \(i \leq s\) and these modules are annihilated by a power of \(I\).
Proof
This is a special case of Lemma 0EFK.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(I \subset A\) be an ideal. Let \(M\) be a finite \(A\)-module. Let \(s\) and \(d\) be integers. Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(\mathfrak p \in V(I)\), no condition,
\(\text{cd}(A, I) \leq d\),
if \(\mathfrak p \not \in V(I)\) and \(V(\mathfrak p) \cap V(I) \not = \{\mathfrak m\}\) then \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > d + s\]
Then there exists an ideal \(J_0 \subset A\) with \(V(J_0) \cap V(I) = \{\mathfrak m\}\) such that for any \(J \subset J_0\) with \(V(J) \cap V(I) = \{\mathfrak m\}\) the map \[R\Gamma_J(M) \longrightarrow R\Gamma_J(M)^\wedge = R\Gamma_\mathfrak m(M)^\wedge\] induces an isomorphism in cohomology in degrees \(\leq s\). Here \({}^\wedge\) denotes derived \(I\)-adic completion.
Proof
This lemma is a special case of Lemma 0EFL since condition (5)(c) is implied by condition (4) as \(\delta_{max} = \delta_{min} = \delta(\mathfrak m)\). We will give the proof of this important special case as it is somewhat easier (fewer things to check).
There is no difference between \(R\Gamma_\mathfrak a\) and \(R\Gamma_{V(\mathfrak a)}\) in our current situation, see Dualizing Complexes, Lemma 0955. Next, we observe that \[R\Gamma_\mathfrak m(M)^\wedge = R\Gamma_I(R\Gamma_J(M))^\wedge = R\Gamma_J(M)^\wedge\] by Dualizing Complexes, Lemmas 0BJC and 0A6W which explains the equality sign in the statement of the lemma.
Observe that the lemma holds for \(s < 0\). This is not a trivial case because it is not a priori clear that \(H^s(R\Gamma_\mathfrak m(M)^\wedge)\) is zero for negative \(s\). However, this vanishing was established in Lemma 0DXJ. We will prove the lemma by induction for \(s \geq 0\).
The assumptions of Lemma 0DXM are satisfied by Local Cohomology, Lemma 0DXF. The lemma for \(s = 0\) follows from Lemma 0DXM and Dualizing Complexes, Lemma 0EEX.
Assume \(s > 0\) and the lemma holds for smaller values of \(s\). Let \(M' \subset M\) be the submodule of elements whose support is condained in \(V(I) \cup V(J)\) for some ideal \(J\) with \(V(J) \cap V(I) = \{\mathfrak m\}\). Then \(M'\) is a finite \(A\)-module. We claim that \[R\Gamma_J(M') \to R\Gamma_\mathfrak m(M')^\wedge\] is an isomorphism for any choice of \(J\). Namely, for any such module there is a short exact sequence \(0 \to M_1 \oplus M_2 \to M' \to N \to 0\) with \(M_1\) annihilated by a power of \(J\), with \(M_2\) annihilated by a power of \(I\) and with \(N\) annihilated by a power of \(\mathfrak m\). In the case of \(M_1\) we see that \(R\Gamma_J(M_1) = M_1\) and since \(M_1\) is a finite \(A\)-module and \(I\)-adically complete we have \(M_1^\wedge = M_1\). Thus the claim holds for \(M_1\). In the case of \(M_2\) we see that \(H^i_J(M_2)\) is annihilated by a power of \(I\) and hence derived complete. Thus the claim for \(M_2\). By the same arguments the claim holds for \(N\) and we conclude that the claim holds. Considering the short exact sequence \(0 \to M' \to M \to M/M' \to 0\) we see that it suffices to prove the lemma for \(M/M'\). This we may assume \(\mathfrak p \in \text{Ass}(M)\) implies \(V(\mathfrak p) \cap V(I) \not = \{\mathfrak m\}\), i.e., \(\mathfrak p\) is a prime as in (4).
Choose an ideal \(J_0\) as in Lemma 0DXM and an integer \(t > 0\) such that \((J_0I)^t\) annihilates \(H^s_J(M)\). Here \(J\) denotes an arbitrary ideal \(J \subset J_0\) with \(V(J) \cap V(I) = \{\mathfrak m\}\). The assumptions of Lemma 0DXK are satisfied for every \(\mathfrak p \in \text{Ass}(M)\) (see previous paragraph). Thus the annihilator \(\mathfrak a \subset A\) of \(H^s(R\Gamma_\mathfrak m(M)^\wedge)\) is not contained in \(\mathfrak p\) for \(\mathfrak p \in \text{Ass}(M)\). Thus we can find an \(f \in \mathfrak a(J_0I)^t\) not in any associated prime of \(M\) which is an annihilator of both \(H^s(R\Gamma_\mathfrak m(M)^\wedge)\) and \(H^s_J(M)\). Then \(f\) is a nonzerodivisor on \(M\) and we can consider the short exact sequence \[0 \to M \xrightarrow{f} M \to M/fM \to 0\] Our choice of \(f\) shows that we obtain \[\xymatrix{ H^{s - 1}_J(M) \ar[d] \ar[r] & H^{s - 1}_J(M/fM) \ar[d] \ar[r] & H^s_J(M) \ar[d] \ar[r] & 0 \\ H^{s - 1}(R\Gamma_\mathfrak m(M)^\wedge) \ar[r] & H^{s - 1}(R\Gamma_\mathfrak m(M/fM)^\wedge) \ar[r] & H^s(R\Gamma_\mathfrak m(M)^\wedge) \ar[r] & 0 }\] for any \(J \subset J_0\) with \(V(J) \cap V(I) = \{\mathfrak m\}\). Thus if we choose \(J\) such that it works for \(M\) and \(M/fM\) and \(s - 1\) (possible by induction hypothesis), then we conclude that the lemma is true.
Algebraization of local cohomology, III
In this section we bootstrap the material in Sections 0EFF and 0EFP to give a stronger result the following situation.
Situation
Here \(A\) is a Noetherian ring. We have an ideal \(I \subset A\), a finite \(A\)-module \(M\), and a subset \(T \subset V(I)\) stable under specialization. We have integers \(s\) and \(d\). We assume
\(A\) has a dualizing complex,
\(\text{cd}(A, I) \leq d\),
given primes \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) with \(\mathfrak p \not \in V(I)\), \(\mathfrak r \in V(I) \setminus T\), \(\mathfrak q \in T\) we have \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s\]
given \(\mathfrak q \in T\) denoting \(A', \mathfrak m', I', M'\) are the usual \(I\)-adic completions of \(A_\mathfrak q, \mathfrak qA_\mathfrak q, I_\mathfrak q, M_\mathfrak q\) we have \[\text{depth}(M'_{\mathfrak p'}) > s\] for all \(\mathfrak p' \in \Spec(A') \setminus V(I')\) with \(V(\mathfrak p') \cap V(I') = \{\mathfrak m'\}\).
The following lemma explains why in Situation 0EFU it suffices to look at triples \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) of primes in (4) even though the actual assumption only involves \(\mathfrak p\) and \(\mathfrak q\).
Lemma
In Situation 0EFU let \(\mathfrak p \subset \mathfrak q\) be primes of \(A\) with \(\mathfrak p \not \in V(I)\) and \(\mathfrak q \in T\). If there does not exist an \(\mathfrak r \in V(I) \setminus T\) with \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) then \(\text{depth}(M_\mathfrak p) > s\).
Proof
Choose \(\mathfrak q' \in T\) with \(\mathfrak p \subset \mathfrak q' \subset \mathfrak q\) such that there is no prime in \(T\) strictly in between \(\mathfrak p\) and \(\mathfrak q'\). To prove the lemma we may and do replace \(\mathfrak q\) by \(\mathfrak q'\). Next, let \(\mathfrak p' \subset A_\mathfrak q\) be the prime corresponding to \(\mathfrak p\). After doing this we obtain that \(V(\mathfrak p') \cap V(IA_\mathfrak q) = \{\mathfrak q A_\mathfrak q\}\) because of the nonexistence of a prime \(\mathfrak r\) as in the lemma. Let \(A', I', \mathfrak m', M'\) be the \(I\)-adic completions of \(A_\mathfrak q, I_\mathfrak q, \mathfrak qA_\mathfrak q, M_\mathfrak q\). Since \(A_\mathfrak q \to A'\) is faithfully flat (Algebra, Lemma 00MC) we can choose \(\mathfrak p'' \subset A'\) lying over \(\mathfrak p'\) with \(\dim(A'_{\mathfrak p''}/\mathfrak p' A'_{\mathfrak p''}) = 0\). Then we see that \[\text{depth}(M'_{\mathfrak p''}) = \text{depth}((M_\mathfrak q \otimes_{A_\mathfrak q} A')_{\mathfrak p''}) = \text{depth}(M_\mathfrak p \otimes_{A_\mathfrak p} A'_{\mathfrak p''}) = \text{depth}(M_\mathfrak p)\] by flatness of \(A \to A'\) and our choice of \(\mathfrak p''\), see Algebra, Lemma 0338. Since \(\mathfrak p''\) lies over \(\mathfrak p'\) we have \(V(\mathfrak p'') \cap V(I') = \{\mathfrak m'\}\). Thus condition (6) in Situation 0EFU implies \(\text{depth}(M'_{\mathfrak p''}) > s\) which finishes the proof.
The following tedious lemma explains the relationships between various collections of conditions one might impose.
Lemma
In Situation 0EFU we have
if \(T' \subset T\) is a smaller specialization stable subset, then \(A, I, T', M\) satisfies the assumptions of Situation 0EFU,
if \(S \subset A\) is a multiplicative subset, then \(S^{-1}A, S^{-1}I, T', S^{-1}M\) satisfies the assumptions of Situation 0EFU where \(T' \subset V(S^{-1}I)\) is the inverse image of \(T\),
the quadruple \(A', I', T', M'\) satisfies the assumptions of Situation 0EFU where \(A', I', M'\) are the usual \(I\)-adic completions of \(A, I, M\) and \(T' \subset V(I')\) is the inverse image of \(T\).
Let \(I \subset \mathfrak a \subset A\) be an ideal such that \(V(\mathfrak a) \subset T\). Then
if \(I\) is contained in the Jacobson radical of \(A\), then all hypotheses of Lemmas 0EFI and 0EFK are satisfied for \(A, I, \mathfrak a, M\),
if \(A\) is complete with respect to \(I\), then all hypotheses except for possibly (5) of Lemma 0EFL are satisfied for \(A, I, \mathfrak a, M\),
if \(A\) is local with maximal ideal \(\mathfrak m = \mathfrak a\), then all hypotheses of Lemmas 0DXM and 0EFQ hold for \(A, \mathfrak m, I, M\),
if \(A\) is local with maximal ideal \(\mathfrak m = \mathfrak a\) and \(I\)-adically complete, then all hypotheses of Lemma 0DXP hold for \(A, \mathfrak m, I, M\),
Proof
Proof of (E). We have to prove assumptions (1), (3), (4), (6) of Situation 0EFU hold for \(A, I, T, M\). Shrinking \(T\) to \(T'\) weakens assumption (6) and strengthens assumption (4). However, if we have \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) with \(\mathfrak p \not \in V(I)\), \(\mathfrak r \in V(I) \setminus T'\), \(\mathfrak q \in T'\) as in assumption (4) for \(A, I, T', M\), then either we can pick \(\mathfrak r \in V(I) \setminus T\) and condition (4) for \(A, I, T, M\) kicks in or we cannot find such an \(\mathfrak r\) in which case we get \(\text{depth}(M_\mathfrak p) > s\) by Lemma 0EID. This proves (4) holds for \(A, I, T', M\) as desired.
Proof of (F). This is straightforward and we omit the details.
Proof of (G). We have to prove assumptions (1), (3), (4), (6) of Situation 0EFU hold for the \(I\)-adic completions \(A', I', T', M'\). Please keep in mind that \(\Spec(A') \to \Spec(A)\) induces an isomorphism \(V(I') \to V(I)\).
Assumption (1): The ring \(A'\) has a dualizing complex, see Dualizing Complexes, Lemma 0BFR.
Assumption (3): Since \(I' = IA'\) this follows from Local Cohomology, Lemma 0DXA.
Assumption (4): If we have primes \(\mathfrak p' \subset \mathfrak r' \subset \mathfrak q'\) in \(A'\) with \(\mathfrak p' \not \in V(I')\), \(\mathfrak r' \in V(I') \setminus T'\), \(\mathfrak q' \in T'\) then their images \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) in the spectrum of \(A\) satisfy \(\mathfrak p \not \in V(I)\), \(\mathfrak r \in V(I) \setminus T\), \(\mathfrak q \in T\). Then we have \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s\] by assumption (4) for \(A, I, T, M\). We have \(\text{depth}(M'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p)\) and \(\text{depth}(M'_{\mathfrak p'}) + \dim((A'/\mathfrak p')_{\mathfrak q'}) = \text{depth}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q)\) by Local Cohomology, Lemma 0EHW. Thus assumption (4) holds for \(A', I', T', M'\).
Assumption (6): Let \(\mathfrak q' \in T'\) lying over the prime \(\mathfrak q \in T\). Then \(A'_{\mathfrak q'}\) and \(A_\mathfrak q\) have isomorphic \(I\)-adic completions and similarly for \(M_\mathfrak q\) and \(M'_{\mathfrak q'}\). Thus assumption (6) for \(A', I', T', M'\) is equivalent to assumption (6) for \(A, I, T, M\).
Proof of (A). We have to check conditions (1), (2), (3), (4), and (6) of Lemmas 0EFI and 0EFK for \((A, I, \mathfrak a, M)\). Warning: the set \(T\) in the statement of these lemmas is not the same as the set \(T\) above.
Condition (1): This holds because we have assumed \(A\) has a dualizing complex in Situation 0EFU.
Condition (2): This is empty.
Condition (3): Let \(\mathfrak p \subset A\) with \(V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)\). Since \(I\) is contained in the Jacobson radical of \(A\) we see that \(V(\mathfrak p) \cap V(I) \not = \emptyset\). Let \(\mathfrak q \in V(\mathfrak p) \cap V(I)\) be a generic point. Since \(\text{cd}(A_\mathfrak q, I_\mathfrak q) \leq d\) (Local Cohomology, Lemma 0DXB) and since \(V(\mathfrak p A_\mathfrak q) \cap V(I_\mathfrak q) = \{\mathfrak q A_\mathfrak q\}\) we get \(\dim((A/\mathfrak p)_\mathfrak q) \leq d\) by Local Cohomology, Lemma 0DXF which proves (3).
Condition (4): Suppose \(\mathfrak p \not \in V(I)\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\). It suffices to show \[\text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s\] If there exists a prime \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) with \(\mathfrak r \in V(I) \setminus T\), then this follows immediately from assumption (4) in Situation 0EFU. If not, then \(\text{depth}(M_\mathfrak p) > s\) by Lemma 0EID.
Condition (6): Let \(\mathfrak p \not \in V(I)\) with \(V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)\). Since \(I\) is contained in the Jacobson radical of \(A\) we see that \(V(\mathfrak p) \cap V(I) \not = \emptyset\). Choose \(\mathfrak q \in V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)\). It is clear there does not exist a prime \(\mathfrak p \subset \mathfrak r \subset \mathfrak q\) with \(\mathfrak r \in V(I) \setminus T\). By Lemma 0EID we have \(\text{depth}(M_\mathfrak p) > s\) which proves (6).
Proof of (B). We have to check conditions (1), (2), (3), (4) of Lemma 0EFL. Warning: the set \(T\) in the statement of this lemma is not the same as the set \(T\) above.
Condition (1): This holds because \(A\) is complete and has a dualizing complex.
Condition (2): This is empty.
Condition (3): This is the same as assumption (3) in Situation 0EFU.
Condition (4): This is the same as assumption (4) in Lemma 0EFI which we proved in (A).
Proof of (C). This is true because the assumptions in Lemmas 0DXM and 0EFQ are the same as the assumptions in Lemmas 0EFI and 0EFK in the local case and we proved these hold in (A).
Proof of (D). This is true because the assumptions in Lemma 0DXP are the same as the assumptions (1), (2), (3), (4) in Lemma 0EFL and we proved these hold in (B).
Lemma
In Situation 0EFU assume \(A\) is local with maximal ideal \(\mathfrak m\) and \(T = \{\mathfrak m\}\). Then \(H^i_\mathfrak m(M) \to \lim H^i_\mathfrak m(M/I^nM)\) is an isomorphism for \(i \leq s\) and these modules are annihilated by a power of \(I\).
Proof
Let \(A', I', \mathfrak m', M'\) be the usual \(I\)-adic completions of \(A, I, \mathfrak m, M\). Recall that we have \(H^i_\mathfrak m(M) \otimes_A A' = H^i_{\mathfrak m'}(M')\) by flatness of \(A \to A'\) and Dualizing Complexes, Lemma 0ALZ. Since \(H^i_\mathfrak m(M)\) is \(\mathfrak m\)-power torsion we have \(H^i_\mathfrak m(M) = H^i_\mathfrak m(M) \otimes_A A'\), see More on Algebra, Lemma 05EC. We conclude that \(H^i_\mathfrak m(M) = H^i_{\mathfrak m'}(M')\). The exact same arguments will show that \(H^i_\mathfrak m(M/I^nM) = H^i_{\mathfrak m'}(M'/(I')^nM')\) for all \(n\) and \(i\).
Lemmas 0DXP, 0DXM, and 0EFQ apply to \(A', \mathfrak m', I', M'\) by Lemma 0EFV parts (C) and (D). Thus we get an isomorphism \[H^i_{\mathfrak m'}(M') \longrightarrow H^i(R\Gamma_{\mathfrak m'}(M')^\wedge)\] for \(i \leq s\) where \({}^\wedge\) is derived \(I'\)-adic completion and these modules are annihilated by a power of \(I'\). By Lemma 0DXJ we obtain isomorphisms \[H^i_{\mathfrak m'}(M') \longrightarrow \lim H^i_{\mathfrak m'}(M'/(I')^nM'))\] for \(i \leq s\). Combined with the already established comparison with local cohomology over \(A\) we conclude the lemma is true.
Lemma
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(M\) be a finite \(A\)-module. Let \(s\) and \(d\) be integers. If we assume
\(A\) has a dualizing complex,
\(\text{cd}(A, I) \leq d\),
if \(\mathfrak p \not \in V(I)\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\) then \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s\) or \(\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s\).
Then \(A, I, V(\mathfrak a), M, s, d\) are as in Situation 0EFU.
Proof
We have to show that assumptions (1), (3), (4), and (6) of Situation 0EFU hold. It is clear that (a) \(\Rightarrow\) (1), (b) \(\Rightarrow\) (3), and (c) \(\Rightarrow\) (4). To finish the proof in the next paragraph we show (6) holds.
Let \(\mathfrak q \in V(\mathfrak a)\). Denote \(A', I', \mathfrak m', M'\) the \(I\)-adic completions of \(A_\mathfrak q, I_\mathfrak q, \mathfrak qA_\mathfrak q, M_\mathfrak q\). Let \(\mathfrak p' \subset A'\) be a nonmaximal prime with \(V(\mathfrak p') \cap V(I') = \{\mathfrak m'\}\). Observe that this implies \(\dim(A'/\mathfrak p') \leq d\) by Local Cohomology, Lemma 0DXF. Denote \(\mathfrak p \subset A\) the image of \(\mathfrak p'\). We have \(\text{depth}(M'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p)\) and \(\text{depth}(M'_{\mathfrak p'}) + \dim(A'/\mathfrak p') = \text{depth}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q)\) by Local Cohomology, Lemma 0EHW. By assumption (c) either we have \(\text{depth}(M'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p) > s\) and we’re done or we have \(\text{depth}(M'_{\mathfrak p'}) + \dim(A'/\mathfrak p') > s + d\) which implies \(\text{depth}(M'_{\mathfrak p'}) > s\) because of the already shown inequality \(\dim(A'/\mathfrak p') \leq d\). In both cases we obtain what we want.
Lemma
In Situation 0EFU the inverse systems \(\{H^i_T(I^nM)\}_{n \geq 0}\) are pro-zero for \(i \leq s\). Moreover, there exists an integer \(m_0\) such that for all \(m \geq m_0\) there exists an integer \(m'(m) \geq m\) such that for \(k \geq m'(m)\) the image of \(H^{s + 1}_T(I^kM) \to H^{s + 1}_T(I^mM)\) maps injectively to \(H^{s + 1}_T(I^{m_0}M)\).
Proof
Fix \(m\). Let \(\mathfrak q \in T\). By Lemmas 0EFV and 0EFR we see that \[H^i_\mathfrak q(M_\mathfrak q) \longrightarrow \lim H^i_\mathfrak q(M_\mathfrak q/I^nM_\mathfrak q)\] is an isomorphism for \(i \leq s\). The inverse systems \(\{H^i_\mathfrak q(I^nM_\mathfrak q)\}_{n \geq 0}\) and \(\{H^i_\mathfrak q(M/I^nM)\}_{n \geq 0}\) satisfy the Mittag-Leffler condition for all \(i\), see Lemma 0DX1. Thus looking at the inverse system of long exact sequences \[0 \to H^0_\mathfrak q(I^nM_\mathfrak q) \to H^0_\mathfrak q(M_\mathfrak q) \to H^0_\mathfrak q(M_\mathfrak q/I^nM_\mathfrak q) \to H^1_\mathfrak q(I^nM_\mathfrak q) \to H^1_\mathfrak q(M_\mathfrak q) \to \ldots\] we conclude (some details omitted) that there exists an integer \(m'(m, \mathfrak q) \geq m\) such that for all \(k \geq m'(m, \mathfrak q)\) the map \(H^i_\mathfrak q(I^kM_\mathfrak q) \to H^i_\mathfrak q(I^mM_\mathfrak q)\) is zero for \(i \leq s\) and the image of \(H^{s + 1}_\mathfrak q(I^kM_\mathfrak q) \to H^{s + 1}_\mathfrak q(I^mM_\mathfrak q)\) is independent of \(k \geq m'(m, \mathfrak q)\) and maps injectively into \(H^{s + 1}_\mathfrak q(M_\mathfrak q)\).
Suppose we can show that \(m'(m, \mathfrak q)\) can be chosen independently of \(\mathfrak q \in T\). Then the lemma follows immediately from Local Cohomology, Lemmas 0EF9 and 0EFA.
Let \(\omega_A^\bullet\) be a dualizing complex. Let \(\delta : \Spec(A) \to \mathbf{Z}\) be the corresponding dimension function. Recall that \(\delta\) attains only a finite number of values, see Dualizing Complexes, Lemma 0A80. Claim: for each \(d \in \mathbf{Z}\) the integer \(m'(m, \mathfrak q)\) can be chosen independently of \(\mathfrak q \in T\) with \(\delta(\mathfrak q) = d\). Clearly the claim implies the lemma by what we said above.
Pick \(\mathfrak q \in T\) with \(\delta(\mathfrak q) = d\). Consider the ext modules \[E(n, j) = \text{Ext}^j_A(I^nM, \omega_A^\bullet)\] A key feature we will use is that these are finite \(A\)-modules. Recall that \((\omega_A^\bullet)_\mathfrak q[-d]\) is a normalized dualizing complex for \(A_\mathfrak q\) by definition of the dimension function associated to a dualizing complex, see Dualizing Complexes, Section 0A7W. The local duality theorem (Dualizing Complexes, Lemma 0AAK) tells us that the \(\mathfrak qA_\mathfrak q\)-adic completion of \(E(n, -d - i)_\mathfrak q\) is Matlis dual to \(H^i_\mathfrak q(I^nM_\mathfrak q)\). Thus the choice of \(m'(m, \mathfrak q)\) for \(i \leq s\) in the first paragraph tells us that for \(k \geq m'(m, \mathfrak q)\) and \(j \geq -d - s\) the map \[E(m, j)_\mathfrak q \to E(k, j)_\mathfrak q\] is zero. Since these modules are finite and nonzero only for a finite number of possible \(j\) (small detail omitted), we can find an open neighbourhood \(W \subset \Spec(A)\) of \(\mathfrak q\) such that \[E(m, j)_{\mathfrak q'} \to E(m'(m, \mathfrak q), j)_{\mathfrak q'}\] is zero for \(j \geq -d - s\) for all \(\mathfrak q' \in W\). Then of course the maps \(E(m, j)_{\mathfrak q'} \to E(k, j)_{\mathfrak q'}\) for \(k \geq m'(m, \mathfrak q)\) are zero as well.
For \(i = s + 1\) corresponding to \(j = - d - s - 1\) we obtain from local duality and the results of the first paragraph that \[K_{k, \mathfrak q} = \Ker(E(m, -d - s - 1)_\mathfrak q \to E(k, -d - s - 1)_\mathfrak q)\] is independent of \(k \geq m'(m, \mathfrak q)\) and that \[E(0, -d - s - 1)_\mathfrak q \to E(m, -d - s - 1)_\mathfrak q/K_{m'(m, \mathfrak q), \mathfrak q}\] is surjective. For \(k \geq m'(m, \mathfrak q)\) set \[K_k = \Ker(E(m, -d - s - 1) \to E(k, -d - s - 1))\] Since \(K_k\) is an increasing sequence of submodules of the finite module \(E(m, -d - s - 1)\) we see that, at the cost of increasing \(m'(m, \mathfrak q)\) a little bit, we may assume \(K_{m'(m, \mathfrak q)} = K_k\) for \(k \geq m'(m, \mathfrak q)\). After shrinking \(W\) further if necessary, we may also assume that \[E(0, -d - s - 1)_{\mathfrak q'} \to E(m, -d - s - 1)_{\mathfrak q'}/K_{m'(m, \mathfrak q), \mathfrak q'}\] is surjective for all \(\mathfrak q' \in W\) (as before use that these modules are finite and that the map is surjective after localization at \(\mathfrak q\)).
Any subset, in particular \(T_d = \{\mathfrak q \in T \text{ with }\delta(\mathfrak q) = d\}\), of the Noetherian topological space \(\Spec(A)\) with the endowed topology is Noetherian and hence quasi-compact. Above we have seen that for every \(\mathfrak q \in T_d\) there is an open neighbourhood \(W\) where \(m'(m, \mathfrak q)\) works for all \(\mathfrak q' \in T_d \cap W\). We conclude that we can find an integer \(m'(m, d)\) such that for all \(\mathfrak q \in T_d\) we have \[E(m, j)_\mathfrak q \to E(m'(m, d), j)_\mathfrak q\] is zero for \(j \geq -d - s\) and with \(K_{m'(m, d)} = \Ker(E(m, -d - s - 1) \to E(m'(m, d), -d - s - 1))\) we have \[K_{m'(m, d), \mathfrak q} = \Ker(E(m, -d - s - 1)_{\mathfrak q} \to E(k, -d - s - 1)_{\mathfrak q})\] for all \(k \geq m'(m, d)\) and the map \[E(0, -d - s - 1)_\mathfrak q \to E(m, -d - s - 1)_\mathfrak q/K_{m'(m, d), \mathfrak q}\] is surjective. Using the local duality theorem again (in the opposite direction) we conclude that the claim is correct. This finishes the proof.
Lemma
In Situation 0EFU there exists an integer \(m_0 \geq 0\) such that
\(H^i_T(M) \to H^i_T(M/I^nM)\) is injective for all \(i \leq s\) and \(n \geq m_0\),
\(\{H^i_T(M/I^nM)\}_{n \geq 0}\) satisfies the Mittag-Leffler condition for \(i < s\).
\(\{H^i_T(I^{m_0}M/I^nM)\}_{n \geq m_0}\) satisfies the Mittag-Leffler condition for \(i \leq s\),
\(H^i_T(M) \to \lim H^i_T(M/I^nM)\) is an isomorphism for \(i < s\),
\(H^s_T(I^{m_0}M) \to \lim H^s_T(I^{m_0}M/I^nM)\) is an isomorphism for \(i \leq s\),
\(H^s_T(M) \to \lim H^s_T(M/I^nM)\) is injective with cokernel killed by \(I^{m_0}\), and
\(R^1\lim H^s_T(M/I^nM)\) is killed by \(I^{m_0}\).
Proof
To prove (1) let \(0 \leq i \leq s\) and choose integers \(m > n > 0\) such that \(H^i_T(I^mM) \to H^i_T(I^nM)\) is zero; this is possibly by Lemma 0EFX. The commutative diagram with exact rows \[\xymatrix{ H^i_T(I^nM) \ar[r] & H^i_T(M) \ar[r] & H^i_T(M/I^nM) \\ H^i_T(I^mM) \ar[r] \ar[u]^0 & H^i_T(M) \ar[r] \ar[u]^1 & H^i_T(M/I^mM) \ar[u] \\ }\] shows that the map \(H^i_T(M) \to H^i_T(M/I^mM)\) is injective. Whence (1) holds for any sufficiently large \(m_0\).
Consider the long exact sequences \[0 \to H^0_T(I^nM) \to H^0_T(M) \to H^0_T(M/I^nM) \to H^1_T(I^nM) \to H^1_T(M) \to \ldots\] Parts (2) and (4) follow from this and Lemma 0EFX.
Let \(m_0\) and \(m'(-)\) be as in Lemma 0EFX. For \(m \geq m_0\) consider the long exact sequence \[H^s_T(I^mM) \to H^s_T(I^{m_0}M) \to H^s_T(I^{m_0}M/I^mM) \to H^{s + 1}_T(I^mM) \to H^1_T(I^{m_0}M)\] Then for \(k \geq m'(m)\) the image of \(H^{s + 1}_T(I^kM) \to H^{s + 1}_T(I^mM)\) maps injectively to \(H^{s + 1}_T(I^{m_0}M)\). Hence the image of \(H^s_T(I^{m_0}M/I^kM) \to H^s_T(I^{m_0}M/I^mM)\) maps to zero in \(H^{s + 1}_T(I^mM)\) for all \(k \geq m'(m)\). We conclude that (3) and (5) hold.
Consider the short exact sequences \(0 \to I^{m_0}M \to M \to M/I^{m_0} M \to 0\) and \(0 \to I^{m_0}M/I^nM \to M/I^nM \to M/I^{m_0} M \to 0\). We obtain a diagram \[\xymatrix{ H^{s - 1}_T(M/I^{m_0}M) \ar[r] & \lim H^s_T(I^{m_0}M/I^nM) \ar[r] & \lim H^s_T(M/I^nM) \ar[r] & H^s_T(M/I^{m_0}M) \\ H^{s - 1}_T(M/I^{m_0}M) \ar[r] \ar@{=}[u] & H^s_T(I^{m_0}M) \ar[r] \ar[u]_{\cong} & H^s_T(M) \ar[r] \ar[u] & H^s_T(M/I^{m_0}M) \ar@{=}[u] }\] whose lower row is exact. The top row is also exact (at the middle two spots) by Homology, Lemma 070B. Part (6) follows.
Write \(B_n = H^s_T(M/I^nM)\). Let \(A_n \subset B_n\) be the image of \(H^s_T(I^{m_0}M/I^nM) \to H^s_T(M/I^nM)\). Then \((A_n)\) satisfies the Mittag-Leffler condition by (3) and Homology, Lemma 02N1. Also \(C_n = B_n/A_n\) is killed by \(I^{m_0}\). Thus \(R^1\lim B_n \cong R^1\lim C_n\) is killed by \(I^{m_0}\) and we get (7).
Theorem
In Situation 0EFU for \(i \leq s\) the inverse system \(\{H^i_T(M/I^nM)\}_{n \geq 0}\) is essentially constant with value \(H^i_T(M)\). In particular, \(\{H^i_T(M/I^nM)\}_{n \geq 0}\) is Mittag-Leffler, \(H^i_T(M)\) is annihilated by a power of \(I\), and \(H^i_T(M) = \lim H^i_T(M/I^nM)\).
Proof
Let \(0 \leq i \leq s\). If we show that \(\{H^i_T(M/I^nM)\}_{n \geq 0}\) is Mittag-Leffler and \(H^i_T(M) = \lim H^i_T(M/I^nM)\), then the injectivity of \(H^i_T(M) \to H^i_T(M/I^nM)\) for \(n \gg 0\) shown in Lemma 0EFY will imply that \(\{H^i_T(M/I^nM)\}_{n \geq 0}\) is essentially constant with value \(H^i_T(M)\). Small detail omitted.
We already know \(\{H^i_T(M/I^nM)\}_{n \geq 0}\) is Mittag-Leffler and \(H^i_T(M) = \lim H^i_T(M/I^nM)\) for \(i < s\) by Lemma 0EFY. We also know these statements hold for \(i = s\) and the module \(I^{m_0}M\) for some \(m_0 > 0\). To finish of the proof we will show that in fact these assertions for \(i = s\) holds for \(M\).
Let \(M' = H^0_I(M)\) and \(M'' = M/M'\) so that we have a short exact sequence \[0 \to M' \to M \to M'' \to 0\] and \(M''\) has \(H^0_I(M') = 0\) by Dualizing Complexes, Lemma 0AW0. By Artin-Rees (Algebra, Lemma 00IN) we get short exact sequences \[0 \to M' \to M/I^n M \to M''/I^n M'' \to 0\] for \(n\) large enough. Consider the long exact sequences \[H^s_T(M') \to H^s_T(M/I^nM) \to H^s_T(M''/I^nM'') \to H^{s + 1}_T(M')\] Now it is a simple matter to see that if we have Mittag-Leffler for the inverse system \(\{H^s_T(M''/I^nM'')\}_{n \geq 0}\) then we have Mittag-Leffler for the inverse system \(\{H^s_T(M/I^nM)\}_{n \geq 0}\). (Note that the ML condition for an inverse system of groups \(G_n\) only depends on the values of the inverse system for sufficiently large \(n\).) Moreover the sequence \[H^s_T(M') \to \lim H^s_T(M/I^nM) \to \lim H^s_T(M''/I^nM'') \to H^{s + 1}_T(M')\] is exact because we have ML in the required spots, see Homology, Lemma 070B. Hence, if \(H^s_T(M'') \to \lim H^s_T(M''/I^nM'')\) is an isomorphism, then \(H^s_T(M) \to \lim H^s_T(M/I^nM)\) is an isomorphism too by the five lemma (Homology, Lemma 05QB). This reduces us to the case discussed in the next paragraph.
Assume that \(H^0_I(M) = 0\). Choose generators \(f_1, \ldots, f_r\) of \(I^{m_0}\) where \(m_0\) is the integer found for \(M\) in Lemma 0EFY. Then we consider the exact sequence \[0 \to M \xrightarrow{f_1, \ldots, f_r} (I^{m_0}M)^{\oplus r} \to Q \to 0\] defining \(Q\). Some observations: the first map is injective exactly because \(H^0_I(M) = 0\). The cokernel \(Q\) of this injection is a finite \(A\)-module such that for every \(1 \leq j \leq r\) we have \(Q_{f_j} \cong (M_{f_j})^{\oplus r - 1}\). In particular, for a prime \(\mathfrak p \subset A\) with \(\mathfrak p \not \in V(I)\) we have \(Q_\mathfrak p \cong (M_\mathfrak p)^{\oplus r - 1}\). Similarly, given \(\mathfrak q \in T\) and \(\mathfrak p' \subset A' = (A_\mathfrak q)^\wedge\) not contained in \(V(IA')\), we have \(Q'_{\mathfrak p'} \cong (M'_{\mathfrak p'})^{\oplus r - 1}\) where \(Q' = (Q_\mathfrak q)^\wedge\) and \(M' = (M_\mathfrak q)^\wedge\). Thus the conditions in Situation 0EFU hold for \(A, I, T, Q\). (Observe that \(Q\) may have nonvanishing \(H^0_I(Q)\) but this won’t matter.)
For any \(n \geq 0\) we set \(F^nM = M \cap I^n(I^{m_0}M)^{\oplus r}\) so that we get short exact sequences \[0 \to F^nM \to I^n(I^{m_0}M)^{\oplus r} \to I^nQ \to 0\] By Artin-Rees (Algebra, Lemma 00IN) there exists a \(c \geq 0\) such that \(I^n M \subset F^nM \subset I^{n - c}M\) for all \(n \geq c\). Let \(m_0\) be the integer and let \(m'(m)\) be the function defined for \(m \geq m_0\) found in Lemma 0EFX applied to \(M\). Note that the integer \(m_0\) is the same as our integer \(m_0\) chosen above (you don’t need to check this: you can just take the maximum of the two integers if you like). Finally, by Lemma 0EFX applied to \(Q\) for every integer \(m\) there exists an integer \(m''(m) \geq m\) such that \(H^s_T(I^kQ) \to H^s_T(I^mQ)\) is zero for all \(k \geq m''(m)\).
Fix \(m \geq m_0\). Choose \(k \geq m'(m''(m + c))\). Choose \(\xi \in H^{s + 1}_T(I^kM)\) which maps to zero in \(H^{s + 1}_T(M)\). We want to show that \(\xi\) maps to zero in \(H^{s + 1}_T(I^mM)\). Namely, this will show that \(\{H^s_T(M/I^nM)\}_{n \geq 0}\) is Mittag-Leffler exactly as in the proof of Lemma 0EFY. Picture to help visualize the argument: \[\xymatrix{ & H^{s + 1}_T(I^kM) \ar[r] \ar[d] & H^{s + 1}_T(I^k(I^{m_0}M)^{\oplus r}) \ar[d] & \\ H^s_T(I^{m''(m + c)}Q) \ar[r]_-\delta \ar[d] & H^{s + 1}_T(F^{m''(m + c)}M) \ar[r] \ar[d] & H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\oplus r}) \\ H^s_T(I^{m + c}Q) \ar[r] & H^{s + 1}_T(F^{m + c}M) \ar[d] & \\ & H^{s + 1}_T(I^mM) }\] The image of \(\xi\) in \(H^{s + 1}_T(I^k(I^{m_0}M)^{\oplus r})\) maps to zero in \(H^{s + 1}_T((I^{m_0}M)^{\oplus r})\) and hence maps to zero in \(H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\oplus r})\) by choice of \(m'(-)\). Thus the image \(\xi' \in H^{s + 1}_T(F^{m''(m + c)}M)\) maps to zero in \(H^{s + 1}_T(I^{m''(m + c)}(I^{m_0}M)^{\oplus r})\) and hence \(\xi' = \delta(\eta)\) for some \(\eta \in H^s_T(I^{m''(m + c)}Q)\). By our choice of \(m''(-)\) we find that \(\eta\) maps to zero in \(H^s_T(I^{m + c}Q)\). This in turn means that \(\xi'\) maps to zero in \(H^{s + 1}_T(F^{m + c}M)\). Since \(F^{m + c}M \subset I^mM\) we conclude.
Finally, we prove the statement on limits. Consider the short exact sequences \[0 \to M/F^nM \to (I^{m_0}M)^{\oplus r}/I^n (I^{m_0}M)^{\oplus r} \to Q/I^nQ \to 0\] We have \(\lim H^s_T(M/I^nM) = \lim H^s_T(M/F^nM)\) as these inverse systems are pro-isomorphic. We obtain a commutative diagram \[\xymatrix{ H^{s - 1}_T(Q) \ar[r] \ar[d] & \lim H^{s - 1}_T(Q/I^nQ) \ar[d] \\ H^s_T(M) \ar[r] \ar[d] & \lim H^s_T(M/I^nM) \ar[d] \\ H^s_T((I^{m_0}M)^{\oplus r}) \ar[r] \ar[d] & \lim H^s_T((I^{m_0}M)^{\oplus r}/I^n(I^{m_0}M)^{\oplus r}) \ar[d] \\ H^s_T(Q) \ar[r] & \lim H^s_T(Q/I^nQ) }\] The right column is exact because we have ML in the required spots, see Homology, Lemma 070B. The lowest horizontal arrow is injective (!) by part (6) of Lemma 0EFY. The horizontal arrow above it is bijective by part (5) of Lemma 0EFY. The arrows in cohomological degrees \(\leq s - 1\) are isomorphisms. Thus we conclude \(H^s_T(M) \to \lim H^s_T(M/I^nM)\) is an isomorphism by the five lemma (Homology, Lemma 05QB). This finishes the proof of the theorem.
Lemma
Let \(I \subset \mathfrak a \subset A\) be ideals of a Noetherian ring \(A\) and let \(M\) be a finite \(A\)-module. Let \(s\) and \(d\) be integers. Suppose that
\(A, I, V(\mathfrak a), M\) satisfy the assumptions of Situation 0EFU for \(s\) and \(d\), and
\(A, I, \mathfrak a, M\) satisfy the conditions of Lemma 0EFL for \(s + 1\) and \(d\) with \(J = \mathfrak a\).
Then there exists an ideal \(J_0 \subset \mathfrak a\) with \(V(J_0) \cap V(I) = V(\mathfrak a)\) such that for any \(J \subset J_0\) with \(V(J) \cap V(I) = V(\mathfrak a)\) the map \[H^{s + 1}_J(M) \longrightarrow \lim H^{s + 1}_\mathfrak a(M/I^nM)\] is an isomorphism.
Proof
Namely, we have the existence of \(J_0\) and the isomorphism \(H^{s + 1}_J(M) = H^{s + 1}(R\Gamma_\mathfrak a(M)^\wedge)\) by Lemma 0EFL, we have a short exact sequence \[0 \to R^1\lim H^s_\mathfrak a(M/I^nM) \to H^{s + 1}(R\Gamma_\mathfrak a(M)^\wedge) \to \lim H^{s + 1}_\mathfrak a(M/I^nM) \to 0\] by Dualizing Complexes, Lemma 0EEW, and the module \(R^1\lim H^s_\mathfrak a(M/I^nM)\) is zero because \(\{H^s_\mathfrak a(M/I^nM)\}_{n \geq 0}\) has Mittag-Leffler by Theorem 0EIE.
Algebraization of formal sections, I
In this section we study the problem of algebraization of formal sections in the local case. Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(I \subset A\) be an ideal. Let \[X = \Spec(A) \supset U = \Spec(A) \setminus \{\mathfrak m\}\] and denote \(Y = V(I)\) the closed subscheme corresponding to \(I\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module. In this section we consider the limits \[\lim_n H^i(U, \mathcal{F}/I^n\mathcal{F})\] This is closely related to the cohomology of the pullback of \(\mathcal{F}\) to the formal completion of \(U\) along \(Y\); however, since we have not yet introduced formal schemes, we cannot use this terminology here.
Lemma
Let \(U\) be the punctured spectrum of a Noetherian local ring \(A\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module. Let \(I \subset A\) be an ideal. Then \[H^i(R\Gamma(U, \mathcal{F})^\wedge) = \lim H^i(U, \mathcal{F}/I^n\mathcal{F})\] for all \(i\) where \(R\Gamma(U, \mathcal{F})^\wedge\) denotes the derived \(I\)-adic completion.
Proof
By Lemmas 0BLX and 0A0K we have \[R\Gamma(U, \mathcal{F})^\wedge = R\Gamma(U, \mathcal{F}^\wedge) = R\Gamma(U, R\lim \mathcal{F}/I^n\mathcal{F})\] Thus we obtain short exact sequences \[0 \to R^1\lim H^{i - 1}(U, \mathcal{F}/I^n\mathcal{F}) \to H^i(R\Gamma(U, \mathcal{F})^\wedge) \to \lim H^i(U, \mathcal{F}/I^n\mathcal{F}) \to 0\] by Cohomology, Lemma 0D60. The \(R^1\lim\) terms vanish because the inverse systems of groups \(H^i(U, \mathcal{F}/I^n\mathcal{F})\) satisfy the Mittag-Leffler condition by Lemma 0DX1.
Theorem
Let \((A, \mathfrak m)\) be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal \(I\). Set \(X = \Spec(A)\), \(Y = V(I)\), and \(U = X \setminus \{\mathfrak m\}\). Let \(\mathcal{F}\) be a coherent sheaf on \(U\). Assume
\(\text{cd}(A, I) \leq d\), i.e., \(H^i(X \setminus Y, \mathcal{G}) = 0\) for \(i \geq d\) and quasi-coherent \(\mathcal{G}\) on \(X\),
for any \(x \in X \setminus Y\) whose closure \(\overline{\{x\}}\) in \(X\) meets \(U \cap Y\) we have \[\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) \geq s \quad\text{or}\quad \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\overline{\{x\}}) > d + s\]
Then there exists an open \(V_0 \subset U\) containing \(U \cap Y\) such that for any open \(V \subset V_0\) containing \(U \cap Y\) the map \[H^i(V, \mathcal{F}) \to \lim H^i(U, \mathcal{F}/I^n\mathcal{F})\] is an isomorphism for \(i < s\). If in addition \(\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\overline{\{x\}}) > s\) for all \(x \in U \cap Y\), then these cohomology groups are finite \(A\)-modules.
Proof
Choose a finite \(A\)-module \(M\) such that \(\mathcal{F}\) is the restriction to \(U\) of the coherent \(\mathcal{O}_X\)-module associated to \(M\), see Local Cohomology, Lemma 0BK0. Then the assumptions of Lemma 0DXP are satisfied. Pick \(J_0\) as in that lemma and set \(V_0 = X \setminus V(J_0)\). Then opens \(V \subset V_0\) containing \(U \cap Y\) correspond \(1\)-to-\(1\) with ideals \(J \subset J_0\) with \(V(J) \cap V(I) = \{\mathfrak m\}\). Moreover, for such a choice we have a distinguished triangle \[R\Gamma_J(M) \to M \to R\Gamma(V, \mathcal{F}) \to R\Gamma_J(M)[1]\] We similarly have a distinguished triangle \[R\Gamma_\mathfrak m(M)^\wedge \to M \to R\Gamma(U, \mathcal{F})^\wedge \to R\Gamma_\mathfrak m(M)^\wedge[1]\] involving derived \(I\)-adic completions. The cohomology groups of \(R\Gamma(U, \mathcal{F})^\wedge\) are equal to the limits in the statement of the theorem by Lemma 0DXI. The canonical map between these triangles and some easy arguments show that our theorem follows from the main Lemma 0DXP (note that we have \(i < s\) here whereas we have \(i \leq s\) in the lemma; this is because of the shift). The finiteness of the cohomology groups (under the additional assumption) follows from Lemma 0DXN.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal \(I\). Set \(X = \Spec(A)\), \(Y = V(I)\), and \(U = X \setminus \{\mathfrak m\}\). Let \(\mathcal{F}\) be a coherent sheaf on \(U\). Assume for any associated point \(x \in U\) of \(\mathcal{F}\) we have \(\dim(\overline{\{x\}}) > \text{cd}(A, I) + 1\) where \(\overline{\{x\}}\) is the closure in \(X\). Then the map \[\colim H^0(V, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\] is an isomorphism of finite \(A\)-modules where the colimit is over opens \(V \subset U\) containing \(U \cap Y\).
Proof
Apply Theorem 0DXQ with \(s = 1\) (we get finiteness too).
Algebraization of formal sections, II
It is a bit difficult to succinctly state all possible consequences of the results in Sections 0EFF and 0EFT for cohomology of coherent sheaves on quasi-affine schemes and their completion with respect to an ideal. This section gives a nonexhaustive list of applications to \(H^0\). The next section contains applications to higher cohomology.
Lemma
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0\) be a short exact sequence of coherent modules on \(U = \Spec(A) \setminus V(\mathfrak a)\). Let \(\mathcal{V}\) be the set of open subschemes \(V \subset U\) containing \(U \cap V(I)\) ordered by reverse inclusion. Consider the commutative diagram \[\xymatrix{ \colim_\mathcal{V} H^0(V, \mathcal{F}') \ar[d] \ar[r] & \colim_\mathcal{V} H^0(V, \mathcal{F}) \ar[d] \ar[r] & \colim_\mathcal{V} H^0(V, \mathcal{F}'') \ar[d] \\ \lim H^0(U, \mathcal{F}'/I^n\mathcal{F}') \ar[r] & \lim H^0(U, \mathcal{F}'/I^n\mathcal{F}) \ar[r] & \lim H^0(U, \mathcal{F}'/I^n\mathcal{F}'') }\] If the left and right downarrows are isomorphisms so is the middle. If the middle and left downarrows are isomorphisms, so is the left.
Proof
The sequences in the diagram are exact in the middle and the first arrow is injective. Thus the final statement follows from an easy diagram chase. For the rest of the proof we assume the left and right downward arrows are isomorphisms. A diagram chase shows that the middle downward arrow is injective. All that remains is to show that it is surjective.
We may choose finite \(A\)-modules \(M\) and \(M'\) such that \(\mathcal{F}\) and \(\mathcal{F}'\) are the restriction of \(\widetilde{M}\) and \(\widetilde{M}'\) to \(U\), see Local Cohomology, Lemma 0BK0. After replacing \(M'\) by \(\mathfrak a^n M'\) for some \(n \geq 0\) we may assume that \(\mathcal{F}' \to \mathcal{F}\) corresponds to a module map \(M' \to M\), see Cohomology of Schemes, Lemma 01YB. After replacing \(M'\) by the image of \(M' \to M\) and setting \(M'' = M/M'\) we see that our short exact sequence corresponds to the restriction of the short exact sequence of coherent modules associated to the short exact sequence \(0 \to M' \to M \to M'' \to 0\) of \(A\)-modules.
Let \(\hat s \in \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\) with image \(\hat s'' \in \lim H^0(U, \mathcal{F}''/I^n\mathcal{F}'')\). By assumption we find \(V \in \mathcal{V}\) and a section \(s'' \in \mathcal{F}''(V)\) mapping to \(\hat s''\). Let \(J \subset A\) be an ideal such that \(V(J) = \Spec(A) \setminus V\). By Cohomology of Schemes, Lemma 01YB after replacing \(J\) by a power, we may assume there is an \(A\)-linear map \(\varphi : J \to M''\) corresponding to \(s''\). We fix this choice of \(J\); in the rest of the proof we will replace \(V\) by a smaller \(V\) in \(\mathcal{V}\), i.e, we will have \(V \cap V(J) = \emptyset\).
Choose a presentation \(A^{\oplus m} \to A^{\oplus n} \to J \to 0\). Denote \(g_1, \ldots, g_n \in J\) the images of the basis vectors of \(A^{\oplus n}\), so that \(J = (g_1, \ldots, g_n)\). Let \(A^{\oplus m} \to A^{\oplus n}\) be given by the matrix \((a_{ji})\) so that \(\sum a_{ji} g_i = 0\), \(j = 1, \ldots, m\). Since \(M \to M''\) is surjective, for each \(i\) we can choose \(m_i \in M\) mapping to \(\varphi(g_i) \in M''\). Then the element \(g_i \hat s - m_i\) of \(\lim H^0(U, \mathcal{F}/I^n\mathcal{F})\) lies in the submodule \(\lim H^0(U, \mathcal{F}'/I^n\mathcal{F}')\). By assumption after shrinking \(V\) we may assume there are \(s'_i \in \mathcal{F}'(V)\), \(i = 1, \ldots, n\) with \(s'_i\) mapping to \(g_i \hat s - m_i\). Set \(s_i = s'_i + m_i\) in \(\mathcal{F}(V)\). Note that \(\sum a_{ji} s_i\) maps to \(\sum a_{ji}g_i\hat s = 0\) by the map \[\colim_\mathcal{V} \mathcal{F}(V') \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\] Since this map is injective (see above), we may after shrinking \(V\) assume that \(\sum a_{ji}s_i = 0\) in \(\mathcal{F}(V)\) for all \(j = 1, \ldots, m\). Then it follows that we obtain an \(A\)-module map \(J \to \mathcal{F}(V)\) sending \(g_i\) to \(s_i\). By the universal property of \(\widetilde{J}\) this \(A\)-module map corresponds to an \(\mathcal{O}_V\)-module map \(\widetilde{J}|_V \to \mathcal{F}\). However, since \(V(J) \cap V = \emptyset\) we have \(\widetilde{J}|_V = \mathcal{O}_V\). Thus we have produced a section \(s \in \mathcal{F}(V)\). We omit the computation that shows that \(s\) maps to \(\hat s\) by the map displayed above.
The following lemma will be superseded by Proposition 0EG2.
Lemma
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(\mathcal{F}\) be a coherent module on \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(x \in \text{Ass}(\mathcal{F})\), \(x \not \in V(I)\), \(\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)\), and \(z \in \overline{\{x\}} \cap V(\mathfrak a)\), then \(\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1\),
one of the following holds:
the restriction of \(\mathcal{F}\) to \(U \setminus V(I)\) is \((S_1)\)
the dimension of \(V(\mathfrak a)\) is at most \(2\)4.
Then we obtain an isomorphism \[\colim H^0(V, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\] where the colimit is over opens \(V \subset U\) containing \(U \cap V(I)\).
Proof
Choose a finite \(A\)-module \(M\) such that \(\mathcal{F}\) is the restriction to \(U\) of the coherent module associated to \(M\), see Local Cohomology, Lemma 0BK0. Set \(d = \text{cd}(A, I)\). Let \(\mathfrak p\) be a prime of \(A\) not contained in \(V(I)\) and let \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\). Then either \(\mathfrak p\) is not an associated prime of \(M\) and hence \(\text{depth}(M_\mathfrak p) \geq 1\) or we have \(\dim((A/\mathfrak p)_\mathfrak q) > d + 1\) by (2). Thus the hypotheses of Lemma 0EFL are satisfied for \(s = 1\) and \(d\); here we use condition (3). Thus we find there exists an ideal \(J_0 \subset \mathfrak a\) with \(V(J_0) \cap V(I) = V(\mathfrak a)\) such that for any \(J \subset J_0\) with \(V(J) \cap V(I) = V(\mathfrak a)\) the maps \[H^i_J(M) \longrightarrow H^i(R\Gamma_\mathfrak a(M)^\wedge)\] are isomorphisms for \(i = 0, 1\). Consider the morphisms of exact triangles \[\xymatrix{ R\Gamma_J(M) \ar[d] \ar[r] & M \ar[r] \ar[d] & R\Gamma(V, \mathcal{F}) \ar[d] \ar[r] & R\Gamma_J(M)[1] \ar[d] \\ R\Gamma_J(M)^\wedge \ar[r] & M \ar[r] & R\Gamma(V, \mathcal{F})^\wedge \ar[r] & R\Gamma_J(M)^\wedge[1] \\ R\Gamma_\mathfrak a(M)^\wedge \ar[r] \ar[u] & M \ar[r] \ar[u] & R\Gamma(U, \mathcal{F})^\wedge \ar[r] \ar[u] & R\Gamma_\mathfrak a(M)^\wedge[1] \ar[u] }\] where \(V = \Spec(A) \setminus V(J)\). Recall that \(R\Gamma_\mathfrak a(M)^\wedge \to R\Gamma_J(M)^\wedge\) is an isomorphism (because \(\mathfrak a\), \(\mathfrak a + I\), and \(J + I\) cut out the same closed subscheme, for example see proof of Lemma 0EFL). Hence \(R\Gamma(U, \mathcal{F})^\wedge = R\Gamma(V, \mathcal{F})^\wedge\). This produces a commutative diagram \[\xymatrix{ 0 \ar[r] & H^0_J(M) \ar[r] \ar[d] & M \ar[r] \ar[d] \ar[r] & \Gamma(V, \mathcal{F}) \ar[d] \ar[r] & H^1_J(M) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & H^0(R\Gamma_J(M)^\wedge) \ar[r] & M \ar[r] & H^0(R\Gamma(V, \mathcal{F})^\wedge) \ar[r] & H^1(R\Gamma_J(M)^\wedge) \ar[r] & 0 \\ 0 \ar[r] & H^0(R\Gamma_\mathfrak a(M)^\wedge) \ar[r] \ar[u] & M \ar[r] \ar[u] & H^0(R\Gamma(U, \mathcal{F})^\wedge) \ar[r] \ar[u] & H^1(R\Gamma_\mathfrak a(M)^\wedge) \ar[r] \ar[u] & 0 }\] with exact rows and isomorphisms for the lower vertical arrows. Hence we obtain an isomorphism \(\Gamma(V, \mathcal{F}) \to H^0(R\Gamma(U, \mathcal{F})^\wedge)\). By Lemmas 0BLX and 0A0K we have \[R\Gamma(U, \mathcal{F})^\wedge = R\Gamma(U, \mathcal{F}^\wedge) = R\Gamma(U, R\lim \mathcal{F}/I^n\mathcal{F})\] and we find \(H^0(R\Gamma(U, \mathcal{F})^\wedge) = \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\) by Cohomology, Lemma 0D60.
Now we bootstrap the preceding lemma to get rid of condition (3).
Proposition
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(\mathcal{F}\) be a coherent module on \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(x \in \text{Ass}(\mathcal{F})\), \(x \not \in V(I)\), \(\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)\), and \(z \in \overline{\{x\}} \cap V(\mathfrak a)\), then \(\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1\).
Then we obtain an isomorphism \[\colim H^0(V, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\] where the colimit is over opens \(V \subset U\) containing \(U \cap V(I)\).
Proof
Let \(T \subset U\) be the set of points \(x\) with \(\overline{\{x\}} \cap V(I) \subset V(\mathfrak a)\). Let \(\mathcal{F} \to \mathcal{F}'\) be the surjection of coherent modules on \(U\) constructed in Local Cohomology, Lemma 0DX3. Since \(\mathcal{F} \to \mathcal{F}'\) is an isomorphism over an open \(V \subset U\) containing \(U \cap V(I)\) it suffices to prove the lemma with \(\mathcal{F}\) replaced by \(\mathcal{F}'\). Hence we may and do assume for \(x \in U\) with \(\overline{\{x\}} \cap V(I) \subset V(\mathfrak a)\) we have \(\text{depth}(\mathcal{F}_x) \geq 1\).
Let \(\mathcal{V}\) be the set of open subschemes \(V \subset U\) containing \(U \cap V(I)\) ordered by reverse inclusion. This is a directed set. We first claim that \[\mathcal{F}(V) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\] is injective for any \(V \in \mathcal{F}\) (and in particular the map of the lemma is injective). Namely, an associated point \(x\) of \(\mathcal{F}\) must have \(\overline{\{x\}} \cap U \cap Y \not = \emptyset\) by the previous paragraph. If \(y \in \overline{\{x\}} \cap U \cap Y\) then \(\mathcal{F}_x\) is a localization of \(\mathcal{F}_y\) and \(\mathcal{F}_y \subset \lim \mathcal{F}_y/I^n \mathcal{F}_y\) by Krull’s intersection theorem (Algebra, Lemma 00IP). This proves the claim as a section \(s \in \mathcal{F}(V)\) in the kernel would have to have empty support, hence would have to be zero.
Choose a finite \(A\)-module \(M\) such that \(\mathcal{F}\) is the restriction of \(\widetilde{M}\) to \(U\), see Local Cohomology, Lemma 0BK0. We may and do assume that \(H^0_\mathfrak a(M) = 0\). Let \(\text{Ass}(M) \setminus V(I) = \{\mathfrak p_1, \ldots, \mathfrak p_n\}\). We will prove the lemma by induction on \(n\). After reordering we may assume that \(\mathfrak p_n\) is a minimal element of the set \(\{\mathfrak p_1, \ldots, \mathfrak p_n\}\) with respect to inclusion, i.e, \(\mathfrak p_n\) is a generic point of the support of \(M\). Set \[M' = H^0_{\mathfrak p_1 \ldots \mathfrak p_{n - 1} I}(M)\] and \(M'' = M/M'\). Let \(\mathcal{F}'\) and \(\mathcal{F}''\) be the coherent \(\mathcal{O}_U\)-modules corresponding to \(M'\) and \(M''\). Dualizing Complexes, Lemma 0AW0 implies that \(M''\) has only one associated prime, namely \(\mathfrak p_n\). Hence \(\mathcal{F}''\) has only one associated point and we see that condition (3)(a) of Lemma 0EIF holds; thus the map \(\colim H^0(V, \mathcal{F}'') \to \lim H^0(U, \mathcal{F}''/I^n\mathcal{F}'')\) is an isomorphism. On the other hand, since \(\mathfrak p_n \not \in V(\mathfrak p_1 \ldots \mathfrak p_{n - 1} I)\) we see that \(\mathfrak p_n\) is not an associated prime of \(M'\). Hence the induction hypothesis applies to \(M'\); note that since \(\mathcal{F}' \subset \mathcal{F}\) the condition \(\text{depth}(\mathcal{F}'_x) \geq 1\) at points \(x\) with \(\overline{\{x\}} \cap V(I) \subset V(\mathfrak a)\) holds, see Algebra, Lemma 00LX. Thus the map \(\colim H^0(V, \mathcal{F}') \to \lim H^0(U, \mathcal{F}'/I^n\mathcal{F}')\) is an isomorphism too. We conclude by Lemma 0H48.
Lemma
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(\mathcal{F}\) be a coherent module on \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(x \in \text{Ass}(\mathcal{F})\), \(x \not \in V(I)\), \(\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)\), and \(z \in V(\mathfrak a) \cap \overline{\{x\}}\), then \(\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1\),
for \(x \in U\) with \(\overline{\{x\}} \cap V(I) \subset V(\mathfrak a)\) we have \(\text{depth}(\mathcal{F}_x) \geq 2\),
Then we obtain an isomorphism \[H^0(U, \mathcal{F}) \longrightarrow \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\]
Proof
Let \(\hat s \in \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\). By Proposition 0EG2 we find that \(\hat s\) is the image of an element \(s \in \mathcal{F}(V)\) for some \(V \subset U\) open containing \(U \cap V(I)\). However, condition (3) shows that \(\text{depth}(\mathcal{F}_x) \geq 2\) for all \(x \in U \setminus V\) and hence we find that \(\mathcal{F}(V) = \mathcal{F}(U)\) by Divisors, Lemma 0E9I and the proof is complete.
Lemma
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a \subset A\) be an element of an ideal of \(A\). Let \(M\) be a finite \(A\)-module. Assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor on \(M\),
\(H^1_\mathfrak a(M/fM)\) is a finite \(A\)-module.
Then with \(U = \Spec(A) \setminus V(\mathfrak a)\) the map \[\colim_V \Gamma(V, \widetilde{M}) \longrightarrow \lim \Gamma(U, \widetilde{M/f^nM})\] is an isomorphism where the colimit is over opens \(V \subset U\) containing \(U \cap V(f)\).
Proof
Set \(\mathcal{F} = \widetilde{M}|_U\). The finiteness of \(H^1_\mathfrak a(M/fM)\) implies that \(H^0(U, \mathcal{F}/f\mathcal{F})\) is finite, see Local Cohomology, Lemma 0BK0. By Cohomology, Lemma 0BLB (which applies as \(f\) is a nonzerodivisor on \(\mathcal{F}\)) we see that \(N = \lim H^0(U, \mathcal{F}/f^n\mathcal{F})\) is a finite \(A\)-module, is \(f\)-torsion free, and \(N/fN \subset H^0(U, \mathcal{F}/f\mathcal{F})\). On the other hand, we have a map \(M \to N\) and a compatible map \[M/fM \longrightarrow H^0(U, \mathcal{F}/f\mathcal{F})\] For \(g \in \mathfrak a\) we see that \((M/fM)_g\) maps isomorphically to \(H^0(U \cap D(f), \mathcal{F}/f\mathcal{F})\) since \(\mathcal{F}/f\mathcal{F}\) is the restriction of \(\widetilde{M/fM}\) to \(U\). We conclude that \(M_g \to N_g\) induces an isomorphism \[M_g/fM_g = (M/fM)_g \to (N/fN)_g = N_g/fN_g\] Since \(f\) is a nonzerodivisor on both \(N\) and \(M\) we conclude that \(M_g \to N_g\) induces an isomorphism on \(f\)-adic completions which in turn implies \(M_g \to N_g\) is an isomorphism in an open neighbourhood of \(V(f) \cap D(g)\). Since \(g \in \mathfrak a\) was arbitrary, we conclude that \(M\) and \(N\) determine isomorphic coherent modules over an open \(V\) as in the statement of the lemma. This finishes the proof.
Proposition
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a \subset A\) be an element of an ideal of \(A\). Let \(\mathcal{F}\) be a coherent module on \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) is \(f\)-adically complete and has a dualizing complex,
if \(x \in \text{Ass}(\mathcal{F})\), \(x \not \in V(f)\), \(\overline{\{x\}} \cap V(f) \not \subset V(\mathfrak a)\), and \(z \in \overline{\{x\}} \cap V(\mathfrak a)\), then \(\dim(\mathcal{O}_{\overline{\{x\}}, z}) > 2\).
Then the map \[\colim_V \Gamma(V, \mathcal{F}) \longrightarrow \lim \Gamma(U, \mathcal{F}/f^n\mathcal{F})\] is an isomorphism where the colimit is over opens \(V \subset U\) containing \(U \cap V(f)\).
Proof
Recall that \(A\) is universally catenary and with Gorenstein formal fibres, see Dualizing Complexes, Lemmas 0AWY and 0A80. Thus we may consider the map \(\mathcal{F} \to \mathcal{F}'\) constructed in Local Cohomology, Lemma 0DX5 for the closed subset \(V(f) \cap U\) of \(U\). Observe that
The kernel and cokernel of \(\mathcal{F} \to \mathcal{F}'\) are supported on \(V(f) \cap U\).
The module \(\mathcal{F}'\) is \(f\)-torsion free as its stalks have depth \(\geq 1\) for all points of \(V(f) \cap U\), i.e., \(\mathcal{F}'\) has no associated points in \(V(f) \cap U\).
If \(y \in V(f) \cap U\) is an associated point of \(\mathcal{F}'/f\mathcal{F}'\), then \(\text{depth}(\mathcal{F}'_y) = 1\) and hence (by the construction of \(\mathcal{F}'\)) there is an immediate specialization \(x \leadsto y\) with \(x \not \in V(f)\) an associated point of \(\mathcal{F}\). It follows that \(y\) cannot have an immediate specialization in \(\Spec(A)\) to a point \(z \in V(\mathfrak a)\) by our assumption (2).
It follows from (3) that \(H^0(U, \mathcal{F}'/f\mathcal{F}')\) is a finite \(A\)-module, see Local Cohomology, Lemma 0BJY.
These observations will allow us to finish the proof.
First, we claim the lemma holds for \(\mathcal{F}'\). Namely, choose a finite \(A\)-module \(M'\) such that \(\mathcal{F}'\) is the restriction to \(U\) of the coherent module associated to \(M'\), see Local Cohomology, Lemma 0BK0. Since \(\mathcal{F}'\) is \(f\)-torsion free, we may assume \(M'\) is \(f\)-torsion free as well. Observation (4) above shows that \(H^1_\mathfrak a(M')\) is a finite \(A\)-module, see Local Cohomology, Lemma 0BK0. Thus the claim by Lemma 0EIH.
Second, we observe that the lemma holds trivially for any coherent \(\mathcal{O}_U\)-module supported on \(V(f) \cap U\). Let \(\mathcal{K}\), resp. \(\mathcal{G}\), resp. \(\mathcal{Q}\) be the kernel, resp. image, resp. cokernel of the map \(\mathcal{F} \to \mathcal{F}'\). The short exact sequence \(0 \to \mathcal{G} \to \mathcal{F}' \to \mathcal{Q} \to 0\) and Lemma 0H48 show that the result holds for \(\mathcal{G}\). Then we do this again with the short exact sequence \(0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{G} \to 0\) to finish the proof.
Proof
The proposition is a special case of Proposition 0EG2.
Lemma
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a \subset A\) be an element of an ideal of \(A\). Let \(M\) be a finite \(A\)-module. Assume
\(A\) is \(f\)-adically complete,
\(H^1_\mathfrak a(M)\) and \(H^2_\mathfrak a(M)\) are annihilated by a power of \(f\).
Then with \(U = \Spec(A) \setminus V(\mathfrak a)\) the map \[\Gamma(U, \widetilde{M}) \longrightarrow \lim \Gamma(U, \widetilde{M/f^nM})\] is an isomorphism.
Proof
We may apply Lemma 0BLD to \(U\) and \(\mathcal{F} = \widetilde{M}|_U\) because \(\mathcal{F}\) is a Noetherian object in the category of coherent \(\mathcal{O}_U\)-modules. Since \(H^1(U, \mathcal{F}) = H^2_\mathfrak a(M)\) (Local Cohomology, Lemma 0BK0) is annihilated by a power of \(f\), we see that its \(f\)-adic Tate module is zero. Hence the lemma shows \(\lim H^0(U, \mathcal{F}/f^n \mathcal{F})\) is equal to the usual \(f\)-adic completion of \(H^0(U, \mathcal{F})\). Consider the short exact sequence \[0 \to M/H^0_\mathfrak a(M) \to H^0(U, \mathcal{F}) \to H^1_\mathfrak a(M) \to 0\] of Local Cohomology, Lemma 0BK0. Since \(M/H^0_\mathfrak a(M)\) is a finite \(A\)-module, it is complete, see Algebra, Lemma 00MA. Since \(H^1_\mathfrak a(M)\) is killed by a power of \(f\), we conclude from Algebra, Lemma 0BNG that \(H^0(U, \mathcal{F})\) is complete as well. This finishes the proof.
Algebraization of formal sections, III
The next section contains a nonexhaustive list of applications of the material on completion of local cohomology to higher cohomology of coherent modules on quasi-affine schemes and their completion with respect to an ideal.
Proposition
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(\mathcal{F}\) be a coherent module on \(U = \Spec(A) \setminus V(\mathfrak a)\). Let \(s \geq 0\). Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(x \in U \setminus V(I)\) then \(\text{depth}(\mathcal{F}_x) > s\) or \[\text{depth}(\mathcal{F}_x) + \dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + s + 1\] for all \(z \in V(\mathfrak a) \cap \overline{\{x\}}\),
one of the following conditions holds:
the restriction of \(\mathcal{F}\) to \(U \setminus V(I)\) is \((S_{s + 1})\), or
the dimension of \(V(\mathfrak a)\) is at most \(2\)5.
Then the maps \[H^i(U, \mathcal{F}) \longrightarrow \lim H^i(U, \mathcal{F}/I^n\mathcal{F})\] are isomorphisms for \(i < s\). Moreover we have an isomorphism \[\colim H^s(V, \mathcal{F}) \longrightarrow \lim H^s(U, \mathcal{F}/I^n\mathcal{F})\] where the colimit is over opens \(V \subset U\) containing \(U \cap V(I)\).
Proof
We may assume \(s > 0\) as the case \(s = 0\) was done in Proposition 0EG2.
Choose a finite \(A\)-module \(M\) such that \(\mathcal{F}\) is the restriction to \(U\) of the coherent module associated to \(M\), see Local Cohomology, Lemma 0BK0. Set \(d = \text{cd}(A, I)\). Let \(\mathfrak p\) be a prime of \(A\) not contained in \(V(I)\) and let \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\). Then either \(\text{depth}(M_\mathfrak p) \geq s + 1 > s\) or we have \(\dim((A/\mathfrak p)_\mathfrak q) > d + s + 1\) by (2). By Lemma 0EFW we conclude that the assumptions of Situation 0EFU are satisfied for \(A, I, V(\mathfrak a), M, s, d\). On the other hand, the hypotheses of Lemma 0EFL are satisfied for \(s + 1\) and \(d\); this is where condition (3) is used.
Applying Lemma 0EFL we find there exists an ideal \(J_0 \subset \mathfrak a\) with \(V(J_0) \cap V(I) = V(\mathfrak a)\) such that for any \(J \subset J_0\) with \(V(J) \cap V(I) = V(\mathfrak a)\) the maps \[H^i_J(M) \longrightarrow H^i(R\Gamma_\mathfrak a(M)^\wedge)\] is an isomorphism for \(i \leq s + 1\).
For \(i \leq s\) the map \(H^i_\mathfrak a(M) \to H^i_J(M)\) is an isomorphism by Lemmas 0EFV and 0EFK. Using the comparison of cohomology and local cohomology (Local Cohomology, Lemma 0DWR) we deduce \(H^i(U, \mathcal{F}) \to H^i(V,\mathcal{F})\) is an isomorphism for \(V = \Spec(A) \setminus V(J)\) and \(i < s\).
By Theorem 0EIE we have \(H^i_\mathfrak a(M) = \lim H^i_\mathfrak a(M/I^nM)\) for \(i \leq s\). By Lemma 0EG0 we have \(H^{s + 1}_\mathfrak a(M) = \lim H^{s + 1}_\mathfrak a(M/I^nM)\).
The isomorphism \(H^0(U, \mathcal{F}) = H^0(V, \mathcal{F}) = \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\) follows from the above and Proposition 0EG2. For \(0 < i < s\) we get the desired isomorphisms \(H^i(U, \mathcal{F}) = H^i(V, \mathcal{F}) = \lim H^i(U, \mathcal{F}/I^n\mathcal{F})\) in the same manner using the relation between local cohomology and cohomology; it is easier than the case \(i = 0\) because for \(i > 0\) we have \[H^i(U, \mathcal{F}) = H^{i + 1}_\mathfrak a(M), \quad H^i(V, \mathcal{F}) = H^{i + 1}_J(M), \quad H^i(R\Gamma(U, \mathcal{F})^\wedge) = H^{i + 1}(R\Gamma_\mathfrak a(M)^\wedge)\] Similarly for the final statement.
Lemma
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a \subset A\) be an element of an ideal of \(A\). Let \(M\) be a finite \(A\)-module. Let \(s \geq 0\). Assume
\(A\) is \(f\)-adically complete,
\(H^i_\mathfrak a(M)\) is annihilated by a power of \(f\) for \(i \leq s + 1\).
Then with \(U = \Spec(A) \setminus V(\mathfrak a)\) the map \[H^i(U, \widetilde{M}) \longrightarrow \lim H^i(U, \widetilde{M/f^nM})\] is an isomorphism for \(i < s\).
Proof
By induction on \(s\). If \(s = 0\), the assertion is empty. If \(s = 1\), then the result is Lemma 0EII. Assume \(s > 1\). By induction it suffices to prove the result for \(i = s - 1 \geq 1\). We may apply Lemma 0BLD to \(U\) and \(\mathcal{F} = \widetilde{M}|_U\) because \(\mathcal{F}\) is a Noetherian object in the category of coherent \(\mathcal{O}_U\)-modules. Observe that \(H^j(U, \mathcal{F}) = H^{j + 1}_\mathfrak a(M)\) for all \(j\) by Local Cohomology, Lemma 0BK0. Thus for \(j = s = (s - 1) + 1\) this is annihilated by a power of \(f\) by assumption. Thus it follows from Lemma 0BLD that \(\lim H^{s - 1}(U, \mathcal{F}/f^n\mathcal{F})\) is the usual \(f\)-adic completion of \(H^{s - 1}(U, \mathcal{F})\). Then again using that this module is killed by a power of \(f\) we see that the completion is simply equal to \(H^{s - 1}(U, \mathcal{F})\) as desired.
Application to connectedness
In this section we discuss Grothendieck’s connectedness theorem and variants; the original version can be found as [SGA2, Exposee XIII, Theorem 2.1]. There is a version called Faltings’ connectedness theorem in the literature; our guess is that this refers to [Faltings-some, Theorem 6]. Let us state and prove the optimal version for complete local rings given in [Varbaro, Theorem 1.6].
Lemma
Let \((A, \mathfrak m)\) be a Noetherian complete local ring. Let \(I\) be a proper ideal of \(A\). Set \(X = \Spec(A)\) and \(Y = V(I)\). Denote
\(d\) the minimal dimension of an irreducible component of \(X\), and
\(c\) the minimal dimension of a closed subset \(Z \subset X\) such that \(X \setminus Z\) is disconnected.
Then for \(Z \subset Y\) closed we have \(Y \setminus Z\) is connected if \(\dim(Z) < \min(c, d - 1) - \text{cd}(A, I)\). In particular, the punctured spectrum of \(A/I\) is connected if \(\text{cd}(A, I) < \min(c, d - 1)\).
Proof
Let us first prove the final assertion. As a first case, if the punctured spectrum of \(A/I\) is empty, then Local Cohomology, Lemma 0DXF shows every irreducible component of \(X\) has dimension \(\leq \text{cd}(A, I)\) and we get \(\min(c, d - 1) - \text{cd}(A, I) < 0\) which implies the lemma holds in this case. Thus we may assume \(U \cap Y\) is nonempty where \(U = X \setminus \{\mathfrak m\}\) is the punctured spectrum of \(A\). We may replace \(A\) by its reduction. Observe that \(A\) has a dualizing complex (Dualizing Complexes, Lemma 0BFR) and that \(A\) is complete with respect to \(I\) (Algebra, Lemma 090T). If we assume \(d - 1 > \text{cd}(A, I)\), then we may apply Lemma 0DXR to see that \[\colim H^0(V, \mathcal{O}_V) \longrightarrow \lim H^0(U, \mathcal{O}_U/I^n\mathcal{O}_U)\] is an isomorphism where the colimit is over opens \(V \subset U\) containing \(U \cap Y\). If \(U \cap Y\) is disconnected, then its \(n\)th infinitesimal neighbourhood in \(U\) is disconnected for all \(n\) and we find the right hand side has a nontrivial idempotent (here we use that \(U \cap Y\) is nonempty). Thus we can find a \(V\) which is disconnected. Set \(Z = X \setminus V\). By Local Cohomology, Lemma 0DXF we see that every irreducible component of \(Z\) has dimension \(\leq \text{cd}(A, I)\). Hence \(c \leq \text{cd}(A, I)\) and this indeed proves the final statement.
We can deduce the statement of the lemma from what we just proved as follows. Suppose that \(Z \subset Y\) closed and \(Y \setminus Z\) is disconnected and \(\dim(Z) = e\). Recall that a connected space is nonempty by convention. Hence we conclude either (a) \(Y = Z\) or (b) \(Y \setminus Z = W_1 \amalg W_2\) with \(W_i\) nonempty, open, and closed in \(Y \setminus Z\). In case (b) we may pick points \(w_i \in W_i\) which are closed in \(U\), see Morphisms, Lemma 02J6. Then we can find \(f_1, \ldots, f_e \in \mathfrak m\) such that \(V(f_1, \ldots, f_e) \cap Z = \{\mathfrak m\}\) and in case (b) we may assume \(w_i \in V(f_1, \ldots, f_e)\). Namely, we can inductively using prime avoidance choose \(f_i\) such that \(\dim V(f_1, \ldots, f_i) \cap Z = e - i\) and such that in case (b) we have \(w_1, w_2 \in V(f_i)\). It follows that the punctured spectrum of \(A/I + (f_1, \ldots, f_e)\) is disconnected (small detail omitted). Since \(\text{cd}(A, I + (f_1, \ldots, f_e)) \leq \text{cd}(A, I) + e\) by Local Cohomology, Lemmas 0ECP and 0DX9 we conclude that \[\text{cd}(A, I) + e \geq \min(c, d - 1)\] by the first part of the proof. This implies \(e \geq \min(c, d - 1) - \text{cd}(A, I)\) which is what we had to show.
Lemma
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(\mathfrak p \subset A\) is a minimal prime not contained in \(V(I)\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\), then \(\dim((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1\),
any nonempty open \(V \subset \Spec(A)\) which contains \(V(I) \setminus V(\mathfrak a)\) is connected6.
Then \(V(I) \setminus V(\mathfrak a)\) is either empty or connected.
Proof
We may replace \(A\) by its reduction. Then we have the inequality in (2) for all associated primes of \(A\). By Proposition 0EG2 we see that \[\colim H^0(V, \mathcal{O}_V) = \lim H^0(T_n, \mathcal{O}_{T_n})\] where the colimit is over the opens \(V\) as in (3) and \(T_n\) is the \(n\)th infinitesimal neighbourhood of \(T = V(I) \setminus V(\mathfrak a)\) in \(U = \Spec(A) \setminus V(\mathfrak a)\). Thus \(T\) is either empty or connected, since if not, then the right hand side would have a nontrivial idempotent and we’ve assumed the left hand side does not. Some details omitted.
Lemma
Let \(A\) be a Noetherian domain which has a dualizing complex and which is complete with respect to a nonzero \(f \in A\). Let \(f \in \mathfrak a \subset A\) be an ideal. Assume every irreducible component of \(Z = V(\mathfrak a)\) has codimension \(> 2\) in \(X = \Spec(A)\), i.e., assume every irreducible component of \(Z\) has codimension \(> 1\) in \(Y = V(f)\). Then \(Y \setminus Z\) is connected.
Proof
This is a special case of Lemma 0EG5 (whose proof relies on Proposition 0EG2). Below we prove it using the easier Proposition 0H49.
Set \(U = X \setminus Z\). By Proposition 0H49 we have an isomorphism \[\colim \Gamma(V, \mathcal{O}_V) \to \lim_n \Gamma(U, \mathcal{O}_U/f^n \mathcal{O}_U)\] where the colimit is over open \(V \subset U\) containing \(U \cap Y\). Hence if \(U \cap Y\) is disconnected, then for some \(V\) there exists a nontrivial idempotent in \(\Gamma(V, \mathcal{O}_V)\). This is impossible as \(V\) is an integral scheme as \(X\) is the spectrum of a domain.
The completion functor
Let \(X\) be a Noetherian scheme. Let \(Y \subset X\) be a closed subscheme with quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\). In this section we consider inverse systems of coherent \(\mathcal{O}_X\)-modules \((\mathcal{F}_n)\) with \(\mathcal{F}_n\) annihilated by \(I^n\) such that the transition maps induce isomorphisms \(\mathcal{F}_{n + 1}/I^n\mathcal{F}_{n + 1} \to \mathcal{F}_n\). The category of these inverse systems was denoted \[\textit{Coh}(X, \mathcal{I})\] in Cohomology of Schemes, Section 0EHN. This category is equivalent to the category of coherent modules on the formal completion of \(X\) along \(Y\); however, since we have not yet introduced formal schemes or coherent modules on them, we cannot use this terminology here. We are particularly interested in the completion functor \[\textit{Coh}(\mathcal{O}_X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] See Cohomology of Schemes, Equation (0880).
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme. Let \(Y_n \subset X\) be the \(n\)th infinitesimal neighbourhood of \(Y\) in \(X\). Consider the following conditions
\(X\) is quasi-affine and \(\Gamma(X, \mathcal{O}_X) \to \lim \Gamma(Y_n, \mathcal{O}_{Y_n})\) is an isomorphism,
\(X\) has an ample invertible module \(\mathcal{L}\) and \(\Gamma(X, \mathcal{L}^{\otimes m}) \to \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})\) is an isomorphism for all \(m \gg 0\),
for every finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) the map \(\Gamma(X, \mathcal{E}) \to \lim \Gamma(Y_n, \mathcal{E}|_{Y_n})\) is an isomorphism, and
the completion functor \(\textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(X, \mathcal{I})\) is fully faithful on the full subcategory of finite locally free objects.
Then (1) \(\Rightarrow\) (2) \(\Rightarrow\) (3) \(\Rightarrow\) (4) and (4) \(\Rightarrow\) (3).
Proof
Proof of (3) \(\Rightarrow\) (4). If \(\mathcal{F}\) and \(\mathcal{G}\) are finite locally free on \(X\), then considering \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})\) and using Cohomology of Schemes, Lemma 0882 we see that (3) implies (4).
Proof of (2) \(\rightarrow\) (3). Namely, let \(\mathcal{L}\) be ample on \(X\) and suppose that \(\mathcal{E}\) is a finite locally free \(\mathcal{O}_X\)-module. We claim we can find a universally exact sequence \[0 \to \mathcal{E} \to (\mathcal{L}^{\otimes p})^{\oplus r} \to (\mathcal{L}^{\otimes q})^{\oplus s}\] for some \(r, s \geq 0\) and \(0 \ll p \ll q\). If this holds, then using the exact sequence \[0 \to \lim \Gamma(\mathcal{E}|_{Y_n}) \to \lim \Gamma((\mathcal{L}^{\otimes p})^{\oplus r}|_{Y_n}) \to \lim \Gamma((\mathcal{L}^{\otimes q})^{\oplus s}|_{Y_n})\] and the isomorphisms in (2) we get the isomorphism in (3). To prove the claim, consider the dual locally free module \(\SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)\) and apply Properties, Proposition 01Q3 to find a surjection \[(\mathcal{L}^{\otimes -p})^{\oplus r} \longrightarrow \SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)\] Taking duals we obtain the first map in the exact sequence (it is universally injective because being a surjection is universal). Repeat with the cokernel to get the second. Some details omitted.
Proof of (1) \(\Rightarrow\) (2). This is true because if \(X\) is quasi-affine then \(\mathcal{O}_X\) is an ample invertible module, see Properties, Lemma 01QE.
We omit the proof of (4) \(\Rightarrow\) (3).
Given a Noetherian scheme and a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) we will say an object \((\mathcal{F}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) is finite locally free if each \(\mathcal{F}_n\) is a finite locally free \(\mathcal{O}_X/\mathcal{I}^n\)-module.
Lemma
Let \(X\) be a Noetherian scheme and let \(Y \subset X\) be a closed subscheme with ideal sheaf \(\mathcal{I} \subset \mathcal{O}_X\). Let \(Y_n \subset X\) be the \(n\)th infinitesimal neighbourhood of \(Y\) in \(X\). Let \(\mathcal{V}\) be the set of open subschemes \(V \subset X\) containing \(Y\) ordered by reverse inclusion.
\(X\) is quasi-affine and \[\colim_\mathcal{V} \Gamma(V, \mathcal{O}_V) \longrightarrow \lim \Gamma(Y_n, \mathcal{O}_{Y_n})\] is an isomorphism,
\(X\) has an ample invertible module \(\mathcal{L}\) and \[\colim_\mathcal{V} \Gamma(V, \mathcal{L}^{\otimes m}) \longrightarrow \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})\] is an isomorphism for all \(m \gg 0\),
for every \(V \in \mathcal{V}\) and every finite locally free \(\mathcal{O}_V\)-module \(\mathcal{E}\) the map \[\colim_{V' \geq V} \Gamma(V', \mathcal{E}|_{V'}) \longrightarrow \lim \Gamma(Y_n, \mathcal{E}|_{Y_n})\] is an isomorphism, and
the completion functor \[\colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects (see explanation above).
Then (1) \(\Rightarrow\) (2) \(\Rightarrow\) (3) \(\Rightarrow\) (4) and (4) \(\Rightarrow\) (3).
Proof
Observe that \(\mathcal{V}\) is a directed set, so the colimits are as in Categories, Section 04AX. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 0EKP; we urge the reader to skip it.
Proof of (3) \(\Rightarrow\) (4). If \(\mathcal{F}\) and \(\mathcal{G}\) are finite locally free on \(V \in \mathcal{V}\), then considering \(\mathcal{H} = \SheafHom_{\mathcal{O}_V}(\mathcal{G}, \mathcal{F})\) and using Cohomology of Schemes, Lemma 0882 we see that (3) implies (4).
Proof of (2) \(\Rightarrow\) (3). Let \(\mathcal{L}\) be ample on \(X\) and suppose that \(\mathcal{E}\) is a finite locally free \(\mathcal{O}_V\)-module for some \(V \in \mathcal{V}\). We claim we can find a universally exact sequence \[0 \to \mathcal{E} \to (\mathcal{L}^{\otimes p})^{\oplus r}|_{V} \to (\mathcal{L}^{\otimes q})^{\oplus s}|_{V}\] for some \(r, s \geq 0\) and \(0 \ll p \ll q\). If this is true, then the isomorphism in (2) will imply the isomorphism in (3). To prove the claim, recall that \(\mathcal{L}|_V\) is ample, see Properties, Lemma 0B3E. Consider the dual locally free module \(\SheafHom_{\mathcal{O}_V}(\mathcal{E}, \mathcal{O}_V)\) and apply Properties, Proposition 01Q3 to find a surjection \[(\mathcal{L}^{\otimes -p})^{\oplus r}|_V \longrightarrow \SheafHom_{\mathcal{O}_V}(\mathcal{E}, \mathcal{O}_V)\] (it is universally injective because being a surjection is universal). Taking duals we obtain the first map in the exact sequence. Repeat with the cokernel to get the second. Some details omitted.
Proof of (1) \(\Rightarrow\) (2). This is true because if \(X\) is quasi-affine then \(\mathcal{O}_X\) is an ample invertible module, see Properties, Lemma 01QE.
We omit the proof of (4) \(\Rightarrow\) (3).
Lemma
Let \(X\) be a Noetherian scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. The functor \[\textit{Coh}(X, \mathcal{I}) \longrightarrow \text{Pro-}\QCoh(\mathcal{O}_X)\] is fully faithful, see Categories, Remark 05PX.
Proof
Let \((\mathcal{F}_n)\) and \((\mathcal{G}_n)\) be objects of \(\textit{Coh}(X, \mathcal{I})\). A morphism of pro-objects \(\alpha\) from \((\mathcal{F}_n)\) to \((\mathcal{G}_n)\) is given by a system of maps \(\alpha_n : \mathcal{F}_{m(n)} \to \mathcal{G}_n\) compatible with the transition maps where \(\mathbf{N} \to \mathbf{N}\), \(n \mapsto m(n)\) is an increasing function (in particular \(m(n) \geq n\)). Since \(\mathcal{F}_n = \mathcal{F}_{m(n)}/\mathcal{I}^n\mathcal{F}_{m(n)}\) and since \(\mathcal{G}_n\) is annihilated by \(\mathcal{I}^n\) we see that \(\alpha_n\) induces a map \(\mathcal{F}_n \to \mathcal{G}_n\).
Next we add some examples of the kind of fully faithfulness result we will be able to prove using the work done earlier in this chapter.
Lemma
Let \(I \subset \mathfrak a\) be ideals of a Noetherian ring \(A\). Let \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
for any associated prime \(\mathfrak p \subset A\) with \(\mathfrak p \not \in V(I)\) and \(V(\mathfrak p) \cap V(I) \not \subset V(\mathfrak a)\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\) we have \(\dim((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1\),
for \(\mathfrak p \subset A\) with \(\mathfrak p \not \in V(I)\) and \(V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)\) we have \(\text{depth}(A_\mathfrak p) \geq 2\).
Then the completion functor \[\textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects.
Proof
By Lemma 0EKP it suffices to show that \[\Gamma(U, \mathcal{O}_U) = \lim \Gamma(U, \mathcal{O}_U/I^n\mathcal{O}_U)\] This follows immediately from Lemma 0EIG.
Lemma
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a \subset A\) be an element of an ideal of \(A\). Let \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) is \(f\)-adically complete,
\(H^1_\mathfrak a(A)\) and \(H^2_\mathfrak a(A)\) are annihilated by a power of \(f\).
Then the completion functor \[\textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects.
Proof
By Lemma 0EKP it suffices to show that \[\Gamma(U, \mathcal{O}_U) = \lim \Gamma(U, \mathcal{O}_U/I^n\mathcal{O}_U)\] This follows immediately from Lemma 0EII.
Lemma
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a\) be an element of an ideal of \(A\). Let \(U = \Spec(A) \setminus V(\mathfrak a)\). Assume
\(A\) has a dualizing complex and is complete with respect to \(f\),
for every prime \(\mathfrak p \subset A\), \(f \not \in \mathfrak p\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\) we have \(\text{depth}(A_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > 2\).
Then the completion functor \[\textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects.
Proof
Follows from Lemma 0EKS and Local Cohomology, Proposition 0EFC.
Lemma
Let \(I \subset \mathfrak a \subset A\) be ideals of a Noetherian ring \(A\). Let \(U = \Spec(A) \setminus V(\mathfrak a)\). Let \(\mathcal{V}\) be the set of open subschemes of \(U\) containing \(U \cap V(I)\) ordered by reverse inclusion. Assume
\(A\) is \(I\)-adically complete and has a dualizing complex,
for any associated prime \(\mathfrak p \subset A\) with \(I \not \subset \mathfrak p\) and \(V(\mathfrak p) \cap V(I) \not \subset V(\mathfrak a)\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\) we have \(\dim((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1\).
Then the completion functor \[\colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects.
Proof
By Lemma 0EK2 it suffices to show that \[\colim_\mathcal{V} \Gamma(V, \mathcal{O}_V) = \lim \Gamma(U, \mathcal{O}_U/I^n\mathcal{O}_U)\] This follows immediately from Proposition 0EG2.
Lemma
Let \(A\) be a Noetherian ring. Let \(f \in \mathfrak a \subset A\) be an element of an ideal of \(A\). Let \(U = \Spec(A) \setminus V(\mathfrak a)\). Let \(\mathcal{V}\) be the set of open subschemes of \(U\) containing \(U \cap V(f)\) ordered by reverse inclusion. Assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
\(H^1_\mathfrak a(A/fA)\) is a finite \(A\)-module.
Then the completion functor \[\colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(U, f\mathcal{O}_U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge\] is fully faithful on the full subcategory of finite locally free objects.
Proof
By Lemma 0EK2 it suffices to show that \[\colim_\mathcal{V} \Gamma(V, \mathcal{O}_V) = \lim \Gamma(U, \mathcal{O}_U/I^n\mathcal{O}_U)\] This follows immediately from Lemma 0EIH.
Lemma
Let \(I \subset \mathfrak a \subset A\) be ideals of a Noetherian ring \(A\). Let \(U = \Spec(A) \setminus V(\mathfrak a)\). Let \(\mathcal{V}\) be the set of open subschemes of \(U\) containing \(U \cap V(I)\) ordered by reverse inclusion. Let \(\mathcal{F}\) and \(\mathcal{G}\) be coherent \(\mathcal{O}_V\)-modules for some \(V \in \mathcal{V}\). The map \[\colim_{V' \geq V} \Hom_V(\mathcal{G}|_{V'}, \mathcal{F}|_{V'}) \longrightarrow \Hom_{\textit{Coh}(U, I\mathcal{O}_U)}(\mathcal{G}^\wedge, \mathcal{F}^\wedge)\] is bijective if the following assumptions hold:
\(A\) is \(I\)-adically complete and has a dualizing complex,
if \(x \in \text{Ass}(\mathcal{F})\), \(x \not \in V(I)\), \(\overline{\{x\}} \cap V(I) \not \subset V(\mathfrak a)\) and \(z \in \overline{\{x\}} \cap V(\mathfrak a)\), then \(\dim(\mathcal{O}_{\overline{\{x\}}, z}) > \text{cd}(A, I) + 1\).
Proof
We may choose coherent \(\mathcal{O}_U\)-modules \(\mathcal{F}'\) and \(\mathcal{G}'\) whose restriction to \(V\) is \(\mathcal{F}\) and \(\mathcal{G}\), see Properties, Lemma 0G41. We may modify our choice of \(\mathcal{F}'\) to ensure that \(\text{Ass}(\mathcal{F}') \subset V\), see for example Local Cohomology, Lemma 0DX3. Thus we may and do replace \(V\) by \(U\) and \(\mathcal{F}\) and \(\mathcal{G}\) by \(\mathcal{F}'\) and \(\mathcal{G}'\). Set \(\mathcal{H} = \SheafHom_{\mathcal{O}_U}(\mathcal{G}, \mathcal{F})\). This is a coherent \(\mathcal{O}_U\)-module. We have \[\Hom_V(\mathcal{G}|_V, \mathcal{F}|_V) = H^0(V, \mathcal{H}) \quad\text{and}\quad \lim H^0(U, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(U, I\mathcal{O}_U)} (\mathcal{G}^\wedge, \mathcal{F}^\wedge)\] See Cohomology of Schemes, Lemma 0882. Thus if we can show that the assumptions of Proposition 0EG2 hold for \(\mathcal{H}\), then the proof is complete. This holds because \(\text{Ass}(\mathcal{H}) \subset \text{Ass}(\mathcal{F})\). See Cohomology of Schemes, Lemma 0EBC.
Algebraization of coherent formal modules, I
The essential surjectivity of the completion functor (see below) was studied systematically in [SGA2], [MRaynaud-book], and [MRaynaud-paper]. We work in the following affine situation.
Situation
Here \(A\) is a Noetherian ring and \(I \subset \mathfrak a \subset A\) are ideals. We set \(X = \Spec(A)\), \(Y = V(I) = \Spec(A/I)\), and \(Z = V(\mathfrak a) = \Spec(A/\mathfrak a)\). Furthermore \(U = X \setminus Z\).
In this section we try to find conditions that guarantee an object of \(\textit{Coh}(U, I\mathcal{O}_U)\) is in the image of the completion functor \(\textit{Coh}(\mathcal{O}_U) \to \textit{Coh}(U, I\mathcal{O}_U)\). See Cohomology of Schemes, Section 0EHN and Section 0EKN.
Lemma
In Situation 0EHC. Consider an inverse system \((M_n)\) of \(A\)-modules such that
\(M_n\) is a finite \(A\)-module,
\(M_n\) is annihilated by \(I^n\),
the kernel and cokernel of \(M_{n + 1}/I^nM_{n + 1} \to M_n\) are \(\mathfrak a\)-power torsion.
Then \((\widetilde{M}_n|_U)\) is in \(\textit{Coh}(U, I\mathcal{O}_U)\). Conversely, every object of \(\textit{Coh}(U, I\mathcal{O}_U)\) arises in this manner.
Proof
We omit the verification that \((\widetilde{M}_n|_U)\) is in \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). By Local Cohomology, Lemma 0BK0 we see that \(\mathcal{F}_n = \widetilde{M_n}\) for some finite \(A/I^n\)-module \(M_n\). After dividing \(M_n\) by \(H^0_\mathfrak a(M_n)\) we may assume \(M_n \subset H^0(U, \mathcal{F}_n)\), see Dualizing Complexes, Lemma 0AW0 and the already referenced lemma. After replacing inductively \(M_{n + 1}\) by the inverse image of \(M_n\) under the map \(M_{n + 1} \to H^0(U, \mathcal{F}_{n + 1}) \to H^0(U, \mathcal{F}_n)\), we may assume \(M_{n + 1}\) maps into \(M_n\). This gives an inverse system \((M_n)\) satisfying (1) and (2) such that \(\mathcal{F}_n = \widetilde{M_n}\). To see that (3) holds, use that \(M_{n + 1}/I^nM_{n + 1} \to M_n\) is a map of finite \(A\)-modules which induces an isomorphism after applying \(\widetilde{\ }\) and restriction to \(U\) (here we use the first referenced lemma one more time).
In Situation 0EHC we can study the completion functor Cohomology of Schemes, Equation (0880) [0EIK]\[\begin{equation} \textit{Coh}(\mathcal{O}_U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \end{equation}\] If \(A\) is \(I\)-adically complete, then this functor is fully faithful on suitable subcategories by our earlier work on algebraization of formal sections, see Section 0EKN and Lemma 0EJ7 for some sample results. Next, let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Still assuming \(A\) is \(I\)-adically complete, we can ask: When is \((\mathcal{F}_n)\) in the essential image of the completion functor displayed above?
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Consider the following conditions:
\((\mathcal{F}_n)\) is in the essential image of the functor (0EIK),
\((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module,
\((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_V\)-module for \(U \cap Y \subset V \subset U\) open,
\((\mathcal{F}_n)\) is the completion of the restriction to \(U\) of a coherent \(\mathcal{O}_X\)-module,
\((\mathcal{F}_n)\) is the restriction to \(U\) of the completion of a coherent \(\mathcal{O}_X\)-module,
there exists an object \((\mathcal{G}_n)\) of \(\textit{Coh}(X, I\mathcal{O}_X)\) whose restriction to \(U\) is \((\mathcal{F}_n)\).
Then conditions (1), (2), (3), (4), and (5) are equivalent and imply (6). If \(A\) is \(I\)-adically complete then condition (6) implies the others.
Proof
Parts (1) and (2) are equivalent, because the completion of a coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\) is by definition the image of \(\mathcal{F}\) under the functor (0EIK). If \(V \subset U\) is an open subscheme containing \(U \cap Y\), then we have \[\textit{Coh}(V, I\mathcal{O}_V) = \textit{Coh}(U, I\mathcal{O}_U)\] since the category of coherent \(\mathcal{O}_V\)-modules supported on \(V \cap Y\) is the same as the category of coherent \(\mathcal{O}_U\)-modules supported on \(U \cap Y\). Thus the completion of a coherent \(\mathcal{O}_V\)-module is an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Having said this the equivalence of (2), (3), (4), and (5) holds because the functors \(\textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_U) \to \textit{Coh}(\mathcal{O}_V)\) are essentially surjective. See Properties, Lemma 0G41.
It is always the case that (5) implies (6). Assume \(A\) is \(I\)-adically complete. Then any object of \(\textit{Coh}(X, I\mathcal{O}_X)\) corresponds to a finite \(A\)-module by Cohomology of Schemes, Lemma 087W. Thus we see that (6) implies (5) in this case.
Example
Let \(k\) be a field. Let \(A = k[x, y][[t]]\) with \(I = (t)\) and \(\mathfrak a = (x, y, t)\). Let us use notation as in Situation 0EHC. Observe that \(U \cap Y = (D(x) \cap Y) \cup (D(y) \cap Y)\) is an affine open covering. For \(n \geq 1\) consider the invertible module \(\mathcal{L}_n\) of \(\mathcal{O}_U/t^n\mathcal{O}_U\) given by glueing \(A_x/t^nA_x\) and \(A_y/t^nA_y\) via the invertible element of \(A_{xy}/t^nA_{xy}\) which is the image of any power series of the form \[u = 1 + \frac{t}{xy} + \sum_{n \geq 2} a_n \frac{t^n}{(xy)^{\varphi(n)}}\] with \(a_n \in k[x, y]\) and \(\varphi(n) \in \mathbf{N}\). Then \((\mathcal{L}_n)\) is an invertible object of \(\textit{Coh}(U, I\mathcal{O}_U)\) which is not the completion of a coherent \(\mathcal{O}_U\)-module \(\mathcal{L}\). We only sketch the argument and we omit most of the details. Let \(y \in U \cap Y\). Then the completion of the stalk \(\mathcal{L}_y\) would be an invertible module hence \(\mathcal{L}_y\) is invertible. Thus there would exist an open \(V \subset U\) containing \(U \cap Y\) such that \(\mathcal{L}|_V\) is invertible. By Divisors, Lemma 0BD9 we find an invertible \(A\)-module \(M\) with \(\widetilde{M}|_V \cong \mathcal{L}|_V\). However the ring \(A\) is a UFD hence we see \(M \cong A\) which would imply \(\mathcal{L}_n \cong \mathcal{O}_U/I^n\mathcal{O}_U\). Since \(\mathcal{L}_2 \not \cong \mathcal{O}_U/I^2\mathcal{O}_U\) by construction we get a contradiction as desired.
Note that if we take \(a_n = 0\) for \(n \geq 2\), then we see that \(\lim H^0(U, \mathcal{L}_n)\) is nonzero: in this case we the function \(x\) on \(D(x)\) and the function \(x + t/y\) on \(D(y)\) glue. On the other hand, if we take \(a_n = 1\) and \(\varphi(n) = 2^n\) or even \(\varphi(n) = n^2\) then the reader can show that \(\lim H^0(U, \mathcal{L}_n)\) is zero; this gives another proof that \((\mathcal{L}_n)\) is not algebraizable in this case.
If in Situation 0EHC the ring \(A\) is not \(I\)-adically complete, then Lemma 0EIL suggests the correct thing is to ask whether \((\mathcal{F}_n)\) is in the essential image of the restriction functor \[\textit{Coh}(X, I\mathcal{O}_X) \longrightarrow \textit{Coh}(U, I\mathcal{O}_U)\] However, we can no longer say that this means \((\mathcal{F}_n)\) is algebraizable. Thus we introduce the following terminology.
Definition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). We say \((\mathcal{F}_n)\) extends to \(X\) if there exists an object \((\mathcal{G}_n)\) of \(\textit{Coh}(X, I\mathcal{O}_X)\) whose restriction to \(U\) is isomorphic to \((\mathcal{F}_n)\).
This notion is equivalent to being algebraizable over the completion.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(A', I', \mathfrak a'\) be the \(I\)-adic completions of \(A, I, \mathfrak a\). Set \(X' = \Spec(A')\) and \(U' = X' \setminus V(\mathfrak a')\). The following are equivalent
\((\mathcal{F}_n)\) extends to \(X\), and
the pullback of \((\mathcal{F}_n)\) to \(U'\) is the completion of a coherent \(\mathcal{O}_{U'}\)-module.
Proof
Recall that \(A \to A'\) is a flat ring map which induces an isomorphism \(A/I \to A'/I'\). See Algebra, Lemmas 00MB and 031C. Thus \(X' \to X\) is a flat morphism inducing an isomorphism \(Y' \to Y\). Thus \(U' \to U\) is a flat morphism which induces an isomorphism \(U' \cap Y' \to U \cap Y\). This implies that in the commutative diagram \[\xymatrix{ \textit{Coh}(X', I\mathcal{O}_{X'}) \ar[r] & \textit{Coh}(U', I\mathcal{O}_{U'}) \\ \textit{Coh}(X, I\mathcal{O}_X) \ar[u] \ar[r] & \textit{Coh}(U, I\mathcal{O}_U) \ar[u] }\] the vertical functors are equivalences. See Cohomology of Schemes, Lemma 0EHQ. The lemma follows formally from this and the results of Lemma 0EIL.
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). To figure out if \((\mathcal{F}_n)\) extends to \(X\) it makes sense to look at the \(A\)-module [0EHD]\[\begin{equation} M = \lim H^0(U, \mathcal{F}_n) \end{equation}\] Observe that \(M\) has a limit topology which is (a priori) coarser than the \(I\)-adic topology since \(M \to H^0(U, \mathcal{F}_n)\) annihilates \(I^nM\). There are canonical maps \[\widetilde{M}|_U \to \widetilde{M/I^nM}|_U \to \widetilde{H^0(U, \mathcal{F}_n)}|_U \to \mathcal{F}_n\] One could hope that \(\widetilde{M}\) restricts to a coherent module on \(U\) and that \((\mathcal{F}_n)\) is the completion of this module. This is naive because this has almost no chance of being true if \(A\) is not complete. But even if \(A\) is \(I\)-adically complete this notion is very difficult to work with. A less naive approach is to consider the following requirement.
Definition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). We say \((\mathcal{F}_n)\) canonically extends to \(X\) if the the inverse system \[\{\widetilde{H^0(U, \mathcal{F}_n)}\}_{n \geq 1}\] in \(\QCoh(\mathcal{O}_X)\) is pro-isomorphic to an object \((\mathcal{G}_n)\) of \(\textit{Coh}(X, I\mathcal{O}_X)\).
We will see in Lemma 0EIR that the condition in Definition 0EIP is stronger than the condition of Definition 0EIM.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). If \((\mathcal{F}_n)\) canonically extends to \(X\), then
\((\widetilde{H^0(U, \mathcal{F}_n)})\) is pro-isomorphic to an object \((\mathcal{G}_n)\) of \(\textit{Coh}(X, I \mathcal{O}_X)\) unique up to unique isomorphism,
the restriction of \((\mathcal{G}_n)\) to \(U\) is isomorphic to \((\mathcal{F}_n)\), i.e., \((\mathcal{F}_n)\) extends to \(X\),
the inverse system \(\{H^0(U, \mathcal{F}_n)\}\) satisfies the Mittag-Leffler condition, and
the module \(M\) in (0EHD) is finite over the \(I\)-adic completion of \(A\) and the limit topology on \(M\) is the \(I\)-adic topology.
Proof
The existence of \((\mathcal{G}_n)\) in (1) follows from Definition 0EIP. The uniqueness of \((\mathcal{G}_n)\) in (1) follows from Lemma 0EIQ. Write \(\mathcal{G}_n = \widetilde{M_n}\). Then \(\{M_n\}\) is an inverse system of finite \(A\)-modules with \(M_n = M_{n + 1}/I^n M_{n + 1}\). By Definition 0EIP the inverse system \(\{H^0(U, \mathcal{F}_n)\}\) is pro-isomorphic to \(\{M_n\}\). Hence we see that the inverse system \(\{H^0(U, \mathcal{F}_n)\}\) satisfies the Mittag-Leffler condition and that \(M = \lim M_n\) (as topological modules). Thus the properties of \(M\) in (4) follow from Algebra, Lemmas 09B8, 031D, and 05GG. Since \(U\) is quasi-affine the canonical maps \[\widetilde{H^0(U, \mathcal{F}_n)}|_U \to \mathcal{F}_n\] are isomorphisms (Properties, Lemma 0EHM). We conclude that \((\mathcal{G}_n|_U)\) and \((\mathcal{F}_n)\) are pro-isomorphic and hence isomorphic by Lemma 0EIQ.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(A \to A'\) be a flat ring map. Set \(X' = \Spec(A')\), let \(U' \subset X'\) be the inverse image of \(U\), and denote \(g : U' \to U\) the induced morphism. Set \((\mathcal{F}'_n) = (g^*\mathcal{F}_n)\), see Cohomology of Schemes, Lemma 0887. If \((\mathcal{F}_n)\) canonically extends to \(X\), then \((\mathcal{F}'_n)\) canonically extends to \(X'\). Moreover, the extension found in Lemma 0EIR for \((\mathcal{F}_n)\) pulls back to the extension for \((\mathcal{F}'_n)\).
Proof
Let \(f : X' \to X\) be the induced morphism. We have \(H^0(U', \mathcal{F}'_n) = H^0(U, \mathcal{F}_n) \otimes_A A'\) by flat base change, see Cohomology of Schemes, Lemma 02KH. Thus if \((\mathcal{G}_n)\) in \(\textit{Coh}(X, I\mathcal{O}_X)\) is pro-isomorphic to \((\widetilde{H^0(U, \mathcal{F}_n)})\), then \((f^*\mathcal{G}_n)\) is pro-isomorphic to \[(f^*\widetilde{H^0(U, \mathcal{F}_n)}) = (\widetilde{H^0(U, \mathcal{F}_n) \otimes_A A'}) = (\widetilde{H^0(U', \mathcal{F}'_n)})\] This finishes the proof.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(M\) be as in (0EHD). Assume
the inverse system \(H^0(U, \mathcal{F}_n)\) has Mittag-Leffler,
the limit topology on \(M\) agrees with the \(I\)-adic topology, and
the image of \(M \to H^0(U, \mathcal{F}_n)\) is a finite \(A\)-module for all \(n\).
Then \((\mathcal{F}_n)\) extends canonically to \(X\). In particular, if \(A\) is \(I\)-adically complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
Since \(H^0(U, \mathcal{F}_n)\) has the Mittag-Leffler condition and since the limit topology on \(M\) is the \(I\)-adic topology we see that \(\{M/I^nM\}\) and \(\{H^0(U, \mathcal{F}_n)\}\) are pro-isomorphic inverse systems of \(A\)-modules. Thus if we set \[\mathcal{G}_n = \widetilde{M/I^n M}\] then we see that to verify the condition in Definition 0EIP it suffices to show that \(M\) is a finite module over the \(I\)-adic completion of \(A\). This follows from the fact that \(M/I^n M\) is finite by condition (c) and the above and Algebra, Lemma 031D.
The following is in some sense the most straightforward possible application Lemma 0EHH above.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(I = (f)\) is a principal ideal for a nonzerodivisor \(f \in \mathfrak a\),
\(\mathcal{F}_n\) is a finite locally free \(\mathcal{O}_U/f^n\mathcal{O}_U\)-module,
\(H^1_\mathfrak a(A/fA)\) and \(H^2_\mathfrak a(A/fA)\) are finite \(A\)-modules.
Then \((\mathcal{F}_n)\) extends canonically to \(X\). In particular, if \(A\) is complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
We will prove this by verifying hypotheses (a), (b), and (c) of Lemma 0EHH.
Since \(\mathcal{F}_n\) is locally free over \(\mathcal{O}_U/f^n\mathcal{O}_U\) we see that we have short exact sequences \(0 \to \mathcal{F}_n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0\) for all \(n\). Thus condition (b) holds by Cohomology, Lemma 0EHA.
As \(f\) is a nonzerodivisor we obtain short exact sequences \[0 \to A/f^nA \xrightarrow{f} A/f^{n + 1}A \to A/fA \to 0\] and we have corresponding short exact sequences \(0 \to \mathcal{F}_n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0\). We will use Local Cohomology, Lemma 0BK0 without further mention. Our assumptions imply that \(H^0(U, \mathcal{O}_U/f\mathcal{O}_U)\) and \(H^1(U, \mathcal{O}_U/f\mathcal{O}_U)\) are finite \(A\)-modules. Hence the same thing is true for \(\mathcal{F}_1\), see Local Cohomology, Lemma 0BLT. Using induction and the short exact sequences we find that \(H^0(U, \mathcal{F}_n)\) are finite \(A\)-modules for all \(n\). In this way we see hypothesis (c) is satisfied.
Finally, as \(H^1(U, \mathcal{F}_1)\) is a finite \(A\)-module we can apply Cohomology, Lemma 0BLC to see hypothesis (a) holds.
Remark
In Lemma 0DXW if \(A\) is universally catenary with Cohen-Macaulay formal fibres (for example if \(A\) has a dualizing complex), then the condition that \(H^1_\mathfrak a(A/fA)\) and \(H^2_\mathfrak a(A/fA)\) are finite \(A\)-modules, is equivalent with \[\text{depth}((A/f)_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > 2\] for all \(\mathfrak p \in V(f) \setminus V(\mathfrak a)\) and \(\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)\) by Local Cohomology, Theorem 0BJV.
For example, if \(A/fA\) is \((S_2)\) and if every irreducible component of \(Z = V(\mathfrak a)\) has codimension \(\geq 3\) in \(Y = \Spec(A/fA)\), then we get the finiteness of \(H^1_\mathfrak a(A/fA)\) and \(H^2_\mathfrak a(A/fA)\). This should be contrasted with the slightly weaker conditions found in Lemma 0DXU (see also Remark 0DXV).
Algebraization of coherent formal modules, II
We continue the discussion started in Section 0DXS. This section can be skipped on a first reading.
Lemma
In Situation 0EHC. Let \((\mathcal{F}_n) \to (\mathcal{F}'_n)\) be a morphism of \(\textit{Coh}(U, I\mathcal{O}_U)\) whose kernel and cokernel are annihilated by a power of \(I\). Then
\((\mathcal{F}_n)\) extends to \(X\) if and only if \((\mathcal{F}'_n)\) extends to \(X\), and
\((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module if and only if \((\mathcal{F}'_n)\) is.
Proof
Part (2) follows immediately from Cohomology of Schemes, Lemma 0889. To see part (1), we first use Lemma 0EIN to reduce to the case where \(A\) is \(I\)-adically complete. However, in that case (1) reduces to (2) by Lemma 0EIL.
The following two lemmas where originally used in the proof of Lemma 0EHH. We keep them here for the reader who is interested to know what intermediate results one can obtain.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). If the inverse system \(H^0(U, \mathcal{F}_n)\) has Mittag-Leffler, then the canonical maps \[\widetilde{M/I^nM}|_U \to \mathcal{F}_n\] are surjective for all \(n\) where \(M\) is as in (0EHD).
Proof
Surjectivity may be checked on the stalk at some point \(y \in Y \setminus Z\). If \(y\) corresponds to the prime \(\mathfrak q \subset A\), then we can choose \(f \in \mathfrak a\), \(f \not \in \mathfrak q\). Then it suffices to show \[M_f \longrightarrow H^0(U, \mathcal{F}_n)_f = H^0(D(f), \mathcal{F}_n)\] is surjective as \(D(f)\) is affine (equality holds by Properties, Lemma 01P7). Since we have the Mittag-Leffler property, we find that \[\Im(M \to H^0(U, \mathcal{F}_n)) = \Im(H^0(U, \mathcal{F}_m) \to H^0(U, \mathcal{F}_n))\] for some \(m \geq n\). Using the long exact sequence of cohomology we see that \[\Coker(H^0(U, \mathcal{F}_m) \to H^0(U, \mathcal{F}_n)) \subset H^1(U, \Ker(\mathcal{F}_m \to \mathcal{F}_n))\] Since \(U = X \setminus V(\mathfrak a)\) this \(H^1\) is \(\mathfrak a\)-power torsion. Hence after inverting \(f\) the cokernel becomes zero.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(M\) be as in (0EHD). Set \[\mathcal{G}_n = \widetilde{M/I^nM}.\] If the limit topology on \(M\) agrees with the \(I\)-adic topology, then \(\mathcal{G}_n|_U\) is a coherent \(\mathcal{O}_U\)-module and the map of inverse systems \[(\mathcal{G}_n|_U) \longrightarrow (\mathcal{F}_n)\] is injective in the abelian category \(\textit{Coh}(U, I\mathcal{O}_U)\).
Proof
Observe that \(\mathcal{G}_n\) is a quasi-coherent \(\mathcal{O}_X\)-module annihilated by \(I^n\) and that \(\mathcal{G}_{n + 1}/I^n\mathcal{G}_{n + 1} = \mathcal{G}_n\). Consider \[M_n = \Im(M \longrightarrow H^0(U, \mathcal{F}_n))\] The assumption says that the inverse systems \((M_n)\) and \((M/I^nM)\) are isomorphic as pro-objects of \(\text{Mod}_A\). Pick \(f \in \mathfrak a\) so \(D(f) \subset U\) is an affine open. Then we have \[(M_n)_f \subset H^0(U, \mathcal{F}_n)_f = H^0(D(f), \mathcal{F}_n)\] Equality holds by Properties, Lemma 01P7. Thus \(\widetilde{M_n}|_U \to \mathcal{F}_n\) is injective. It follows that \(\widetilde{M_n}|_U\) is a coherent module (Cohomology of Schemes, Lemma 01Y1). Since \(M \to M/I^nM\) is surjective and factors as \(M_{n'} \to M/I^nM\) for some \(n' \geq n\) we find that \(\mathcal{G}_n|_U\) is coherent as the quotient of a coherent module. Combined with the initical remarks of the proof we conclude that \((\mathcal{G}_n|_U)\) indeed forms an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Finally, to show the injectivity of the map it suffices to show that \[\lim (M/I^nM)_f = \lim H^0(D(f), \mathcal{G}_n) \to \lim H^0(D(f), \mathcal{F}_n)\] is injective, see Cohomology of Schemes, Lemmas 087X and 087W. The injectivity of \(\lim (M_n)_f \to \lim H^0(D(f), \mathcal{F}_n)\) is clear (see above) and by our remark on pro-systems we have \(\lim (M_n)_f = \lim (M/I^nM)_f\). This finishes the proof.
A distance function
Let \(Y\) be a Noetherian scheme and let \(Z \subset Y\) be a closed subset. We define a function [0EIX]\[\begin{equation} \delta^Y_Z = \delta_Z : Y \longrightarrow \mathbf{Z}_{\geq 0} \cup \{\infty\} \end{equation}\] which measures the “distance” of a point of \(Y\) from \(Z\). For an informal discussion, please see Remark 0EJ0. Let \(y \in Y\). We set \(\delta_Z(y) = \infty\) if \(y\) is contained in a connected component of \(Y\) which does not meet \(Z\). If \(y\) is contained in a connected component of \(Y\) which meets \(Z\), then we can find \(k \geq 0\) and a system \[V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \supset V_k \subset W_k\] of integral closed subschemes of \(Y\) such that \(V_0 \subset Z\) and \(y \in W_k\) is the generic point. Set \(c_i = \text{codim}(V_i, W_i)\) for \(i = 0, \ldots, k\) and \(b_i = \text{codim}(V_{i + 1}, W_i)\) for \(i = 0, \ldots, k - 1\). For such a system we set \[\delta(V_0, W_0, V_1, \ldots, W_k) = k + \max_{i = 0, 1, \ldots, k} (c_i + c_{i + 1} + \ldots + c_k - b_i - b_{i + 1} - \ldots - b_{k - 1})\] This is \(\geq k\) as we can take \(i = k\) and we have \(c_k \geq 0\). Finally, we set \[\delta_Z(y) = \min \delta(V_0, W_0, V_1, \ldots, W_k)\] where the minimum is over all systems of integral closed subschemes of \(Y\) as above.
Lemma
Let \(Y\) be a Noetherian scheme and let \(Z \subset Y\) be a closed subset.
For \(y \in Y\) we have \(\delta_Z(y) = 0 \Leftrightarrow y \in Z\).
The subsets \(\{y \in Y \mid \delta_Z(y) \leq k\}\) are stable under specialization.
For \(y \in Y\) and \(z \in \overline{\{y\}} \cap Z\) we have \(\dim(\mathcal{O}_{\overline{\{y\}}, z}) \geq \delta_Z(y)\).
If \(\delta\) is a dimension function on \(Y\), then \(\delta_Z(y) \geq \delta(y) - \delta_{max}\) where \(\delta_{max}\) is the maximum value of \(\delta\) on \(Z\).
If \(Y = \Spec(A)\) is the spectrum of a catenary Noetherian local ring with maximal ideal \(\mathfrak m\) and \(Z = \{\mathfrak m\}\), then \(\delta_Z(y) = \dim(\overline{\{y\}})\).
If \(Y' \subset Y\) is an open subscheme, then \(\delta^{Y'}_{Y' \cap Z}(y') \geq \delta^Y_Z(y')\) for \(y' \in Y'\).
Assume \(Y\) is catenary. Then
Let \(y' \leadsto y\) be an immediate specialization of points of \(Y\). If \(Y\) is catenary, then \(\delta_Z(y') \leq \delta_Z(y) + 1\).
Given a pattern of specializations \[\xymatrix{ & y'_0 \ar@{~>}[ld] \ar@{~>}[rd] & & y'_1 \ar@{~>}[ld] & \ldots & y'_{k - 1} \ar@{~>}[rd] & \\ y_0 & & y_1 & & \ldots & & y_k = y }\] between points of \(Y\) with \(y_0 \in Z\) and \(y_i' \leadsto y_i\) an immediate specialization, then \(\delta_Z(y_k) \leq k\).
Proof
Proof of (1). If \(y \in Z\), then we can take \(k = 0\) and \(V_0 = W_0 = \overline{\{y\}}\) and we get \(\delta(V_0, W_0) = 0\) so \(\delta_Z(y) = 0\). If \(y \not \in Z\), then for every system \(V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_k\) for \(y\) we either have \(k = 0\) and \(V_0 \not = W_0\) or \(k > 0\). In both cases \(\delta(V_0, W_0, \ldots, W_k) > 0\). Hence \(\delta_Z(y) > 0\).
Proof of (2). Let \(y \leadsto y'\) be a nontrivial specialization and let \(V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_k\) is a system for \(y\). Here there are two cases. Case I: \(V_k = W_k\), i.e., \(c_k = 0\). In this case we can set \(V'_k = W'_k = \overline{\{y'\}}\). An easy computation shows that \(\delta(V_0, W_0, \ldots, V'_k, W'_k) \leq \delta(V_0, W_0, \ldots, V_k, W_k)\) because only \(b_{k - 1}\) is changed into a bigger integer. Case II: \(V_k \not = W_k\), i.e., \(c_k > 0\). In this case, setting \(V_{k + 1} = W_{k + 1} = \overline{\{y'\}}\) we see that \(V_0 \subset W_0 \supset \ldots \subset W_k \supset V_{k + 1} \subset W_{k + 1}\) is a system for \(y'\). Then \(c_{k + 1} = 0\) and \(b_k > 0\) so we get \[\begin{align*} & \delta(V_0, \ldots, W_{k + 1}) \\ & = k + 1 + \max_{i = 0, 1, \ldots, k + 1} (c_i + c_{i + 1} + \ldots + c_k + c_{k + 1} - b_i - b_{i + 1} - \ldots - b_{k - 1} - b_k) \\ & = k + 1 + \max_{i = 0, 1, \ldots, k + 1} (c_i + c_{i + 1} + \ldots + c_k - b_i - b_{i + 1} - \ldots - b_{k - 1} - b_k) \\ & \leq k + \max_{i = 0, 1, \ldots, k} (c_i + c_{i + 1} + \ldots + c_k - b_i - b_{i + 1} - \ldots - b_{k - 1}) \\ & = \delta(V_0, \ldots, W_k) \end{align*}\] The inequality holds because \(c_k > 0\) and \(b_k > 0\) which in particular implies that \(\delta(V_0, \ldots, W_k) \geq k + c_k \geq k + 1\).
Proof of (3). Given \(y \in Y\) and \(z \in \overline{\{y\}} \cap Z\) we get the system \[V_0 = \overline{\{z\}} \subset W_0 = \overline{\{y\}}\] and \(c_0 = \text{codim}(V_0, W_0) = \dim(\mathcal{O}_{\overline{\{y\}}, z})\) by Properties, Lemma 02IZ. Thus we see that \(\delta(V_0, W_0) = 0 + c_0 = c_0\) which proves what we want.
Proof of (4). Let \(\delta\) be a dimension function on \(Y\). Let \(V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_k\) be a system for \(y\). Let \(y'_i \in W_i\) and \(y_i \in V_i\) be the generic points, so \(y_0 \in Z\) and \(y_k = y\). Then we see that \[\delta(y_i) - \delta(y_{i - 1}) = \delta(y'_{i - 1}) - \delta(y_{i - 1}) - \delta(y'_{i - 1}) + \delta(y_i) = c_{i - 1} - b_{i - 1}\] Finally, we have \(\delta(y'_k) - \delta(y_{k - 1}) = c_k\). Thus we see that \[\delta(y) - \delta(y_0) = c_0 + \ldots + c_k - b_0 - \ldots - b_{k - 1}\] We conclude \(\delta(V_0, W_0, \ldots, W_k) \geq k + \delta(y) - \delta(y_0)\) which proves what we want.
Proof of (5). The function \(\delta(y) = \dim(\overline{\{y\}})\) is a dimension function. Hence \(\delta(y) \leq \delta_Z(y)\) by part (4). By part (3) we have \(\delta_Z(y) \leq \delta(y)\) and we are done.
Proof of (6). This is clear as their are fewer systems to consider in the computation of \(\delta^{Y'}_{Y' \cap Z}\).
Proof of (7). Let \(V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_k\) be a system for \(y\). Set \(W'_k = \overline{\{y'\}}\). Since \(Y\) is catenary, we see that \(\text{codim}(V_k, W'_k) = \text{codim}(V_k, W_k) + 1\). It follows easily that \(\delta(V_0, \ldots, V_k, W'_k) \leq \delta(V_0, \ldots, V_k, W_k) + 1\) which proves what we want.
Proof of (8): combine (7) and (2).
Lemma
Let \(Y\) be a universally catenary Noetherian scheme. Let \(Z \subset Y\) be a closed subscheme. Let \(f : Y' \to Y\) be a finite type morphism all of whose fibres have dimension \(\leq e\). Set \(Z' = f^{-1}(Z)\). Then \[\delta_Z(y) \leq \delta_{Z'}(y') + e - \text{trdeg}_{\kappa(y)}(\kappa(y'))\] for \(y' \in Y'\) with image \(y \in Y\).
Proof
If \(\delta_{Z'}(y') = \infty\), then there is nothing to prove. If \(\delta_{Z'}(y') < \infty\), then we choose a system of integral closed subschemes \[V'_0 \subset W'_0 \supset V'_1 \subset W'_1 \supset \ldots \subset W'_k\] of \(Y'\) with \(V'_0 \subset Z'\) and \(y'\) the generic point of \(W'_k\) such that \(\delta_{Z'}(y') = \delta(V'_0, W'_0, \ldots, W'_k)\). Denote \[V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_k\] the scheme theoretic images of the above schemes in \(Y\). Observe that \(y\) is the generic point of \(W_k\) and that \(V_0 \subset Z\). For each \(i\) we look at the diagram \[\xymatrix{ V'_i \ar[r] \ar[d] & W'_i \ar[d] & V'_{i + 1} \ar[l] \ar[d] \\ V_i \ar[r] & W_i & V_{i + 1} \ar[l] }\] Denote \(n_i\) the relative dimension of \(V'_i/V_i\) and \(m_i\) the relative dimension of \(W'_i/W_i\); more precisely these are the transcendence degrees of the corresponding extensions of the function fields. Set \(c_i = \text{codim}(V_i, W_i)\), \(c'_i = \text{codim}(V'_i, W'_i)\), \(b_i = \text{codim}(V_{i + 1}, W_i)\), and \(b'_i = \text{codim}(V'_{i + 1}, W'_i)\). By the dimension formula we have \[c_i = c'_i + n_i - m_i \quad\text{and}\quad b_i = b'_i + n_{i + 1} - m_i\] See Morphisms, Lemma 02JU. Hence \(c_i - b_i = c'_i - b'_i + n_i - n_{i + 1}\). Thus we see that \[\begin{align*} & c_i + c_{i + 1} + \ldots + c_k - b_i - b_{i + 1} - \ldots - b_{k - 1} \\ & = c'_i + c'_{i + 1} + \ldots + c'_k - b'_i - b'_{i + 1} - \ldots - b'_{k - 1} + n_i - n_k + c_k - c'_k \\ & = c'_i + c'_{i + 1} + \ldots + c'_k - b'_i - b'_{i + 1} - \ldots - b'_{k - 1} + n_i - m_k \end{align*}\] Thus we see that \[\begin{align*} \max_{i = 0, \ldots, k} & (c_i + c_{i + 1} + \ldots + c_k - b_i - b_{i + 1} - \ldots - b_{k - 1}) \\ & = \max_{i = 0, \ldots, k} (c'_i + c'_{i + 1} + \ldots + c'_k - b'_i - b'_{i + 1} - \ldots - b'_{k - 1} + n_i - m_k) \\ & = \max_{i = 0, \ldots, k} (c'_i + c'_{i + 1} + \ldots + c'_k - b'_i - b'_{i + 1} - \ldots - b'_{k - 1} + n_i) - m_k \\ & \leq \max_{i = 0, \ldots, k} (c'_i + c'_{i + 1} + \ldots + c'_k - b'_i - b'_{i + 1} - \ldots - b'_{k - 1}) + e - m_k \end{align*}\] Since \(m_k = \text{trdeg}_{\kappa(y)}(\kappa(y'))\) we conclude that \[\delta(V_0, W_0, \ldots, W_k) \leq \delta(V'_0, W'_0, \ldots, W'_k) + e - \text{trdeg}_{\kappa(y)}(\kappa(y'))\] as desired.
Remark
Let \(Y\) be a catenary Noetherian scheme and let \(Z \subset Y\) be a closed subset. By Lemma 0EIY we have \[\delta_Z(y) \leq \min \left\{ k \middle| \begin{matrix} \text{ there exist specializations in }Y \\ y_0 \leftarrow y'_0 \rightarrow y_1 \leftarrow y'_1 \rightarrow \ldots \leftarrow y'_{k - 1} \rightarrow y_k = y \\ \text{ with }y_0 \in Z\text{ and }y_i' \leadsto y_i \text{ immediate} \end{matrix} \right\}\] We claim that if \(Y\) is of finite type over a field, then equality holds. If we ever need this result we will formulate a precise result and prove it here. However, in general if we define \(\delta_Z\) by the right hand side of this inequality, then we don’t know if Lemma 0EIZ remains true.
Example
Let \(k\) be a field and \(Y = \mathbf{A}^n_k\). Denote \(\delta : Y \to \mathbf{Z}_{\geq 0}\) the usual dimension function.
If \(Z = \{z\}\) for some closed point \(z\), then
\(\delta_Z(y) = \delta(y)\) if \(y \leadsto z\) and
\(\delta_Z(y) = \delta(y) + 1\) if \(y \not \leadsto z\).
If \(Z\) is a closed subvariety and \(W = \overline{\{y\}}\), then
\(\delta_Z(y) = 0\) if \(W \subset Z\),
\(\delta_Z(y) = \dim(W) - \dim(Z)\) if \(Z\) is contained in \(W\),
\(\delta_Z(y) = 1\) if \(\dim(W) \leq \dim(Z)\) and \(W \not \subset Z\),
\(\delta_Z(y) = \dim(W) - \dim(Z) + 1\) if \(\dim(W) > \dim(Z)\) and \(Z \not \subset W\).
A generalization of case (1) is if \(Y\) is of finite type over a field and \(Z = \{z\}\) is a closed point. Then \(\delta_Z(y) = \delta(y) + t\) where \(t\) is the minimum length of a chain of curves connecting \(z\) to a closed point of \(\overline{\{y\}}\).
Algebraization of coherent formal modules, III
We continue the discussion started in Sections 0DXS and 0EIT. We will use the distance function of Section 0EIW to formulate a some natural conditions on coherent formal modules in Situation 0EHC.
In Situation 0EHC given a point \(y \in U \cap Y\) we can consider the \(I\)-adic completion \[\mathcal{O}_{X, y}^\wedge = \lim \mathcal{O}_{X, y}/I^n\mathcal{O}_{X, y}\] This is a Noetherian local ring complete with respect to \(I\mathcal{O}_{X, y}^\wedge\) with maximal ideal \(\mathfrak m_y^\wedge\), see Algebra, Section 0BNH. Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let us define the “stalk” of \((\mathcal{F}_n)\) at \(y\) by the formula \[\mathcal{F}_y^\wedge = \lim \mathcal{F}_{n, y}\] This is a finite module over \(\mathcal{O}_{X, y}^\wedge\). See Algebra, Lemmas 09B8 and 031D.
Definition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(a, b\) be integers. Let \(\delta^Y_Z\) be as in (0EIX). We say \((\mathcal{F}_n)\) satisfies the \((a, b)\)-inequalities if for \(y \in U \cap Y\) and a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\)
if \(V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) \not = \{\mathfrak m_y^\wedge\}\), then \[\text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \delta^Y_Z(y) \geq a \quad\text{or}\quad \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > b\]
if \(V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) = \{\mathfrak m_y^\wedge\}\), then \[\text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \delta^Y_Z(y) > a\]
We say \((\mathcal{F}_n)\) satisfies the strict \((a, b)\)-inequalities if for \(y \in U \cap Y\) and a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\) we have \[\text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \delta^Y_Z(y) > a \quad\text{or}\quad \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > b\]
Here are some elementary observations.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(a, b\) be integers.
If \((\mathcal{F}_n)\) is annihilated by a power of \(I\), then \((\mathcal{F}_n)\) satisfies the \((a, b)\)-inequalities for any \(a, b\).
If \((\mathcal{F}_n)\) satisfies the \((a + 1, b)\)-inequalities, then \((\mathcal{F}_n)\) satisfies the strict \((a, b)\)-inequalities.
If \(\text{cd}(A, I) \leq d\) and \(A\) has a dualizing complex, then
\((\mathcal{F}_n)\) satisfies the \((s, s + d)\)-inequalities if and only if for all \(y \in U \cap Y\) the tuple \(\mathcal{O}_{X, y}^\wedge, I\mathcal{O}_{X, y}^\wedge, \{\mathfrak m_y^\wedge\}, \mathcal{F}_y^\wedge, s - \delta^Y_Z(y), d\) is as in Situation 0EFU.
If \((\mathcal{F}_n)\) satisfies the strict \((s, s + d)\)-inequalities, then \((\mathcal{F}_n)\) satisfies the \((s, s + d)\)-inequalities.
Proof
Immediate except for part (4) which is a consequence of Lemma 0EFW and the translation in (3).
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). If \(\text{cd}(A, I) = 1\), then \(\mathcal{F}\) satisfies the \((2, 3)\)-inequalities if and only if \[\text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > 3\] for all \(y \in U \cap Y\) and \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\).
Proof
Observe that for a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\), \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\) we have \(V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) = \{\mathfrak m_y^\wedge\} \Leftrightarrow \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = 1\) as \(\text{cd}(A, I) = 1\). See Local Cohomology, Lemmas 0DXA and 0DXF. OK, consider the three numbers \(\alpha = \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) \geq 0\), \(\beta = \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) \geq 1\), and \(\gamma = \delta^Y_Z(y) \geq 1\). Then we see Definition 0EJ3 requires
if \(\beta > 1\), then \(\alpha + \gamma \geq 2\) or \(\alpha + \beta + \gamma > 3\), and
if \(\beta = 1\), then \(\alpha + \gamma > 2\).
It is trivial to see that this is equivalent to \(\alpha + \beta + \gamma > 3\).
In the rest of this section, which we suggest the reader skip on a first reading, we will show that, when \(A\) is \(I\)-adically complete, the category of \((\mathcal{F}_n)\) of \(\textit{Coh}(U, I\mathcal{O}_U)\) which extend to \(X\) and satisfy the strict \((1, 1 + \text{cd}(A, I))\)-inequalities is equivalent to a full subcategory of the category of coherent \(\mathcal{O}_U\)-modules.
Lemma
In Situation 0EHC let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module and \(d \geq 1\). Assume
\(A\) is \(I\)-adically complete, has a dualizing complex, and \(\text{cd}(A, I) \leq d\),
the completion \(\mathcal{F}^\wedge\) of \(\mathcal{F}\) satisfies the strict \((1, 1 + d)\)-inequalities.
Let \(x \in X\) be a point. Let \(W = \overline{\{x\}}\). If \(W \cap Y\) has an irreducible component contained in \(Z\) and one which is not, then \(\text{depth}(\mathcal{F}_x) \geq 1\).
Proof
Let \(W \cap Y = W_1 \cup \ldots \cup W_n\) be the decomposition into irreducible components. By assumption, after renumbering, we can find \(0 < m < n\) such that \(W_1, \ldots, W_m \subset Z\) and \(W_{m + 1}, \ldots, W_n \not \subset Z\). We conclude that \[W \cap Y \setminus \left((W_1 \cup \ldots \cup W_m) \cap (W_{m + 1} \cup \ldots \cup W_n)\right)\] is disconnected. By Lemma 0EG5 we can find \(1 \leq i \leq m < j \leq n\) and \(z \in W_i \cap W_j\) such that \(\dim(\mathcal{O}_{W, z}) \leq d + 1\). Choose an immediate specialization \(y \leadsto z\) with \(y \in W_j\), \(y \not \in Z\); existence of \(y\) follows from Properties, Lemma 02IM. Observe that \(\delta^Y_Z(y) = 1\) and \(\dim(\mathcal{O}_{W, y}) \leq d\). Let \(\mathfrak p \subset \mathcal{O}_{X, y}\) be the prime corresponding to \(x\). Let \(\mathfrak p' \subset \mathcal{O}_{X, y}^\wedge\) be a minimal prime over \(\mathfrak p\mathcal{O}_{X, y}^\wedge\). Then we have \[\text{depth}(\mathcal{F}_x) = \text{depth}((\mathcal{F}^\wedge_y)_{\mathfrak p'}) \quad\text{and}\quad \dim(\mathcal{O}_{W, y}) = \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p')\] See Algebra, Lemma 0338 and Local Cohomology, Lemma 0EHW. Now we read off the conclusion from the inequalities given to us.
Lemma
In Situation 0EHC let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module and \(d \geq 1\). Assume
\(A\) is \(I\)-adically complete, has a dualizing complex, and \(\text{cd}(A, I) \leq d\),
the completion \(\mathcal{F}^\wedge\) of \(\mathcal{F}\) satisfies the strict \((1, 1+ d)\)-inequalities, and
for \(x \in U\) with \(\overline{\{x\}} \cap Y \subset Z\) we have \(\text{depth}(\mathcal{F}_x) \geq 2\).
Then \(H^0(U, \mathcal{F}) \to \lim H^0(U, \mathcal{F}/I^n\mathcal{F})\) is an isomorphism.
Proof
We will prove this by showing that Lemma 0EIG applies. Thus we let \(x \in \text{Ass}(\mathcal{F})\) with \(x \not \in Y\). Set \(W = \overline{\{x\}}\). By condition (3) we see that \(W \cap Y \not \subset Z\). By Lemma 0EJ5 we see that no irreducible component of \(W \cap Y\) is contained in \(Z\). Thus if \(z \in W \cap Z\), then there is an immediate specialization \(y \leadsto z\), \(y \in W \cap Y\), \(y \not \in Z\). For existence of \(y\) use Properties, Lemma 02IM. Then \(\delta^Y_Z(y) = 1\) and the assumption implies that \(\dim(\mathcal{O}_{W, y}) > d\). Hence \(\dim(\mathcal{O}_{W, z}) > 1 + d\) and we win.
Lemma
In Situation 0EHC let \(\mathcal{F}\) be a coherent \(\mathcal{O}_U\)-module and \(d \geq 1\). Assume
\(A\) is \(I\)-adically complete, has a dualizing complex, and \(\text{cd}(A, I) \leq d\),
the completion \(\mathcal{F}^\wedge\) of \(\mathcal{F}\) satisfies the strict \((1, 1 + d)\)-inequalities, and
for \(x \in U\) with \(\overline{\{x\}} \cap Y \subset Z\) we have \(\text{depth}(\mathcal{F}_x) \geq 2\).
Then the map \[\Hom_U(\mathcal{G}, \mathcal{F}) \longrightarrow \Hom_{\textit{Coh}(U, I\mathcal{O}_U)}(\mathcal{G}^\wedge, \mathcal{F}^\wedge)\] is bijective for every coherent \(\mathcal{O}_U\)-module \(\mathcal{G}\).
Proof
Set \(\mathcal{H} = \SheafHom_{\mathcal{O}_U}(\mathcal{G}, \mathcal{F})\). Using Cohomology of Schemes, Lemma 0EBC or More on Algebra, Lemma 0AV5 we see that the completion of \(\mathcal{H}\) satisfies the strict \((1, 1 + d)\)-inequalities and that for \(x \in U\) with \(\overline{\{x\}} \cap Y \subset Z\) we have \(\text{depth}(\mathcal{H}_x) \geq 2\). Details omitted. Thus by Lemma 0EJ6 we have \[\Hom_U(\mathcal{G}, \mathcal{F}) = H^0(U, \mathcal{H}) = \lim H^0(U, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(U, I\mathcal{O}_U)} (\mathcal{G}^\wedge, \mathcal{F}^\wedge)\] See Cohomology of Schemes, Lemma 0882 for the final equality.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\) and \(d \geq 1\). Assume
\(A\) is \(I\)-adically complete, has a dualizing complex, and \(\text{cd}(A, I) \leq d\),
\((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module,
\((\mathcal{F}_n)\) satisfies the strict \((1, 1 + d)\)-inequalities.
Then there exists a unique coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\) whose completion is \((\mathcal{F}_n)\) such that for \(x \in U\) with \(\overline{\{x\}} \cap Y \subset Z\) we have \(\text{depth}(\mathcal{F}_x) \geq 2\).
Proof
Choose a coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\) whose completion is \((\mathcal{F}_n)\). Let \(T = \{x \in U \mid \overline{\{x\}} \cap Y \subset Z\}\). We will construct \(\mathcal{F}\) by applying Local Cohomology, Lemma 0EI3 with \(\mathcal{F}\) and \(T\). Then uniqueness will follow from the mapping property of Lemma 0EJ7.
Since \(T\) is stable under specialization in \(U\) the only thing to check is the following. If \(x' \leadsto x\) is an immediate specialization of points of \(U\) with \(x \in T\) and \(x' \not \in T\), then \(\text{depth}(\mathcal{F}_{x'}) \geq 1\). Set \(W = \overline{\{x\}}\) and \(W' = \overline{\{x'\}}\). Since \(x' \not \in T\) we see that \(W' \cap Y\) is not contained in \(Z\). If \(W' \cap Y\) contains an irreducible component contained in \(Z\), then we are done by Lemma 0EJ5. If not, we choose an irreducible component \(W_1\) of \(W \cap Y\) and an irreducible component \(W'_1\) of \(W' \cap Y\) with \(W_1 \subset W'_1\). Let \(z \in W_1\) be the generic point. Let \(y \leadsto z\), \(y \in W'_1\) be an immediate specialization with \(y \not \in Z\); existence of \(y\) follows from \(W'_1 \not \subset Z\) (see above) and Properties, Lemma 02IM. Then we have the following \(z \in Z\), \(x \leadsto z\), \(x' \leadsto y \leadsto z\), \(y \in Y \setminus Z\), and \(\delta^Y_Z(y) = 1\). By Local Cohomology, Lemma 0DXF and the fact that \(z\) is a generic point of \(W \cap Y\) we have \(\dim(\mathcal{O}_{W, z}) \leq d\). Since \(x' \leadsto x\) is an immediate specialization we have \(\dim(\mathcal{O}_{W', z}) \leq d + 1\). Since \(y \not = z\) we conclude \(\dim(\mathcal{O}_{W', y}) \leq d\). If \(\text{depth}(\mathcal{F}_{x'}) = 0\) then we would get a contradiction with assumption (3); details about passage from \(\mathcal{O}_{X, y}\) to its completion omitted. This finishes the proof.
Algebraization of coherent formal modules, IV
In this section we prove two stronger versions of Lemma 0DXW in the local case, namely, Lemmas 0DXU and 0EJ9. Although these lemmas will be obsoleted by the more general Proposition 0EJJ, their proofs are significantly easier.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) is local and \(\mathfrak a = \mathfrak m\) is the maximal ideal,
\(A\) has a dualizing complex,
\(I = (f)\) is a principal ideal for a nonzerodivisor \(f \in \mathfrak m\),
\(\mathcal{F}_n\) is a finite locally free \(\mathcal{O}_U/f^n\mathcal{O}_U\)-module,
if \(\mathfrak p \in V(f) \setminus \{\mathfrak m\}\), then \(\text{depth}((A/f)_\mathfrak p) + \dim(A/\mathfrak p) > 1\), and
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
Then \((\mathcal{F}_n)\) extends canonically to \(X\). In particular, if \(A\) is complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
We will prove this by verifying hypotheses (a), (b), and (c) of Lemma 0EHH.
Since \(\mathcal{F}_n\) is locally free over \(\mathcal{O}_U/f^n\mathcal{O}_U\) we see that we have short exact sequences \(0 \to \mathcal{F}_n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0\) for all \(n\). Thus condition (b) holds by Cohomology, Lemma 0EHA.
By induction on \(n\) and the short exact sequences \(0 \to A/f^n \to A/f^{n + 1} \to A/f \to 0\) we see that the associated primes of \(A/f^nA\) agree with the associated primes of \(A/fA\). Since the associated points of \(\mathcal{F}_n\) correspond to the associated primes of \(A/f^nA\) not equal to \(\mathfrak m\) by assumption (3), we conclude that \(M_n = H^0(U, \mathcal{F}_n)\) is a finite \(A\)-module by (5) and Local Cohomology, Proposition 0BK3. Thus hypothesis (c) holds.
To finish the proof it suffices to show that there exists an \(n > 1\) such that the image of \[H^1(U, \mathcal{F}_n) \longrightarrow H^1(U, \mathcal{F}_1)\] has finite length as an \(A\)-module. Namely, this will imply hypothesis (a) by Cohomology, Lemma 0DXG. The image is independent of \(n\) for \(n\) large enough by Lemma 0DX1. Let \(\omega_A^\bullet\) be a normalized dualizing complex for \(A\). By the local duality theorem and Matlis duality (Dualizing Complexes, Lemma 0AAK and Proposition 08Z9) our claim is equivalent to: the image of \[\text{Ext}^{-2}_A(M_1, \omega_A^\bullet) \to \text{Ext}^{-2}_A(M_n, \omega_A^\bullet)\] has finite length for \(n \gg 1\). The modules in question are finite \(A\)-modules supported at \(V(f)\). Thus it suffices to show that this map is zero after localization at a prime \(\mathfrak q\) containing \(f\) and different from \(\mathfrak m\). Let \(\omega_{A_\mathfrak q}^\bullet\) be a normalized dualizing complex on \(A_\mathfrak q\) and recall that \(\omega_{A_\mathfrak q}^\bullet = (\omega_A^\bullet)_\mathfrak q[\dim(A/\mathfrak q)]\) by Dualizing Complexes, Lemma 0A7Z. Using the local structure of \(\mathcal{F}_n\) given in (4) we find that it suffices to show the vanishing of \[\text{Ext}^{-2 + \dim(A/\mathfrak q)}_{A_\mathfrak q}( A_\mathfrak q/f, \omega_{A_\mathfrak q}^\bullet) \to \text{Ext}^{-2 + \dim(A/\mathfrak q)}_{A_\mathfrak q}( A_\mathfrak q/f^n, \omega_{A_\mathfrak q}^\bullet)\] for \(n\) large enough. If \(\dim(A/\mathfrak q) > 3\), then this is immediate from Local Cohomology, Lemma 0DWZ. For the other cases we will use the long exact sequence \[\ldots \xrightarrow{f^n} H^{-1}(\omega_{A_\mathfrak q}^\bullet) \to \text{Ext}^{-1}_{A_\mathfrak q}( A_\mathfrak q/f^n, \omega_{A_\mathfrak q}^\bullet) \to H^0(\omega_{A_\mathfrak q}^\bullet) \xrightarrow{f^n} H^0(\omega_{A_\mathfrak q}^\bullet) \to \text{Ext}^0_{A_\mathfrak q}( A_\mathfrak q/f^n, \omega_{A_\mathfrak q}^\bullet) \to 0\] If \(\dim(A/\mathfrak q) = 2\), then \(H^0(\omega_{A_\mathfrak q}^\bullet) = 0\) because \(\text{depth}(A_\mathfrak q) \geq 1\) as \(f\) is a nonzerodivisor. Thus the long exact sequence shows the condition is that \[f^{n - 1} : H^{-1}(\omega_{A_\mathfrak q}^\bullet)/f \to H^{-1}(\omega_{A_\mathfrak q}^\bullet)/f^n\] is zero. Now \(H^{-1}(\omega^\bullet_\mathfrak q)\) is a finite module supported in the primes \(\mathfrak p \subset A_\mathfrak q\) such that \(\text{depth}(A_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) \leq 1\). Since \(\dim((A/\mathfrak p)_\mathfrak q) = \dim(A/\mathfrak p) - 2\) condition (6) tells us these primes are contained in \(V(f)\). Thus the desired vanishing for \(n\) large enough. Finally, if \(\dim(A/\mathfrak q) = 1\), then condition (5) combined with the fact that \(f\) is a nonzerodivisor insures that \(A_\mathfrak q\) has depth at least \(2\). Hence \(H^0(\omega_{A_\mathfrak q}^\bullet) = H^{-1}(\omega_{A_\mathfrak q}^\bullet) = 0\) and the long exact sequence shows the claim is equivalent to the vanishing of \[f^{n - 1} : H^{-2}(\omega_{A_\mathfrak q}^\bullet)/f \to H^{-2}(\omega_{A_\mathfrak q}^\bullet)/f^n\] Now \(H^{-2}(\omega^\bullet_\mathfrak q)\) is a finite module supported in the primes \(\mathfrak p \subset A_\mathfrak q\) such that \(\text{depth}(A_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) \leq 2\). By condition (6) all of these primes are contained in \(V(f)\). Thus the desired vanishing for \(n\) large enough.
Remark
Let \((A, \mathfrak m)\) be a complete Noetherian normal local domain of dimension \(\geq 4\) and let \(f \in \mathfrak m\) be nonzero. Then assumptions (1), (2), (3), (5), and (6) of Lemma 0DXU are satisfied. Thus vectorbundles on the formal completion of \(U\) along \(U \cap V(f)\) can be algebraized. In Lemma 0EJ9 we will generalize this to more general coherent formal modules; please also compare with Remark 0EJC.
Lemma
In Situation 0EHC let \((M_n)\) be an inverse system of \(A\)-modules as in Lemma 0DXT and let \((\mathcal{F}_n)\) be the corresponding object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Let \(d \geq \text{cd}(A, I)\) and \(s \geq 0\) be integers. With notation as above assume
\(A\) is local with maximal ideal \(\mathfrak m = \mathfrak a\),
\(A\) has a dualizing complex, and
\((\mathcal{F}_n)\) satisfies the \((s, s + d)\)-inequalities (Definition 0EJ3).
Let \(E\) be an injective hull of the residue field of \(A\). Then for \(i \leq s\) there exists a finite \(A\)-module \(N\) annihilated by a power of \(I\) and for \(n \gg 0\) compatible maps \[H^i_\mathfrak m(M_n) \to \Hom_A(N, E)\] whose cokernels are finite length \(A\)-modules and whose kernels \(K_n\) form an inverse system such that \(\Im(K_{n''} \to K_{n'})\) has finite length for \(n'' \gg n' \gg 0\).
Proof
Let \(\omega_A^\bullet\) be a normalized dualizing complex. Then \(\delta^Y_Z = \delta\) is the dimension function associated with this dualizing complex. Observe that \(\Ext^{-i}_A(M_n, \omega_A^\bullet)\) is a finite \(A\)-module annihilated by \(I^n\). Fix \(0 \leq i \leq s\). Below we will find \(n_1 > n_0 > 0\) such that if we set \[N = \Im(\Ext^{-i}_A(M_{n_0}, \omega_A^\bullet) \to \Ext^{-i}_A(M_{n_1}, \omega_A^\bullet))\] then the kernels of the maps \[N \to \Ext^{-i}_A(M_n, \omega_A^\bullet),\quad n \geq n_1\] are finite length \(A\)-modules and the cokernels \(Q_n\) form a system such that \(\Im(Q_{n'} \to Q_{n''})\) has finite length for \(n'' \gg n' \gg n_1\). This is equivalent to the statement that the system \(\{\Ext^{-i}_A(M_n, \omega_A^\bullet)\}_{n \geq 1}\) is essentially constant in the quotient of the category of finite \(A\)-modules modulo the Serre subcategory of finite length \(A\)-modules. By the local duality theorem (Dualizing Complexes, Lemma 0AAK) and Matlis duality (Dualizing Complexes, Proposition 08Z9) we conclude that there are maps \[H^i_\mathfrak m(M_n) \to \Hom_A(N, E),\quad n \geq n_1\] as in the statement of the lemma.
Pick \(f \in \mathfrak m\). Let \(B = A_f^\wedge\) be the \(I\)-adic completion of the localization \(A_f\). Recall that \(\omega_{A_f}^\bullet = \omega_A^\bullet \otimes_A A_f\) and \(\omega_B^\bullet = \omega_A^\bullet \otimes_A B\) are dualizing complexes (Dualizing Complexes, Lemma 0A7G and 0DWD). Let \(M\) be the finite \(B\)-module \(\lim M_{n, f}\) (compare with discussion in Cohomology of Schemes, Lemma 087W). Then \[\Ext^{-i}_A(M_n, \omega_A^\bullet)_f = \Ext^{-i}_{A_f}(M_{n, f}, \omega_{A_f}^\bullet) = \Ext^{-i}_B(M/I^n M, \omega_B^\bullet)\] Since \(\mathfrak m\) can be generated by finitely many \(f \in \mathfrak m\) it suffices to show that for each \(f\) the system \[\{\Ext^{-i}_B(M/I^n M, \omega_B^\bullet)\}_{n \geq 1}\] is essentially constant. Some details omitted.
Let \(\mathfrak q \subset IB\) be a prime ideal. Then \(\mathfrak q\) corresponds to a point \(y \in U \cap Y\). Observe that \(\delta(\mathfrak q) = \dim(\overline{\{y\}})\) is also the value of the dimension function associated to \(\omega_B^\bullet\) (we omit the details; use that \(\omega_B^\bullet\) is gotten from \(\omega_A^\bullet\) by tensoring up with \(B\)). Assumption (3) guarantees via Lemma 0EJ4 that Lemma 0EFR applies to \(B_\mathfrak q, IB_\mathfrak q, \mathfrak qB_\mathfrak q, M_\mathfrak q\) with \(s\) replaced by \(s - \delta(y)\). We obtain that \[H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}(M_\mathfrak q) = \lim H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}( (M/I^nM)_\mathfrak q)\] and this module is annihilated by a power of \(I\). By Lemma 0EHB we find that the inverse systems \(H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}((M/I^nM)_\mathfrak q)\) are essentially constant with value \(H^{i - \delta(\mathfrak q)}_{\mathfrak qB_\mathfrak q}(M_\mathfrak q)\). Since \((\omega_B^\bullet)_\mathfrak q[-\delta(\mathfrak q)]\) is a normalized dualizing complex on \(B_\mathfrak q\) the local duality theorem shows that the system \[\Ext^{-i}_B(M/I^n M, \omega_B^\bullet)_\mathfrak q\] is essentially constant with value \(\Ext^{-i}_B(M, \omega_B^\bullet)_\mathfrak q\).
To finish the proof we globalize as in the proof of Lemma 0EFX; the argument here is easier because we know the value of our system already. Namely, consider the maps \[\alpha_n : \Ext^{-i}_B(M/I^n M, \omega_B^\bullet) \longrightarrow \Ext^{-i}_B(M, \omega_B^\bullet)\] for varying \(n\). By the above, for every \(\mathfrak q\) we can find an \(n\) such that \(\alpha_n\) is surjective after localization at \(\mathfrak q\). Since \(B\) is Noetherian and \(\Ext^{-i}_B(M, \omega_B^\bullet)\) a finite module, we can find an \(n\) such that \(\alpha_n\) is surjective. For any \(n\) such that \(\alpha_n\) is surjective, given a prime \(\mathfrak q \in V(IB)\) we can find an \(n' > n\) such that \(\Ker(\alpha_n)\) maps to zero in \(\Ext^{-i}(M/I^{n'}M, \omega_B^\bullet)\) at least after localizing at \(\mathfrak q\). Since \(\Ker(\alpha_n)\) is a finite \(A\)-module and since supports of sections are quasi-compact, we can find an \(n'\) such that \(\Ker(\alpha_n)\) maps to zero in \(\Ext^{-i}(M/I^{n'}M, \omega_B^\bullet)\). In this way we see that \(\Ext^{-i}(M/I^n M, \omega_B^\bullet)\) is essentially constant with value \(\Ext^{-i}(M, \omega_B^\bullet)\). This finishes the proof.
Here is a more general version of Lemma 0DXU.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) is local and \(\mathfrak a = \mathfrak m\) is the maximal ideal,
\(A\) has a dualizing complex,
\(I = (f)\) is a principal ideal,
\((\mathcal{F}_n)\) satisfies the \((2, 3)\)-inequalities.
Then \((\mathcal{F}_n)\) extends to \(X\). In particular, if \(A\) is \(I\)-adically complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
Recall that \(\textit{Coh}(U, I\mathcal{O}_U)\) is an abelian category, see Cohomology of Schemes, Lemma 087X. Over affine opens of \(U\) the object \((\mathcal{F}_n)\) corresponds to a finite module over a Noetherian ring (Cohomology of Schemes, Lemma 087W). Thus the kernels of the maps \(f^N : (\mathcal{F}_n) \to (\mathcal{F}_n)\) stabilize for \(N\) large enough. By Lemmas 0EIU and 0EIL in order to prove the lemma we may replace \((\mathcal{F}_n)\) by the image of such a map. Thus we may assume \(f\) is injective on \((\mathcal{F}_n)\). After this replacement the equivalent conditions of Lemma 0EH9 hold for the inverse system \((\mathcal{F}_n)\) on \(U\). We will use this without further mention in the rest of the proof.
We will check hypotheses (a), (b), and (c) of Lemma 0EHH. Hypothesis (b) holds by Cohomology, Lemma 0EHA.
Pick an inverse system of modules \(\{M_n\}\) as in Lemma 0DXT. We may assume \(H^0_\mathfrak m(M_n) = 0\) by replacing \(M_n\) by \(M_n/H^0_\mathfrak m(M_n)\) if necessary. Then we obtain short exact sequences \[0 \to M_n \to H^0(U, \mathcal{F}_n) \to H^1_\mathfrak m(M_n) \to 0\] for all \(n\). Let \(E\) be an injective hull of the residue field of \(A\). By Lemma 0EHK and our current assumption (4) we can choose, an integer \(m \geq 0\), finite \(A\)-modules \(N_1\) and \(N_2\) annihilated by \(f^c\) for some \(c \geq 0\) and compatible systems of maps \[H^i_\mathfrak m(M_n) \to \Hom_A(N_i, E), \quad i = 1, 2\] for \(n \geq m\) with the properties stated in the lemma.
We know that \(M = \lim H^0(U, \mathcal{F}_n)\) is an \(A\)-module whose limit topology is the \(f\)-adic topology. Thus, given \(n\), the module \(M/f^nM\) is a subquotient of \(H^0(U, \mathcal{F}_N)\) for some \(N \gg n\). Looking at the information obtained above we see that \(f^cM/f^nM\) is a finite \(A\)-module. Since \(f\) is a nonzerodivisor on \(M\) we conclude that \(M/f^{n - c}M\) is a finite \(A\)-module. In this way we see that hypothesis (c) of Lemma 0EHH holds.
Next, we study the module \[Ob = \lim H^1(U, \mathcal{F}_n) = \lim H^2_\mathfrak m(M_n)\] For \(n \geq m\) let \(K_n\) be the kernel of the map \(H^2_\mathfrak m(M_n) \to \Hom_A(N_2, E)\). Set \(K = \lim K_n\). We obtain an exact sequence \[0 \to K \to Ob \to \Hom_A(N_2, E)\] By the above the limit topology on \(Ob = \lim H^2_\mathfrak m(M_n)\) is the \(f\)-adic topology. Since \(N_2\) is annihilated by \(f^c\) we conclude the same is true for the limit topology on \(K = \lim K_n\). Thus \(K/fK\) is a subquotient of \(K_n\) for \(n \gg 1\). However, since \(\{K_n\}\) is pro-isomorphic to an inverse system of finite length \(A\)-modules (by the conclusion of Lemma 0EHK) we conclude that \(K/fK\) is a subquotient of a finite length \(A\)-module. It follows that \(K\) is a finite \(A\)-module, see Algebra, Lemma 031D. (In fact, we even see that \(\dim(\text{Supp}(K)) = 1\) but we will not need this.)
Given \(n \geq 1\) consider the boundary map \[\delta_n : H^0(U, \mathcal{F}_n) \longrightarrow \lim_N H^1(U, f^n\mathcal{F}_N) \xrightarrow{f^{-n}} Ob\] (the second map is an isomorphism) coming from the short exact sequences \[0 \to f^n\mathcal{F}_N \to \mathcal{F}_N \to \mathcal{F}_n \to 0\] For each \(n\) set \[P_n = \Im(H^0(U, \mathcal{F}_{n + m}) \to H^0(U, \mathcal{F}_n))\] where \(m\) is as above. Observe that \(\{P_n\}\) is an inverse system and that the map \(f : \mathcal{F}_n \to \mathcal{F}_{n + 1}\) on global sections maps \(P_n\) into \(P_{n + 1}\). If \(p \in P_n\), then \(\delta_n(p) \in K \subset Ob\) because \(\delta_n(p)\) maps to zero in \(H^1(U, f^n\mathcal{F}_{n + m}) = H^2_\mathfrak m(M_m)\) and the composition of \(\delta_n\) and \(Ob \to \Hom_A(N_2, E)\) factors through \(H^2_\mathfrak m(M_m)\) by our choice of \(m\). Hence \[\bigoplus\nolimits_{n \geq 0} \Im(P_n \to Ob)\] is a finite graded \(A[T]\)-module where \(T\) acts via multiplication by \(f\). Namely, it is a graded submodule of \(K[T]\) and \(K\) is finite over \(A\). Arguing as in the proof of Cohomology, Lemma 0GYK7 we find that the inverse system \(\{P_n\}\) satisfies ML. Since \(\{P_n\}\) is pro-isomorphic to \(\{H^0(U, \mathcal{F}_n)\}\) we conclude that \(\{H^0(U, \mathcal{F}_n)\}\) has ML. Thus hypothesis (a) of Lemma 0EHH holds and the proof is complete.
We can unwind condition of Lemma 0EJ9 as follows.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) is local with maximal ideal \(\mathfrak a = \mathfrak m\),
\(\text{cd}(A, I) = 1\).
Then \((\mathcal{F}_n)\) satisfies the \((2, 3)\)-inequalities if and only if for all \(y \in U \cap Y\) with \(\dim(\{y\}) = 1\) and every prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\), \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\) we have \[\text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 2\]
Proof
We will use Lemma 0EKW without further mention. In particular, we see the condition is necessary. Conversely, suppose the condition is true. Note that \(\delta^Y_Z(y) = \dim(\overline{\{y\}})\) by Lemma 0EIY. Let us write \(\delta\) for this function. Let \(y \in U \cap Y\). If \(\delta(y) > 2\), then the inequality of Lemma 0EKW holds. Finally, suppose \(\delta(y) = 2\). We have to show that \[\text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 1\] Choose a specialization \(y \leadsto y'\) with \(\delta(y') = 1\). Then there is a ring map \(\mathcal{O}_{X, y'}^\wedge \to \mathcal{O}_{X, y}^\wedge\) which identifies the target with the completion of the localization of \(\mathcal{O}_{X, y'}^\wedge\) at a prime \(\mathfrak q\) with \(\dim(\mathcal{O}_{X, y'}^\wedge/\mathfrak q) = 1\). Moreover, we then obtain \[\mathcal{F}_y^\wedge = \mathcal{F}_{y'}^\wedge \otimes_{\mathcal{O}_{X, y'}^\wedge} \mathcal{O}_{X, y}^\wedge\] Let \(\mathfrak p' \subset \mathcal{O}_{X, y'}^\wedge\) be the image of \(\mathfrak p\). By Local Cohomology, Lemma 0EHW we have \[\begin{align*} \text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) & = \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \dim((\mathcal{O}_{X, y}^\wedge/\mathfrak p)_{\mathfrak p'}) \\ & = \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p') - 1 \end{align*}\] the last equality because the specialization is immediate. Thus the lemma is prove by the assumed inequality for \(y', \mathfrak p'\).
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) is local with maximal ideal \(\mathfrak a = \mathfrak m\),
\(A\) has a dualizing complex,
\(\text{cd}(A, I) = 1\),
for \(y \in U \cap Y\) the module \(\mathcal{F}_y^\wedge\) is finite locally free outside \(V(I\mathcal{O}_{X, y}^\wedge)\), for example if \(\mathcal{F}_n\) is a finite locally free \(\mathcal{O}_U/I^n\mathcal{O}_U\)-module, and
one of the following is true
\(A_f\) is \((S_2)\) and every irreducible component of \(X\) not contained in \(Y\) has dimension \(\geq 4\), or
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
Then \((\mathcal{F}_n)\) satisfies the \((2, 3)\)-inequalities.
Proof
We will use the criterion of Lemma 0EJA. Let \(y \in U \cap Y\) with \(\dim(\overline{\{y\}} = 1\) and let \(\mathfrak p\) be a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\). Condition (4) shows that \(\text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) = \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p)\). Thus we have to prove \[\text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 2\] Let \(\mathfrak p_0 \subset A\) be the image of \(\mathfrak p\). Let \(\mathfrak q \subset A\) be the prime corresponding to \(y\). By Local Cohomology, Lemma 0EHW we have \[\begin{align*} \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) & = \text{depth}(A_{\mathfrak p_0}) + \dim((A/\mathfrak p_0)_\mathfrak q) \\ & = \text{depth}(A_{\mathfrak p_0}) + \dim(A/\mathfrak p_0) - 1 \end{align*}\] If (5)(a) holds, then we get that this is \[\geq \min(2, \dim(A_{\mathfrak p_0})) + \dim(A/\mathfrak p_0) - 1\] Note that in any case \(\dim(A/\mathfrak p_0) \geq 2\). Hence if we get \(2\) for the minimum, then we are done. If not we get \[\dim(A_{\mathfrak p_0}) + \dim(A/\mathfrak p_0) - 1 \geq 4 - 1\] because every component of \(\Spec(A)\) passing through \(\mathfrak p_0\) has dimension \(\geq 4\). If (5)(b) holds, then we win immediately.
Remark
Let \((A, \mathfrak m)\) be a Noetherian local ring which has a dualizing complex and is complete with respect to \(f \in \mathfrak m\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, f\mathcal{O}_U)\) where \(U\) is the punctured spectrum of \(A\). Set \(Y = V(f) \subset X = \Spec(A)\). If for \(y \in U \cap V(f)\) closed in \(U\), i.e., with \(\dim(\overline{\{y\}}) = 1\), we assume the \(\mathcal{O}_{X, y}^\wedge\)-module \(\mathcal{F}_y^\wedge\) satisfies the following two conditions
\(\mathcal{F}_y^\wedge[1/f]\) is \((S_2)\) as a \(\mathcal{O}_{X, y}^\wedge[1/f]\)-module, and
for \(\mathfrak p \in \text{Ass}(\mathcal{F}_y^\wedge[1/f])\) we have \(\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) \geq 3\).
Then \((\mathcal{F}_n)\) is the completion of a coherent module on \(U\). This follows from Lemmas 0EJ9 and 0EJA.
Improving coherent formal modules
Let \(X\) be a Noetherian scheme. Let \(Y \subset X\) be a closed subscheme with quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(X, \mathcal{I})\). In this section we construct maps \((\mathcal{F}_n) \to (\mathcal{F}'_n)\) similar to the maps constructed in Local Cohomology, Section 0DX2 for coherent modules. For a point \(y \in Y\) we set \[\mathcal{O}_{X, y}^\wedge = \lim \mathcal{O}_{X, y}/\mathcal{I}^n_y, \quad \mathcal{I}_y^\wedge = \lim \mathcal{I}_y/\mathcal{I}^n_y \quad\text{and}\quad \mathfrak m_y^\wedge = \lim \mathfrak m_y/\mathcal{I}_y^n\] Then \(\mathcal{O}_{X, y}^\wedge\) is a Noetherian local ring with maximal ideal \(\mathfrak m_y^\wedge\) complete with respect to \(\mathcal{I}_y^\wedge = \mathcal{I}_y\mathcal{O}_{X, y}^\wedge\). We also set \[\mathcal{F}_y^\wedge = \lim \mathcal{F}_{n, y}\] Then \(\mathcal{F}_y^\wedge\) is a finite module over \(\mathcal{O}_{X, y}^\wedge\) with \(\mathcal{F}_y^\wedge/(\mathcal{I}_y^\wedge)^n\mathcal{F}_y^\wedge = \mathcal{F}_{n, y}\) for all \(n\), see Algebra, Lemmas 09B8 and 031D.
Lemma
In the situation above assume \(X\) locally has a dualizing complex. Let \(T \subset Y\) be a subset stable under specialization. Assume for \(y \in T\) and for a nonmaximal prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(V(\mathfrak p) \cap V(\mathcal{I}^\wedge_y) = \{\mathfrak m_y^\wedge\}\) we have \[\text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) > 0\] Then there exists a canonical map \((\mathcal{F}_n) \to (\mathcal{F}_n')\) of inverse systems of coherent \(\mathcal{O}_X\)-modules with the following properties
for \(y \in T\) we have \(\text{depth}(\mathcal{F}'_{n, y}) \geq 1\),
\((\mathcal{F}'_n)\) is isomorphic as a pro-system to an object \((\mathcal{G}_n)\) of \(\textit{Coh}(X, \mathcal{I})\),
the induced morphism \((\mathcal{F}_n) \to (\mathcal{G}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) is surjective with kernel annihilated by a power of \(\mathcal{I}\).
Proof
For every \(n\) we let \(\mathcal{F}_n \to \mathcal{F}'_n\) be the surjection constructed in Local Cohomology, Lemma 0DX3. Since this is the quotient of \(\mathcal{F}_n\) by the subsheaf of sections supported on \(T\) we see that we get canonical maps \(\mathcal{F}'_{n + 1} \to \mathcal{F}'_n\) such that we obtain a map \((\mathcal{F}_n) \to (\mathcal{F}_n')\) of inverse systems of coherent \(\mathcal{O}_X\)-modules. Property (1) holds by construction.
To prove properties (2) and (3) we may assume that \(X = \Spec(A_0)\) is affine and \(A_0\) has a dualizing complex. Let \(I_0 \subset A_0\) be the ideal corresponding to \(Y\). Let \(A, I\) be the \(I_0\)-adic completions of \(A_0, I_0\). For later use we observe that \(A\) has a dualizing complex (Dualizing Complexes, Lemma 0BFR). Let \(M\) be the finite \(A\)-module corresponding to \((\mathcal{F}_n)\), see Cohomology of Schemes, Lemma 087W. Then \(\mathcal{F}_n\) corresponds to \(M_n = M/I^nM\). Recall that \(\mathcal{F}'_n\) corresponds to the quotient \(M'_n = M_n / H^0_T(M_n)\), see Local Cohomology, Lemma 0DX3 and its proof.
Set \(s = 0\) and \(d = \text{cd}(A, I)\). We claim that \(A, I, T, M, s, d\) satisfy assumptions (1), (3), (4), (6) of Situation 0EFU. Namely, (1) and (3) are immediate from the above, (4) is the empty condition as \(s = 0\), and (6) is the assumption we made in the statement of the lemma.
By Theorem 0EIE we see that \(\{H^0_T(M_n)\}\) is essentially constant with value \(H^0_T(M)\). Thus the limit of the short exact sequences \(0 \to H^0_T(M_n) \to M_n \to M'_n \to 0\) is the short exact sequence \[0 \to H^0_T(M) \to M \to \lim M'_n \to 0\] Set \(M' = \lim M'_n = M/H^0_T(M)\). This is a finite \(A\)-module. Let \((\mathcal{G}_n)\) be the object of \(\textit{Coh}(X, \mathcal{I})\) corresponding to \(M'\). To finish the proof we have to show that the canonical map \(\{M'/I^nM'\} \to \{M'_n\}\) is a pro-isomorphism. This is equivalent to saying that \(\{H^0_T(M) + I^nM\} \to \{\Ker(M \to M'_n)\}\) is a pro-isomorphism. Dividing by \(I^nM\) it suffices to show that \(\{H^0_T(M)/H^0_T(M) \cap I^nM\} \to \{H^0_T(M_n)\}\) is a pro-isomorphism. Since \(H^0_T(M)\) is annihilated by a power of \(I\), by Artin-Rees we see that \(H^0_T(M) \cap I^nM = 0\) for all \(n \gg 0\). Thus we have the desired pro-isomorphism as we’ve seen above that \(\{H^0_T(M_n)\}\) is essentially constant with value \(H^0_T(M)\).
Lemma
In the situation above assume \(X\) locally has a dualizing complex. Let \(T' \subset T \subset Y\) be subsets stable under specialization. Let \(d \geq 0\) be an integer. Assume
affine locally we have \(X = \Spec(A_0)\) and \(Y = V(I_0)\) and \(\text{cd}(A_0, I_0) \leq d\),
for \(y \in T\) and a nonmaximal prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(V(\mathfrak p) \cap V(\mathcal{I}_y^\wedge) = \{\mathfrak m_y^\wedge\}\) we have \[\text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) > 0\]
for \(y \in T'\) and for a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(\mathcal{I}_y^\wedge)\) and \(V(\mathfrak p) \cap V(\mathcal{I}_y^\wedge) \not = \{\mathfrak m_y^\wedge\}\) we have \[\text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) \geq 1 \quad\text{or}\quad \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 1 + d\]
for \(y \in T'\) and a nonmaximal prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(V(\mathfrak p) \cap V(\mathcal{I}_y^\wedge) = \{\mathfrak m_y^\wedge\}\) we have \[\text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge_y)_\mathfrak p) > 1\]
if \(y \leadsto y'\) is an immediate specialization and \(y' \in T'\), then \(y \in T\).
Then there exists a canonical map \((\mathcal{F}_n) \to (\mathcal{F}_n'')\) of inverse systems of coherent \(\mathcal{O}_X\)-modules with the following properties
for \(y \in T\) we have \(\text{depth}(\mathcal{F}''_{n, y}) \geq 1\),
for \(y' \in T'\) we have \(\text{depth}(\mathcal{F}''_{n, y'}) \geq 2\),
\((\mathcal{F}''_n)\) is isomorphic as a pro-system to an object \((\mathcal{H}_n)\) of \(\textit{Coh}(X, \mathcal{I})\),
the induced morphism \((\mathcal{F}_n) \to (\mathcal{H}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) has kernel and cokernel annihilated by a power of \(\mathcal{I}\).
Proof
As in Lemma 0EJE and its proof for every \(n\) we let \(\mathcal{F}_n \to \mathcal{F}'_n\) be the surjection constructed in Local Cohomology, Lemma 0DX3 using \(T\). Next, we let \(\mathcal{F}'_n \to \mathcal{F}''_n\) be the injection constructed in Local Cohomology, Lemma 0EI4 and its proof. The constructions show that we get canonical maps \(\mathcal{F}''_{n + 1} \to \mathcal{F}''_n\) such that we obtain maps \[(\mathcal{F}_n) \longrightarrow (\mathcal{F}_n') \longrightarrow (\mathcal{F}''_n)\] of inverse systems of coherent \(\mathcal{O}_X\)-modules. Properties (1) and (2) hold by construction.
To prove properties (3) and (4) we may assume that \(X = \Spec(A_0)\) is affine and \(A_0\) has a dualizing complex. Let \(I_0 \subset A_0\) be the ideal corresponding to \(Y\). Let \(A, I\) be the \(I_0\)-adic completions of \(A_0, I_0\). For later use we observe that \(A\) has a dualizing complex (Dualizing Complexes, Lemma 0BFR). Let \(M\) be the finite \(A\)-module corresponding to \((\mathcal{F}_n)\), see Cohomology of Schemes, Lemma 087W. Then \(\mathcal{F}_n\) corresponds to \(M_n = M/I^nM\). Recall that \(\mathcal{F}'_n\) corresponds to the quotient \(M'_n = M_n / H^0_T(M_n)\). Also, recall that \(M' = \lim M'_n\) is the quotient of \(M\) by \(H^0_T(M)\), that \(H^0_T(M)\) is \(I\)-power torsion, and that \(\{M'_n\}\) and \(\{M'/I^nM'\}\) are isomorphic as pro-systems. Finally, we see that \(\mathcal{F}''_n\) corresponds to an extension \[0 \to M'_n \to M''_n \to H^1_{T'}(M'_n) \to 0\] see proof of Local Cohomology, Lemma 0EI4.
Set \(s = 1\). We claim that \(A, I, T', M', s, d\) satisfy assumptions (1), (3), (4), (6) of Situation 0EFU. Namely, (1) and (3) are immediate, (4) is implied by (c), and (6) follows from (d). We omit the proofs of (c) \(\Rightarrow\) (4) and (d) \(\Rightarrow\) (6) .
By Theorem 0EIE we see that \(\{H^1_{T'}(M'/I^nM')\}\) is essentially constant with value \(H^1_{T'}(M')\). We deduce \(\{H^1_{T'}(M'_n)\}\) is essentially constant with value \(H^1_{T'}(M')\). Thus the limit of the short exact sequences displayed above is the short exact sequence \[0 \to M' \to \lim M''_n \to H^1_{T'}(M') \to 0\] Set \(M'' = \lim M''_n\). It follows from Local Cohomology, Proposition 0EFD that \(H^1_{T'}(M')\) and hence \(M''\) are finite \(A\)-modules (in verifying the conditions use that \(M'\) has depth at least \(1\) at primes in \(T - T'\)). Let \((\mathcal{H}_n)\) be the object of \(\textit{Coh}(X, \mathcal{I})\) corresponding to the finite \(A\)-module \(M''\). To finish the proof we have to show that the canonical map \(\{M''/I^nM''\} \to \{M''_n\}\) is a pro-isomorphism. Since we already know that \(\{M'/I^nM'\}\) is pro-isomorphic to \(\{M'_n\}\) the reader verifies (omitted) this is equivalent to asking \(\{H^1_{T'}(M')/I^nH^1_{T'}(M')\} \to \{H^1_{T'}(M'_n)\}\) to be a pro-isomorphism. Since \(H^1_{T'}(M')/I^nH^1_{T'}(M') = H^1_{T'}(M')\) for \(n\) large enough, this follows because \(\{H^1_{T'}(M'_n)\}\) is essentially constant with value \(H^1_{T'}(M')\) as seen above.
Lemma
In Situation 0EHC assume that \(A\) has a dualizing complex. Let \(d \geq \text{cd}(A, I)\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume \((\mathcal{F}_n)\) satisfies the \((2, 2 + d)\)-inequalities, see Definition 0EJ3. Then there exists a canonical map \((\mathcal{F}_n) \to (\mathcal{F}_n'')\) of inverse systems of coherent \(\mathcal{O}_U\)-modules with the following properties
\(\text{depth}(\mathcal{F}''_{n, y}) + \delta^Y_Z(y) \geq 3\) for all \(y \in U \cap Y\),
\((\mathcal{F}''_n)\) is isomorphic as a pro-system to an object \((\mathcal{H}_n)\) of \(\textit{Coh}(U, I\mathcal{O}_U)\),
the induced morphism \((\mathcal{F}_n) \to (\mathcal{H}_n)\) of \(\textit{Coh}(U, I\mathcal{O}_U)\) has kernel and cokernel annihilated by a power of \(I\),
the modules \(H^0(U, \mathcal{F}''_n)\) and \(H^1(U, \mathcal{F}''_n)\) are finite \(A\)-modules for all \(n\).
Proof
The existence and properties (1), (2), (3) follow from Lemma 0EJF applied to \(U\), \(U \cap Y\), \(T = \{y \in U \cap Y : \delta^Y_Z(y) \leq 2\}\), \(T' = \{y \in U \cap Y : \delta^Y_Z(y) \leq 1\}\), and \((\mathcal{F}_n)\). The finiteness of the modules \(H^0(U, \mathcal{F}''_n)\) and \(H^1(U, \mathcal{F}''_n)\) follows from Local Cohomology, Lemma 0BJY and the elementary properties of the function \(\delta^Y_Z(-)\) proved in Lemma 0EIY.
Algebraization of coherent formal modules, V
In this section we prove our most general results on algebraization of coherent formal modules. We first prove it in case the ideal has cohomological dimension \(1\). Then we apply this to a blowup to prove a more general result.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) has a dualizing complex and \(\text{cd}(A, I) = 1\),
\((\mathcal{F}_n)\) is pro-isomorphic to an inverse system \((\mathcal{F}_n'')\) of coherent \(\mathcal{O}_U\)-modules such that \(\text{depth}(\mathcal{F}''_{n, y}) + \delta^Y_Z(y) \geq 3\) for all \(y \in U \cap Y\).
Then \((\mathcal{F}_n)\) extends canonically to \(X\), see Definition 0EIP.
Proof
We will check hypotheses (a), (b), and (c) of Lemma 0EHH. Before we start, let us point out that the modules \(H^0(U, \mathcal{F}''_n)\) and \(H^1(U, \mathcal{F}''_n)\) are finite \(A\)-modules for all \(n\) by Local Cohomology, Lemma 0BJY.
Observe that for each \(p \geq 0\) the limit topology on \(\lim H^p(U, \mathcal{F}_n)\) is the \(I\)-adic topology by Lemma 0EH7. In particular, hypothesis (b) holds.
We know that \(M = \lim H^0(U, \mathcal{F}_n)\) is an \(A\)-module whose limit topology is the \(I\)-adic topology. Thus, given \(n\), the module \(M/I^nM\) is a subquotient of \(H^0(U, \mathcal{F}_N)\) for some \(N \gg n\). Since the inverse system \(\{H^0(U, \mathcal{F}_N)\}\) is pro-isomorphic to an inverse system of finite \(A\)-modules, namely \(\{H^0(U, \mathcal{F}''_N)\}\), we conclude that \(M/I^nM\) is finite. It follows that \(M\) is finite, see Algebra, Lemma 031D. In particular hypothesis (c) holds.
For each \(n \geq 0\) let us write \(Ob_n = \lim_N H^1(U, I^n\mathcal{F}_N)\). A special case is \(Ob = Ob_0 = \lim_N H^1(U, \mathcal{F}_N)\). Arguing exactly as in the previous paragraph we find that \(Ob\) is a finite \(A\)-module. (In fact, we also know that \(Ob/I Ob\) is annihilated by a power of \(\mathfrak a\), but it seems somewhat difficult to use this.)
We set \(\mathcal{F} = \lim \mathcal{F}_n\), we pick generators \(f_1, \ldots, f_r\) of \(I\), we pick \(c \geq 1\), and we choose \(\Phi_\mathcal{F}\) as in Lemma 0EH6. We will use the results of Lemma 0EI8 without further mention. In particular, for each \(n \geq 1\) there are maps \[\delta_n : H^0(U, \mathcal{F}_n) \longrightarrow H^1(U, I^n\mathcal{F}) \longrightarrow Ob_n\] The first comes from the short exact sequence \(0 \to I^n\mathcal{F} \to \mathcal{F} \to \mathcal{F}_n \to 0\) and the second from \(I^n\mathcal{F} = \lim I^n\mathcal{F}_N\). We will later use that if \(\delta_n(s) = 0\) for \(s \in H^0(U, \mathcal{F}_n)\) then we can for each \(n' \geq n\) find \(s' \in H^0(U, \mathcal{F}_{n'})\) mapping to \(s\). Observe that there are commutative diagrams \[\xymatrix{ H^0(U, \mathcal{F}_{nc}) \ar[r] \ar[dd] & H^1(U, I^{nc}\mathcal{F}) \ar[dd] \ar[rd]^{\Phi_\mathcal{F}} \\ & & \bigoplus_{e_1 + \ldots + e_r = n} H^1(U, \mathcal{F}) \cdot T_1^{e_1} \ldots T_r^{e_r} \ar[ld] \\ H^0(U, \mathcal{F}_n) \ar[r] & H^1(U, I^n\mathcal{F}) }\] We conclude that the obstruction map \(H^0(U, \mathcal{F}_n) \to Ob_n\) sends the image of \(H^0(U, \mathcal{F}_{nc}) \to H^0(U, \mathcal{F}_n)\) into the submodule \[Ob'_n = \Im\left( \bigoplus\nolimits_{e_1 + \ldots + e_r = n} Ob \cdot T_1^{e_1} \ldots T_r^{e_r} \to Ob_n \right)\] where on the summand \(Ob \cdot T_1^{e_1} \ldots T_r^{e_r}\) we use the map on cohomology coming from the reductions modulo powers of \(I\) of the multiplication map \(f_1^{e_1} \ldots f_r^{e_r} : \mathcal{F} \to I^n\mathcal{F}\). By construction \[\bigoplus\nolimits_{n \geq 0} Ob'_n\] is a finite graded module over the Rees algebra \(\bigoplus_{n \geq 0} I^n\). For each \(n\) we set \[M_n = \{s \in H^0(U, \mathcal{F}_n) \mid \delta_n(s) \in Ob'_n\}\] Observe that \(\{M_n\}\) is an inverse system and that \(f_j : \mathcal{F}_n \to \mathcal{F}_{n + 1}\) on global sections maps \(M_n\) into \(M_{n + 1}\). By exactly the same argument as in the proof of Cohomology, Lemma 0GYK we find that \(\{M_n\}\) is ML. Namely, because the Rees algebra is Noetherian we can choose a finite number of homogeneous generators of the form \(\delta_{n_j}(z_j)\) with \(z_j \in M_{n_j}\) for the graded submodule \(\bigoplus_{n \geq 0} \Im(M_n \to Ob'_n)\). Then if \(k = \max(n_j)\) we find that for \(n \geq k\) and any \(z \in M_n\) we can find \(a_j \in I^{n - n_j}\) such that \(z - \sum a_j z_j\) is in the kernel of \(\delta_n\) and hence in the image of \(M_{n'}\) for all \(n' \geq n\) (because the vanishing of \(\delta_n\) means that we can lift \(z - \sum a_j z_j\) to an element \(z' \in H^0(U, \mathcal{F}_{n'c})\) for all \(n' \ge n\) and then the image of \(z'\) in \(H^0(U, \mathcal{F}_{n'})\) is in \(M_{n'}\) by what we proved above). Thus \(\Im(M_n \to M_{n - k}) = \Im(M_{n'} \to M_{n - k})\) for all \(n' \geq n\).
Choose \(n\). By the Mittag-Leffler property of \(\{M_n\}\) we just established we can find an \(n' \geq n\) such that the image of \(M_{n'} \to M_n\) is the same as the image of \(M' \to M_n\). By the above we see that the image of \(M' \to M_n\) contains the image of \(H^0(U, \mathcal{F}_{n'c}) \to H^0(U, \mathcal{F}_n)\). Thus we see that \(\{M_n\}\) and \(\{H^0(U, \mathcal{F}_n)\}\) are pro-isomorphic. Therefore \(\{H^0(U, \mathcal{F}_n)\}\) has ML and we finally conclude that hypothesis (a) holds. This concludes the proof.
Proposition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) has a dualizing complex and \(\text{cd}(A, I) = 1\),
\((\mathcal{F}_n)\) satisfies the \((2, 3)\)-inequalities, see Definition 0EJ3.
Then \((\mathcal{F}_n)\) extends to \(X\). In particular, if \(A\) is \(I\)-adically complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
By Lemma 0EIU we may replace \((\mathcal{F}_n)\) by the object \((\mathcal{H}_n)\) of \(\textit{Coh}(U, I\mathcal{O}_U)\) found in Lemma 0EJG. Thus we may assume that \((\mathcal{F}_n)\) is pro-isomorphic to an inverse system \((\mathcal{F}_n'')\) with the properties mentioned in Lemma 0EJG. In Lemma 0EJI we proved that \((\mathcal{F}_n)\) canonically extends to \(X\). The final statement follows from Lemma 0EIR.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) has a dualizing complex,
all fibres of the blowing up \(b : X' \to X\) of \(I\) have dimension \(\leq d - 1\),
one of the following is true
\((\mathcal{F}_n)\) satisfies the \((d + 1, d + 2)\)-inequalities (Definition 0EJ3), or
for \(y \in U \cap Y\) and a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\) we have \[\text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > d + 2\]
Then \((\mathcal{F}_n)\) extends to \(X\).
Proof
Let \(Y' \subset X'\) be the exceptional divisor. Let \(Z' \subset Y'\) be the inverse image of \(Z \subset Y\). Then \(U' = X' \setminus Z'\) is the inverse image of \(U\). With \(\delta^{Y'}_{Z'}\) as in (0EIX) we set \[T' = \{y' \in Y' \mid \delta^{Y'}_{Z'}(y') = 1\} \subset T = \{y' \in Y' \mid \delta^{Y'}_{Z'}(y') = 1\text{ or }2\}\] By Lemma 0EIY parts (1) and (2) these are specialization stable subsets of \(U' \cap Y' = Y' \setminus Z'\). Consider the object \((b|_{U'}^*\mathcal{F}_n)\) of \(\textit{Coh}(U', I\mathcal{O}_{U'})\), see Cohomology of Schemes, Lemma 0887. For \(y' \in U' \cap Y'\) let us denote \[\mathcal{F}_{y'}^\wedge = \lim (b|_{U'}^*\mathcal{F}_n)_{y'}\] the “stalk” of this pullback at \(y'\). We claim that conditions (a), (b), (c), (d), and (e) of Lemma 0EJF hold for the object \((b|_{U'}^*\mathcal{F}_n)\) on \(U'\) with \(d\) replaced by \(1\) and the subsets \(T' \subset T \subset U' \cap Y'\). Condition (a) holds because \(Y'\) is an effective Cartier divisor and hence locally cut out by \(1\) equation. Condition (e) holds by Lemma 0EIY part (7). To prove (b), (c), and (d) we need some preparation.
Let \(y' \in U' \cap Y'\) and let \(\mathfrak p' \subset \mathcal{O}_{X', y'}^\wedge\) be a prime ideal not contained in \(V(I\mathcal{O}_{X', y'}^\wedge)\). Denote \(y = b(y') \in U \cap Y\). Choose \(f \in I\) such that \(y'\) is contained in the spectrum of the affine blowup algebra \(A[\frac{I}{f}]\), see Divisors, Lemma 0804. For any \(A\)-algebra \(B\) denote \(B' = B[\frac{IB}{f}]\) the corresponding affine blowup algebra. Denote \(I\)-adic completion by \({\ }^\wedge\). By our choice of \(f\) we get a ring map \((\mathcal{O}_{X, y}^\wedge)' \to \mathcal{O}_{X', y'}^\wedge\). If we let \(\mathfrak q' \subset (\mathcal{O}_{X, y}^\wedge)'\) be the inverse image of \(\mathfrak m_{y'}^\wedge\), then we see that \(((\mathcal{O}_{X, y}^\wedge)'_{\mathfrak q'})^\wedge = \mathcal{O}_{X', y'}^\wedge\). Let \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) be the corresponding prime. At this point we have a commutative diagram \[\xymatrix{ \mathcal{O}_{X, y}^\wedge \ar[d] \ar[r] & (\mathcal{O}_{X, y}^\wedge)' \ar[d]_\alpha \ar[r] & (\mathcal{O}_{X, y}^\wedge)'_{\mathfrak q'} \ar[d] \ar[r]_\beta & \mathcal{O}_{X', y'}^\wedge \ar[d] \\ \mathcal{O}_{X, y}^\wedge/\mathfrak p \ar[r] & (\mathcal{O}_{X, y}^\wedge/\mathfrak p)' \ar[r] & (\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'} \ar[r]^\gamma & ((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'})^\wedge \ar[d] \\ & & & \mathcal{O}_{X', y'}^\wedge/\mathfrak p' }\] whose vertical arrows are surjective. By More on Algebra, Lemma 07NV and the dimension formula (Algebra, Lemma 02IJ) we have \[\dim(((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'})^\wedge) = \dim((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'}) = \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) - \text{trdeg}(\kappa(y')/\kappa(y))\] Tracing through the definitions of pullbacks, stalks, localizations, and completions we find \[(\mathcal{F}_y^\wedge)_{\mathfrak p} \otimes_{(\mathcal{O}_{X, y}^\wedge)_\mathfrak p} (\mathcal{O}_{X', y'}^\wedge)_{\mathfrak p'} = (\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}\] Details omitted. The ring maps \(\beta\) and \(\gamma\) in the diagram are flat with Gorenstein (hence Cohen-Macaulay) fibres, as these are completions of rings having a dualizing complex. See Dualizing Complexes, Lemmas 0BJN and 0AWY and the discussion in More on Algebra, Section 0BIR. Observe that \((\mathcal{O}_{X, y}^\wedge)_\mathfrak p = (\mathcal{O}_{X, y}^\wedge)'_{\tilde{\mathfrak p}}\) where \(\tilde{\mathfrak p}\) is the kernel of \(\alpha\) in the diagram. On the other hand, \((\mathcal{O}_{X, y}^\wedge)'_{\tilde{\mathfrak p}} \to (\mathcal{O}_{X', y'}^\wedge)_{\mathfrak p'}\) is flat with CM fibres by the above. Whence \((\mathcal{O}_{X, y}^\wedge)_\mathfrak p \to (\mathcal{O}_{X', y'}^\wedge)_{\mathfrak p'}\) is flat with CM fibres. Using Algebra, Lemma 0338 we see that \[\text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) = \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(F_\mathfrak r)\] where \(F\) is the generic formal fibre of \((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'}\) and \(\mathfrak r\) is the prime corresponding to \(\mathfrak p'\). Since \((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'}\) is a universally catenary local domain, its \(I\)-adic completion is equidimensional and (universally) catenary by Ratliff’s theorem (More on Algebra, Proposition 0AW6). It then follows that \[\dim(((\mathcal{O}_{X, y}^\wedge/\mathfrak p)'_{\mathfrak q'})^\wedge) = \dim(F_\mathfrak r) + \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p')\] Combined with Lemma 0EIZ we get [0EJL]\[\begin{equation} \begin{aligned} & \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \delta^{Y'}_{Z'}(y') \\ & = \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(F_\mathfrak r) + \delta^{Y'}_{Z'}(y') \\ & \geq \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) + \dim(F_\mathfrak r) + \text{trdeg}(\kappa(y')/\kappa(y)) - (d - 1) \\ & = \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) - (d - 1) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) - \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') \end{aligned} \end{equation}\] Please keep in mind that \(\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) \geq \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p')\). Rewriting this we get [0EJM]\[\begin{equation} \begin{aligned} & \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') + \delta^{Y'}_{Z'}(y') \\ & \geq \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) - (d - 1) \end{aligned} \end{equation}\] This inequality will allow us to check the remaining conditions.
Conditions (b) and (d) of Lemma 0EJF. Assume \(V(\mathfrak p') \cap V(I\mathcal{O}_{X', y'}^\wedge) = \{\mathfrak m_{y'}^\wedge\}\). This implies that \(\dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') = 1\) because \(Z'\) is an effective Cartier divisor. The combination of (b) and (d) is equivalent with \[\text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \delta^{Y'}_{Z'}(y') > 2\] If \((\mathcal{F}_n)\) satisfies the inequalities in (3)(b) then we immediately conclude this is true by applying (0EJM). If \((\mathcal{F}_n)\) satisfies (3)(a), i.e., the \((d + 1, d + 2)\)-inequalities, then we see that in any case \[\text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) \geq d + 1 \quad\text{or}\quad \text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > d + 2\] Looking at (0EJL) and (0EJM) above this gives what we want except possibly if \(\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = 1\). However, if \(\dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = 1\), then we have \(V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge) = \{\mathfrak m_y^\wedge\}\) and we see that actually \[\text{depth}((\mathcal{F}_y^\wedge)_{\mathfrak p}) + \delta^Y_Z(y) > d + 1\] as \((\mathcal{F}_n)\) satisfies the \((d + 1, d + 2)\)-inequalities and we conclude again.
Condition (c) of Lemma 0EJF. Assume \(V(\mathfrak p') \cap V(I\mathcal{O}_{X', y'}^\wedge) \not = \{\mathfrak m_{y'}^\wedge\}\). Then condition (c) is equivalent to \[\text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \delta^{Y'}_{Z'}(y') \geq 2 \quad\text{or}\quad \text{depth}((\mathcal{F}_{y'}^\wedge)_{\mathfrak p'}) + \dim(\mathcal{O}_{X', y'}^\wedge/\mathfrak p') + \delta^{Y'}_{Z'}(y') > 3\] If \((\mathcal{F}_n)\) satisfies the inequalities in (3)(b) then we see the second of the two displayed inequalities holds true by applying (0EJM). If \((\mathcal{F}_n)\) satisfies (3)(a), i.e., the \((d + 1, d + 2)\)-inequalities, then this follows immediately from (0EJL) and (0EJM). This finishes the proof of our claim.
Choose \((b|_{U'}^*\mathcal{F}_n) \to (\mathcal{F}_n'')\) and \((\mathcal{H}_n)\) in \(\textit{Coh}(U', I\mathcal{O}_{U'})\) as in Lemma 0EJF. For any affine open \(W \subset X'\) observe that \(\delta^{W \cap Y'}_{W \cap Z'}(y') \geq \delta^{Y'}_{Z'}(y')\) by Lemma 0EIY part (6). Hence we see that \((\mathcal{H}_n|_W)\) satisfies the assumptions of Lemma 0EJI. Thus \((\mathcal{H}_n|_W)\) extends canonically to \(W\). Let \((\mathcal{G}_{W, n})\) in \(\textit{Coh}(W, I\mathcal{O}_W)\) be the canonical extension as in Lemma 0EIR. By Lemma 0EIS we see that for \(W' \subset W\) there is a unique isomorphism \[(\mathcal{G}_{W, n}|_{W'}) \longrightarrow (\mathcal{G}_{W', n})\] compatible with the given isomorphisms \((\mathcal{G}_{W, n}|_{W \cap U}) \cong (\mathcal{H}_n|_{W \cap U})\). We conclude that there exists an object \((\mathcal{G}_n)\) of \(\textit{Coh}(X', I\mathcal{O}_{X'})\) whose restriction to \(U\) is isomorphic to \((\mathcal{H}_n)\).
If \(A\) is \(I\)-adically complete we can finish the proof as follows. By Grothendieck’s existence theorem (Cohomology of Schemes, Lemma 0885) we see that \((\mathcal{G}_n)\) is the completion of a coherent \(\mathcal{O}_{X'}\)-module. Then by Cohomology of Schemes, Lemma 0889 we see that \((b|_{U'}^*\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_{U'}\)-module \(\mathcal{F}'\). By Cohomology of Schemes, Lemma 088B we see that there is a map \[(\mathcal{F}_n) \longrightarrow ((b|_{U'})_*\mathcal{F}')^\wedge\] whose kernel and cokernel is annihilated by a power of \(I\). Then finally, we win by applying Lemma 0EIU.
If \(A\) is not complete, then, before starting the proof, we may replace \(A\) by its completion, see Lemma 0EIN. After completion the assumptions still hold: this is immediate for condition (3), follows from Dualizing Complexes, Lemma 0BFR for condition (1), and from Divisors, Lemma 0805 for condition (2). Thus the complete case implies the general case.
Proposition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) has a dualizing complex,
\(V(I) = V(f_1, \ldots, f_d)\) for some \(d \geq 1\) and \(f_1, \ldots, f_d \in A\),
one of the following is true
\((\mathcal{F}_n)\) satisfies the \((d + 1, d + 2)\)-inequalities (Definition 0EJ3), or
for \(y \in U \cap Y\) and a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\) we have \[\text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) > d + 2\]
Then \((\mathcal{F}_n)\) extends to \(X\). In particular, if \(A\) is \(I\)-adically complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
We may assume \(I = (f_1, \ldots, f_d)\), see Cohomology of Schemes, Lemma 0EHR. Then we see that all fibres of the blowup of \(X\) in \(I\) have dimension at most \(d - 1\). Thus we get the extension from Lemma 0EJK. The final statement follows from Lemma 0EIL.
Please compare the next lemma with Remarks 0EHI, 0DXV, 0EJC, and 0EJT.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) is a local ring which has a dualizing complex,
all irreducible components of \(X\) have the same dimension,
the scheme \(X \setminus Y\) is Cohen-Macaulay,
\(I\) is generated by \(d\) elements,
\(\dim(X) - \dim(Z) > d + 2\), and
for \(y \in U \cap Y\) the module \(\mathcal{F}_y^\wedge\) is finite locally free outside \(V(I\mathcal{O}_{X, y}^\wedge)\), for example if \(\mathcal{F}_n\) is a finite locally free \(\mathcal{O}_U/I^n\mathcal{O}_U\)-module.
Then \((\mathcal{F}_n)\) extends to \(X\). In particular if \(A\) is \(I\)-adically complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
We will show that the hypotheses (1), (2), (3)(b) of Proposition 0EJN are satisfied. This is clear for (1) and (2).
Let \(y \in U \cap Y\) and let \(\mathfrak p\) be a prime \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) with \(\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)\). The last condition shows that \(\text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) = \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p)\). Since \(X \setminus Y\) is Cohen-Macaulay we see that \((\mathcal{O}_{X, y}^\wedge)_\mathfrak p\) is Cohen-Macaulay. Thus we see that \[\begin{align*} & \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) \\ & = \dim((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) \\ & = \dim(\mathcal{O}_{X, y}^\wedge) + \delta^Y_Z(y) \end{align*}\] The final equality because \(\mathcal{O}_{X, y}\) is equidimensional by the second condition. Let \(\delta(y) = \dim(\overline{\{y\}})\). This is a dimension function as \(A\) is a catenary local ring. By Lemma 0EIY we have \(\delta^Y_Z(y) \geq \delta(y) - \dim(Z)\). Since \(X\) is equidimensional we get \[\dim(\mathcal{O}_{X, y}^\wedge) + \delta^Y_Z(y) \geq \dim(\mathcal{O}_{X, y}^\wedge) + \delta(y) - \dim(Z) = \dim(X) - \dim(Z)\] Thus we get the desired inequality and we win.
Remark
We are unable to prove or disprove the analogue of Proposition 0EJN where the assumption that \(I\) has \(d\) generators is replaced with the assumption \(\text{cd}(A, I) \leq d\). If you know a proof or have a counter example, please email stacks.project@gmail.com. Another obvious question is to what extend the conditions in Proposition 0EJN are necessary.
Algebraization of coherent formal modules, VI
In this section we add a few more easier to prove cases.
Proposition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
there exist \(f_1, \ldots, f_d \in I\) such that for \(y \in U \cap Y\) the ideal \(I\mathcal{O}_{X, y}\) is generated by \(f_1, \ldots, f_d\) and \(f_1, \ldots, f_d\) form a \(\mathcal{F}_y^\wedge\)-regular sequence,
\(H^0(U, \mathcal{F}_1)\) and \(H^1(U, \mathcal{F}_1)\) are finite \(A\)-modules.
Then \((\mathcal{F}_n)\) extends canonically to \(X\). In particular, if \(A\) is complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
We will prove this by verifying hypotheses (a), (b), and (c) of Lemma 0EHH. For every \(n\) we have a short exact sequence \[0 \to I^n\mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1} \to \mathcal{F}_n \to 0\] Since \(f_1, \ldots, f_d\) forms a regular sequence (and hence quasi-regular, see Algebra, Lemma 00LN) on each of the “stalks” \(\mathcal{F}_y^\wedge\) and since we have \(I\mathcal{F}_n = (f_1, \ldots, f_d)\mathcal{F}_n\) for all \(n\), we find that \[I^n\mathcal{F}_{n + 1} = \bigoplus\nolimits_{e_1 + \ldots + e_d = n} \mathcal{F}_1 \cdot f_1^{e_1} \ldots f_d^{e_d}\] by checking on stalks. Using the assumption of finiteness of \(H^0(U, \mathcal{F}_1)\) and induction, we first conclude that \(M_n = H^0(U, \mathcal{F}_n)\) is a finite \(A\)-module for all \(n\). In this way we see that condition (c) of Lemma 0EHH holds. We also see that \[\bigoplus\nolimits_{n \geq 0} H^1(U, I^n\mathcal{F}_{n + 1})\] is a finite graded \(R = \bigoplus I^n/I^{n +1}\)-module. By Cohomology, Lemma 0GYK we conclude that condition (a) of Lemma 0EHH is satisfied. Finally, condition (b) of Lemma 0EHH is satisfied because \(\bigoplus H^0(U, I^n\mathcal{F}_{n + 1})\) is a finite graded \(R\)-module and we can apply Cohomology, Lemma 0GYM.
Remark
In the situation of Proposition 0EJS if we assume \(A\) has a dualizing complex, then the condition that \(H^0(U, \mathcal{F}_1)\) and \(H^1(U, \mathcal{F}_1)\) are finite is equivalent to \[\text{depth}(\mathcal{F}_{1, y}) + \dim(\mathcal{O}_{\overline{\{y\}}, z}) > 2\] for all \(y \in U \cap Y\) and \(z \in Z \cap \overline{\{y\}}\). See Local Cohomology, Lemma 0BJY. This holds for example if \(\mathcal{F}_1\) is a finite locally free \(\mathcal{O}_{U \cap Y}\)-module, \(Y\) is \((S_2)\), and \(\text{codim}(Z', Y') \geq 3\) for every pair of irreducible components \(Y'\) of \(Y\), \(Z'\) of \(Z\) with \(Z' \subset Y'\).
Proposition
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume there is Noetherian local ring \((R, \mathfrak m)\) and a ring map \(R \to A\) such that
\(I = \mathfrak m A\),
for \(y \in U \cap Y\) the stalk \(\mathcal{F}_y^\wedge\) is \(R\)-flat,
\(H^0(U, \mathcal{F}_1)\) and \(H^1(U, \mathcal{F}_1)\) are finite \(A\)-modules.
Then \((\mathcal{F}_n)\) extends canonically to \(X\). In particular, if \(A\) is complete, then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_U\)-module.
Proof
The proof is exactly the same as the proof of Proposition 0EJS. Namely, if \(\kappa = R/\mathfrak m\) then for \(n \geq 0\) there is an isomorphism \[I^n \mathcal{F}_{n + 1} \cong \mathcal{F}_1 \otimes_\kappa \mathfrak m^n/\mathfrak m^{n + 1}\] and the right hand side is a finite direct sum of copies of \(\mathcal{F}_1\). This can be checked by looking at stalks. Everything else is exactly the same.
Remark
Proposition 0EJU is a local version of [Baranovsky, Theorem 2.10 (i)]. It is straightforward to deduce the global results from the local one; we will sketch the argument. Namely, suppose \((R, \mathfrak m)\) is a complete Noetherian local ring and \(X \to \Spec(R)\) is a proper morphism. For \(n \geq 1\) set \(X_n = X \times_{\Spec(R)} \Spec(R/\mathfrak m^n)\). Let \(Z \subset X_1\) be a closed subset of the special fibre. Set \(U = X \setminus Z\) and denote \(j : U \to X\) the inclusion morphism. Suppose given an object \[(\mathcal{F}_n) \text{ of } \textit{Coh}(U, \mathfrak m\mathcal{O}_U)\] which is flat over \(R\) in the sense that \(\mathcal{F}_n\) is flat over \(R/\mathfrak m^n\) for all \(n\). Assume that \(j_*\mathcal{F}_1\) and \(R^1j_*\mathcal{F}_1\) are coherent modules. Then affine locally on \(X\) we get a canonical extension of \((\mathcal{F}_n)\) by Proposition 0EJU and formation of this extension commutes with localization (by Lemma 0DXW). Thus we get a canonical global object \((\mathcal{G}_n)\) of \(\textit{Coh}(X, \mathfrak m\mathcal{O}_X)\) whose restriction of \(U\) is \((\mathcal{F}_n)\). By Grothendieck’s existence theorem (Cohomology of Schemes, Proposition 088C) we see there exists a coherent \(\mathcal{O}_X\)-module \(\mathcal{G}\) whose completion is \((\mathcal{G}_n)\). In this way we see that \((\mathcal{F}_n)\) is algebraizable, i.e., it is the completion of a coherent \(\mathcal{O}_U\)-module.
We add that the coherence of \(j_*\mathcal{F}_1\) and \(R^1j_*\mathcal{F}_1\) is a condition on the special fibre. Namely, if we denote \(j_1 : U_1 \to X_1\) the special fibre of \(j : U \to X\), then we can think of \(\mathcal{F}_1\) as a coherent sheaf on \(U_1\) and we have \(j_*\mathcal{F}_1 = j_{1, *}\mathcal{F}_1\) and \(R^1j_*\mathcal{F}_1 = R^1j_{1, *}\mathcal{F}_1\). Hence for example if \(X_1\) is \((S_2)\) and irreducible, we have \(\dim(X_1) - \dim(Z) \geq 3\), and \(\mathcal{F}_1\) is a locally free \(\mathcal{O}_{U_1}\)-module, then \(j_{1, *}\mathcal{F}_1\) and \(R^1j_{1, *}\mathcal{F}_1\) are coherent modules.
Application to the completion functor
In this section we just combine some already obtained results in order to conveniently reference them. There are many (stronger) results we could state here.
Lemma
In Situation 0EHC assume
\(A\) has a dualizing complex and is \(I\)-adically complete,
\(I = (f)\) generated by a single element,
\(A\) is local with maximal ideal \(\mathfrak a = \mathfrak m\),
one of the following is true
\(A_f\) is \((S_2)\) and for \(\mathfrak p \subset A\), \(f \not \in \mathfrak p\) minimal we have \(\dim(A/\mathfrak p) \geq 4\), or
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
Then with \(U_0 = U \cap V(f)\) the completion functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_U)\] is an equivalence on the full subcategories of finite locally free objects.
Proof
It follows from Lemma 0EKU that the functor is fully faithful (details omitted). Let us prove essential surjectivity. Let \((\mathcal{F}_n)\) be a finite locally free object of \(\textit{Coh}(U, f\mathcal{O}_U)\). By either Lemma 0EJ9 or Proposition 0EJJ there exists a coherent \(\mathcal{O}_U\)-module \(\mathcal{F}\) such that \((\mathcal{F}_n)\) is the completion of \(\mathcal{F}\). Namely, for the application of either result the only thing to check is that \((\mathcal{F}_n)\) satisfies the \((2, 3)\)-inequalities. This is done in Lemma 0EJB. If \(y \in U_0\), then the \(f\)-adic completion of the stalk \(\mathcal{F}_y\) is isomorphic to a finite free module over the \(f\)-adic completion of \(\mathcal{O}_{U, y}\). Hence \(\mathcal{F}\) is finite locally free in an open neighbourhood \(U'\) of \(U_0\). This finishes the proof.
Lemma
In Situation 0EHC assume
\(I = (f)\) is principal,
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
\(H^1_\mathfrak a(A/fA)\) and \(H^2_\mathfrak a(A/fA)\) are finite \(A\)-modules.
Then with \(U_0 = U \cap V(f)\) the completion functor \[\colim_{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_U)\] is an equivalence on the full subcategories of finite locally free objects.
Proof
The functor is fully faithful by Lemma 0EKV. Essential surjectivity follows from Lemma 0DXW.
Coherent triples
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\) be a nonzerodivisor. Set \(X = \Spec(A)\), \(X_0 = \Spec(A/fA)\), \(U = X \setminus V(\mathfrak m)\), and \(U_0 = U \cap X_0\). We say \((\mathcal{F}, \mathcal{F}_0, \alpha)\) is a coherent triple if we have
\(\mathcal{F}\) is a coherent \(\mathcal{O}_U\)-module such that \(f : \mathcal{F} \to \mathcal{F}\) is injective,
\(\mathcal{F}_0\) is a coherent \(\mathcal{O}_{X_0}\)-module,
\(\alpha : \mathcal{F}/f\mathcal{F} \to \mathcal{F}_0|_{U_0}\) is an isomorphism.
There is an obvious notion of a morphism of coherent triples which turns the collection of all coherent triples into a category.
The category of coherent triples is additive but not abelian. However, it is clear what a short exact sequence of coherent triples is.
Given two coherent triples \((\mathcal{F}, \mathcal{F}_0, \alpha)\) and \((\mathcal{G}, \mathcal{G}_0, \beta)\) it may not be the case that \((\mathcal{F} \otimes_{\mathcal{O}_U} \mathcal{G}, \mathcal{F}_0 \otimes_{\mathcal{O}_{X_0}} \mathcal{G}_0, \alpha \otimes \beta)\) is a coherent triple8. However, if the stalks \(\mathcal{G}_x\) are free for all \(x \in U_0\), then this does hold.
We will say the coherent triple \((\mathcal{G}, \mathcal{G}_0, \beta)\) is locally free, resp. invertible if \(\mathcal{G}\) and \(\mathcal{G}_0\) are locally free, resp. invertible modules. In this case tensoring with \((\mathcal{G}, \mathcal{G}_0, \beta)\) makes sense (see above) and turns short exact sequences of coherent triples into short exact sequences of coherent triples.
Lemma
For any coherent triple \((\mathcal{F}, \mathcal{F}_0, \alpha)\) there exists a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}'\) such that \(f : \mathcal{F}' \to \mathcal{F}'\) is injective, an isomorphism \(\alpha' : \mathcal{F}'|_U \to \mathcal{F}\), and a map \(\alpha'_0 : \mathcal{F}'/f\mathcal{F}' \to \mathcal{F}_0\) such that \(\alpha \circ (\alpha' \bmod f) = \alpha'_0|_{U_0}\).
Proof
Choose a finite \(A\)-module \(M\) such that \(\mathcal{F}\) is the restriction to \(U\) of the coherent \(\mathcal{O}_X\)-module associated to \(M\), see Local Cohomology, Lemma 0BK0. Since \(\mathcal{F}\) is \(f\)-torsion free, we may replace \(M\) by its quotient by \(f\)-power torsion. On the other hand, let \(M_0 = \Gamma(X_0, \mathcal{F}_0)\) so that \(\mathcal{F}_0\) is the coherent \(\mathcal{O}_{X_0}\)-module associated to the finite \(A/fA\)-module \(M_0\). By Cohomology of Schemes, Lemma 01YB there exists an \(n\) such that the isomorphism \(\alpha_0\) corresponds to an \(A/fA\)-module homomorphism \(\mathfrak m^n M/fM \to M_0\) (whose kernel and cokernel are annihilated by a power of \(\mathfrak m\), but we don’t need this). Thus if we take \(M' = \mathfrak m^n M\) and we let \(\mathcal{F}'\) be the coherent \(\mathcal{O}_X\)-module associated to \(M'\), then the lemma is clear.
Let \((\mathcal{F}, \mathcal{F}_0, \alpha)\) be a coherent triple. Choose \(\mathcal{F}', \alpha', \alpha'_0\) as in Lemma 0F23. Set [0F24]\[\begin{equation} \chi(\mathcal{F}, \mathcal{F}_0, \alpha) = \text{length}_A(\Coker(\alpha'_0)) - \text{length}_A(\Ker(\alpha'_0)) \end{equation}\] The expression on the right makes sense as \(\alpha'_0\) is an isomorphism over \(U_0\) and hence its kernel and coherent are coherent modules supported on \(\{\mathfrak m\}\) which therefore have finite length (Algebra, Lemma 00L5).
Lemma
The quantity \(\chi(\mathcal{F}, \mathcal{F}_0, \alpha)\) in (0F24) does not depend on the choice of \(\mathcal{F}', \alpha', \alpha'_0\) as in Lemma 0F23.
Proof
Let \(\mathcal{F}', \alpha', \alpha'_0\) and \(\mathcal{F}'', \alpha'', \alpha''_0\) be two such choices. For \(n > 0\) set \(\mathcal{F}'_n = \mathfrak m^n \mathcal{F}'\). By Cohomology of Schemes, Lemma 01YB for some \(n\) there exists an \(\mathcal{O}_X\)-module map \(\mathcal{F}'_n \to \mathcal{F}''\) agreeing with the identification \(\mathcal{F}''|_U = \mathcal{F}'|_U\) determined by \(\alpha'\) and \(\alpha''\). Then the diagram \[\xymatrix{ \mathcal{F}'_n/f\mathcal{F}'_n \ar[r] \ar[d] & \mathcal{F}'/f\mathcal{F}' \ar[d]^{\alpha_0'} \\ \mathcal{F}''/f\mathcal{F}'' \ar[r]^{\alpha_0''} & \mathcal{F}_0 }\] is commutative after restricting to \(U_0\). Hence by Cohomology of Schemes, Lemma 01YB it is commutative after restricting to \(\mathfrak m^l(\mathcal{F}'_n/f\mathcal{F}'_n)\) for some \(l > 0\). Since \(\mathcal{F}'_{n + l}/f\mathcal{F}'_{n + l} \to \mathcal{F}'_n/f\mathcal{F}'_n\) factors through \(\mathfrak m^l(\mathcal{F}'_n/f\mathcal{F}'_n)\) we see that after replacing \(n\) by \(n + l\) the diagram is commutative. In other words, we have found a third choice \(\mathcal{F}''', \alpha''', \alpha'''_0\) such that there are maps \(\mathcal{F}''' \to \mathcal{F}''\) and \(\mathcal{F}''' \to \mathcal{F}'\) over \(X\) compatible with the maps over \(U\) and \(X_0\). This reduces us to the case discussed in the next paragraph.
Assume we have a map \(\mathcal{F}'' \to \mathcal{F}'\) over \(X\) compatible with \(\alpha', \alpha''\) over \(U\) and with \(\alpha'_0, \alpha''_0\) over \(X_0\). Observe that \(\mathcal{F}'' \to \mathcal{F}'\) is injective as it is an isomorphism over \(U\) and since \(f : \mathcal{F}'' \to \mathcal{F}''\) is injective. Clearly \(\mathcal{F}'/\mathcal{F}''\) is supported on \(\{\mathfrak m\}\) hence has finite length. We have the maps of coherent \(\mathcal{O}_{X_0}\)-modules \[\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}' \xrightarrow{\alpha'_0} \mathcal{F}_0\] whose composition is \(\alpha''_0\) and which are isomorphisms over \(U_0\). Elementary homological algebra gives a \(6\)-term exact sequence \[\begin{matrix} 0 \to \Ker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}') \to \Ker(\alpha''_0) \to \Ker(\alpha'_0) \to \\ \Coker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}') \to \Coker(\alpha''_0) \to \Coker(\alpha'_0) \to 0 \end{matrix}\] By additivity of lengths (Algebra, Lemma 00IV) we find that it suffices to show that \[\text{length}_A( \Coker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}')) - \text{length}_A( \Ker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}')) = 0\] This follows from applying the snake lemma to the diagram \[\xymatrix{ 0 \ar[r] & \mathcal{F}'' \ar[r]_f \ar[d] & \mathcal{F}'' \ar[r] \ar[d] & \mathcal{F}''/f\mathcal{F}'' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}' \ar[r]^f & \mathcal{F}' \ar[r] & \mathcal{F}'/f\mathcal{F}' \ar[r] & 0 }\] and the fact that \(\mathcal{F}'/\mathcal{F}''\) has finite length.
Lemma
We have \(\chi(\mathcal{G}, \mathcal{G}_0, \beta) = \chi(\mathcal{F}, \mathcal{F}_0, \alpha) + \chi(\mathcal{H}, \mathcal{H}_0, \gamma)\) if \[0 \to (\mathcal{F}, \mathcal{F}_0, \alpha) \to (\mathcal{G}, \mathcal{G}_0, \beta) \to (\mathcal{H}, \mathcal{H}_0, \gamma) \to 0\] is a short exact sequence of coherent triples.
Proof
Choose \(\mathcal{G}', \beta', \beta'_0\) as in Lemma 0F23 for the triple \((\mathcal{G}, \mathcal{G}_0, \beta)\). Denote \(j : U \to X\) the inclusion morphism. Let \(\mathcal{F}' \subset \mathcal{G}'\) be the kernel of the composition \[\mathcal{G}' \xrightarrow{\beta'} j_*\mathcal{G} \to j_*\mathcal{H}\] Observe that \(\mathcal{H}' = \mathcal{G}'/\mathcal{F}'\) is a coherent subsheaf of \(j_*\mathcal{H}\) and hence \(f : \mathcal{H}' \to \mathcal{H}'\) is injective. Hence by the snake lemma we obtain a short exact sequence \[0 \to \mathcal{F}'/f\mathcal{F}' \to \mathcal{G}'/f\mathcal{G}' \to \mathcal{H}'/f\mathcal{H}' \to 0\] We have isomorphisms \(\alpha' : \mathcal{F}'|_U \to \mathcal{F}\), \(\beta' : \mathcal{G}'|_U \to \mathcal{G}\), and \(\gamma' : \mathcal{H}'|_U \to \mathcal{H}\) by construction. To finish the proof we’ll need to construct maps \(\alpha'_0 : \mathcal{F}'/f\mathcal{F}' \to \mathcal{F}_0\) and \(\gamma'_0 : \mathcal{H}'/f\mathcal{H}' \to \mathcal{H}_0\) as in Lemma 0F23 and fitting into a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{F}'/f\mathcal{F}' \ar[r] \ar@{..>}[d]^{\alpha'_0} & \mathcal{G}'/f\mathcal{G}' \ar[r] \ar[d]^{\beta'_0} & \mathcal{H}'/f\mathcal{H}' \ar[r] \ar@{..>}[d]^{\gamma'_0} & 0 \\ 0 \ar[r] & \mathcal{F}_0 \ar[r] & \mathcal{G}_0 \ar[r] & \mathcal{H}_0 \ar[r] & 0 }\] However, this may not be possible with our initial choice of \(\mathcal{G}'\). From the displayed diagram we see the obstruction is exactly the composition \[\delta : \mathcal{F}'/f\mathcal{F}' \to \mathcal{G}'/f\mathcal{G}' \xrightarrow{\beta'_0} \mathcal{G}_0 \to \mathcal{H}_0\] Note that the restriction of \(\delta\) to \(U_0\) is zero by our choice of \(\mathcal{F}'\) and \(\mathcal{H}'\). Hence by Cohomology of Schemes, Lemma 01YB there exists an \(k > 0\) such that \(\delta\) vanishes on \(\mathfrak m^k \cdot (\mathcal{F}'/f\mathcal{F}')\). For \(n > k\) set \(\mathcal{G}'_n = \mathfrak m^n \mathcal{G}'\), \(\mathcal{F}'_n = \mathcal{G}'_n \cap \mathcal{F}'\), and \(\mathcal{H}'_n = \mathcal{G}'_n/\mathcal{F}'_n\). Observe that \(\beta'_0\) can be composed with \(\mathcal{G}'_n/f\mathcal{G}'_n \to \mathcal{G}'/f\mathcal{G}'\) to give a map \(\beta'_{n, 0} : \mathcal{G}'_n/f\mathcal{G}'_n \to \mathcal{G}_0\) as in Lemma 0F23. By Artin-Rees (Algebra, Lemma 00IN) we may choose \(n\) such that \(\mathcal{F}'_n \subset \mathfrak m^k \mathcal{F}'\). As above the maps \(f : \mathcal{F}'_n \to \mathcal{F}'_n\), \(f : \mathcal{G}'_n \to \mathcal{G}'_n\), and \(f : \mathcal{H}'_n \to \mathcal{H}'_n\) are injective and as above using the snake lemma we obtain a short exact sequence \[0 \to \mathcal{F}'_n/f\mathcal{F}'_n \to \mathcal{G}'_n/f\mathcal{G}'_n \to \mathcal{H}'_n/f\mathcal{H}'_n \to 0\] As above we have isomorphisms \(\alpha'_n : \mathcal{F}'_n|_U \to \mathcal{F}\), \(\beta'_n : \mathcal{G}'_n|_U \to \mathcal{G}\), and \(\gamma'_n : \mathcal{H}'_n|_U \to \mathcal{H}\). We consider the obstruction \[\delta_n : \mathcal{F}'_n/f\mathcal{F}'_n \to \mathcal{G}'_n/f\mathcal{G}'_n \xrightarrow{\beta'_{n, 0}} \mathcal{G}_0 \to \mathcal{H}_0\] as before. However, the commutative diagram \[\xymatrix{ \mathcal{F}'_n/f\mathcal{F}'_n \ar[r] \ar[d] & \mathcal{G}'_n/f\mathcal{G}'_n \ar[r]_{\beta'_{n, 0}} \ar[d] & \mathcal{G}_0 \ar[r] \ar[d] & \mathcal{H}_0 \ar[d] \\ \mathcal{F}'/f\mathcal{F}' \ar[r] & \mathcal{G}'/f\mathcal{G}' \ar[r]^{\beta'_0} & \mathcal{G}_0 \ar[r] & \mathcal{H}_0 }\] our choice of \(n\) and our observation about \(\delta\) show that \(\delta_n = 0\). This produces the desired maps \(\alpha'_{n, 0} : \mathcal{F}'_n/f\mathcal{F}'_n \to \mathcal{F}_0\), and \(\gamma'_{n, 0} : \mathcal{H}'_n/f\mathcal{H}'_n \to \mathcal{H}_0\). OK, so we may use \(\mathcal{F}'_n, \alpha'_n, \alpha'_{n, 0}\), \(\mathcal{G}'_n, \beta'_n, \beta'_{n, 0}\), and \(\mathcal{H}'_n, \gamma'_n, \gamma'_{n, 0}\) to compute \(\chi(\mathcal{F}, \mathcal{F}_0, \alpha)\), \(\chi(\mathcal{G}, \mathcal{G}_0, \beta)\), and \(\chi(\mathcal{H}, \mathcal{H}_0, \gamma)\). Now finally the lemma follows from an application of the snake lemma to \[\xymatrix{ 0 \ar[r] & \mathcal{F}'_n/f\mathcal{F}'_n \ar[r] \ar[d] & \mathcal{G}'_n/f\mathcal{G}'_n \ar[r] \ar[d] & \mathcal{H}'_n/f\mathcal{H}'_n \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_0 \ar[r] & \mathcal{G}_0 \ar[r] & \mathcal{H}_0 \ar[r] & 0 }\] and additivity of lengths (Algebra, Lemma 00IV).
Proposition
Let \((\mathcal{F}, \mathcal{F}_0, \alpha)\) be a coherent triple. Let \((\mathcal{L}, \mathcal{L}_0, \lambda)\) be an invertible coherent triple. If \(\mathcal{F} = 0\), then the function below is constant. If \(\mathcal{F} \not = 0\), then the function \[\mathbf{Z} \longrightarrow \mathbf{Z},\quad n \longmapsto \chi((\mathcal{F}, \mathcal{F}_0, \alpha) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n})\] is a polynomial of degree \(\leq \dim(\text{Supp}(\mathcal{F}))\).
More precisely, if \(\mathcal{F} = 0\), then the function is constant. If \(\mathcal{F}\) has finite support in \(U\), then the function is constant. If the support of \(\mathcal{F}\) in \(U\) has dimension \(1\), i.e., the closure of the support of \(\mathcal{F}\) in \(X\) has dimension \(2\), then the function is linear, etc.
Proof
We will prove this by induction on the dimension of the support of \(\mathcal{F}\).
The base case is when \(\mathcal{F} = 0\). Then either \(\mathcal{F}_0\) is zero or its support is \(\{\mathfrak m\}\). In this case we have \[(\mathcal{F}, \mathcal{F}_0, \alpha) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n} = (0, \mathcal{F}_0 \otimes \mathcal{L}_0^{\otimes n}, 0) \cong (0, \mathcal{F}_0, 0)\] Thus the function of the lemma is constant with value equal to the length of \(\mathcal{F}_0\).
Induction step. Assume the support of \(\mathcal{F}\) is nonempty. Let \(\mathcal{G}_0 \subset \mathcal{F}_0\) denote the submodule of sections supported on \(\{\mathfrak m\}\). Then we get a short exact sequence \[0 \to (0, \mathcal{G}_0, 0) \to (\mathcal{F}, \mathcal{F}_0, \alpha) \to (\mathcal{F}, \mathcal{F}_0/\mathcal{G}_0, \alpha) \to 0\] This sequence remains exact if we tensor by the invertible coherent triple \((\mathcal{L}, \mathcal{L}_0, \lambda)\), see discussion above. Thus by additivity of \(\chi\) (Lemma 0F26) and the base case explained above, it suffices to prove the induction step for \((\mathcal{F}, \mathcal{F}_0/\mathcal{G}_0, \alpha)\). In this way we see that we may assume \(\mathfrak m\) is not an associated point of \(\mathcal{F}_0\).
Let \(T = \text{Ass}(\mathcal{F}) \cup \text{Ass}(\mathcal{F}/f\mathcal{F})\). Since \(U\) is quasi-affine, we can find \(s \in \Gamma(U, \mathcal{L})\) which does not vanish at any \(u \in T\), see Properties, Lemma 0F20. After multiplying \(s\) by a suitable element of \(\mathfrak m\) we may assume \(\lambda(s \bmod f) = s_0|_{U_0}\) for some \(s_0 \in \Gamma(X_0, \mathcal{L}_0)\); details omitted. We obtain a morphism \[(s, s_0) : (\mathcal{O}_U, \mathcal{O}_{X_0}, 1) \longrightarrow (\mathcal{L}, \mathcal{L}_0, \lambda)\] in the category of coherent triples. Let \(\mathcal{G} = \Coker(s : \mathcal{F} \to \mathcal{F} \otimes \mathcal{L})\) and \(\mathcal{G}_0 = \Coker(s_0 : \mathcal{F}_0 \to \mathcal{F}_0 \otimes \mathcal{L}_0)\). Observe that \(s_0 : \mathcal{F}_0 \to \mathcal{F}_0 \otimes \mathcal{L}_0\) is injective as it is injective on \(U_0\) by our choice of \(s\) and as \(\mathfrak m\) isn’t an associated point of \(\mathcal{F}_0\). It follows that there exists an isomorphism \(\beta : \mathcal{G}/f\mathcal{G} \to \mathcal{G}_0|_{U_0}\) such that we obtain a short exact sequence \[0 \to (\mathcal{F}, \mathcal{F}_0, \alpha) \to (\mathcal{F}, \mathcal{F}_0, \alpha) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda) \to (\mathcal{G}, \mathcal{G}_0, \beta) \to 0\] By induction on the dimension of the support we know the proposition holds for the coherent triple \((\mathcal{G}, \mathcal{G}_0, \beta)\). Using the additivity of Lemma 0F26 we see that \[n \longmapsto \chi((\mathcal{F}, \mathcal{F}_0, \alpha) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n + 1}) - \chi((\mathcal{F}, \mathcal{F}_0, \alpha) \otimes (\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n})\] is a polynomial. We conclude by a variant of Algebra, Lemma 00JZ for functions defined for all integers (details omitted).
Lemma
Assume \(\text{depth}(A) \geq 3\) or equivalently \(\text{depth}(A/fA) \geq 2\). Let \((\mathcal{L}, \mathcal{L}_0, \lambda)\) be an invertible coherent triple. Then \[\chi(\mathcal{L}, \mathcal{L}_0, \lambda) = \text{length}_A \Coker(\Gamma(U, \mathcal{L}) \to \Gamma(U_0, \mathcal{L}_0))\] and in particular this is \(\geq 0\). Moreover, \(\chi(\mathcal{L}, \mathcal{L}_0, \lambda) = 0\) if and only if \(\mathcal{L} \cong \mathcal{O}_U\).
Proof
The equivalence of the depth conditions follows from Algebra, Lemma 090R. By the depth condition we see that \(\Gamma(U, \mathcal{O}_U) = A\) and \(\Gamma(U_0, \mathcal{O}_{U_0}) = A/fA\), see Dualizing Complexes, Lemma 0AVZ and Local Cohomology, Lemma 0BK0. Using Local Cohomology, Lemma 0BLT we find that \(M = \Gamma(U, \mathcal{L})\) is a finite \(A\)-module. This in turn implies \(\text{depth}(M) \geq 2\) for example by part (4) of Local Cohomology, Lemma 0BK0 or by Divisors, Lemma 0EY0. Also, we have \(\mathcal{L}_0 \cong \mathcal{O}_{X_0}\) as \(X_0\) is a local scheme. Hence we also see that \(M_0 = \Gamma(X_0, \mathcal{L}_0) = \Gamma(U_0, \mathcal{L}_0|_{U_0})\) and that this module is isomorphic to \(A/fA\).
By the above \(\mathcal{F}' = \widetilde{M}\) is a coherent \(\mathcal{O}_X\)-module whose restriction to \(U\) is isomorphic to \(\mathcal{L}\). The isomorphism \(\lambda : \mathcal{L}/f\mathcal{L} \to \mathcal{L}_0|_{U_0}\) determines a map \(M/fM \to M_0\) on global sections which is an isomorphism over \(U_0\). Since \(\text{depth}(M) \geq 2\) we see that \(H^0_\mathfrak m(M/fM) = 0\) and it follows that \(M/fM \to M_0\) is injective. Thus by definition \[\chi(\mathcal{L}, \mathcal{L}_0, \lambda) = \text{length}_A \Coker(M/fM \to M_0)\] which gives the first statement of the lemma.
Finally, if this length is \(0\), then \(M \to M_0\) is surjective. Hence we can find \(s \in M = \Gamma(U, \mathcal{L})\) mapping to a trivializing section of \(\mathcal{L}_0\). Consider the finite \(A\)-modules \(K\), \(Q\) defined by the exact sequence \[0 \to K \to A \xrightarrow{s} M \to Q \to 0\] The supports of \(K\) and \(Q\) do not meet \(U_0\) because \(s\) is nonzero at points of \(U_0\). Using Algebra, Lemma 00LX we see that \(\text{depth}(K) \geq 2\) (observe that \(As \subset M\) has \(\text{depth} \geq 1\) as a submodule of \(M\)). Thus the support of \(K\) if nonempty has dimension \(\geq 2\) by Algebra, Lemma 00LK. This contradicts \(\text{Supp}(M) \cap V(f) \subset \{\mathfrak m\}\) unless \(K = 0\). When \(K = 0\) we find that \(\text{depth}(Q) \geq 2\) and we conclude \(Q = 0\) as before. Hence \(A \cong M\) and \(\mathcal{L}\) is trivial.
Invertible modules on punctured spectra, I
In this section we prove some local Lefschetz theorems for the Picard group. Some of the ideas are taken from [Kollar-pic], [Bhatt-local], and [Kollar-map-pic].
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\) be a nonzerodivisor and assume that \(\text{depth}(A/fA) \geq 2\), or equivalently \(\text{depth}(A) \geq 3\). Let \(U\), resp. \(U_0\) be the punctured spectrum of \(A\), resp. \(A/fA\). The map \[\Pic(U) \to \Pic(U_0)\] is injective on torsion.
Proof
Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_U\)-module. Observe that \(\mathcal{L}\) maps to \(0\) in \(\Pic(U_0)\) if and only if we can extend \(\mathcal{L}\) to an invertible coherent triple \((\mathcal{L}, \mathcal{L}_0, \lambda)\) as in Section 0F22. By Proposition 0F27 the function \[n \longmapsto \chi((\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n})\] is a polynomial. By Lemma 0F28 the value of this polynomial is zero if and only if \(\mathcal{L}^{\otimes n}\) is trivial. Thus if \(\mathcal{L}\) is torsion, then this polynomial has infinitely many zeros, hence is identically zero, hence \(\mathcal{L}\) is trivial.
Proposition
Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\). Assume
\(A\) has a dualizing complex,
\(f\) is a nonzerodivisor,
\(\text{depth}(A/fA) \geq 2\), or equivalently \(\text{depth}(A) \geq 3\),
if \(f \in \mathfrak p \subset A\) is a prime ideal with \(\dim(A/\mathfrak p) = 2\), then \(\text{depth}(A_\mathfrak p) \geq 2\).
Let \(U\), resp. \(U_0\) be the punctured spectrum of \(A\), resp. \(A/fA\). The map \[\Pic(U) \to \Pic(U_0)\] is injective. Finally, if (1), (2), (3), \(A\) is \((S_2)\), and \(\dim(A) \geq 4\), then (4) holds.
Proof
Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_U\)-module. Observe that \(\mathcal{L}\) maps to \(0\) in \(\Pic(U_0)\) if and only if we can extend \(\mathcal{L}\) to an invertible coherent triple \((\mathcal{L}, \mathcal{L}_0, \lambda)\) as in Section 0F22. By Proposition 0F27 the function \[n \longmapsto \chi((\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n})\] is a polynomial \(P\). By Lemma 0F28 we have \(P(n) \geq 0\) for all \(n \in \mathbf{Z}\) with equality if and only if \(\mathcal{L}^{\otimes n}\) is trivial. In particular \(P(0) = 0\) and \(P\) is either identically zero and we win or \(P\) has even degree \(\geq 2\).
Set \(M = \Gamma(U, \mathcal{L})\) and \(M_0 = \Gamma(X_0, \mathcal{L}_0) = \Gamma(U_0, \mathcal{L}_0)\). Then \(M\) is a finite \(A\)-module of depth \(\geq 2\) and \(M_0 \cong A/fA\), see proof of Lemma 0F28. Note that \(H^2_\mathfrak m(M)\) is finite \(A\)-module by Local Cohomology, Lemma 0BPY and the fact that \(H^i_\mathfrak m(A) = 0\) for \(i = 0, 1, 2\) since \(\text{depth}(A) \geq 3\). Consider the short exact sequence \[0 \to M/fM \to M_0 \to Q \to 0\] Lemma 0F28 tells us \(Q\) has finite length equal to \(\chi(\mathcal{L}, \mathcal{L}_0, \lambda)\). We obtain \(Q = H^1_\mathfrak m(M/fM)\) and \(H^i_\mathfrak m(M/fM) = H^i_\mathfrak m(M_0) \cong H^i_\mathfrak m(A/fA)\) for \(i > 1\) from the long exact sequence of local cohomology associated to the displayed short exact sequence. Consider the long exact sequence of local cohomology associated to the sequence \(0 \to M \to M \to M/fM \to 0\). It starts with \[0 \to Q \to H^2_\mathfrak m(M) \to H^2_\mathfrak m(M) \to H^2_\mathfrak m(A/fA)\] Using additivity of lengths we see that \(\chi(\mathcal{L}, \mathcal{L}_0, \lambda)\) is equal to the length of the image of \(H^2_\mathfrak m(M) \to H^2_\mathfrak m(A/fA)\).
Let prove the lemma in a special case to elucidate the rest of the proof. Namely, assume for a moment that \(H^2_\mathfrak m(A/fA)\) is a finite length module. Then we would have \(P(1) \leq \text{length}_A H^2_\mathfrak m(A/fA)\). The exact same argument applied to \(\mathcal{L}^{\otimes n}\) shows that \(P(n) \leq \text{length}_A H^2_\mathfrak m(A/fA)\) for all \(n\). Thus \(P\) cannot have positive degree and we win. In the rest of the proof we will modify this argument to give a linear upper bound for \(P(n)\) which suffices.
Let us study the map \(H^2_\mathfrak m(M) \to H^2_\mathfrak m(M_0) \cong H^2_\mathfrak m(A/fA)\). Choose a normalized dualizing complex \(\omega_A^\bullet\) for \(A\). By local duality (Dualizing Complexes, Lemma 0AAK) this map is Matlis dual to the map \[\text{Ext}^{-2}_A(M, \omega_A^\bullet) \longleftarrow \text{Ext}^{-2}_A(M_0, \omega_A^\bullet)\] whose image therefore has the same (finite) length. The support (if nonempty) of the finite \(A\)-module \(\text{Ext}^{-2}_A(M_0, \omega_A^\bullet)\) consists of \(\mathfrak m\) and a finite number of primes \(\mathfrak p_1, \ldots, \mathfrak p_r\) containing \(f\) with \(\dim(A/\mathfrak p_i) = 1\). Namely, by Local Cohomology, Lemma 0DWZ the support is contained in the set of primes \(\mathfrak p \subset A\) with \(\text{depth}_{A_\mathfrak p}(M_{0, \mathfrak p}) + \dim(A/\mathfrak p) \leq 2\). Thus it suffices to show there is no prime \(\mathfrak p\) containing \(f\) with \(\dim(A/\mathfrak p) = 2\) and \(\text{depth}_{A_\mathfrak p}(M_{0, \mathfrak p}) = 0\). However, because \(M_{0, \mathfrak p} \cong (A/fA)_\mathfrak p\) this would give \(\text{depth}(A_\mathfrak p) = 1\) which contradicts assumption (4). Choose a section \(t \in \Gamma(U, \mathcal{L}^{\otimes -1})\) which does not vanish in the points \(\mathfrak p_1, \ldots, \mathfrak p_r\), see Properties, Lemma 0F20. Multiplication by \(t\) on global sections determines a map \(t : M \to A\) which defines an isomorphism \(M_{\mathfrak p_i} \to A_{\mathfrak p_i}\) for \(i = 1, \ldots, r\). Denote \(t_0 = t|_{U_0}\) the corresponding section of \(\Gamma(U_0, \mathcal{L}_0^{\otimes -1})\) which similarly determines a map \(t_0 : M_0 \to A/fA\) compatible with \(t\). We conclude that there is a commutative diagram \[\xymatrix{ \text{Ext}^{-2}_A(M, \omega_A^\bullet) & \text{Ext}^{-2}_A(M_0, \omega_A^\bullet) \ar[l] \\ \text{Ext}^{-2}_A(A, \omega_A^\bullet) \ar[u]^t & \text{Ext}^{-2}_A(A/fA, \omega_A^\bullet) \ar[l] \ar[u]_{t_0} }\] It follows that the length of the image of the top horizontal map is at most the length of \(\text{Ext}^{-2}_A(A/fA, \omega_A^\bullet)\) plus the length of the cokernel of \(t_0\).
However, if we replace \(\mathcal{L}\) by \(\mathcal{L}^n\) for \(n > 1\), then we can use \[t^n : M_n = \Gamma(U, \mathcal{L}^{\otimes n}) \longrightarrow \Gamma(U, \mathcal{O}_U) = A\] instead of \(t\). This replaces \(t_0\) by its \(n\)th power. Thus the length of the image of the map \(\text{Ext}^{-2}_A(M_n, \omega_A^\bullet) \leftarrow \text{Ext}^{-2}_A(M_{n, 0}, \omega_A^\bullet)\) is at most the length of \(\text{Ext}^{-2}_A(A/fA, \omega_A^\bullet)\) plus the length of the cokernel of \[t_0^n : \text{Ext}^{-2}_A(A/fA, \omega_A^\bullet) \longrightarrow \text{Ext}^{-2}_A(M_{n, 0}, \omega_A^\bullet)\] Via the isomorphism \(M_0 \cong A/fA\) the map \(t_0\) becomes \(g : A/fA \to A/fA\) for some \(g \in A/fA\) and via the corresponding isomorphisms \(M_{n, 0} \cong A/fA\) the map \(t_0^n\) becomes \(g^n : A/fA \to A/fA\). Thus the length of the cokernel above is the length of the quotient of \(\text{Ext}^{-2}_A(A/fA, \omega_A^\bullet)\) by \(g^n\). Since \(\text{Ext}^{-2}_A(A/fA, \omega_A^\bullet)\) is a finite \(A\)-module with support \(T\) of dimension \(1\) and since \(V(g) \cap T\) consists of the closed point by our choice of \(t\) this length grows linearly in \(n\) by Algebra, Lemma 00L8.
To finish the proof we prove the final assertion. Assume \(f \in \mathfrak m \subset A\) satisfies (1), (2), (3), \(A\) is \((S_2)\), and \(\dim(A) \geq 4\). Condition (1) implies \(A\) is catenary, see Dualizing Complexes, Lemma 0A80. Then \(\Spec(A)\) is equidimensional by Local Cohomology, Lemma 0FIW. Thus \(\dim(A_\mathfrak p) + \dim(A/\mathfrak p) \geq 4\) for every prime \(\mathfrak p\) of \(A\). Then \(\text{depth}(A_\mathfrak p) \geq \min(2, \dim(A_\mathfrak p)) \geq \min(2, 4 - \dim(A/\mathfrak p))\) and hence (4) holds.
Remark
In SGA2 we find the following result. Let \((A, \mathfrak m)\) be a Noetherian local ring. Let \(f \in \mathfrak m\). Assume \(A\) is a quotient of a regular ring, the element \(f\) is a nonzerodivisor, and
if \(\mathfrak p \subset A\) is a prime ideal with \(\dim(A/\mathfrak p) = 1\), then \(\text{depth}(A_\mathfrak p) \geq 2\), and
\(\text{depth}(A/fA) \geq 3\), or equivalently \(\text{depth}(A) \geq 4\).
Let \(U\), resp. \(U_0\) be the punctured spectrum of \(A\), resp. \(A/fA\). Then the map \[\Pic(U) \to \Pic(U_0)\] is injective. This is [SGA2, Exposee XI, Lemma 3.16]9. This result from SGA2 follows from Proposition 0F2B because
Invertible modules on punctured spectra, II
Next we turn to surjectivity in local Lefschetz for the Picard group. First to extend an invertible module on \(U_0\) to an open neighbourhood we have the following simple criterion.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring and \(f \in \mathfrak m\). Assume
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
\(H^1_\mathfrak m(A/fA)\) and \(H^2_\mathfrak m(A/fA)\) are finite \(A\)-modules, and
\(H^3_\mathfrak m(A/fA) = 0\)10.
Let \(U\), resp. \(U_0\) be the punctured spectrum of \(A\), resp. \(A/fA\). Then \[\colim_{U_0 \subset U' \subset U\text{ open}} \Pic(U') \longrightarrow \Pic(U_0)\] is surjective.
Proof
Let \(U_0 \subset U_n \subset U\) be the \(n\)th infinitesimal neighbourhood of \(U_0\). Observe that the ideal sheaf of \(U_n\) in \(U_{n + 1}\) is isomorphic to \(\mathcal{O}_{U_0}\) as \(U_0 \subset U\) is the principal closed subscheme cut out by the nonzerodivisor \(f\). Hence we have an exact sequence of abelian groups \[\Pic(U_{n + 1}) \to \Pic(U_n) \to H^2(U_0, \mathcal{O}_{U_0}) = H^3_\mathfrak m(A/fA) = 0\] see More on Morphisms, Lemma 0C6R. Thus every invertible \(\mathcal{O}_{U_0}\)-module is the restriction of an invertible coherent formal module, i.e., an invertible object of \(\textit{Coh}(U, f\mathcal{O}_U)\). We conclude by applying Lemma 0EKZ.
Remark
Let \((A, \mathfrak m)\) be a Noetherian local ring and \(f \in \mathfrak m\). The conclusion of Lemma 0F2D holds if we assume
\(A\) has a dualizing complex,
\(A\) is \(f\)-adically complete,
\(f\) is a nonzerodivisor,
one of the following is true
\(A_f\) is \((S_2)\) and for \(\mathfrak p \subset A\), \(f \not \in \mathfrak p\) minimal we have \(\dim(A/\mathfrak p) \geq 4\), or
if \(\mathfrak p \not \in V(f)\) and \(V(\mathfrak p) \cap V(f) \not = \{\mathfrak m\}\), then \(\text{depth}(A_\mathfrak p) + \dim(A/\mathfrak p) > 3\).
\(H^3_{\mathfrak m}(A/fA) = 0\).
The proof is exactly the same as the proof of Lemma 0F2D using Lemma 0EKY instead of Lemma 0EKZ. Two points need to be made here: (a) it seems hard to find examples where one knows \(H^3_{\mathfrak m}(A/fA) = 0\) without assuming \(\text{depth}(A/fA) \geq 4\), and (b) the proof of Lemma 0EKY is a good deal harder than the proof of Lemma 0EKZ.
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring and \(f \in \mathfrak m\). Assume
the conditions of Lemma 0F2D hold, and
for every maximal ideal \(\mathfrak p \subset A_f\) the punctured spectrum of \((A_f)_\mathfrak p\) has trivial Picard group.
Let \(U\), resp. \(U_0\) be the punctured spectrum of \(A\), resp. \(A/fA\). Then \[\Pic(U) \longrightarrow \Pic(U_0)\] is surjective.
Proof
Let \(\mathcal{L}_0 \in \Pic(U_0)\). By Lemma 0F2D there exists an open \(U_0 \subset U' \subset U\) and \(\mathcal{L}' \in \Pic(U')\) whose restriction to \(U_0\) is \(\mathcal{L}_0\). Since \(U' \supset U_0\) we see that \(U \setminus U'\) consists of points corresponding to prime ideals \(\mathfrak p_1, \ldots, \mathfrak p_n\) as in (2). By assumption we can find invertible modules \(\mathcal{L}'_i\) on \(\Spec(A_{\mathfrak p_i})\) agreeing with \(\mathcal{L}'\) over the punctured spectrum \(U' \times_U \Spec(A_{\mathfrak p_i})\) since trivial invertible modules always extend. By Limits, Lemma 0F21 applied \(n\) times we see that \(\mathcal{L}'\) extends to an invertible module on \(U\).
Lemma
Let \((A, \mathfrak m)\) be a Noetherian local ring of depth \(\geq 2\). Let \(A^\wedge\) be its completion. Let \(U\), resp. \(U^\wedge\) be the punctured spectrum of \(A\), resp. \(A^\wedge\). Then \(\Pic(U) \to \Pic(U^\wedge)\) is injective.
Proof
Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_U\)-module with pullback \(\mathcal{L}^\wedge\) on \(U^\wedge\). We have \(H^0(U, \mathcal{O}_U) = A\) by our assumption on depth and Dualizing Complexes, Lemma 0AVZ and Local Cohomology, Lemma 0BK0. Thus \(\mathcal{L}\) is trivial if and only if \(M = H^0(U, \mathcal{L})\) is isomorphic to \(A\) as an \(A\)-module. (Details omitted.) Since \(A \to A^\wedge\) is flat we have \(M \otimes_A A^\wedge = \Gamma(U^\wedge, \mathcal{L}^\wedge)\) by flat base change, see Cohomology of Schemes, Lemma 02KH. Finally, it is easy to see that \(M \cong A\) if and only if \(M \otimes_A A^\wedge \cong A^\wedge\).
Lemma
Let \((A, \mathfrak m)\) be a regular local ring. Then the Picard group of the punctured spectrum of \(A\) is trivial.
Proof
Combine Divisors, Lemma 0BD9 with More on Algebra, Lemma 0AG0.
Now we can bootstrap the earlier results to prove that Picard groups are trivial for punctured spectra of complete intersections of dimension \(\geq 4\). Recall that a Noetherian local ring is called a complete intersection if its completion is the quotient of a regular local ring by the ideal generated by a regular sequence. See the discussion in Divided Power Algebra, Section 09PY.
Proposition
Let \((A, \mathfrak m)\) be a Noetherian local ring. If \(A\) is a complete intersection of dimension \(\geq 4\), then the Picard group of the punctured spectrum of \(A\) is trivial.
Proof
By Lemma 0F2G we may assume that \(A\) is a complete local ring. By assumption we can write \(A = B/(f_1, \ldots, f_r)\) where \(B\) is a complete regular local ring and \(f_1, \ldots, f_r\) is a regular sequence. We will finish the proof by induction on \(r\). The base case is \(r = 0\) which follows from Lemma 0F2H.
Assume that \(A = B/(f_1, \ldots, f_r)\) and that the proposition holds for \(r - 1\). Set \(A' = B/(f_1, \ldots, f_{r - 1})\) and apply Lemma 0F2F to \(f_r \in A'\). This is permissible:
condition (1) of Lemma 0F2D holds because our local rings are complete,
condition (2) of Lemma 0F2D holds holds as \(f_1, \ldots, f_r\) is a regular sequence,
condition (3) and (4) of Lemma 0F2D hold as \(A = A'/f_r A'\) is Cohen-Macaulay of dimension \(\dim(A) \geq 4\),
condition (2) of Lemma 0F2F holds by induction hypothesis as \(\dim((A'_{f_r})_\mathfrak p) \geq 4\) for a maximal prime \(\mathfrak p\) of \(A'_{f_r}\) and as \((A'_{f_r})_\mathfrak p = B_\mathfrak q/(f_1, \ldots, f_{r - 1})\) for some prime ideal \(\mathfrak q \subset B\) and \(B_\mathfrak q\) is regular.
This finishes the proof.
Example
The dimension bound in Proposition 0F2I is sharp. For example the Picard group of the punctured spectrum of \(A = k[[x, y, z, w]]/(xy - zw)\) is nontrivial. Namely, the ideal \(I = (x, z)\) cuts out an effective Cartier divisor \(D\) on the punctured spectrum \(U\) of \(A\) as it is easy to see that \(I_x, I_y, I_z, I_w\) are invertible ideals in \(A_x, A_y, A_z, A_w\). But on the other hand, \(A/I\) has depth \(\geq 1\) (in fact \(2\)), hence \(I\) has depth \(\geq 2\) (in fact \(3\)), hence \(I = \Gamma(U, \mathcal{O}_U(-D))\). Thus if \(\mathcal{O}_U(-D)\) were trivial, then we’d have \(I \cong \Gamma(U, \mathcal{O}_U) = A\) which isn’t true as \(I\) isn’t generated by \(1\) element.
Example
Proposition 0F2I cannot be extended to quotients \[A = B/(f_1, \ldots, f_r)\] where \(B\) is regular and \(\dim(B) - r \geq 4\). In other words, the condition that \(f_1, \ldots, f_r\) be a regular sequence is (in general) needed for vanishing of the Picard group of the punctured spectrum of \(A\). Namely, let \(k\) be a field and set \[A = k[[a, b, x, y, z, u, v, w]]/(a^3, b^3, xa^2 + yab + zb^2, w^2)\] Observe that \(A = A_0[w]/(w^2)\) with \(A_0 = k[[a, b, x, y, z, u, v]]/(a^3, b^3, xa^2 + yab + zb^2)\). We will show below that \(A_0\) has depth \(2\). Denote \(U\) the punctured spectrum of \(A\) and \(U_0\) the punctured spectrum of \(A_0\). Observe there is a short exact sequence \(0 \to A_0 \to A \to A_0 \to 0\) where the first arrow is given by multiplication by \(w\). By More on Morphisms, Lemma 0C6R we find that there is an exact sequence \[H^0(U, \mathcal{O}_U^*) \to H^0(U_0, \mathcal{O}_{U_0}^*) \to H^1(U_0, \mathcal{O}_{U_0}) \to \Pic(U)\] Since the depth of \(A_0\) and hence \(A\) is \(2\) we see that \(H^0(U_0, \mathcal{O}_{U_0}) = A_0\) and \(H^0(U, \mathcal{O}_U) = A\) and that \(H^1(U_0, \mathcal{O}_{U_0})\) is nonzero, see Dualizing Complexes, Lemma 0AVZ and Local Cohomology, Lemma 0DWR. Thus the last arrow displayed above is nonzero and we conclude that \(\Pic(U)\) is nonzero.
To show that \(A_0\) has depth \(2\) it suffices to show that \(A_1 = k[[a, b, x, y, z]]/(a^3, b^3, xa^2 + yab + zb^2)\) has depth \(0\). This is true because \(a^2b^2\) maps to a nonzero element of \(A_1\) which is annihilated by each of the variables \(a, b, x, y, z\). For example \(ya^2b^2 = (yab)(ab) = - (xa^2 + zb^2)(ab) = -xa^3b - yab^3 = 0\) in \(A_1\). The other cases are similar.
Application to Lefschetz theorems
In this section we discuss the relation between coherent sheaves on a projective scheme \(P\) and coherent modules on formal completion along an ample divisor \(Q\).
Let \(k\) be a field. Let \(P\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_P\)-module. Let \(s \in \Gamma(P, \mathcal{L})\) be a section11 and let \(Q = Z(s)\) be the zero scheme, see Divisors, Definition 02OQ. For all \(n \geq 1\) we denote \(Q_n = Z(s^n)\) the \(n\)th infinitesimal neighbourhood of \(Q\). If \(\mathcal{F}\) is a coherent \(\mathcal{O}_P\)-module, then we denote \(\mathcal{F}_n = \mathcal{F}|_{Q_n}\) the restriction, i.e., the pullback of \(\mathcal{F}\) by the closed immersion \(Q_n \to P\).
Proposition
In the situation above assume there exists an integer \(\sigma\) such that for all points \(p \in P \setminus Q\) we have \[\text{depth}(\mathcal{F}_p) + \dim(\overline{\{p\}}) > \sigma\] Then the map \[H^i(P, \mathcal{F}) \longrightarrow \lim H^i(Q_n, \mathcal{F}_n)\] is an isomorphism for \(0 \leq i < \sigma\).
Proof
We will use More on Morphisms, Lemma 0EKI and we will use the notation used and results found More on Morphisms, Section 0EKF without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case \(A = \bigoplus_{m \geq 0} \Gamma(P, \mathcal{L}^{\otimes m})\) is a finite type \(k\)-algebra all of whose graded parts are finite dimensional \(k\)-vector spaces, see Cohomology of Schemes, Lemma 0B5T.
We may and do think of \(s\) as an element \(f \in A_1 \subset A\), i.e., a homogeneous element of degree \(1\) of \(A\). Denote \(Y = V(f) \subset X\) the closed subscheme defined by \(f\). Then \(U \cap Y = (\pi|_U)^{-1}(Q)\) scheme theoretically. Recall the notation \(\mathcal{F}_U = \pi^*\mathcal{F}|_U = (\pi|_U)^*\mathcal{F}\). This is a coherent \(\mathcal{O}_U\)-module. Choose a finite \(A\)-module \(M\) such that \(\mathcal{F}_U = \widetilde{M}|_U\) (for existence see Local Cohomology, Lemma 0BK0). We claim that \(H^i_Z(M)\) is annihilated by a power of \(f\) for \(i \leq \sigma + 1\).
To prove the claim we will apply Local Cohomology, Proposition 0EFC. Translating into geometry we see that it suffices to prove for \(u \in U\), \(u \not \in Y\) and \(z \in \overline{\{u\}} \cap Z\) that \[\text{depth}(\mathcal{F}_{U, u}) + \dim(\mathcal{O}_{\overline{\{u\}}, z}) > \sigma + 1\] This requires only a small amount of thought.
Observe that \(Z = \Spec(A_0)\) is a finite set of closed points of \(X\) because \(A_0\) is a finite dimensional \(k\)-algebra. (The reader who would like \(Z\) to be a singleton can replace the finite \(k\)-algebra \(A_0\) by \(k\); it won’t affect anything else in the proof.)
The morphism \(\pi : L \to P\) and its restriction \(\pi|_U : U \to P\) are smooth of relative dimension \(1\). Let \(u \in U\), \(u \not \in Y\) and \(z \in \overline{\{u\}} \cap Z\). Let \(p = \pi(u) \in P \setminus Q\) be its image. Then either \(u\) is a generic point of the fibre of \(\pi\) over \(p\) or a closed point of the fibre. If \(u\) is a generic point of the fibre, then \(\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_p)\) and \(\dim(\overline{\{u\}}) = \dim(\overline{\{p\}}) + 1\). If \(u\) is a closed point of the fibre, then \(\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_p) + 1\) and \(\dim(\overline{\{u\}}) = \dim(\overline{\{p\}})\). In both cases we have \(\dim(\overline{\{u\}}) = \dim(\mathcal{O}_{\overline{\{u\}}, z})\) because every point of \(Z\) is closed. Thus the desired inequality follows from the assumption in the statement of the lemma.
Let \(A'\) be the \(f\)-adic completion of \(A\). So \(A \to A'\) is flat by Algebra, Lemma 00MB. Denote \(U' \subset X' = \Spec(A')\) the inverse image of \(U\) and similarly for \(Y'\) and \(Z'\). Let \(\mathcal{F}'\) on \(U'\) be the pullback of \(\mathcal{F}_U\) and let \(M' = M \otimes_A A'\). By flat base change for local cohomology (Local Cohomology, Lemma 0EF5) we have \[H^i_{Z'}(M') = H^i_Z(M) \otimes_A A'\] and we find that for \(i \leq \sigma + 1\) these are annihilated by a power of \(f\). Consider the diagram \[\xymatrix{ & H^i(U, \mathcal{F}_U) \ar[ld] \ar[d] \ar[r] & \lim H^i(U, \mathcal{F}_U/f^n\mathcal{F}_U) \ar@{=}[d] \\ H^i(U, \mathcal{F}_U) \otimes_A A' \ar@{=}[r] & H^i(U', \mathcal{F}') \ar[r] & \lim H^i(U', \mathcal{F}'/f^n\mathcal{F}') }\] The lower horizontal arrow is an isomorphism for \(i < \sigma\) by Lemma 0EKM and the torsion property we just proved. The horizontal equal sign is flat base change (Cohomology of Schemes, Lemma 02KH) and the vertical equal sign is because \(U \cap Y\) and \(U' \cap Y'\) as well as their \(n\)th infinitesimal neighbourhoods are mapped isomorphically onto each other (as we are completing with respect to \(f\)).
Applying More on Morphisms, Equation (0EKH) we have compatible direct sum decompositions \[\lim H^i(U, \mathcal{F}_U/f^n\mathcal{F}_U) = \lim \left( \bigoplus\nolimits_{m \in \mathbf{Z}} H^i(Q_n, \mathcal{F}_n \otimes \mathcal{L}^{\otimes m}) \right)\] and \[H^i(U, \mathcal{F}_U) = \bigoplus\nolimits_{m \in \mathbf{Z}} H^i(P, \mathcal{F} \otimes \mathcal{L}^{\otimes m})\] Thus we conclude by Algebra, Lemma 0EKD.
Lemma
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\). Let \(Y = Z(s)\) be the zero scheme of \(s\) with \(n\)th infinitesimal neighbourhood \(Y_n = Z(s^n)\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Assume that for all \(x \in X \setminus Y\) we have \[\text{depth}(\mathcal{F}_x) + \dim(\overline{\{x\}}) > 1\] Then \(\Gamma(V, \mathcal{F}) \to \lim \Gamma(Y_n, \mathcal{F}|_{Y_n})\) is an isomorphism for any open subscheme \(V \subset X\) containing \(Y\).
Proof
By Proposition 0EL1 this is true for \(V = X\). Thus it suffices to show that the map \(\Gamma(V, \mathcal{F}) \to \lim \Gamma(Y_n, \mathcal{F}|_{Y_n})\) is injective. If \(\sigma \in \Gamma(V, \mathcal{F})\) maps to zero, then its support is disjoint from \(Y\) (details omitted; hint: use Krull’s intersection theorem). Then the closure \(T \subset X\) of \(\text{Supp}(\sigma)\) is disjoint from \(Y\). Whence \(T\) is proper over \(k\) (being closed in \(X\)) and affine (being closed in the affine scheme \(X \setminus Y\), see Morphisms, Lemma 0EKE) and hence finite over \(k\) (Morphisms, Lemma 01WN). Thus \(T\) is a finite set of closed points of \(X\). Thus \(\text{depth}(\mathcal{F}_x) \geq 2\) is at least \(1\) for \(x \in T\) by our assumption. We conclude that \(\Gamma(V, \mathcal{F}) \to \Gamma(V \setminus T, \mathcal{F})\) is injective and \(\sigma = 0\) as desired.
Example
Let \(k\) be a field and let \(X\) be a proper variety over \(k\). Let \(Y \subset X\) be an effective Cartier divisor such that \(\mathcal{O}_X(Y)\) is ample and denote \(Y_n\) its \(n\)th infinitesimal neighbourhood. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. Here are some special cases of Proposition 0EL1.
If \(X\) is a curve, we don’t learn anything.
If \(X\) is a Cohen-Macaulay (for example normal) surface, then \[H^0(X, \mathcal{E}) \to \lim H^0(Y_n, \mathcal{E}|_{Y_n})\] is an isomorphism.
If \(X\) is a Cohen-Macaulay threefold, then \[H^0(X, \mathcal{E}) \to \lim H^0(Y_n, \mathcal{E}|_{Y_n}) \quad\text{and}\quad H^1(X, \mathcal{E}) \to \lim H^1(Y_n, \mathcal{E}|_{Y_n})\] are isomorphisms.
Presumably the pattern is clear. If \(X\) is a normal threefold, then we can conclude the result for \(H^0\) but not for \(H^1\).
Before we prove the next main result, we need a lemma.
Lemma
In Situation 0EHC let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(U, I\mathcal{O}_U)\). Assume
\(A\) is a graded ring, \(\mathfrak a = A_+\), and \(I\) is a homogeneous ideal,
\((\mathcal{F}_n) = (\widetilde{M_n}|_U)\) where \((M_n)\) is an inverse system of graded \(A\)-modules, and
\((\mathcal{F}_n)\) extends canonically to \(X\).
Then there is a finite graded \(A\)-module \(N\) such that
the inverse systems \((N/I^nN)\) and \((M_n)\) are pro-isomorphic in the category of graded \(A\)-modules modulo \(A_+\)-power torsion modules, and
\((\mathcal{F}_n)\) is the completion of of the coherent module associated to \(N\).
Proof
Let \((\mathcal{G}_n)\) be the canonical extension as in Lemma 0EIR. The grading on \(A\) and \(M_n\) determines an action \[a : \mathbf{G}_m \times X \longrightarrow X\] of the group scheme \(\mathbf{G}_m\) on \(X\) such that \((\widetilde{M_n})\) becomes an inverse system of \(\mathbf{G}_m\)-equivariant quasi-coherent \(\mathcal{O}_X\)-modules, see Groupoids, Example 0EKJ. Since \(\mathfrak a\) and \(I\) are homogeneous ideals the closed subschemes \(Z\), \(Y\) and the open subscheme \(U\) are \(\mathbf{G}_m\)-invariant closed and open subschemes. The restriction \((\mathcal{F}_n)\) of \((\widetilde{M_n})\) is an inverse system of \(\mathbf{G}_m\)-equivariant coherent \(\mathcal{O}_U\)-modules. In other words, \((\mathcal{F}_n)\) is a \(\mathbf{G}_m\)-equivariant coherent formal module, in the sense that there is an isomorphism \[\alpha : (a^*\mathcal{F}_n) \longrightarrow (p^*\mathcal{F}_n)\] over \(\mathbf{G}_m \times U\) satisfying a suitable cocycle condition. Since \(a\) and \(p\) are flat morphisms of affine schemes, by Lemma 0EIS we conclude that there exists a unique isomorphism \[\beta : (a^*\mathcal{G}_n) \longrightarrow (p^*\mathcal{G}_n)\] over \(\mathbf{G}_m \times X\) restricting to \(\alpha\) on \(\mathbf{G}_m \times U\). The uniqueness guarantees that \(\beta\) satisfies the corresponding cocycle condition. In this way each \(\mathcal{G}_n\) becomes a \(\mathbf{G}_m\)-equivariant coherent \(\mathcal{O}_X\)-module in a manner compatible with transition maps.
By Groupoids, Lemma 0EKL we see that \(\mathcal{G}_n\) with its \(\mathbf{G}_m\)-equivariant structure corresponds to a graded \(A\)-module \(N_n\). The transition maps \(N_{n + 1} \to N_n\) are graded module maps. Note that \(N_n\) is a finite \(A\)-module and \(N_n = N_{n + 1}/I^n N_{n + 1}\) because \((\mathcal{G}_n)\) is an object of \(\textit{Coh}(X, I\mathcal{O}_X)\). Let \(N\) be the finite graded \(A\)-module foud in Algebra, Lemma 0EKC. Then \(N_n = N/I^nN\), whence \((\mathcal{G}_n)\) is the completion of the coherent module associated to \(N\), and a fortiori we see that (b) is true.
To see (a) we have to unwind the situation described above a bit more. First, observe that the kernel and cokernel of \(M_n \to H^0(U, \mathcal{F}_n)\) is \(A_+\)-power torsion (Local Cohomology, Lemma 0BK0). Observe that \(H^0(U, \mathcal{F}_n)\) comes with a natural grading such that these maps and the transition maps of the system are graded \(A\)-module map; for example we can use that \((U \to X)_*\mathcal{F}_n\) is a \(\mathbf{G}_m\)-equivariant module on \(X\) and use Groupoids, Lemma 0EKL. Next, recall that \((N_n)\) and \((H^0(U, \mathcal{F}_n))\) are pro-isomorphic by Definition 0EIP and Lemma 0EIR. We omit the verification that the maps defining this pro-isomorphism are graded module maps. Thus \((N_n)\) and \((M_n)\) are pro-isomorphic in the category of graded \(A\)-modules modulo \(A_+\)-power torsion modules.
Let \(k\) be a field. Let \(P\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_P\)-module. Let \(s \in \Gamma(P, \mathcal{L})\) be a section and let \(Q = Z(s)\) be the zero scheme, see Divisors, Definition 02OQ. Let \(\mathcal{I} \subset \mathcal{O}_P\) be the ideal sheaf of \(Q\). We will use \(\textit{Coh}(P, \mathcal{I})\) to denote the category of coherent formal modules introduced in Cohomology of Schemes, Section 0EHN.
Proposition
In the situation above let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(P, \mathcal{I})\). Assume for all \(q \in Q\) and for all primes \(\mathfrak p \in \mathcal{O}_{P, q}^\wedge\), \(\mathfrak p \not \in V(\mathcal{I}_q^\wedge)\) we have \[\text{depth}((\mathcal{F}_q^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{P, q}^\wedge/\mathfrak p) + \dim(\overline{\{q\}}) > 2\] Then \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_P\)-module.
Proof
By Cohomology of Schemes, Lemma 0889 to prove the lemma, we may replace \((\mathcal{F}_n)\) by an object differing from it by \(\mathcal{I}\)-torsion (see below for more precision). Let \(T' = \{q \in Q \mid \dim(\overline{\{q\}}) = 0\}\) and \(T = \{q \in Q \mid \dim(\overline{\{q\}}) \leq 1\}\). The assumption in the proposition is exactly that \(Q \subset P\), \((\mathcal{F}_n)\), and \(T' \subset T \subset Q\) satisfy the conditions of Lemma 0EJF with \(d = 1\); besides trivial manipulations of inequalities, use that \(V(\mathfrak p) \cap V(\mathcal{I}^\wedge_y) = \{\mathfrak m^\wedge_y\} \Leftrightarrow \dim(\mathcal{O}_{P, q}^\wedge/\mathfrak p) = 1\) as \(\mathcal{I}_y^\wedge\) is generated by \(1\) element. Combining these two remarks, we may replace \((\mathcal{F}_n)\) by the object \((\mathcal{H}_n)\) of \(\textit{Coh}(P, \mathcal{I})\) found in Lemma 0EJF. Thus we may and do assume \((\mathcal{F}_n)\) is pro-isomorphic to an inverse system \((\mathcal{F}_n'')\) of coherent \(\mathcal{O}_P\)-modules such that \(\text{depth}(\mathcal{F}''_{n, q}) + \dim(\overline{\{q\}}) \geq 2\) for all \(q \in Q\).
We will use More on Morphisms, Lemma 0EKI and we will use the notation used and results found More on Morphisms, Section 0EKF without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case \(A = \bigoplus_{m \geq 0} \Gamma(P, \mathcal{L}^{\otimes m})\) is a finite type \(k\)-algebra all of whose graded parts are finite dimensional \(k\)-vector spaces, see Cohomology of Schemes, Lemma 0B5T.
By Cohomology of Schemes, Lemma 0887 the pull back by \(\pi|_U : U \to P\) is an object \((\pi|_U^*\mathcal{F}_n)\) of \(\textit{Coh}(U, f\mathcal{O}_U)\) which is pro-isomorphic to the inverse system \((\pi|_U^*\mathcal{F}_n'')\) of coherent \(\mathcal{O}_U\)-modules. We claim \[\text{depth}(\pi|_U^*\mathcal{F}''_{n, y}) + \delta_Z^Y(y) \geq 3\] for all \(y \in U \cap Y\). Since all the points of \(Z\) are closed, we see that \(\delta_Z^Y(y) \geq \dim(\overline{\{y\}})\) for all \(y \in U \cap Y\), see Lemma 0EIY. Let \(q \in Q\) be the image of \(y\). Since the morphism \(\pi : U \to P\) is smooth of relative dimension \(1\) we see that either \(y\) is a closed point of a fibre of \(\pi\) or a generic point. Thus we see that \[\text{depth}(\pi^*\mathcal{F}''_{n, y}) + \delta_Z^Y(y) \geq \text{depth}(\pi^*\mathcal{F}''_{n, y}) + \dim(\overline{\{y\}}) = \text{depth}(\mathcal{F}''_{n, q}) + \dim(\overline{\{q\}}) + 1\] because either the depth goes up by \(1\) or the dimension. This proves the claim.
By Lemma 0EJI we conclude that \((\pi|_U^*\mathcal{F}_n)\) canonically extends to \(X\). Observe that \[M_n = \Gamma(U, \pi|_U^*\mathcal{F}_n) = \bigoplus\nolimits_{m \in \mathbf{Z}} \Gamma(P, \mathcal{F}_n \otimes_{\mathcal{O}_P} \mathcal{L}^{\otimes m})\] is canonically a graded \(A\)-module, see More on Morphisms, Equation (0EKH). By Properties, Lemma 0EHM we have \(\pi|_U^*\mathcal{F}_n = \widetilde{M_n}|_U\). Thus we may apply Lemma 0EL4 to find a finite graded \(A\)-module \(N\) such that \((M_n)\) and \((N/I^nN)\) are pro-isomorphic in the category of graded \(A\)-modules modulo \(A_+\)-torsion modules. Let \(\mathcal{F}\) be the coherent \(\mathcal{O}_P\)-module associated to \(N\), see Cohomology of Schemes, Proposition 0BXF. The same proposition tells us that \((\mathcal{F}/\mathcal{I}^n\mathcal{F})\) is pro-isomorphic to \((\mathcal{F}_n)\). Since both are objects of \(\textit{Coh}(P, \mathcal{I})\) we win by Lemma 0EIQ.
Example
Let \(k\) be a field and let \(X\) be a proper variety over \(k\). Let \(Y \subset X\) be an effective Cartier divisor such that \(\mathcal{O}_X(Y)\) is ample and denote \(\mathcal{I} \subset \mathcal{O}_X\) the corresponding sheaf of ideals. Let \((\mathcal{E}_n)\) an object of \(\textit{Coh}(X, \mathcal{I})\) with \(\mathcal{E}_n\) finite locally free. Here are some special cases of Proposition 0EL5.
If \(X\) is a curve or a surface, we don’t learn anything.
If \(X\) is a Cohen-Macaulay threefold, then \((\mathcal{E}_n)\) is the completion of a coherent \(\mathcal{O}_X\)-module \(\mathcal{E}\).
More generally, if \(\dim(X) \geq 3\) and \(X\) is \((S_3)\), then \((\mathcal{E}_n)\) is the completion of a coherent \(\mathcal{O}_X\)-module \(\mathcal{E}\).
Of course, if \(\mathcal{E}\) exists, then \(\mathcal{E}\) is finite locally free in an open neighbourhood of \(Y\).
Proposition
Let \(k\) be a field. Let \(X\) be a proper scheme over \(k\). Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module and let \(s \in \Gamma(X, \mathcal{L})\). Let \(Y = Z(s)\) be the zero scheme of \(s\) and denote \(\mathcal{I} \subset \mathcal{O}_X\) the corresponding sheaf of ideals. Let \(\mathcal{V}\) be the set of open subschemes of \(X\) containing \(Y\) ordered by reverse inclusion. Assume that for all \(x \in X \setminus Y\) we have \[\text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2\] Then the completion functor \[\colim_\mathcal{V} \textit{Coh}(\mathcal{O}_V) \longrightarrow \textit{Coh}(X, \mathcal{I})\] is an equivalence on the full subcategories of finite locally free objects.
Proof
To prove fully faithfulness it suffices to prove that \[\colim_\mathcal{V} \Gamma(V, \mathcal{L}^{\otimes m}) \longrightarrow \lim \Gamma(Y_n, \mathcal{L}^{\otimes m}|_{Y_n})\] is an isomorphism for all \(m\), see Lemma 0EK2. This follows from Lemma 0EL2.
Essential surjectivity. Let \((\mathcal{F}_n)\) be a finite locally free object of \(\textit{Coh}(X, \mathcal{I})\). Then for \(y \in Y\) we have \(\mathcal{F}_y^\wedge = \lim \mathcal{F}_{n, y}\) is is a finite free \(\mathcal{O}_{X, y}^\wedge\)-module. Let \(\mathfrak p \subset \mathcal{O}_{X, y}^\wedge\) be a prime with \(\mathfrak p \not \in V(\mathcal{I}_y^\wedge)\). Then \(\mathfrak p\) lies over a prime \(\mathfrak p_0 \subset \mathcal{O}_{X, y}\) which corresponds to a specialization \(x \leadsto y\) with \(x \not \in Y\). By Local Cohomology, Lemma 0EHW and some dimension theory (see Varieties, Section 06LF) we have \[\text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) = \text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) - \dim(\overline{\{y\}})\] Thus our assumptions imply the assumptions of Proposition 0EL5 are satisfied and we find that \((\mathcal{F}_n)\) is the completion of a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\). It then follows that \(\mathcal{F}_y\) is finite free for all \(y \in Y\) and hence \(\mathcal{F}\) is finite locally free in an open neighbourhood \(V\) of \(Y\). This finishes the proof.
For example \(H^q(K)\) for \(K\) pseudo-coherent on our locally Noetherian \(X\).↩︎
Our method forces this additional condition. We will return to this (insert future reference).↩︎
For example if \(M\) satisfies Serre’s condition \((S_s)\) on the complement of \(V(I) \cup T\).↩︎
In the sense that the difference of the maximal and minimal values on \(V(\mathfrak a)\) of a dimension function on \(\Spec(A)\) is at most \(2\).↩︎
In the sense that the difference of the maximal and minimal values on \(V(\mathfrak a)\) of a dimension function on \(\Spec(A)\) is at most \(2\).↩︎
For example if \(A\) is a domain.↩︎
Choose homogeneous generators of the form \(\delta_{n_j}(p_j)\) for the displayed module. Then if \(k = \max(n_j)\) we find that for \(n \geq k\) and any \(p \in P_n\) we can find \(a_j \in A\) such that \(p - \sum a_j f^{n - n_j} p_j\) is in the kernel of \(\delta_n\) and hence in the image of \(P_{n'}\) for all \(n' \geq n\). Thus \(\Im(P_n \to P_{n - k}) = \Im(P_{n'} \to P_{n - k})\) for all \(n' \geq n\).↩︎
Namely, it isn’t necessarily the case that \(f\) is injective on \(\mathcal{F} \otimes_{\mathcal{O}_U} \mathcal{G}\).↩︎
Condition (a) follows from condition (b), see Algebra, Lemma 0FCC.↩︎
Observe that (3) and (4) hold if \(\text{depth}(A/fA) \geq 4\), or equivalently \(\text{depth}(A) \geq 5\).↩︎
We do not require \(s\) to be a regular section. Correspondingly, \(Q\) is only a locally principal closed subscheme of \(P\) and not necessarily an effective Cartier divisor.↩︎