Cohomology of Schemes

Introduction

In this chapter we first prove a number of results on the cohomology of quasi-coherent sheaves. A fundamental reference is [EGA]. Having done this we will elaborate on cohomology of coherent sheaves in the Noetherian setting. See [FAC].

Čech cohomology of quasi-coherent sheaves

Let \(X\) be a scheme. Let \(U \subset X\) be an affine open. Recall that a standard open covering of \(U\) is a covering of the form \(\mathcal{U} : U = \bigcup_{i = 1}^n D(f_i)\) where \(f_1, \ldots, f_n \in \Gamma(U, \mathcal{O}_X)\) generate the unit ideal, see Schemes, Definition 01HT.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{U} : U = \bigcup_{i = 1}^n D(f_i)\) be a standard open covering of an affine open of \(X\). Then \(\check{H}^p(\mathcal{U}, \mathcal{F}) = 0\) for all \(p > 0\).

Proof

Write \(U = \Spec(A)\) for some ring \(A\). In other words, \(f_1, \ldots, f_n\) are elements of \(A\) which generate the unit ideal of \(A\). Write \(\mathcal{F}|_U = \widetilde{M}\) for some \(A\)-module \(M\). Clearly the Čech complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) is identified with the complex \[\prod\nolimits_{i_0} M_{f_{i_0}} \to \prod\nolimits_{i_0i_1} M_{f_{i_0}f_{i_1}} \to \prod\nolimits_{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \to \ldots\] We are asked to show that the extended complex [01XA]\[\begin{equation} 0 \to M \to \prod\nolimits_{i_0} M_{f_{i_0}} \to \prod\nolimits_{i_0i_1} M_{f_{i_0}f_{i_1}} \to \prod\nolimits_{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \to \ldots \end{equation}\] (whose truncation we have studied in Algebra, Lemma 00EK) is exact. It suffices to show that (01XA) is exact after localizing at a prime \(\mathfrak p\), see Algebra, Lemma 00HN. In fact we will show that the extended complex localized at \(\mathfrak p\) is homotopic to zero.

There exists an index \(i\) such that \(f_i \not \in \mathfrak p\). Choose and fix such an element \(i_{\text{fix}}\). Note that \(M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}\). Similarly for a localization at a product \(f_{i_0} \ldots f_{i_p}\) and \(\mathfrak p\) we can drop any \(f_{i_j}\) for which \(i_j = i_{\text{fix}}\). Let us define a homotopy \[h : \prod\nolimits_{i_0 \ldots i_{p + 1}} M_{f_{i_0} \ldots f_{i_{p + 1}}, \mathfrak p} \longrightarrow \prod\nolimits_{i_0 \ldots i_p} M_{f_{i_0} \ldots f_{i_p}, \mathfrak p}\] by the rule \[h(s)_{i_0 \ldots i_p} = s_{i_{\text{fix}} i_0 \ldots i_p}\] (This is “dual” to the homotopy in the proof of Cohomology, Lemma 01EM.) In other words, \(h : \prod_{i_0} M_{f_{i_0}, \mathfrak p} \to M_\mathfrak p\) is projection onto the factor \(M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}\) and in general the map \(h\) equal projection onto the factors \(M_{f_{i_{\text{fix}}} f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p} = M_{f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p}\). We compute \[\begin{align*} (dh + hd)(s)_{i_0 \ldots i_p} & = \sum\nolimits_{j = 0}^p (-1)^j h(s)_{i_0 \ldots \hat i_j \ldots i_p} + d(s)_{i_{\text{fix}} i_0 \ldots i_p}\\ & = \sum\nolimits_{j = 0}^p (-1)^j s_{i_{\text{fix}} i_0 \ldots \hat i_j \ldots i_p} + s_{i_0 \ldots i_p} + \sum\nolimits_{j = 0}^p (-1)^{j + 1} s_{i_{\text{fix}} i_0 \ldots \hat i_j \ldots i_p} \\ & = s_{i_0 \ldots i_p} \end{align*}\] This proves the identity map is homotopic to zero as desired.

The following lemma says in particular that for any affine scheme \(X\) and any quasi-coherent sheaf \(\mathcal{F}\) on \(X\) we have \[H^p(X, \mathcal{F}) = 0\] for all \(p > 0\).

Lemma

Let \(X\) be a scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. For any affine open \(U \subset X\) we have \(H^p(U, \mathcal{F}) = 0\) for all \(p > 0\).

Proof

We are going to apply Cohomology, Lemma 01EW. As our basis \(\mathcal{B}\) for the topology of \(X\) we are going to use the affine opens of \(X\). As our set \(\text{Cov}\) of open coverings we are going to use the standard open coverings of affine opens of \(X\). Next we check that conditions (1), (2) and (3) of Cohomology, Lemma 01EW hold. Note that the intersection of standard opens in an affine is another standard open. Hence property (1) holds. The coverings form a cofinal system of open coverings of any element of \(\mathcal{B}\), see Schemes, Lemma 01HS. Hence (2) holds. Finally, condition (3) of the lemma follows from Lemma 01X9.

Here is a relative version of the vanishing of cohomology of quasi-coherent sheaves on affines.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(f\) is affine then \(R^if_*\mathcal{F} = 0\) for all \(i > 0\).

Proof

According to Cohomology, Lemma 01E4 the sheaf \(R^if_*\mathcal{F}\) is the sheaf associated to the presheaf \(V \mapsto H^i(f^{-1}(V), \mathcal{F}|_{f^{-1}(V)})\). By assumption, whenever \(V\) is affine we have that \(f^{-1}(V)\) is affine, see Morphisms, Definition 01S6. By Lemma 01XB we conclude that \(H^i(f^{-1}(V), \mathcal{F}|_{f^{-1}(V)}) = 0\) whenever \(V\) is affine. Since \(S\) has a basis consisting of affine opens we win.

Lemma

Let \(f : X \to S\) be an affine morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(H^i(X, \mathcal{F}) = H^i(S, f_*\mathcal{F})\) for all \(i \geq 0\).

Proof

Follows from Lemma 01XC and the Leray spectral sequence. See Cohomology, Lemma 01F4.

The following two lemmas explain when Čech cohomology can be used to compute cohomology of quasi-coherent modules.

Lemma

Let \(X\) be a scheme. The following are equivalent

1. \(X\) has affine diagonal \(\Delta : X \to X \times X\),

2. for \(U, V \subset X\) affine open, the intersection \(U \cap V\) is affine, and

3. there exists an open covering \(\mathcal{U} : X = \bigcup_{i \in I} U_i\) such that \(U_{i_0 \ldots i_p}\) is affine open for all \(p \ge 0\) and all \(i_0, \ldots, i_p \in I\).

In particular this holds if \(X\) is separated.

Proof

Assume \(X\) has affine diagonal. Let \(U, V \subset X\) be affine opens. Then \(U \cap V = \Delta^{-1}(U \times V)\) is affine. Thus (2) holds. It is immediate that (2) implies (3). Conversely, if there is a covering of \(X\) as in (3), then \(X \times X = \bigcup U_i \times U_{i'}\) is an affine open covering, and we see that \(\Delta^{-1}(U_i \times U_{i'}) = U_i \cap U_{i'}\) is affine. Then \(\Delta\) is an affine morphism by Morphisms, Lemma 01S8. The final assertion follows from Schemes, Lemma 01KP.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{U} : X = \bigcup_{i \in I} U_i\) be an open covering such that \(U_{i_0 \ldots i_p}\) is affine open for all \(p \ge 0\) and all \(i_0, \ldots, i_p \in I\). In this case for any quasi-coherent sheaf \(\mathcal{F}\) we have \[\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F})\] as \(\Gamma(X, \mathcal{O}_X)\)-modules for all \(p\).

Proof

In view of Lemma 01XB this is a special case of Cohomology, Lemma 01ET.

Vanishing of cohomology

We have seen that on an affine scheme the higher cohomology groups of any quasi-coherent sheaf vanish (Lemma 01XB). It turns out that this also characterizes affine schemes. We give two versions.

Lemma

Let \(X\) be a scheme. Assume that

1. \(X\) is quasi-compact,

2. for every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) we have \(H^1(X, \mathcal{I}) = 0\).

Then \(X\) is affine.

Proof

Let \(x \in X\) be a closed point. Let \(U \subset X\) be an affine open neighbourhood of \(x\). Write \(U = \Spec(A)\) and let \(\mathfrak m \subset A\) be the maximal ideal corresponding to \(x\). Set \(Z = X \setminus U\) and \(Z' = Z \cup \{x\}\). By Schemes, Lemma 01J3 there are quasi-coherent sheaves of ideals \(\mathcal{I}\), resp. \(\mathcal{I}'\) cutting out the reduced closed subschemes \(Z\), resp. \(Z'\). Consider the short exact sequence \[0 \to \mathcal{I}' \to \mathcal{I} \to \mathcal{I}/\mathcal{I}' \to 0.\] Since \(x\) is a closed point of \(X\) and \(x \not \in Z\) we see that \(\mathcal{I}/\mathcal{I}'\) is supported at \(x\). In fact, the restriction of \(\mathcal{I}/\mathcal{I'}\) to \(U\) corresponds to the \(A\)-module \(A/\mathfrak m\). Hence we see that \(\Gamma(X, \mathcal{I}/\mathcal{I'}) = A/\mathfrak m\). Since by assumption \(H^1(X, \mathcal{I}') = 0\) we see there exists a global section \(f \in \Gamma(X, \mathcal{I})\) which maps to the element \(1 \in A/\mathfrak m\) as a section of \(\mathcal{I}/\mathcal{I'}\). Clearly we have \(x \in X_f \subset U\). This implies that \(X_f = D(f_A)\) where \(f_A\) is the image of \(f\) in \(A = \Gamma(U, \mathcal{O}_X)\). In particular \(X_f\) is affine.

Consider the union \(W = \bigcup X_f\) over all \(f \in \Gamma(X, \mathcal{O}_X)\) such that \(X_f\) is affine. Obviously \(W\) is open in \(X\). By the arguments above every closed point of \(X\) is contained in \(W\). The closed subset \(X \setminus W\) of \(X\) is also quasi-compact (see Topology, Lemma 005C). Hence it has a closed point if it is nonempty (see Topology, Lemma 005E). This would contradict the fact that all closed points are in \(W\). Hence we conclude \(X = W\).

Choose finitely many \(f_1, \ldots, f_n \in \Gamma(X, \mathcal{O}_X)\) such that \(X = X_{f_1} \cup \ldots \cup X_{f_n}\) and such that each \(X_{f_i}\) is affine. This is possible as we’ve seen above. By Properties, Lemma 01QF to finish the proof it suffices to show that \(f_1, \ldots, f_n\) generate the unit ideal in \(\Gamma(X, \mathcal{O}_X)\). Consider the short exact sequence \[\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{O}_X^{\oplus n} \ar[rr]^{f_1, \ldots, f_n} & & \mathcal{O}_X \ar[r] & 0 }\] The arrow defined by \(f_1, \ldots, f_n\) is surjective since the opens \(X_{f_i}\) cover \(X\). We let \(\mathcal{F}\) be the kernel of this surjective map. Observe that \(\mathcal{F}\) has a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_n = \mathcal{F}\] so that each subquotient \(\mathcal{F}_i/\mathcal{F}_{i - 1}\) is isomorphic to a quasi-coherent sheaf of ideals. Namely we can take \(\mathcal{F}_i\) to be the intersection of \(\mathcal{F}\) with the first \(i\) direct summands of \(\mathcal{O}_X^{\oplus n}\). The assumption of the lemma implies that \(H^1(X, \mathcal{F}_i/\mathcal{F}_{i - 1}) = 0\) for all \(i\). This implies that \(H^1(X, \mathcal{F}_2) = 0\) because it is sandwiched between \(H^1(X, \mathcal{F}_1)\) and \(H^1(X, \mathcal{F}_2/\mathcal{F}_1)\). Continuing like this we deduce that \(H^1(X, \mathcal{F}) = 0\). Therefore we conclude that the map \[\xymatrix{ \bigoplus\nolimits_{i = 1, \ldots, n} \Gamma(X, \mathcal{O}_X) \ar[rr]^{f_1, \ldots, f_n} & & \Gamma(X, \mathcal{O}_X) }\] is surjective as desired.

Note that if \(X\) is a Noetherian scheme then every quasi-coherent sheaf of ideals is automatically a coherent sheaf of ideals and a finite type quasi-coherent sheaf of ideals. Hence the preceding lemma and the next lemma both apply in this case.

Lemma

Let \(X\) be a scheme. Assume that

1. \(X\) is quasi-compact,

2. \(X\) is quasi-separated, and

3. \(H^1(X, \mathcal{I}) = 0\) for every quasi-coherent sheaf of ideals \(\mathcal{I}\) of finite type.

Then \(X\) is affine.

Proof

By Properties, Lemma 01PG every quasi-coherent sheaf of ideals is a directed colimit of quasi-coherent sheaves of ideals of finite type. By Cohomology, Lemma 01FF taking cohomology on \(X\) commutes with directed colimits. Hence we see that \(H^1(X, \mathcal{I}) = 0\) for every quasi-coherent sheaf of ideals on \(X\). In other words we see that Lemma 01XF applies.

We can use the arguments given above to find a sufficient condition to see when an invertible sheaf is ample. However, we warn the reader that this condition is not necessary.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume that

1. \(X\) is quasi-compact,

2. for every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) there exists an \(n \geq 1\) such that \(H^1(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0\).

Then \(\mathcal{L}\) is ample.

Proof

This is proved in exactly the same way as Lemma 01XF. Let \(x \in X\) be a closed point. Let \(U \subset X\) be an affine open neighbourhood of \(x\) such that \(\mathcal{L}|_U \cong \mathcal{O}_U\). Write \(U = \Spec(A)\) and let \(\mathfrak m \subset A\) be the maximal ideal corresponding to \(x\). Set \(Z = X \setminus U\) and \(Z' = Z \cup \{x\}\). By Schemes, Lemma 01J3 there are quasi-coherent sheaves of ideals \(\mathcal{I}\), resp. \(\mathcal{I}'\) cutting out the reduced closed subschemes \(Z\), resp. \(Z'\). Consider the short exact sequence \[0 \to \mathcal{I}' \to \mathcal{I} \to \mathcal{I}/\mathcal{I}' \to 0.\] For every \(n \geq 1\) we obtain a short exact sequence \[0 \to \mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to \mathcal{I}/\mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to 0.\] By our assumption we may pick \(n\) such that \(H^1(X, \mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = 0\). Since \(x\) is a closed point of \(X\) and \(x \not \in Z\) we see that \(\mathcal{I}/\mathcal{I}'\) is supported at \(x\). In fact, the restriction of \(\mathcal{I}/\mathcal{I'}\) to \(U\) corresponds to the \(A\)-module \(A/\mathfrak m\). Since \(\mathcal{L}\) is trivial on \(U\) we see that the restriction of \(\mathcal{I}/\mathcal{I}' \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\) to \(U\) also corresponds to the \(A\)-module \(A/\mathfrak m\). Hence we see that \(\Gamma(X, \mathcal{I}/\mathcal{I'} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = A/\mathfrak m\). By our choice of \(n\) we see there exists a global section \(s \in \Gamma(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\) which maps to the element \(1 \in A/\mathfrak m\). Clearly we have \(x \in X_s \subset U\) because \(s\) vanishes at points of \(Z\). This implies that \(X_s = D(f)\) where \(f \in A\) is the image of \(s\) in \(A \cong \Gamma(U, \mathcal{L}^{\otimes n})\). In particular \(X_s\) is affine.

Consider the union \(W = \bigcup X_s\) over all \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) for \(n \geq 1\) such that \(X_s\) is affine. Obviously \(W\) is open in \(X\). By the arguments above every closed point of \(X\) is contained in \(W\). The closed subset \(X \setminus W\) of \(X\) is also quasi-compact (see Topology, Lemma 005C). Hence it has a closed point if it is nonempty (see Topology, Lemma 005E). This would contradict the fact that all closed points are in \(W\). Hence we conclude \(X = W\). This means that \(\mathcal{L}\) is ample by Properties, Definition 01PS.

There is a variant of Lemma 0B5P with finite type ideal sheaves which we will formulate and prove here if we ever need it.

Lemma

Let \(f : X \to Y\) be a quasi-compact morphism with \(X\) and \(Y\) quasi-separated. If \(R^1f_*\mathcal{I} = 0\) for every quasi-coherent sheaf of ideals \(\mathcal{I}\) on \(X\), then \(f\) is affine.

Proof

Let \(V \subset Y\) be an affine open subscheme. We have to show that \(U = f^{-1}(V)\) is affine. The inclusion morphism \(V \to Y\) is quasi-compact by Schemes, Lemma 03GI. Hence the base change \(U \to X\) is quasi-compact, see Schemes, Lemma 01K5. Thus any quasi-coherent sheaf of ideals \(\mathcal{I}\) on \(U\) extends to a quasi-coherent sheaf of ideals on \(X\), see Properties, Lemma 01PE. Since the formation of \(R^1f_*\) is local on \(Y\) (Cohomology, Section 01E0) we conclude that \(R^1(U \to V)_*\mathcal{I} = 0\) by the assumption in the lemma. Hence by the Leray Spectral sequence (Cohomology, Lemma 01F2) we conclude that \(H^1(U, \mathcal{I}) = H^1(V, (U \to V)_*\mathcal{I})\). Since \((U \to V)_*\mathcal{I}\) is quasi-coherent by Schemes, Lemma 01LC, we have \(H^1(V, (U \to V)_*\mathcal{I}) = 0\) by Lemma 01XB. Thus we find that \(U\) is affine by Lemma 01XF.

Quasi-coherence of higher direct images

We have seen that the higher cohomology groups of a quasi-coherent module on an affine are zero. For (quasi-)separated quasi-compact schemes \(X\) this implies vanishing of cohomology groups of quasi-coherent sheaves beyond a certain degree. However, it may not be the case that \(X\) has finite cohomological dimension, because that is defined in terms of vanishing of cohomology of all \(\mathcal{O}_X\)-modules.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(P\) be a property of the quasi-compact opens of \(X\). Assume that

1. \(P\) holds for every affine open of \(X\),

2. if \(U\) is quasi-compact open, \(V\) affine open, \(P\) holds for \(U\), \(V\), and \(U \cap V\), then \(P\) holds for \(U \cup V\).

Then \(P\) holds for every quasi-compact open of \(X\) and in particular for \(X\).

Proof

First we argue by induction that \(P\) holds for separated quasi-compact opens \(W \subset X\). Namely, such an open can be written as \(W = U_1 \cup \ldots \cup U_n\) and we can do induction on \(n\) using property (2) with \(U = U_1 \cup \ldots \cup U_{n - 1}\) and \(V = U_n\). This is allowed because \(U \cap V = (U_1 \cap U_n) \cup \ldots \cup (U_{n - 1} \cap U_n)\) is also a union of \(n - 1\) affine open subschemes by Schemes, Lemma 01KP applied to the affine opens \(U_i\) and \(U_n\) of \(W\). Having said this, for any quasi-compact open \(W \subset X\) we can do induction on the number of affine opens needed to cover \(W\) using the same trick as before and using that the quasi-compact open \(U_i \cap U_n\) is separated as an open subscheme of the affine scheme \(U_n\).

Lemma

Let \(X\) be a quasi-compact scheme with affine diagonal (for example if \(X\) is separated). Let \(t = t(X)\) be the minimal number of affine opens needed to cover \(X\). Then \(H^n(X, \mathcal{F}) = 0\) for all \(n \geq t\) and all quasi-coherent sheaves \(\mathcal{F}\).

Proof

First proof. By induction on \(t\). If \(t = 1\) the result follows from Lemma 01XB. If \(t > 1\) write \(X = U \cup V\) with \(V\) affine open and \(U = U_1 \cup \ldots \cup U_{t - 1}\) a union of \(t - 1\) open affines. Note that in this case \(U \cap V = (U_1 \cap V) \cup \ldots (U_{t - 1} \cap V)\) is also a union of \(t - 1\) affine open subschemes. Namely, since the diagonal is affine, the intersection of two affine opens is affine, see Lemma 0BDX. We apply the Mayer-Vietoris long exact sequence \[0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots\] see Cohomology, Lemma 01EB. By induction we see that the groups \(H^i(U, \mathcal{F})\), \(H^i(V, \mathcal{F})\), \(H^i(U \cap V, \mathcal{F})\) are zero for \(i \geq t - 1\). It follows immediately that \(H^i(X, \mathcal{F})\) is zero for \(i \geq t\).

Second proof. Let \(\mathcal{U} : X = \bigcup_{i = 1}^t U_i\) be a finite affine open covering. Since \(X\) is has affine diagonal the multiple intersections \(U_{i_0 \ldots i_p}\) are all affine, see Lemma 0BDX. By Lemma 01XD the Čech cohomology groups \(\check{H}^p(\mathcal{U}, \mathcal{F})\) agree with the cohomology groups. By Cohomology, Lemma 01FM the Čech cohomology groups may be computed using the alternating Čech complex \(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})\). As the covering consists of \(t\) elements we see immediately that \(\check{\mathcal{C}}_{alt}^p(\mathcal{U}, \mathcal{F}) = 0\) for all \(p \geq t\). Hence the result follows.

Lemma

Let \(X\) be a quasi-compact scheme with affine diagonal (for example if \(X\) is separated). Then

1. given a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exists an embedding \(\mathcal{F} \to \mathcal{F}'\) of quasi-coherent \(\mathcal{O}_X\)-modules such that \(H^p(X, \mathcal{F}') = 0\) for all \(p \geq 1\), and

2. \(\{H^n(X, -)\}_{n \geq 0}\) is a universal \(\delta\)-functor from \(\QCoh(\mathcal{O}_X)\) to \(\textit{Ab}\).

Proof

Let \(X = \bigcup U_i\) be a finite affine open covering. Set \(U = \coprod U_i\) and denote \(j : U \to X\) the morphism inducing the given open immersions \(U_i \to X\). Since \(U\) is an affine scheme and \(X\) has affine diagonal, the morphism \(j\) is affine, see Morphisms, Lemma 01SG. For every \(\mathcal{O}_X\)-module \(\mathcal{F}\) there is a canonical map \(\mathcal{F} \to j_*j^*\mathcal{F}\). This map is injective as can be seen by checking on stalks: if \(x \in U_i\), then we have a factorization \[\mathcal{F}_x \to (j_*j^*\mathcal{F})_x \to (j^*\mathcal{F})_{x'} = \mathcal{F}_x\] where \(x' \in U\) is the point \(x\) viewed as a point of \(U_i \subset U\). Now if \(\mathcal{F}\) is quasi-coherent, then \(j^*\mathcal{F}\) is quasi-coherent on the affine scheme \(U\) hence has vanishing higher cohomology by Lemma 01XB. Then \(H^p(X, j_*j^*\mathcal{F}) = 0\) for \(p > 0\) by Lemma 089W as \(j\) is affine. This proves (1). Finally, we see that the map \(H^p(X, \mathcal{F}) \to H^p(X, j_*j^*\mathcal{F})\) is zero and part (2) follows from Homology, Lemma 010T.

Lemma

Let \(X\) be a quasi-compact quasi-separated scheme. Let \(X = U_1 \cup \ldots \cup U_n\) be an open covering with each \(U_i\) quasi-compact and separated (for example affine). Set \[d = \max\nolimits_{I \subset \{1, \ldots, n\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_i) - 1\right)\] where \(t(U)\) is the minimal number of affines needed to cover the scheme \(U\). Then \(H^p(X, \mathcal{F}) = 0\) for all \(p \geq d\) and all quasi-coherent sheaves \(\mathcal{F}\).

Proof

Note that since \(X\) is quasi-separated and \(U_i\) quasi-compact the numbers \(t(\bigcap_{i \in I} U_i)\) are finite. Proof using induction on \(n\). If \(n = 1\) then the result follows from Lemma 01XI. If \(n > 1\), write \(X = U \cup V\) with \(U = U_1 \cup \ldots \cup U_{n - 1}\) and \(V = U_n\). We apply the Mayer-Vietoris long exact sequence \[0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots\] see Cohomology, Lemma 01EB. To finish the proof for \(q \geq d\) we will show that \(H^q(V, \mathcal{F})\), \(H^q(U, \mathcal{F})\), and \(H^{q - 1}(U \cap V, \mathcal{F})\) vanish. By the case \(n = 1\) we have \(H^q(V, \mathcal{F}) = 0\) for \(q \geq t(V) = t(U_n)\). Since \(t(V) \leq d\) this proves what we want. By induction hypothesis we have \(H^q(U, \mathcal{F}) = 0\) for \[q \geq \max\nolimits_{I \subset \{1, \ldots, n - 1\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_i) - 1\right)\] Since the integer on the right is less than or equal to \(d\), this proves what we want. Finally we may use our induction hypothesis for the open \(U \cap V = (U_1 \cap U_n) \cup \ldots \cup (U_{n - 1} \cap U_n)\) to get the vanishing of \(H^q(U \cap V, \mathcal{F}) = 0\) for \[q \geq \max\nolimits_{I \subset \{1, \ldots, n - 1\}} \left(|I| + t(U_n \cap \bigcap\nolimits_{i \in I} U_i) - 1\right)\] Since the integer on the right is strictly less than \(d\) the lemma follows.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume that \(f\) is quasi-separated and quasi-compact.

1. For any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the higher direct images \(R^pf_*\mathcal{F}\) are quasi-coherent on \(S\).

2. If \(S\) is quasi-compact, there exists an integer \(n = n(X, S, f)\) such that \(R^pf_*\mathcal{F} = 0\) for all \(p \geq n\) and any quasi-coherent sheaf \(\mathcal{F}\) on \(X\).

3. In fact, if \(S\) is quasi-compact we can find \(n = n(X, S, f)\) such that for every morphism of schemes \(S' \to S\) we have \(R^p(f')_*\mathcal{F}' = 0\) for \(p \geq n\) and any quasi-coherent sheaf \(\mathcal{F}'\) on \(X'\). Here \(f' : X' = S' \times_S X \to S'\) is the base change of \(f\).

Proof

We first prove (1). Note that under the hypotheses of the lemma the sheaf \(R^0f_*\mathcal{F} = f_*\mathcal{F}\) is quasi-coherent by Schemes, Lemma 01LC. Using Cohomology, Lemma 01E5 we see that forming higher direct images commutes with restriction to open subschemes. Since being quasi-coherent is local on \(S\) we reduce to the case discussed in the next paragraph.

Proof of (1) in case \(S\) is affine. We will use the induction principle. Since \(f\) quasi-compact and quasi-separated we see that \(X\) is quasi-compact and quasi-separated. For \(U \subset X\) quasi-compact open and \(a = f|_U\) we let \(P(U)\) be the property that \(R^pa_*\mathcal{F}\) is quasi-coherent on \(S\) for all quasi-coherent modules \(\mathcal{F}\) on \(U\) and all \(p \geq 0\). Since \(P(X)\) is (1), it suffices the prove conditions (1) and (2) of Lemma 08DR hold. If \(U\) is affine, then \(P(U)\) holds because \(R^pa_*\mathcal{F} = 0\) for \(p \geq 1\) (by Lemma 01XC and Morphisms, Lemma 01SH) and we’ve already observed the result holds for \(p = 0\) in the first paragraph. Next, let \(U \subset X\) be a quasi-compact open, \(V \subset X\) an affine open, and assume \(P(U)\), \(P(V)\), \(P(U \cap V)\) hold. Let \(a = f|_U\), \(b = f|_V\), \(c = f|_{U \cap V}\), and \(g = f|_{U \cup V}\). Then for any quasi-coherent \(\mathcal{O}_{U \cup V}\)-module \(\mathcal{F}\) we have the relative Mayer-Vietoris sequence \[0 \to g_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1g_*\mathcal{F} \to \ldots\] see Cohomology, Lemma 01EC. By \(P(U)\), \(P(V)\), \(P(U \cap V)\) we see that \(R^pa_*(\mathcal{F}|_U)\), \(R^pb_*(\mathcal{F}|_V)\) and \(R^pc_*(\mathcal{F}|_{U \cap V})\) are all quasi-coherent. Using the results on quasi-coherent sheaves in Schemes, Section 01LA this implies that each of the sheaves \(R^pg_*\mathcal{F}\) is quasi-coherent since it sits in the middle of a short exact sequence with a cokernel of a map between quasi-coherent sheaves on the left and a kernel of a map between quasi-coherent sheaves on the right. Whence \(P(U \cup V)\) and the proof of (1) is complete.

Next, we prove (3) and a fortiori (2). Choose a finite affine open covering \(S = \bigcup_{j = 1, \ldots m} S_j\). For each \(j\) choose a finite affine open covering \(f^{-1}(S_j) = \bigcup_{i = 1, \ldots t_j} U_{ji}\). Let \[d_j = \max\nolimits_{I \subset \{1, \ldots, t_j\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_{ji})\right)\] be the integer found in Lemma 071L. We claim that \(n(X, S, f) = \max d_j\) works.

Namely, let \(S' \to S\) be a morphism of schemes and let \(\mathcal{F}'\) be a quasi-coherent sheaf on \(X' = S' \times_S X\). We want to show that \(R^pf'_*\mathcal{F}' = 0\) for \(p \geq n(X, S, f)\). Since this question is local on \(S'\) we may assume that \(S'\) is affine and maps into \(S_j\) for some \(j\). Then \(X' = S' \times_{S_j} f^{-1}(S_j)\) is covered by the open affines \(S' \times_{S_j} U_{ji}\), \(i = 1, \ldots t_j\) and the intersections \[\bigcap\nolimits_{i \in I} S' \times_{S_j} U_{ji} = S' \times_{S_j} \bigcap\nolimits_{i \in I} U_{ji}\] are covered by the same number of affines as before the base change. Applying Lemma 071L we get \(H^p(X', \mathcal{F}') = 0\). By the first part of the proof we already know that each \(R^qf'_*\mathcal{F}'\) is quasi-coherent hence has vanishing higher cohomology groups on our affine scheme \(S'\), thus we see that \(H^0(S', R^pf'_*\mathcal{F}') = H^p(X', \mathcal{F}') = 0\) by Cohomology, Lemma 01F4. Since \(R^pf'_*\mathcal{F}'\) is quasi-coherent we conclude that \(R^pf'_*\mathcal{F}' = 0\).

Lemma

Let \(f : X \to S\) be a morphism of schemes. Assume that \(f\) is quasi-separated and quasi-compact. Assume \(S\) is affine. For any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \[H^q(X, \mathcal{F}) = H^0(S, R^qf_*\mathcal{F})\] for all \(q \in \mathbf{Z}\).

Proof

Consider the Leray spectral sequence \(E_2^{p, q} = H^p(S, R^qf_*\mathcal{F})\) converging to \(H^{p + q}(X, \mathcal{F})\), see Cohomology, Lemma 01F2. By Lemma 01XJ we see that the sheaves \(R^qf_*\mathcal{F}\) are quasi-coherent. By Lemma 01XB we see that \(E_2^{p, q} = 0\) when \(p > 0\). Hence the spectral sequence degenerates at \(E_2\) and we win. See also Cohomology, Lemma 01F4 (2) for the general principle.

Cohomology and base change, I

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Suppose further that \(g : S' \to S\) is any morphism of schemes. Denote \(X' = X_{S'} = S' \times_S X\) the base change of \(X\) and denote \(f' : X' \to S'\) the base change of \(f\). Also write \(g' : X' \to X\) the projection, and set \(\mathcal{F}' = (g')^*\mathcal{F}\). Here is a diagram representing the situation: [02KF]\[\begin{equation} \vcenter{ \xymatrix{ \mathcal{F}' = (g')^*\mathcal{F} & X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f & \mathcal{F} \\ Rf'_*\mathcal{F}' & S' \ar[r]^g & S & Rf_*\mathcal{F} } } \end{equation}\] Here is the simplest case of the base change property we have in mind.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(f\) is affine. In this case \(f_*\mathcal{F} \cong Rf_*\mathcal{F}\) is a quasi-coherent sheaf, and for every base change diagram (02KF) we have \[g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}.\]

Proof

The vanishing of higher direct images is Lemma 01XC. The statement is local on \(S\) and \(S'\). Hence we may assume \(X = \Spec(A)\), \(S = \Spec(R)\), \(S' = \Spec(R')\) and \(\mathcal{F} = \widetilde{M}\) for some \(A\)-module \(M\). We use Schemes, Lemma 01I9 to describe pullbacks and pushforwards of \(\mathcal{F}\). Namely, \(X' = \Spec(R' \otimes_R A)\) and \(\mathcal{F}'\) is the quasi-coherent sheaf associated to \((R' \otimes_R A) \otimes_A M\). Thus we see that the lemma boils down to the equality \[(R' \otimes_R A) \otimes_A M = R' \otimes_R M\] as \(R'\)-modules.

In many situations it is sufficient to know about the following special case of cohomology and base change. It follows immediately from the stronger results in Section 071M, but since it is so important it deserves its own proof.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module with pullback \(\mathcal{F}' = (g')^*\mathcal{F}\). Assume that \(g\) is flat and that \(f\) is quasi-compact and quasi-separated. For any \(i \geq 0\)

1. the base change map of Cohomology, Lemma 02N7 is an isomorphism \[g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}',\]

2. if \(S = \Spec(A)\) and \(S' = \Spec(B)\), then \(H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')\).

Proof

Using Cohomology, Lemma 02N7 in (1) is allowed since \(g'\) is flat by Morphisms, Lemma 01U9. Having said this, part (1) follows from part (2). Namely, part (1) is local on \(S'\) and hence we may assume \(S\) and \(S'\) are affine. In other words, we have \(S = \Spec(A)\) and \(S' = \Spec(B)\) as in (2). Then since \(R^if_*\mathcal{F}\) is quasi-coherent (Lemma 01XJ), it is the quasi-coherent \(\mathcal{O}_S\)-module associated to the \(A\)-module \(H^0(S, R^if_*\mathcal{F}) = H^i(X, \mathcal{F})\) (equality by Lemma 01XK). Similarly, \(R^if'_*\mathcal{F}'\) is the quasi-coherent \(\mathcal{O}_{S'}\)-module associated to the \(B\)-module \(H^i(X', \mathcal{F}')\). Since pullback by \(g\) corresponds to \(- \otimes_A B\) on modules (Schemes, Lemma 01I9) we see that it suffices to prove (2).

Let \(A \to B\) be a flat ring homomorphism. Let \(X\) be a quasi-compact and quasi-separated scheme over \(A\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Set \(X_B = X \times_{\Spec(A)} \Spec(B)\) and denote \(\mathcal{F}_B\) the pullback of \(\mathcal{F}\). We are trying to show that the map \[H^i(X, \mathcal{F}) \otimes_A B \longrightarrow H^i(X_B, \mathcal{F}_B)\] (given by the reference in the statement of the lemma) is an isomorphism.

In case \(X\) is separated, choose an affine open covering \(\mathcal{U} : X = U_1 \cup \ldots \cup U_t\) and recall that \[\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F}),\] see Lemma 01XD. If \(\mathcal{U}_B : X_B = (U_1)_B \cup \ldots \cup (U_t)_B\) we obtain by base change, then it is still the case that each \((U_i)_B\) is affine and that \(X_B\) is separated. Thus we obtain \[\check{H}^p(\mathcal{U}_B, \mathcal{F}_B) = H^p(X_B, \mathcal{F}_B).\] We have the following relation between the Čech complexes \[\check{\mathcal{C}}^\bullet(\mathcal{U}_B, \mathcal{F}_B) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A B\] as follows from Lemma 02KG. Since \(A \to B\) is flat, the same thing remains true on taking cohomology.

In case \(X\) is quasi-separated, choose an affine open covering \(\mathcal{U} : X = U_1 \cup \ldots \cup U_t\). We will use the Čech-to-cohomology spectral sequence Cohomology, Lemma 01ES. The reader who wishes to avoid this spectral sequence can use Mayer-Vietoris and induction on \(t\) as in the proof of Lemma 01XJ. The spectral sequence has \(E_2\)-page \(E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))\) and converges to \(H^{p + q}(X, \mathcal{F})\). Similarly, we have a spectral sequence with \(E_2\)-page \(E_2^{p, q} = \check{H}^p(\mathcal{U}_B, \underline{H}^q(\mathcal{F}_B))\) which converges to \(H^{p + q}(X_B, \mathcal{F}_B)\). Since the intersections \(U_{i_0 \ldots i_p}\) are quasi-compact and separated, the result of the second paragraph of the proof gives \(\check{H}^p(\mathcal{U}_B, \underline{H}^q(\mathcal{F}_B)) = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) \otimes_A B\). Using that \(A \to B\) is flat we conclude that \(H^i(X, \mathcal{F}) \otimes_A B \to H^i(X_B, \mathcal{F}_B)\) is an isomorphism for all \(i\) and we win.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ Y \ar[d]_{g} \ar[r]_h & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) }\] Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module with pullback \(\mathcal{G} = h^*\mathcal{F}\). If \(B\) is a finite locally free \(A\)-module, then \(H^i(X, \mathcal{F}) \otimes_A B = H^i(Y, \mathcal{G})\).

Warning: Do not use this lemma unless you understand the difference between this and Lemma 02KH.

Proof

In case \(X\) is separated, choose an affine open covering \(\mathcal{U} : X = \bigcup_{i \in I} U_i\) and recall that \[\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(X, \mathcal{F}),\] see Lemma 01XD. Let \(\mathcal{V} : Y = \bigcup_{i \in I} g^{-1}(U_i)\) be the corresponding affine open covering of \(Y\). The opens \(V_i = g^{-1}(U_i) = U_i \times_{\Spec(A)} \Spec(B)\) are affine and \(Y\) is separated. Thus we obtain \[\check{H}^p(\mathcal{V}, \mathcal{G}) = H^p(Y, \mathcal{G}).\] We claim the map of Čech complexes \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A B \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})\] is an isomorphism. Namely, as \(B\) is finitely presented as an \(A\)-module we see that tensoring with \(B\) over \(A\) commutes with products, see Algebra, Proposition 059K. Thus it suffices to show that the maps \(\Gamma(U_{i_0 \ldots i_p}, \mathcal{F}) \otimes_A B \to \Gamma(V_{i_0 \ldots i_p}, \mathcal{G})\) are isomorphisms which follows from Lemma 02KG. Since \(A \to B\) is flat, the same thing remains true on taking cohomology.

In the general case we argue in exactly the same way using affine open covering \(\mathcal{U} : X = \bigcup_{i \in I} U_i\) and the corresponding covering \(\mathcal{V} : Y = \bigcup_{i \in I} V_i\) with \(V_i = g^{-1}(U_i)\) as above. We will use the Čech-to-cohomology spectral sequence Cohomology, Lemma 01ES. The spectral sequence has \(E_2\)-page \(E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))\) and converges to \(H^{p + q}(X, \mathcal{F})\). Similarly, we have a spectral sequence with \(E_2\)-page \(E_2^{p, q} = \check{H}^p(\mathcal{V}, \underline{H}^q(\mathcal{G}))\) which converges to \(H^{p + q}(Y, \mathcal{G})\). Since the intersections \(U_{i_0 \ldots i_p}\) are separated, the result of the previous paragraph gives isomorphisms \(\Gamma(U_{i_0 \ldots i_p}, \underline{H}^q(\mathcal{F})) \otimes_A B \to \Gamma(V_{i_0 \ldots i_p}, \underline{H}^q(\mathcal{G}))\). Using that \(- \otimes_A B\) commutes with products and is exact, we conclude that \(\check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) \otimes_A B \to \check{H}^p(\mathcal{V}, \underline{H}^q(\mathcal{G}))\) is an isomorphism. Using that \(A \to B\) is flat we conclude that \(H^i(X, \mathcal{F}) \otimes_A B \to H^i(Y, \mathcal{G})\) is an isomorphism for all \(i\) and we win.

Colimits and higher direct images

General results of this nature can be found in Cohomology, Section 01FE, Sheaves, Lemma 009F, and Modules, Lemma 01BS.

Lemma

Let \(f : X \to S\) be a quasi-compact and quasi-separated morphism of schemes. Let \(\mathcal{F} = \colim \mathcal{F}_i\) be a filtered colimit of abelian sheaves on \(X\). Then for any \(p \geq 0\) we have \[R^pf_*\mathcal{F} = \colim R^pf_*\mathcal{F}_i.\]

Proof

We will apply Cohomology, Lemma 0H7A. Since the affine opens form a basis for the topology of \(S\), it suffices to show that for \(U \subset S\) affine open we have \(H^p(f^{-1}U, \mathcal{F}) = \colim H^p(f^{-1}U, \mathcal{F}_i)\). Since \(f^{-1}U\) is quasi-compact and quasi-separated we conclude using Cohomology, Lemma 01FF. (Because the basis of affine opens in \(f^{-1}U\) satisfies the assumptions of that lemma.)

Cohomology and base change, II

Let \(f : X \to S\) be a morphism of schemes and let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(f\) is quasi-compact and quasi-separated we would like to represent \(Rf_*\mathcal{F}\) by a complex of quasi-coherent sheaves on \(S\). This follows from the fact that the sheaves \(R^if_*\mathcal{F}\) are quasi-coherent if \(S\) is quasi-compact and has affine diagonal, using that \(D_\QCoh(S)\) is equivalent to \(D(\QCoh(\mathcal{O}_S))\), see Derived Categories of Schemes, Proposition 08DB.

In this section we will use a different approach which produces an explicit complex having a good base change property. The construction is particularly easy if \(f\) and \(S\) are separated, or more generally have affine diagonal. Since this is the case which by far the most often used we treat it separately.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(X\) is quasi-compact and \(X\) and \(S\) have affine diagonal (e.g., if \(X\) and \(S\) are separated). In this case we can compute \(Rf_*\mathcal{F}\) as follows:

1. Choose a finite affine open covering \(\mathcal{U} : X = \bigcup_{i = 1, \ldots, n} U_i\).

2. For \(i_0, \ldots, i_p \in \{1, \ldots, n\}\) denote \(f_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to S\) the restriction of \(f\) to the intersection \(U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}\).

3. Set \(\mathcal{F}_{i_0 \ldots i_p}\) equal to the restriction of \(\mathcal{F}\) to \(U_{i_0 \ldots i_p}\).

4. Set \[\check{\mathcal{C}}^p(\mathcal{U}, f, \mathcal{F}) = \bigoplus\nolimits_{i_0 \ldots i_p} f_{i_0 \ldots i_p *} \mathcal{F}_{i_0 \ldots i_p}\] and define differentials \(d : \check{\mathcal{C}}^p(\mathcal{U}, f, \mathcal{F}) \to \check{\mathcal{C}}^{p + 1}(\mathcal{U}, f, \mathcal{F})\) as in Cohomology, Equation (01EE).

Then the complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})\) is a complex of quasi-coherent sheaves on \(S\) which comes equipped with an isomorphism \[\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf_*\mathcal{F}\] in \(D^{+}(S)\). This isomorphism is functorial in the quasi-coherent sheaf \(\mathcal{F}\).

Proof

Consider the resolution \(\mathcal{F} \to {\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})\) of Cohomology, Lemma 02FU. We have an equality of complexes \(\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = f_*{\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})\) of quasi-coherent \(\mathcal{O}_S\)-modules. The morphisms \(j_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to X\) and the morphisms \(f_{i_0 \ldots i_p} : U_{i_0 \ldots i_p} \to S\) are affine by Morphisms, Lemma 01SG and Lemma 0BDX. Hence \(R^qj_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}\) as well as \(R^qf_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}\) are zero for \(q > 0\) (Lemma 01XC). Using \(f \circ j_{i_0 \ldots i_p} = f_{i_0 \ldots i_p}\) and the spectral sequence of Cohomology, Lemma 01F6 we conclude that \(R^qf_*(j_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}) = 0\) for \(q > 0\). Since the terms of the complex \({\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})\) are finite direct sums of the sheaves \(j_{i_0 \ldots i_p *}\mathcal{F}_{i_0 \ldots i_p}\) we conclude using Leray’s acyclicity lemma (Derived Categories, Lemma 015E) that \[Rf_* \mathcal{F} = f_*{\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})\] as desired.

Next, we are going to consider what happens if we do a base change.

Lemma

With notation as in diagram (02KF). Assume \(f : X \to S\) and \(\mathcal{F}\) satisfy the hypotheses of Lemma 01XL. Choose a finite affine open covering \(\mathcal{U} : X = \bigcup U_i\) of \(X\). There is a canonical isomorphism \[g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf'_*\mathcal{F}'\] in \(D^{+}(S')\). Moreover, if \(S' \to S\) is affine, then in fact \[g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}')\] with \(\mathcal{U}' : X' = \bigcup U_i'\) where \(U_i' = (g')^{-1}(U_i) = U_{i, S'}\) is also affine.

Proof

In fact we may define \(U_i' = (g')^{-1}(U_i) = U_{i, S'}\) no matter whether \(S'\) is affine over \(S\) or not. Let \(\mathcal{U}' : X' = \bigcup U_i'\) be the induced covering of \(X'\). In this case we claim that \[g^*\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}')\] with \(\check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}')\) defined in exactly the same manner as in Lemma 01XL. This is clear from the case of affine morphisms (Lemma 02KG) by working locally on \(S'\). Moreover, exactly as in the proof of Lemma 01XL one sees that there is an isomorphism \[\check{\mathcal{C}}^\bullet(\mathcal{U}', f', \mathcal{F}') \longrightarrow Rf'_*\mathcal{F}'\] in \(D^{+}(S')\) since the morphisms \(U_i' \to X'\) and \(U_i' \to S'\) are still affine (being base changes of affine morphisms). Details omitted.

The lemma above says that the complex \[\mathcal{K}^\bullet = \check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})\] is a bounded below complex of quasi-coherent sheaves on \(S\) which universally computes the higher direct images of \(f : X \to S\). This is something about this particular complex and it is not preserved by replacing \(\check{\mathcal{C}}^\bullet(\mathcal{U}, f, \mathcal{F})\) by a quasi-isomorphic complex in general! In other words, this is not a statement that makes sense in the derived category. The reason is that the pullback \(g^*\mathcal{K}^\bullet\) is not equal to the derived pullback \(Lg^*\mathcal{K}^\bullet\) of \(\mathcal{K}^\bullet\) in general!

Here is a more general case where we can prove this statement. We remark that the condition of \(S\) being separated is harmless in most applications, since this is usually used to prove some local property of the total derived image. The proof is significantly more involved and uses hypercoverings; it is a nice example of how you can use them sometimes.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Assume that \(f\) is quasi-compact and quasi-separated and that \(S\) is quasi-compact and separated. There exists a bounded below complex \(\mathcal{K}^\bullet\) of quasi-coherent \(\mathcal{O}_S\)-modules with the following property: For every morphism \(g : S' \to S\) the complex \(g^*\mathcal{K}^\bullet\) is a representative for \(Rf'_*\mathcal{F}'\) with notation as in diagram (02KF).

Proof

(If \(f\) is separated as well, please see Lemma 01XM.) The assumptions imply in particular that \(X\) is quasi-compact and quasi-separated as a scheme. Let \(\mathcal{B}\) be the set of affine opens of \(X\). By Hypercoverings, Lemma 01H7 we can find a hypercovering \(K = (I, \{U_i\})\) such that each \(I_n\) is finite and each \(U_i\) is an affine open of \(X\). By Hypercoverings, Lemma 01GY there is a spectral sequence with \(E_2\)-page \[E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F}))\] converging to \(H^{p + q}(X, \mathcal{F})\). Note that \(\check{H}^p(K, \underline{H}^q(\mathcal{F}))\) is the \(p\)th cohomology group of the complex \[\prod\nolimits_{i \in I_0} H^q(U_i, \mathcal{F}) \to \prod\nolimits_{i \in I_1} H^q(U_i, \mathcal{F}) \to \prod\nolimits_{i \in I_2} H^q(U_i, \mathcal{F}) \to \ldots\] Since each \(U_i\) is affine we see that this is zero unless \(q = 0\) in which case we obtain \[\prod\nolimits_{i \in I_0} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_1} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_2} \mathcal{F}(U_i) \to \ldots\] Thus we conclude that \(R\Gamma(X, \mathcal{F})\) is computed by this complex.

For any \(n\) and \(i \in I_n\) denote \(f_i : U_i \to S\) the restriction of \(f\) to \(U_i\). As \(S\) is separated and \(U_i\) is affine this morphism is affine. Consider the complex of quasi-coherent sheaves \[\mathcal{K}^\bullet = ( \prod\nolimits_{i \in I_0} f_{i, *}\mathcal{F}|_{U_i} \to \prod\nolimits_{i \in I_1} f_{i, *}\mathcal{F}|_{U_i} \to \prod\nolimits_{i \in I_2} f_{i, *}\mathcal{F}|_{U_i} \to \ldots )\] on \(S\). As in Hypercoverings, Lemma 01GY we obtain a map \(\mathcal{K}^\bullet \to Rf_*\mathcal{F}\) in \(D(\mathcal{O}_S)\) by choosing an injective resolution of \(\mathcal{F}\) (details omitted). Consider any affine scheme \(V\) and a morphism \(g : V \to S\). Then the base change \(X_V\) has a hypercovering \(K_V = (I, \{U_{i, V}\})\) obtained by base change. Moreover, \(g^*f_{i, *}\mathcal{F} = f_{i, V, *}(g')^*\mathcal{F}|_{U_{i, V}}\). Thus the arguments above prove that \(\Gamma(V, g^*\mathcal{K}^\bullet)\) computes \(R\Gamma(X_V, (g')^*\mathcal{F})\). This finishes the proof of the lemma as it suffices to prove the equality of complexes Zariski locally on \(S'\).

The following lemma is a variant to flat base change.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_{S'}\)-module flat over \(S\). Assume \(f\) is quasi-compact and quasi-separated. For any \(i \geq 0\) there is an identification \[\mathcal{G} \otimes_{\mathcal{O}_{S'}} g^*R^if_*\mathcal{F} = R^if'_*\left((f')^*\mathcal{G} \otimes_{\mathcal{O}_{X'}} (g')^*\mathcal{F}\right)\]

Proof

Let us construct a map from left to right. First, we have the base change map \(Lg^*Rf_*\mathcal{F} \to Rf'_*L(g')^*\mathcal{F}\). There is also the adjunction map \(\mathcal{G} \to Rf'_*L(f')^*\mathcal{G}\). Using the relative cup product We obtain \[\begin{align*} \mathcal{G} \otimes_{\mathcal{O}_{S'}}^\mathbf{L} Lg^*Rf_*\mathcal{F} & \to Rf'_*L(f')^*\mathcal{G} \otimes_{\mathcal{O}_{S'}}^\mathbf{L} Rf'_*L(g')^*\mathcal{F} \\ & \to Rf'_*\left(L(f')^*\mathcal{G} \otimes_{\mathcal{O}_{X'}}^\mathbf{L} L(g')^*\mathcal{F}\right) \\ & \to Rf'_*\left((f')^*\mathcal{G} \otimes_{\mathcal{O}_{X'}} (g')^*\mathcal{F}\right) \end{align*}\] where for the middle arrow we used the relative cup product, see Cohomology, Remark 0B68. The source of the composition is \[\mathcal{G} \otimes_{\mathcal{O}_{S'}}^\mathbf{L} Lg^*Rf_*\mathcal{F} = \mathcal{G} \otimes_{g^{-1}\mathcal{O}_S}^\mathbf{L} g^{-1}Rf_*\mathcal{F}\] by Cohomology, Lemma 08DE. Since \(\mathcal{G}\) is flat as a sheaf of \(g^{-1}\mathcal{O}_S\)-modules and since \(g^{-1}\) is an exact functor, this is a complex whose \(i\)th cohomology sheaf is \(\mathcal{G} \otimes_{g^{-1}\mathcal{O}_S} g^{-1}R^if_*\mathcal{F} = \mathcal{G} \otimes_{\mathcal{O}_{S'}} g^*R^if_*\mathcal{F}\). In this way we obtain global maps from left to right in the equality of the lemma. To show this map is an isomorphism we may work locally on \(S'\). Thus we may and do assume that \(S\) and \(S'\) are affine schemes.

Proof in case \(S\) and \(S'\) are affine. Say \(S = \Spec(A)\) and \(S' = \Spec(B)\) and say \(\mathcal{G}\) corresponds to the \(B\)-module \(N\) which is assumed to be \(A\)-flat. Since \(S\) is affine, \(X\) is quasi-compact and quasi-separated. We will use a hypercovering argument to finish the proof; if \(X\) is separated or has affine diagonal, then you can use a Čech covering. Let \(\mathcal{B}\) be the set of affine opens of \(X\). By Hypercoverings, Lemma 01H7 we can find a hypercovering \(K = (I, \{U_i\})\) of \(X\) such that each \(I_n\) is finite and each \(U_i\) is an affine open of \(X\). By Hypercoverings, Lemma 01GY there is a spectral sequence with \(E_2\)-page \[E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F}))\] converging to \(H^{p + q}(X, \mathcal{F})\). Since each \(U_i\) is affine and \(\mathcal{F}\) is quasi-coherent the value of \(\underline{H}^q(\mathcal{F})\) is zero on \(U_i\) for \(q > 0\). Thus the spectral sequence degenerates and we conclude that the cohomology modules \(H^q(X, \mathcal{F})\) are computed by \[\prod\nolimits_{i \in I_0} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_1} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_2} \mathcal{F}(U_i) \to \ldots\] Next, note that the base change of our hypercovering to \(S'\) is a hypercovering of \(X' = S' \times_S X\). The schemes \(S' \times_S U_i\) are affine too and we have \[\left((f')^*\mathcal{G} \otimes_{\mathcal{O}_{S'}} (g')^*\mathcal{F}\right) (S' \times_S U_i) = N \otimes_A \mathcal{F}(U_i)\] In this way we conclude that the cohomology modules \(H^q(X', (f')^*\mathcal{G} \otimes_{\mathcal{O}_{S'}} (g')^*\mathcal{F})\) are computed by \[N \otimes_A \left( \prod\nolimits_{i \in I_0} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_1} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_2} \mathcal{F}(U_i) \to \ldots \right)\] Since \(N\) is flat over \(A\), we conclude that \[H^q(X', (f')^*\mathcal{G} \otimes_{\mathcal{O}_{S'}} (g')^*\mathcal{F}) = N \otimes_A H^q(X, \mathcal{F})\] Since this is the translation into algebra of the statement we had to show the proof is complete.

Cohomology of projective space

In this section we compute the cohomology of the twists of the structure sheaf on \(\mathbf{P}^n_S\) over a scheme \(S\). Recall that \(\mathbf{P}^n_S\) was defined as the fibre product \(\mathbf{P}^n_S = S \times_{\Spec(\mathbf{Z})} \mathbf{P}^n_{\mathbf{Z}}\) in Constructions, Definition 01NF. It was shown to be equal to \[\mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n])\] in Constructions, Lemma 01OE. In particular, projective space is a particular case of a projective bundle. If \(S = \Spec(R)\) is affine then we have \[\mathbf{P}^n_S = \mathbf{P}^n_R = \text{Proj}(R[T_0, \ldots, T_n]).\] All these identifications are compatible and compatible with the constructions of the twisted structure sheaves \(\mathcal{O}_{\mathbf{P}^n_S}(d)\).

Before we state the result we need some notation. Let \(R\) be a ring. Recall that \(R[T_0, \ldots, T_n]\) is a graded \(R\)-algebra where each \(T_i\) is homogeneous of degree \(1\). Denote \((R[T_0, \ldots, T_n])_d\) the degree \(d\) summand. It is a finite free \(R\)-module of rank \(\binom{n + d}{d}\) when \(d \geq 0\) and zero else. It has a basis consisting of monomials \(T_0^{e_0} \ldots T_n^{e_n}\) with \(\sum e_i = d\). We will also use the following notation: \(R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\) denotes the \(\mathbf{Z}\)-graded ring with \(\frac{1}{T_i}\) in degree \(-1\). In particular the \(\mathbf{Z}\)-graded \(R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\) module \[\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\] which shows up in the statement below is zero in degrees \(\geq -n\), is free on the generator \(\frac{1}{T_0 \ldots T_n}\) in degree \(-n - 1\) and is free of rank \(\binom{-d - 1}{n}\) for \(d \leq -n - 1\).

Lemma

Let \(R\) be a ring. Let \(n \geq 0\) be an integer. We have \[H^q(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n_R}(d)) = \left\{ \begin{matrix} (R[T_0, \ldots, T_n])_d & \text{if} & q = 0,\ d \geq 0 \\ \left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d & \text{if} & q = n,\ d < 0 \\ 0 & \text{if} & q, n, d\text{ not as above} \end{matrix} \right.\] as \(R\)-modules.

Proof

We will use the standard affine open covering \[\mathcal{U} : \mathbf{P}^n_R = \bigcup\nolimits_{i = 0}^n D_{+}(T_i)\] to compute the cohomology using the Čech complex. This is permissible by Lemma 01XD since any intersection of finitely many affine \(D_{+}(T_i)\) is also a standard affine open (see Constructions, Section 01M3). In fact, we can use the alternating or ordered Čech complex according to Cohomology, Lemmas 01FJ and 01FM.

The ordering we will use on \(\{0, \ldots, n\}\) is the usual one. Hence the complex we are looking at has terms \[\check{\mathcal{C}}_{ord}^p(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) = \bigoplus\nolimits_{i_0 < \ldots < i_p} (R[T_0, \ldots, T_n, \frac{1}{T_{i_0} \ldots T_{i_p}}])_d\] Moreover, the maps are given by the usual formula \[d(s)_{i_0 \ldots i_{p + 1}} = \sum\nolimits_{j = 0}^{p + 1} (-1)^j s_{i_0 \ldots \hat i_j \ldots i_{p + 1}}\] see Cohomology, Section 01FG. Note that each term of this complex has a natural \(\mathbf{Z}^{n + 1}\)-grading. Namely, we get this by declaring a monomial \(T_0^{e_0} \ldots T_n^{e_n}\) to be homogeneous with weight \((e_0, \ldots, e_n) \in \mathbf{Z}^{n + 1}\). It is clear that the differential given above respects the grading. In a formula we have \[\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) = \bigoplus\nolimits_{\vec{e} \in \mathbf{Z}^{n + 1}} \check{\mathcal{C}}^\bullet(\vec{e})\] where not all summands on the right hand side occur (see below). Hence in order to compute the cohomology modules of the complex it suffices to compute the cohomology of the graded pieces and take the direct sum at the end.

Fix \(\vec{e} = (e_0, \ldots, e_n) \in \mathbf{Z}^{n + 1}\). In order for this weight to occur in the complex above we need to assume \(e_0 + \ldots + e_n = d\) (if not then it occurs for a different twist of the structure sheaf of course). Assuming this, set \[NEG(\vec{e}) = \{i \in \{0, \ldots, n\} \mid e_i < 0\}.\] With this notation the weight \(\vec{e}\) summand \(\check{\mathcal{C}}^\bullet(\vec{e})\) of the Čech complex above has the following terms \[\check{\mathcal{C}}^p(\vec{e}) = \bigoplus\nolimits_{i_0 < \ldots < i_p, \ NEG(\vec{e}) \subset \{i_0, \ldots, i_p\}} R \cdot T_0^{e_0} \ldots T_n^{e_n}\] In other words, the terms corresponding to \(i_0 < \ldots < i_p\) such that \(NEG(\vec{e})\) is not contained in \(\{i_0 \ldots i_p\}\) are zero. The differential of the complex \(\check{\mathcal{C}}^\bullet(\vec{e})\) is still given by the exact same formula as above.

Suppose that \(NEG(\vec{e}) = \{0, \ldots, n\}\), i.e., that all exponents \(e_i\) are negative. In this case the complex \(\check{\mathcal{C}}^\bullet(\vec{e})\) has only one term, namely \(\check{\mathcal{C}}^n(\vec{e}) = R \cdot \frac{1}{T_0^{-e_0} \ldots T_n^{-e_n}}\). Hence in this case \[H^q(\check{\mathcal{C}}^\bullet(\vec{e})) = \left\{ \begin{matrix} R \cdot \frac{1}{T_0^{-e_0} \ldots T_n^{-e_n}} & \text{if} & q = n \\ 0 & \text{if} & \text{else} \end{matrix} \right.\] The direct sum of all of these terms clearly gives the value \[\left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d\] in degree \(n\) as stated in the lemma. Moreover these terms do not contribute to cohomology in other degrees (also in accordance with the statement of the lemma).

Assume \(NEG(\vec{e}) = \emptyset\). In this case the complex \(\check{\mathcal{C}}^\bullet(\vec{e})\) has a summand \(R\) corresponding to all \(i_0 < \ldots < i_p\). Let us compare the complex \(\check{\mathcal{C}}^\bullet(\vec{e})\) to another complex. Namely, consider the affine open covering \[\mathcal{V} : \Spec(R) = \bigcup\nolimits_{i \in \{0, \ldots, n\}} V_i\] where \(V_i = \Spec(R)\) for all \(i\). Consider the alternating Čech complex \[\check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)})\] By the same reasoning as above this computes the cohomology of the structure sheaf on \(\Spec(R)\). Hence we see that \(H^p( \check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)}) ) = R\) if \(p = 0\) and is \(0\) whenever \(p > 0\). For these facts, see Lemma 01X9 and its proof. Note that also \(\check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)})\) has a summand \(R\) for every \(i_0 < \ldots < i_p\) and has exactly the same differential as \(\check{\mathcal{C}}^\bullet(\vec{e})\). In other words these complexes are isomorphic complexes and hence have the same cohomology. We conclude that \[H^q(\check{\mathcal{C}}^\bullet(\vec{e})) = \left\{ \begin{matrix} R \cdot T_0^{e_0} \ldots T_n^{e_n} & \text{if} & q = 0 \\ 0 & \text{if} & \text{else} \end{matrix} \right.\] in the case that \(NEG(\vec{e}) = \emptyset\). The direct sum of all of these terms clearly gives the value \[(R[T_0, \ldots, T_n])_d\] in degree \(0\) as stated in the lemma. Moreover these terms do not contribute to cohomology in other degrees (also in accordance with the statement of the lemma).

To finish the proof of the lemma we have to show that the complexes \(\check{\mathcal{C}}^\bullet(\vec{e})\) are acyclic when \(NEG(\vec{e})\) is neither empty nor equal to \(\{0, \ldots, n\}\). Pick an index \(i_{\text{fix}} \not \in NEG(\vec{e})\) (such an index exists). Consider the map \[h : \check{\mathcal{C}}^{p + 1}(\vec{e}) \to \check{\mathcal{C}}^p(\vec{e})\] given by the rule that for \(i_0 < \ldots < i_p\) we have \[h(s)_{i_0 \ldots i_p} = \left\{ \begin{matrix} 0 & \text{if} & p \not \in \{0, \ldots, n - 1\} \\ 0 & \text{if} & i_{\text{fix}} \in \{i_0, \ldots, i_p\} \\ s_{i_{\text{fix}} i_0 \ldots i_p} & \text{if} & i_{\text{fix}} < i_0 \\ (-1)^a s_{i_0 \ldots i_{a - 1} i_{\text{fix}} i_a \ldots i_p} & \text{if} & i_{a - 1} < i_{\text{fix}} < i_a \\ (-1)^{p + 1} s_{i_0 \ldots i_p i_{\text{fix}}} & \text{if} & i_p < i_{\text{fix}} \end{matrix} \right.\] Please compare with the proof of Lemma 01X9. This makes sense because we have \[NEG(\vec{e}) \subset \{i_0, \ldots, i_p\} \Leftrightarrow NEG(\vec{e}) \subset \{i_{\text{fix}}, i_0, \ldots, i_p\}\] The exact same (combinatorial) computation¹ as in the proof of Lemma 01X9 shows that \[(hd + dh)(s)_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}\] Hence we see that the identity map of the complex \(\check{\mathcal{C}}^\bullet(\vec{e})\) is homotopic to zero which implies that it is acyclic.

In the following lemma we are going to use the pairing of free \(R\)-modules \[R[T_0, \ldots, T_n] \times \frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}] \longrightarrow R\] which is defined by the rule \[(f, g) \longmapsto \text{coefficient of } \frac{1}{T_0 \ldots T_n} \text{ in }fg.\] In other words, the basis element \(T_0^{e_0} \ldots T_n^{e_n}\) pairs with the basis element \(T_0^{d_0} \ldots T_n^{d_n}\) to give \(1\) if and only if \(e_i + d_i = -1\) for all \(i\), and pairs to zero in all other cases. Using this pairing we get an identification \[\left(\frac{1}{T_0 \ldots T_n} R[\frac{1}{T_0}, \ldots, \frac{1}{T_n}]\right)_d = \Hom_R((R[T_0, \ldots, T_n])_{-n - 1 - d}, R)\] Thus we can reformulate the result of Lemma 01XT as saying that [01XU]\[\begin{equation} H^q(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n_R}(d)) = \left\{ \begin{matrix} (R[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \Hom_R((R[T_0, \ldots, T_n])_{-n - 1 - d}, R) & \text{if} & q = n \end{matrix} \right. \end{equation}\]

Lemma

The identifications of Equation (01XU) are compatible with base change w.r.t. ring maps \(R \to R'\). Moreover, for any \(f \in R[T_0, \ldots, T_n]\) homogeneous of degree \(m\) the map multiplication by \(f\) \[\mathcal{O}_{\mathbf{P}^n_R}(d) \longrightarrow \mathcal{O}_{\mathbf{P}^n_R}(d + m)\] induces the map on the cohomology group via the identifications of Equation (01XU) which is multiplication by \(f\) for \(H^0\) and the contragredient of multiplication by \(f\) \[(R[T_0, \ldots, T_n])_{-n - 1 - (d + m)} \longrightarrow (R[T_0, \ldots, T_n])_{-n - 1 - d}\] on \(H^n\).

Proof

Suppose that \(R \to R'\) is a ring map. Let \(\mathcal{U}\) be the standard affine open covering of \(\mathbf{P}^n_R\), and let \(\mathcal{U}'\) be the standard affine open covering of \(\mathbf{P}^n_{R'}\). Note that \(\mathcal{U}'\) is the pullback of the covering \(\mathcal{U}\) under the canonical morphism \(\mathbf{P}^n_{R'} \to \mathbf{P}^n_R\). Hence there is a map of Čech complexes \[\gamma : \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) \longrightarrow \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}', \mathcal{O}_{\mathbf{P}_{R'}}(d))\] which is compatible with the map on cohomology by Cohomology, Lemma 01FD. It is clear from the computations in the proof of Lemma 01XT that this map of Čech complexes is compatible with the identifications of the cohomology groups in question. (Namely the basis elements for the Čech complex over \(R\) simply map to the corresponding basis elements for the Čech complex over \(R'\).) Whence the first statement of the lemma.

Now fix the ring \(R\) and consider two homogeneous polynomials \(f, g \in R[T_0, \ldots, T_n]\) both of the same degree \(m\). Since cohomology is an additive functor, it is clear that the map induced by multiplication by \(f + g\) is the same as the sum of the maps induced by multiplication by \(f\) and the map induced by multiplication by \(g\). Moreover, since cohomology is a functor, a similar result holds for multiplication by a product \(fg\) where \(f, g\) are both homogeneous (but not necessarily of the same degree). Hence to verify the second statement of the lemma it suffices to prove this when \(f = x \in R\) or when \(f = T_i\). In the case of multiplication by an element \(x \in R\) the result follows since every cohomology groups or complex in sight has the structure of an \(R\)-module or complex of \(R\)-modules. Finally, we consider the case of multiplication by \(T_i\) as a \(\mathcal{O}_{\mathbf{P}^n_R}\)-linear map \[\mathcal{O}_{\mathbf{P}^n_R}(d) \longrightarrow \mathcal{O}_{\mathbf{P}^n_R}(d + 1)\] The statement on \(H^0\) is clear. For the statement on \(H^n\) consider multiplication by \(T_i\) as a map on Čech complexes \[\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) \longrightarrow \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d + 1))\] We are going to use the notation introduced in the proof of Lemma 01XT. We consider the effect of multiplication by \(T_i\) in terms of the decompositions \[\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d)) = \bigoplus\nolimits_{\vec{e} \in \mathbf{Z}^{n + 1}, \ \sum e_i = d} \check{\mathcal{C}}^\bullet(\vec{e})\] and \[\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{O}_{\mathbf{P}_R}(d + 1)) = \bigoplus\nolimits_{\vec{e} \in \mathbf{Z}^{n + 1}, \ \sum e_i = d + 1} \check{\mathcal{C}}^\bullet(\vec{e})\] It is clear that it maps the subcomplex \(\check{\mathcal{C}}^\bullet(\vec{e})\) to the subcomplex \(\check{\mathcal{C}}^\bullet(\vec{e} + \vec{b}_i)\) where \(\vec{b}_i = (0, \ldots, 0, 1, 0, \ldots, 0))\) the \(i\)th basis vector. In other words, it maps the summand of \(H^n\) corresponding to \(\vec{e}\) with \(e_i < 0\) and \(\sum e_i = d\) to the summand of \(H^n\) corresponding to \(\vec{e} + \vec{b}_i\) (which is zero if \(e_i + b_i \geq 0\)). It is easy to see that this corresponds exactly to the action of the contragredient of multiplication by \(T_i\) as a map \[(R[T_0, \ldots, T_n])_{-n - 1 - (d + 1)} \longrightarrow (R[T_0, \ldots, T_n])_{-n - 1 - d}\] This proves the lemma.

Before we state the relative version we need some notation. Namely, recall that \(\mathcal{O}_S[T_0, \ldots, T_n]\) is a graded \(\mathcal{O}_S\)-module where each \(T_i\) is homogeneous of degree \(1\). Denote \((\mathcal{O}_S[T_0, \ldots, T_n])_d\) the degree \(d\) summand. It is a finite locally free sheaf of rank \(\binom{n + d}{d}\) on \(S\).

Lemma

Let \(S\) be a scheme. Let \(n \geq 0\) be an integer. Consider the structure morphism \[f : \mathbf{P}^n_S \longrightarrow S.\] We have \[R^qf_*(\mathcal{O}_{\mathbf{P}^n_S}(d)) = \left\{ \begin{matrix} (\mathcal{O}_S[T_0, \ldots, T_n])_d & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \SheafHom_{\mathcal{O}_S}( (\mathcal{O}_S[T_0, \ldots, T_n])_{- n - 1 - d}, \mathcal{O}_S) & \text{if} & q = n \end{matrix} \right.\]

Proof

Omitted. Hint: This follows since the identifications in (01XU) are compatible with affine base change by Lemma 01XV.

Next we state the version for projective bundles associated to finite locally free sheaves. Let \(S\) be a scheme. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_S\)-module of constant rank \(n + 1\), see Modules, Section 01C5. In this case we think of \(\text{Sym}(\mathcal{E})\) as a graded \(\mathcal{O}_S\)-module where \(\mathcal{E}\) is the graded part of degree \(1\). And \(\text{Sym}^d(\mathcal{E})\) is the degree \(d\) summand. It is a finite locally free sheaf of rank \(\binom{n + d}{d}\) on \(S\). Recall that our normalization is that \[\pi : \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_S(\text{Sym}(\mathcal{E})) \longrightarrow S\] and that there are natural maps \(\text{Sym}^d(\mathcal{E}) \to \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)\).

Lemma

Let \(S\) be a scheme. Let \(n \geq 1\). Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_S\)-module of constant rank \(n + 1\). Consider the structure morphism \[\pi : \mathbf{P}(\mathcal{E}) \longrightarrow S.\] We have \[R^q\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) = \left\{ \begin{matrix} \text{Sym}^d(\mathcal{E}) & \text{if} & q = 0 \\ 0 & \text{if} & q \not = 0, n \\ \SheafHom_{\mathcal{O}_S}( \text{Sym}^{- n - 1 - d}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{n + 1}\mathcal{E}, \mathcal{O}_S) & \text{if} & q = n \end{matrix} \right.\] These identifications are compatible with base change and isomorphism between locally free sheaves.

Proof

Consider the canonical map \[\pi^*\mathcal{E} \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\] and twist down by \(1\) to get \[\pi^*(\mathcal{E})(-1) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}\] This is a surjective map from a locally free rank \(n + 1\) sheaf onto the structure sheaf. Hence the corresponding Koszul complex is exact (More on Algebra, Lemma 0626). In other words there is an exact complex \[0 \to \pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1) \to \ldots \to \pi^*(\wedge^i\mathcal{E})(-i) \to \ldots \to \pi^*\mathcal{E}(-1) \to \mathcal{O}_{\mathbf{P}(\mathcal{E})} \to 0\] We will think of the term \(\pi^*(\wedge^i\mathcal{E})(-i)\) as being in degree \(-i\). We are going to compute the higher direct images of this acyclic complex using the first spectral sequence of Derived Categories, Lemma 015J. Namely, we see that there is a spectral sequence with terms \[E_1^{p, q} = R^q\pi_*\left(\pi^*(\wedge^{-p}\mathcal{E})(p)\right)\] converging to zero! By the projection formula (Cohomology, Lemma 01E8) we have \[E_1^{p, q} = \wedge^{-p} \mathcal{E} \otimes_{\mathcal{O}_S} R^q\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(p)\right).\] Note that locally on \(S\) the sheaf \(\mathcal{E}\) is trivial, i.e., isomorphic to \(\mathcal{O}_S^{\oplus n + 1}\), hence locally on \(S\) the morphism \(\mathbf{P}(\mathcal{E}) \to S\) can be identified with \(\mathbf{P}^n_S \to S\). Hence locally on \(S\) we can use the result of Lemmas 01XT, 01XV, or 01XW. It follows that \(E_1^{p, q} = 0\) unless \((p, q)\) is \((0, 0)\) or \((-n - 1, n)\). The nonzero terms are \[\begin{align*} E_1^{0, 0} & = \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})} = \mathcal{O}_S \\ E_1^{-n - 1, n} & = R^n\pi_*\left(\pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1)\right) = \wedge^{n + 1}\mathcal{E} \otimes_{\mathcal{O}_S} R^n\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) \end{align*}\] Hence there can only be one nonzero differential in the spectral sequence namely the map \(d_{n + 1}^{-n - 1, n} : E_{n + 1}^{-n - 1, n} \to E_{n + 1}^{0, 0}\) which has to be an isomorphism (because the spectral sequence converges to the \(0\) sheaf). Thus \(E_1^{p, q} = E_{n + 1}^{p, q}\) and we obtain a canonical isomorphism \[\wedge^{n + 1}\mathcal{E} \otimes_{\mathcal{O}_S} R^n\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) = R^n\pi_*\left(\pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1)\right) \xrightarrow{d_{n + 1}^{-n - 1, n}} \mathcal{O}_S\] Since \(\wedge^{n + 1}\mathcal{E}\) is an invertible sheaf, this implies that \(R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\) is invertible as well and canonically isomorphic to the inverse of \(\wedge^{n + 1}\mathcal{E}\). In other words we have proved the case \(d = - n - 1\) of the lemma.

Working locally on \(S\) we see immediately from the computation of cohomology in Lemmas 01XT, 01XV, or 01XW the statements on vanishing of the lemma. Moreover the result on \(R^0\pi_*\) is clear as well, since there are canonical maps \(\text{Sym}^d(\mathcal{E}) \to \pi_* \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)\) for all \(d\). It remains to show that the description of \(R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)\) is correct for \(d < -n - 1\). In order to do this we consider the map \[\pi^*(\text{Sym}^{-d - n - 1}(\mathcal{E})) \otimes_{\mathcal{O}_{\mathbf{P}(\mathcal{E})}} \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\] Applying \(R^n\pi_*\) and the projection formula (see above) we get a map \[\text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes_{\mathcal{O}_S} R^n\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) \longrightarrow R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1) = (\wedge^{n + 1}\mathcal{E})^{\otimes -1}\] (the last equality we have shown above). Again by the local calculations of Lemmas 01XT, 01XV, or 01XW it follows that this map induces a perfect pairing between \(R^n\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d))\) and \(\text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes \wedge^{n + 1}(\mathcal{E})\) as desired.

Coherent sheaves on locally Noetherian schemes

We have defined the notion of a coherent module on any ringed space in Modules, Section 01BU. Although it is possible to consider coherent sheaves on non-Noetherian schemes we will always assume the base scheme is locally Noetherian when we consider coherent sheaves. Here is a characterization of coherent sheaves on locally Noetherian schemes.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{F}\) is coherent,

2. \(\mathcal{F}\) is a quasi-coherent, finite type \(\mathcal{O}_X\)-module,

3. \(\mathcal{F}\) is a finitely presented \(\mathcal{O}_X\)-module,

4. for any affine open \(\Spec(A) = U \subset X\) we have \(\mathcal{F}|_U = \widetilde M\) with \(M\) a finite \(A\)-module, and

5. there exists an affine open covering \(X = \bigcup U_i\), \(U_i = \Spec(A_i)\) such that each \(\mathcal{F}|_{U_i} = \widetilde M_i\) with \(M_i\) a finite \(A_i\)-module.

In particular \(\mathcal{O}_X\) is coherent, any invertible \(\mathcal{O}_X\)-module is coherent, and more generally any finite locally free \(\mathcal{O}_X\)-module is coherent.

Proof

The implications (1) \(\Rightarrow\) (2) and (1) \(\Rightarrow\) (3) hold in general, see Modules, Lemma 01BW. If \(\mathcal{F}\) is finitely presented then \(\mathcal{F}\) is quasi-coherent, see Modules, Lemma 01BO. Hence also (3) \(\Rightarrow\) (2).

Assume \(\mathcal{F}\) is a quasi-coherent, finite type \(\mathcal{O}_X\)-module. By Properties, Lemma 01PB we see that on any affine open \(\Spec(A) = U \subset X\) we have \(\mathcal{F}|_U = \widetilde M\) with \(M\) a finite \(A\)-module. Since \(A\) is Noetherian we see that \(M\) has a finite resolution \[A^{\oplus m} \to A^{\oplus n} \to M \to 0.\] Hence \(\mathcal{F}\) is of finite presentation by Properties, Lemma 01PC. In other words (2) \(\Rightarrow\) (3).

By Modules, Lemma 01BZ it suffices to show that \(\mathcal{O}_X\) is coherent in order to show that (3) implies (1). Thus we have to show: given any open \(U \subset X\) and any finite collection of sections \(f_i \in \mathcal{O}_X(U)\), \(i = 1, \ldots, n\) the kernel of the map \(\bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{O}_U\) is of finite type. Since being of finite type is a local property it suffices to check this in a neighbourhood of any \(x \in U\). Thus we may assume \(U = \Spec(A)\) is affine. In this case \(f_1, \ldots, f_n \in A\) are elements of \(A\). Since \(A\) is Noetherian, see Properties, Lemma 01OW the kernel \(K\) of the map \(\bigoplus_{i = 1, \ldots, n} A \to A\) is a finite \(A\)-module. See for example Algebra, Lemma 00IK. As the functor \(\widetilde{ }\) is exact, see Schemes, Lemma 01HV we get an exact sequence \[\widetilde K \to \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{O}_U\] and by Properties, Lemma 01PB again we see that \(\widetilde K\) is of finite type. We conclude that (1), (2) and (3) are all equivalent.

It follows from Properties, Lemma 01PB that (2) implies (4). It is trivial that (4) implies (5). The discussion in Schemes, Section 01LA show that (5) implies that \(\mathcal{F}\) is quasi-coherent and it is clear that (5) implies that \(\mathcal{F}\) is of finite type. Hence (5) implies (2) and we win.

Lemma

Let \(X\) be a locally Noetherian scheme. The category of coherent \(\mathcal{O}_X\)-modules is abelian. More precisely, the kernel and cokernel of a map of coherent \(\mathcal{O}_X\)-modules are coherent. Any extension of coherent sheaves is coherent.

Proof

This is a restatement of Modules, Lemma 01BY in a particular case.

The following lemma does not always hold for the category of coherent \(\mathcal{O}_X\)-modules on a general ringed space \(X\).

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Any quasi-coherent submodule of \(\mathcal{F}\) is coherent. Any quasi-coherent quotient module of \(\mathcal{F}\) is coherent.

Proof

We may assume that \(X\) is affine, say \(X = \Spec(A)\). Properties, Lemma 01OW implies that \(A\) is Noetherian. Lemma 01XZ turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite \(A\)-module is a finite \(A\)-module, see for example Algebra, Lemma 00IK.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. The \(\mathcal{O}_X\)-modules \(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}\) and \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) are coherent.

Proof

It is shown in Modules, Lemma 01CQ that \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) is coherent. The result for tensor products is Modules, Lemma 01CE

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Let \(\varphi : \mathcal{G} \to \mathcal{F}\) be a homomorphism of \(\mathcal{O}_X\)-modules. Let \(x \in X\).

1. If \(\mathcal{F}_x = 0\) then there exists an open neighbourhood \(U \subset X\) of \(x\) such that \(\mathcal{F}|_U = 0\).

2. If \(\varphi_x : \mathcal{G}_x \to \mathcal{F}_x\) is injective, then there exists an open neighbourhood \(U \subset X\) of \(x\) such that \(\varphi|_U\) is injective.

3. If \(\varphi_x : \mathcal{G}_x \to \mathcal{F}_x\) is surjective, then there exists an open neighbourhood \(U \subset X\) of \(x\) such that \(\varphi|_U\) is surjective.

4. If \(\varphi_x : \mathcal{G}_x \to \mathcal{F}_x\) is bijective, then there exists an open neighbourhood \(U \subset X\) of \(x\) such that \(\varphi|_U\) is an isomorphism.

Proof

See Modules, Lemmas 01B8, 01B9, and 01C0.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Let \(x \in X\). Suppose \(\psi : \mathcal{G}_x \to \mathcal{F}_x\) is a map of \(\mathcal{O}_{X, x}\)-modules. Then there exists an open neighbourhood \(U \subset X\) of \(x\) and a map \(\varphi : \mathcal{G}|_U \to \mathcal{F}|_U\) such that \(\varphi_x = \psi\).

Proof

In view of Lemma 01XZ this is a reformulation of Modules, Lemma 01CP.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Then \(\text{Supp}(\mathcal{F})\) is closed, and \(\mathcal{F}\) comes from a coherent sheaf on the scheme theoretic support of \(\mathcal{F}\), see Morphisms, Definition 05JV.

Proof

Let \(i : Z \to X\) be the scheme theoretic support of \(\mathcal{F}\) and let \(\mathcal{G}\) be the finite type quasi-coherent sheaf on \(Z\) such that \(i_*\mathcal{G} \cong \mathcal{F}\). Since \(Z = \text{Supp}(\mathcal{F})\) we see that the support is closed. The scheme \(Z\) is locally Noetherian by Morphisms, Lemmas 01T5 and 01T6. Finally, \(\mathcal{G}\) is a coherent \(\mathcal{O}_Z\)-module by Lemma 01XZ

Lemma

Let \(i : Z \to X\) be a closed immersion of locally Noetherian schemes. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaf of ideals cutting out \(Z\). The functor \(i_*\) induces an equivalence between the category of coherent \(\mathcal{O}_X\)-modules annihilated by \(\mathcal{I}\) and the category of coherent \(\mathcal{O}_Z\)-modules.

Proof

The functor is fully faithful by Morphisms, Lemma 01QY. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module annihilated by \(\mathcal{I}\). By Morphisms, Lemma 01QY we can write \(\mathcal{F} = i_*\mathcal{G}\) for some quasi-coherent sheaf \(\mathcal{G}\) on \(Z\). By Modules, Lemma 01C4 we see that \(\mathcal{G}\) is of finite type. Hence \(\mathcal{G}\) is coherent by Lemma 01XZ. Thus the functor is also essentially surjective as desired.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Assume \(f\) is finite and \(Y\) locally Noetherian. Then \(R^pf_*\mathcal{F} = 0\) for \(p > 0\) and \(f_*\mathcal{F}\) is coherent if \(\mathcal{F}\) is coherent.

Proof

The higher direct images vanish by Lemma 01XC and because a finite morphism is affine (by definition). Note that the assumptions imply that also \(X\) is locally Noetherian (see Morphisms, Lemma 01T6) and hence the statement makes sense. Let \(\Spec(A) = V \subset Y\) be an affine open subset. By Morphisms, Definition 01WH we see that \(f^{-1}(V) = \Spec(B)\) with \(A \to B\) finite. Lemma 01XZ turns the statement of the lemma into the following algebra fact: If \(M\) is a finite \(B\)-module, then \(M\) is also finite viewed as a \(A\)-module, see Algebra, Lemma 00GJ.

In the situation of the lemma also the higher direct images are coherent since they vanish. We will show that this is always the case for a proper morphism between locally Noetherian schemes (Proposition 02O5).

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be a coherent sheaf with \(\dim(\text{Supp}(\mathcal{F})) \leq 0\). Then \(\mathcal{F}\) is generated by global sections and \(H^i(X, \mathcal{F}) = 0\) for \(i > 0\).

Proof

By Lemma 01Y5 we see that \(\mathcal{F} = i_*\mathcal{G}\) where \(i : Z \to X\) is the inclusion of the scheme theoretic support of \(\mathcal{F}\) and where \(\mathcal{G}\) is a coherent \(\mathcal{O}_Z\)-module. Since the dimension of \(Z\) is \(0\), we see \(Z\) is a disjoint union of affines (Properties, Lemma 0AAX). Hence \(\mathcal{G}\) is globally generated and the higher cohomology groups of \(\mathcal{G}\) are zero (Lemma 01XB). Hence \(\mathcal{F} = i_*\mathcal{G}\) is globally generated. Since the cohomologies of \(\mathcal{F}\) and \(\mathcal{G}\) agree (Lemma 089W applies as a closed immersion is affine) we conclude that the higher cohomology groups of \(\mathcal{F}\) are zero.

Lemma

Let \(X\) be a scheme. Let \(j : U \to X\) be the inclusion of an open. Let \(T \subset X\) be a closed subset contained in \(U\). If \(\mathcal{F}\) is a coherent \(\mathcal{O}_U\)-module with \(\text{Supp}(\mathcal{F}) \subset T\), then \(j_*\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module.

Proof

Consider the open covering \(X = U \cup (X \setminus T)\). Then \(j_*\mathcal{F}|_U = \mathcal{F}\) is coherent and \(j_*\mathcal{F}|_{X \setminus T} = 0\) is also coherent. Hence \(j_*\mathcal{F}\) is coherent.

Coherent sheaves on Noetherian schemes

In this section we mention some properties of coherent sheaves on Noetherian schemes.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. The ascending chain condition holds for quasi-coherent submodules of \(\mathcal{F}\). In other words, given any sequence \[\mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F}\] of quasi-coherent submodules, then \(\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots\) for some \(n \geq 0\).

Proof

Choose a finite affine open covering. On each member of the covering we get stabilization by Algebra, Lemma 00IK. Hence the lemma follows.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals corresponding to a closed subscheme \(Z \subset X\). Then there is some \(n \geq 0\) such that \(\mathcal{I}^n\mathcal{F} = 0\) if and only if \(\text{Supp}(\mathcal{F}) \subset Z\) (set theoretically).

Proof

This follows immediately from Algebra, Lemma 00L6 because \(X\) has a finite covering by spectra of Noetherian rings.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Let \(\mathcal{G} \subset \mathcal{F}\) be a quasi-coherent subsheaf. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Then there exists a \(c \geq 0\) such that for all \(n \geq c\) we have \[\mathcal{I}^{n - c}(\mathcal{I}^c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^n\mathcal{F} \cap \mathcal{G}\]

Proof

This follows immediately from Algebra, Lemma 00IN because \(X\) has a finite covering by spectra of Noetherian rings.

Lemma

Let \(X\) be a Noetherian scheme. Every quasi-coherent \(\mathcal{O}_X\)-module is the filtered colimit of its coherent submodules.

Proof

This is a reformulation of Properties, Lemma 01PG in view of the fact that a finite type quasi-coherent \(\mathcal{O}_X\)-module is coherent by Lemma 01XZ.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{G}\) be a coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Denote \(Z \subset X\) the corresponding closed subscheme and set \(U = X \setminus Z\). There is a canonical isomorphism \[\colim_n \Hom_{\mathcal{O}_X}(\mathcal{I}^n\mathcal{G}, \mathcal{F}) \longrightarrow \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U).\] In particular we have an isomorphism \[\colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}) \longrightarrow \Gamma(U, \mathcal{F}).\]

Proof

We first prove the second map is an isomorphism. It is injective by Properties, Lemma 01PQ. Since \(\mathcal{F}\) is the union of its coherent submodules, see Properties, Lemma 01PG (and Lemma 01XZ) we may and do assume that \(\mathcal{F}\) is coherent to prove surjectivity. Let \(\mathcal{F}_n\) denote the quasi-coherent subsheaf of \(\mathcal{F}\) consisting of sections annihilated by \(\mathcal{I}^n\), see Properties, Lemma 01PQ. Since \(\mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots\) we see that \(\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots\) for some \(n \geq 0\) by Lemma 01Y8. Set \(\mathcal{H} = \mathcal{F}_n\) for this \(n\). By Artin-Rees (Lemma 01YA) there exists an \(c \geq 0\) such that \(\mathcal{I}^m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^{m - c}\mathcal{H}\). Picking \(m = n + c\) we get \(\mathcal{I}^m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^n\mathcal{H} = 0\). Thus if we set \(\mathcal{F}' = \mathcal{I}^m\mathcal{F}\) then we see that \(\mathcal{F}' \cap \mathcal{F}_n = 0\) and \(\mathcal{F}'|_U = \mathcal{F}|_U\). Note in particular that the subsheaf \((\mathcal{F}')_N\) of sections annihilated by \(\mathcal{I}^N\) is zero for all \(N \geq 0\). Hence by Properties, Lemma 01PQ we deduce that the top horizontal arrow in the following commutative diagram is a bijection: \[\xymatrix{ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}') \ar[r] \ar[d] & \Gamma(U, \mathcal{F}') \ar[d] \\ \colim_n \Hom_{\mathcal{O}_X}( \mathcal{I}^n, \mathcal{F}) \ar[r] & \Gamma(U, \mathcal{F}) }\] Since also the right vertical arrow is a bijection we conclude that the bottom horizontal arrow is surjective as desired.

Next, we prove the first arrow of the lemma is a bijection. By Lemma 01XZ the sheaf \(\mathcal{G}\) is of finite presentation and hence the sheaf \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})\) is quasi-coherent, see Schemes, Section 01LA. By definition we have \[\mathcal{H}(U) = \Hom_{\mathcal{O}_U}(\mathcal{G}|_U, \mathcal{F}|_U)\] Pick a \(\psi\) in the right hand side of the first arrow of the lemma, i.e., \(\psi \in \mathcal{H}(U)\). The result just proved applies to \(\mathcal{H}\) and hence there exists an \(n \geq 0\) and an \(\varphi : \mathcal{I}^n \to \mathcal{H}\) which recovers \(\psi\) on restriction to \(U\). By Modules, Lemma 01CN \(\varphi\) corresponds to a map \[\varphi : \mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G} \longrightarrow \mathcal{F}.\] This is almost what we want except that the source of the arrow is the tensor product of \(\mathcal{I}^n\) and \(\mathcal{G}\) and not the product. We will show that, at the cost of increasing \(n\), the difference is irrelevant. Consider the short exact sequence \[0 \to \mathcal{K} \to \mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{I}^n\mathcal{G} \to 0\] where \(\mathcal{K}\) is defined as the kernel. Note that \(\mathcal{I}^n\mathcal{K} = 0\) (proof omitted). By Artin-Rees again we see that \[\mathcal{K} \cap \mathcal{I}^m(\mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G}) = 0\] for some \(m\) large enough. In other words we see that \[\mathcal{I}^m(\mathcal{I}^n \otimes_{\mathcal{O}_X} \mathcal{G}) \longrightarrow \mathcal{I}^{n + m}\mathcal{G}\] is an isomorphism. Let \(\varphi'\) be the restriction of \(\varphi\) to this submodule thought of as a map \(\mathcal{I}^{m + n}\mathcal{G} \to \mathcal{F}\). Then \(\varphi'\) gives an element of the left hand side of the first arrow of the lemma which maps to \(\psi\) via the arrow. In other words we have proved surjectivity of the arrow. We omit the proof of injectivity.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Let \(U \subset X\) be open and let \(\varphi : \mathcal{F}|_U \to \mathcal{G}|_U\) be an \(\mathcal{O}_U\)-module map. Then there exists a coherent submodule \(\mathcal{F}' \subset \mathcal{F}\) agreeing with \(\mathcal{F}\) over \(U\) such that \(\varphi\) extends to \(\varphi' : \mathcal{F}' \to \mathcal{G}\).

Proof

Let \(\mathcal{I} \subset \mathcal{O}_X\) be the coherent sheaf of ideals cutting out the reduced induced scheme structure on \(X \setminus U\). If \(X\) is Noetherian, then Lemma 01YB tells us that we can take \(\mathcal{F}' = \mathcal{I}^n\mathcal{F}\) for some \(n\). The general case will follow from this using Zorn’s lemma.

Consider the set of triples \((U', \mathcal{F}', \varphi')\) where \(U \subset U' \subset X\) is open, \(\mathcal{F}' \subset \mathcal{F}|_{U'}\) is a coherent subsheaf agreeing with \(\mathcal{F}\) over \(U\), and \(\varphi' : \mathcal{F}' \to \mathcal{G}|_{U'}\) restricts to \(\varphi\) over \(U\). We say \((U'', \mathcal{F}'', \varphi'') \geq (U', \mathcal{F}', \varphi')\) if and only if \(U'' \supset U'\), \(\mathcal{F}''|_{U'} = \mathcal{F}'\), and \(\varphi''|_{U'} = \varphi'\). It is clear that if we have a totally ordered collection of triples \((U_i, \mathcal{F}_i, \varphi_i)\), then we can glue the \(\mathcal{F}_i\) to a subsheaf \(\mathcal{F}'\) of \(\mathcal{F}\) over \(U' = \bigcup U_i\) and extend \(\varphi\) to a map \(\varphi' : \mathcal{F}' \to \mathcal{G}|_{U'}\). Hence any totally ordered subset of triples has an upper bound. Finally, suppose that \((U', \mathcal{F}', \varphi')\) is any triple but \(U' \not = X\). Then we can choose an affine open \(W \subset X\) which is not contained in \(U'\). By the result of the first paragraph we can extend the subsheaf \(\mathcal{F}'|_{W \cap U'}\) and the restriction \(\varphi'|_{W \cap U'}\) to some subsheaf \(\mathcal{F}'' \subset \mathcal{F}|_W\) and map \(\varphi'' : \mathcal{F}'' \to \mathcal{G}|_W\). Of course the agreement between \((\mathcal{F}', \varphi')\) and \((\mathcal{F}'', \varphi'')\) over \(W \cap U'\) exactly means that we can extend this to a triple \((U' \cup W, \mathcal{F}''', \varphi''')\). Hence any maximal triple \((U', \mathcal{F}', \varphi')\) (which exist by Zorn’s lemma) must have \(U' = X\) and the proof is complete.

Depth

In this section we talk a little bit about depth and property \((S_k)\) for coherent modules on locally Noetherian schemes. Note that we have already discussed this notion for locally Noetherian schemes in Properties, Section 033P.

Definition

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Let \(k \geq 0\) be an integer.

1. We say \(\mathcal{F}\) has depth \(k\) at a point \(x\) of \(X\) if \(\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) = k\).

2. We say \(X\) has depth \(k\) at a point \(x\) of \(X\) if \(\text{depth}(\mathcal{O}_{X, x}) = k\).

3. We say \(\mathcal{F}\) has property \((S_k)\) if \[\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) \geq \min(k, \dim(\text{Supp}(\mathcal{F}_x)))\] for all \(x \in X\).

4. We say \(X\) has property \((S_k)\) if \(\mathcal{O}_X\) has property \((S_k)\).

Any coherent sheaf satisfies condition \((S_0)\). Condition \((S_1)\) is equivalent to having no embedded associated points, see Divisors, Lemma 0346.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules and \(x \in X\).

1. If \(\mathcal{G}_x\) has depth \(\geq 1\), then \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x\) has depth \(\geq 1\).

2. If \(\mathcal{G}_x\) has depth \(\geq 2\), then \(\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x\) has depth \(\geq 2\).

Proof

Observe that \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) is a coherent \(\mathcal{O}_X\)-module by Lemma 01Y2. Coherent modules are of finite presentation (Lemma 01XZ) hence taking stalks commutes with taking \(\SheafHom\) and \(\Hom\), see Modules, Lemma 01CP. Thus we reduce to the case of finite modules over local rings which is More on Algebra, Lemma 0AV5.

Lemma

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules.

1. If \(\mathcal{G}\) has property \((S_1)\), then \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) has property \((S_1)\).

2. If \(\mathcal{G}\) has property \((S_2)\), then \(\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\) has property \((S_2)\).

Proof

Follows immediately from Lemma 0EBC and the definitions.

We have seen in Properties, Lemma 0342 that a locally Noetherian scheme is Cohen-Macaulay if and only if \((S_k)\) holds for all \(k\). Thus it makes sense to introduce the following definition, which is equivalent to the condition that all stalks are Cohen-Macaulay modules.

Definition

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. We say \(\mathcal{F}\) is Cohen-Macaulay if and only if \((S_k)\) holds for all \(k \geq 0\).

Lemma

Let \(X\) be a regular scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{F}\) is Cohen-Macaulay and \(\text{Supp}(\mathcal{F}) = X\),

2. \(\mathcal{F}\) is finite locally free of rank \(> 0\).

Proof

Let \(x \in X\). If (2) holds, then \(\mathcal{F}_x\) is a free \(\mathcal{O}_{X, x}\)-module of rank \(> 0\). Hence \(\text{depth}(\mathcal{F}_x) = \dim(\mathcal{O}_{X, x})\) because a regular local ring is Cohen-Macaulay (Algebra, Lemma 00NQ). Conversely, if (1) holds, then \(\mathcal{F}_x\) is a maximal Cohen-Macaulay module over \(\mathcal{O}_{X, x}\) (Algebra, Definition 00NF). Hence \(\mathcal{F}_x\) is free by Algebra, Lemma 00NT.

Devissage of coherent sheaves

Let \(X\) be a Noetherian scheme. Consider an integral closed subscheme \(i : Z \to X\). It is often convenient to consider coherent sheaves of the form \(i_*\mathcal{G}\) where \(\mathcal{G}\) is a coherent sheaf on \(Z\). In particular we are interested in these sheaves when \(\mathcal{G}\) is a torsion free rank \(1\) sheaf. For example \(\mathcal{G}\) could be a nonzero sheaf of ideals on \(Z\), or even more specifically \(\mathcal{G} = \mathcal{O}_Z\).

Throughout this section we will use that a coherent sheaf is the same thing as a finite type quasi-coherent sheaf and that a quasi-coherent subquotient of a coherent sheaf is coherent, see Section 01XY. The support of a coherent sheaf is closed, see Modules, Lemma 01BA.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Suppose that \(\text{Supp}(\mathcal{F}) = Z \cup Z'\) with \(Z\), \(Z'\) closed. Then there exists a short exact sequence of coherent sheaves \[0 \to \mathcal{G}' \to \mathcal{F} \to \mathcal{G} \to 0\] with \(\text{Supp}(\mathcal{G}') \subset Z'\) and \(\text{Supp}(\mathcal{G}) \subset Z\).

Proof

Let \(\mathcal{I} \subset \mathcal{O}_X\) be the sheaf of ideals defining the reduced induced closed subscheme structure on \(Z\), see Schemes, Lemma 01J3. Consider the subsheaves \(\mathcal{G}'_n = \mathcal{I}^n\mathcal{F}\) and the quotients \(\mathcal{G}_n = \mathcal{F}/\mathcal{I}^n\mathcal{F}\). For each \(n\) we have a short exact sequence \[0 \to \mathcal{G}'_n \to \mathcal{F} \to \mathcal{G}_n \to 0\] For every point \(x\) of \(Z' \setminus Z\) we have \(\mathcal{I}_x = \mathcal{O}_{X, x}\) and hence \(\mathcal{G}_{n, x} = 0\). Thus we see that \(\text{Supp}(\mathcal{G}_n) \subset Z\). Note that \(X \setminus Z'\) is a Noetherian scheme. Hence by Lemma 01Y9 there exists an \(n\) such that \(\mathcal{G}'_n|_{X \setminus Z'} = \mathcal{I}^n\mathcal{F}|_{X \setminus Z'} = 0\). For such an \(n\) we see that \(\text{Supp}(\mathcal{G}'_n) \subset Z'\). Thus setting \(\mathcal{G}' = \mathcal{G}'_n\) and \(\mathcal{G} = \mathcal{G}_n\) works.

Lemma

Let \(X\) be a Noetherian scheme. Let \(i : Z \to X\) be an integral closed subscheme. Let \(\xi \in Z\) be the generic point. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Assume that \(\mathcal{F}_\xi\) is annihilated by \(\mathfrak m_\xi\). Then there exist an integer \(r \geq 0\) and a coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) and an injective map of coherent sheaves \[i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F}\] which is an isomorphism in a neighbourhood of \(\xi\).

Proof

Let \(\mathcal{J} \subset \mathcal{O}_X\) be the ideal sheaf of \(Z\). Let \(\mathcal{F}' \subset \mathcal{F}\) be the subsheaf of local sections of \(\mathcal{F}\) which are annihilated by \(\mathcal{J}\). It is a quasi-coherent sheaf by Properties, Lemma 01PO. Moreover, \(\mathcal{F}'_\xi = \mathcal{F}_\xi\) because \(\mathcal{J}_\xi = \mathfrak m_\xi\) and part (3) of Properties, Lemma 01PO. By Lemma 01Y3 we see that \(\mathcal{F}' \to \mathcal{F}\) induces an isomorphism in a neighbourhood of \(\xi\). Hence we may replace \(\mathcal{F}\) by \(\mathcal{F}'\) and assume that \(\mathcal{F}\) is annihilated by \(\mathcal{J}\).

Assume \(\mathcal{J}\mathcal{F} = 0\). By Lemma 087T we can write \(\mathcal{F} = i_*\mathcal{G}\) for some coherent sheaf \(\mathcal{G}\) on \(Z\). Suppose we can find a morphism \(\mathcal{I}^{\oplus r} \to \mathcal{G}\) which is an isomorphism in a neighbourhood of the generic point \(\xi\) of \(Z\). Then applying \(i_*\) (which is left exact) we get the result of the lemma. Hence we have reduced to the case \(X = Z\).

Suppose \(Z = X\) is an integral Noetherian scheme with generic point \(\xi\). Note that \(\mathcal{O}_{X, \xi} = \kappa(\xi)\) is the function field of \(X\) in this case. Since \(\mathcal{F}_\xi\) is a finite \(\mathcal{O}_\xi\)-module we see that \(r = \dim_{\kappa(\xi)} \mathcal{F}_\xi\) is finite. Hence the sheaves \(\mathcal{O}_X^{\oplus r}\) and \(\mathcal{F}\) have isomorphic stalks at \(\xi\). By Lemma 01Y4 there exists a nonempty open \(U \subset X\) and a morphism \(\psi : \mathcal{O}_X^{\oplus r}|_U \to \mathcal{F}|_U\) which is an isomorphism at \(\xi\), and hence an isomorphism in a neighbourhood of \(\xi\) by Lemma 01Y3. By Schemes, Lemma 01J3 there exists a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) whose associated closed subscheme \(Y \subset X\) is the complement of \(U\). By Lemma 01YB there exists an \(n \geq 0\) and a morphism \(\mathcal{I}^n(\mathcal{O}_X^{\oplus r}) \to \mathcal{F}\) which recovers our \(\psi\) over \(U\). Since \(\mathcal{I}^n(\mathcal{O}_X^{\oplus r}) = (\mathcal{I}^n)^{\oplus r}\) we get a map as in the lemma. It is injective because \(X\) is integral and it is injective at the generic point of \(X\) (easy proof omitted).

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). There exists a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] by coherent subsheaves such that for each \(j = 1, \ldots, m\) there exist an integral closed subscheme \(Z_j \subset X\) and a nonzero coherent sheaf of ideals \(\mathcal{I}_j \subset \mathcal{O}_{Z_j}\) such that \[\mathcal{F}_j/\mathcal{F}_{j - 1} \cong (Z_j \to X)_* \mathcal{I}_j\]

Proof

Consider the collection \[\mathcal{T} = \left\{ \begin{matrix} Z \subset X \text{ closed such that there exists a coherent sheaf } \mathcal{F} \\ \text{ with } \text{Supp}(\mathcal{F}) = Z \text{ for which the lemma is wrong} \end{matrix} \right\}\] We are trying to show that \(\mathcal{T}\) is empty. If not, then because \(X\) is Noetherian we can choose a minimal element \(Z \in \mathcal{T}\). This means that there exists a coherent sheaf \(\mathcal{F}\) on \(X\) whose support is \(Z\) and for which the lemma does not hold. Clearly \(Z \not = \emptyset\) since the only sheaf whose support is empty is the zero sheaf for which the lemma does hold (with \(m = 0\)).

If \(Z\) is not irreducible, then we can write \(Z = Z_1 \cup Z_2\) with \(Z_1, Z_2\) closed and strictly smaller than \(Z\). Then we can apply Lemma 01YD to get a short exact sequence of coherent sheaves \[0 \to \mathcal{G}_1 \to \mathcal{F} \to \mathcal{G}_2 \to 0\] with \(\text{Supp}(\mathcal{G}_i) \subset Z_i\). By minimality of \(Z\) each of \(\mathcal{G}_i\) has a filtration as in the statement of the lemma. By considering the induced filtration on \(\mathcal{F}\) we arrive at a contradiction. Hence we conclude that \(Z\) is irreducible.

Suppose \(Z\) is irreducible. Let \(\mathcal{J}\) be the sheaf of ideals cutting out the reduced induced closed subscheme structure of \(Z\), see Schemes, Lemma 01J3. By Lemma 01Y9 we see there exists an \(n \geq 0\) such that \(\mathcal{J}^n\mathcal{F} = 0\). Hence we obtain a filtration \[0 = \mathcal{J}^n\mathcal{F} \subset \mathcal{J}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{J}\mathcal{F} \subset \mathcal{F}\] each of whose successive subquotients is annihilated by \(\mathcal{J}\). Hence if each of these subquotients has a filtration as in the statement of the lemma then also \(\mathcal{F}\) does. In other words we may assume that \(\mathcal{J}\) does annihilate \(\mathcal{F}\).

In the case where \(Z\) is irreducible and \(\mathcal{J}\mathcal{F} = 0\) we can apply Lemma 01YE. This gives a short exact sequence \[0 \to i_*(\mathcal{I}^{\oplus r}) \to \mathcal{F} \to \mathcal{Q} \to 0\] where \(\mathcal{Q}\) is defined as the quotient. Since \(\mathcal{Q}\) is zero in a neighbourhood of \(\xi\) by the lemma just cited we see that the support of \(\mathcal{Q}\) is strictly smaller than \(Z\). Hence we see that \(\mathcal{Q}\) has a filtration of the desired type by minimality of \(Z\). But then clearly \(\mathcal{F}\) does too, which is our final contradiction.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\). Assume

1. For any short exact sequence of coherent sheaves \[0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0\] if \(\mathcal{F}_i\), \(i = 1, 2\) have property \(\mathcal{P}\) then so does \(\mathcal{F}\).

2. For every integral closed subscheme \(Z \subset X\) and every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) we have \(\mathcal{P}\) for \(i_*\mathcal{I}\).

Then property \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Proof

First note that if \(\mathcal{F}\) is a coherent sheaf with a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] by coherent subsheaves such that each of \(\mathcal{F}_i/\mathcal{F}_{i - 1}\) has property \(\mathcal{P}\), then so does \(\mathcal{F}\). This follows from the property (1) for \(\mathcal{P}\). On the other hand, by Lemma 01YF we can filter any \(\mathcal{F}\) with successive subquotients as in (2). Hence the lemma follows.

Lemma

Let \(X\) be a Noetherian scheme. Let \(Z_0 \subset X\) be an irreducible closed subset with generic point \(\xi\). Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\) with support contained in \(Z_0\) such that

1. For any short exact sequence of coherent sheaves if two out of three of them have property \(\mathcal{P}\) then so does the third.

2. For every integral closed subscheme \(Z \subset Z_0 \subset X\), \(Z \not = Z_0\) and every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) we have \(\mathcal{P}\) for \((Z \to X)_*\mathcal{I}\).

3. There exists some coherent sheaf \(\mathcal{G}\) on \(X\) such that

   1. \(\text{Supp}(\mathcal{G}) = Z_0\),

   2. \(\mathcal{G}_\xi\) is annihilated by \(\mathfrak m_\xi\),

   3. \(\dim_{\kappa(\xi)} \mathcal{G}_\xi = 1\), and

   4. property \(\mathcal{P}\) holds for \(\mathcal{G}\).

Then property \(\mathcal{P}\) holds for every coherent sheaf \(\mathcal{F}\) on \(X\) whose support is contained in \(Z_0\).

Proof

First note that if \(\mathcal{F}\) is a coherent sheaf with support contained in \(Z_0\) with a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] by coherent subsheaves such that each of \(\mathcal{F}_i/\mathcal{F}_{i - 1}\) has property \(\mathcal{P}\), then so does \(\mathcal{F}\). Or, if \(\mathcal{F}\) has property \(\mathcal{P}\) and all but one of the \(\mathcal{F}_i/\mathcal{F}_{i - 1}\) has property \(\mathcal{P}\) then so does the last one. This follows from assumption (1).

As a first application we conclude that any coherent sheaf whose support is strictly contained in \(Z_0\) has property \(\mathcal{P}\). Namely, such a sheaf has a filtration (see Lemma 01YF) whose subquotients have property \(\mathcal{P}\) according to (2).

Let \(\mathcal{G}\) be as in (3). By Lemma 01YE there exist a sheaf of ideals \(\mathcal{I}\) on \(Z_0\), an integer \(r \geq 1\), and a short exact sequence \[0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0\] where the support of \(\mathcal{Q}\) is strictly contained in \(Z_0\). By (3)(c) we see that \(r = 1\). Since \(\mathcal{Q}\) has property \(\mathcal{P}\) too we conclude that \((Z_0 \to X)_*\mathcal{I}\) has property \(\mathcal{P}\).

Next, suppose that \(\mathcal{I}' \not = 0\) is another quasi-coherent sheaf of ideals on \(Z_0\). Then we can consider the intersection \(\mathcal{I}'' = \mathcal{I}' \cap \mathcal{I}\) and we get two short exact sequences \[0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I} \to \mathcal{Q} \to 0\] and \[0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I}' \to \mathcal{Q}' \to 0.\] Note that the support of the coherent sheaves \(\mathcal{Q}\) and \(\mathcal{Q}'\) are strictly contained in \(Z_0\). Hence \(\mathcal{Q}\) and \(\mathcal{Q}'\) have property \(\mathcal{P}\) (see above). Hence we conclude using (1) that \((Z_0 \to X)_*\mathcal{I}''\) and \((Z_0 \to X)_*\mathcal{I}'\) both have \(\mathcal{P}\) as well.

The final step of the proof is to note that any coherent sheaf \(\mathcal{F}\) on \(X\) whose support is contained in \(Z_0\) has a filtration (see Lemma 01YF again) whose subquotients all have property \(\mathcal{P}\) by what we just said.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\) such that

1. For any short exact sequence of coherent sheaves if two out of three of them have property \(\mathcal{P}\) then so does the third.

2. For every integral closed subscheme \(Z \subset X\) with generic point \(\xi\) there exists some coherent sheaf \(\mathcal{G}\) such that

   1. \(\text{Supp}(\mathcal{G}) = Z\),

   2. \(\mathcal{G}_\xi\) is annihilated by \(\mathfrak m_\xi\),

   3. \(\dim_{\kappa(\xi)} \mathcal{G}_\xi = 1\), and

   4. property \(\mathcal{P}\) holds for \(\mathcal{G}\).

Then property \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Proof

According to Lemma 01YG it suffices to show that for all integral closed subschemes \(Z \subset X\) and all quasi-coherent ideal sheaves \(\mathcal{I} \subset \mathcal{O}_Z\) we have \(\mathcal{P}\) for \((Z \to X)_*\mathcal{I}\). If this fails, then since \(X\) is Noetherian there is a minimal integral closed subscheme \(Z_0 \subset X\) such that \(\mathcal{P}\) fails for \((Z_0 \to X)_*\mathcal{I}_0\) for some quasi-coherent sheaf of ideals \(\mathcal{I}_0 \subset \mathcal{O}_{Z_0}\), but \(\mathcal{P}\) does hold for \((Z \to X)_*\mathcal{I}\) for all integral closed subschemes \(Z \subset Z_0\), \(Z \not = Z_0\) and quasi-coherent ideal sheaves \(\mathcal{I} \subset \mathcal{O}_Z\). Since we have the existence of \(\mathcal{G}\) for \(Z_0\) by part (2), according to Lemma 01YH this cannot happen.

Lemma

Let \(X\) be a Noetherian scheme. Let \(Z_0 \subset X\) be an irreducible closed subset with generic point \(\xi\). Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\) such that

1. For any short exact sequence of coherent sheaves \[0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0\] if \(\mathcal{F}_i\), \(i = 1, 2\) have property \(\mathcal{P}\) then so does \(\mathcal{F}\).

2. If \(\mathcal{P}\) holds for \(\mathcal{F}^{\oplus r}\) for some \(r \geq 1\), then it holds for \(\mathcal{F}\).

3. For every integral closed subscheme \(Z \subset Z_0 \subset X\), \(Z \not = Z_0\) and every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Z\) we have \(\mathcal{P}\) for \((Z \to X)_*\mathcal{I}\).

4. There exists some coherent sheaf \(\mathcal{G}\) such that

   1. \(\text{Supp}(\mathcal{G}) = Z_0\),

   2. \(\mathcal{G}_\xi\) is annihilated by \(\mathfrak m_\xi\), and

   3. for every quasi-coherent sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_X\) such that \(\mathcal{J}_\xi = \mathcal{O}_{X, \xi}\) there exists a quasi-coherent subsheaf \(\mathcal{G}' \subset \mathcal{J}\mathcal{G}\) with \(\mathcal{G}'_\xi = \mathcal{G}_\xi\) and such that \(\mathcal{P}\) holds for \(\mathcal{G}'\).

Then property \(\mathcal{P}\) holds for every coherent sheaf \(\mathcal{F}\) on \(X\) whose support is contained in \(Z_0\).

Proof

Note that if \(\mathcal{F}\) is a coherent sheaf with a filtration \[0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F}\] by coherent subsheaves such that each of \(\mathcal{F}_i/\mathcal{F}_{i - 1}\) has property \(\mathcal{P}\), then so does \(\mathcal{F}\). This follows from assumption (1).

As a first application we conclude that any coherent sheaf whose support is strictly contained in \(Z_0\) has property \(\mathcal{P}\). Namely, such a sheaf has a filtration (see Lemma 01YF) whose subquotients have property \(\mathcal{P}\) according to (3).

Let us denote \(i : Z_0 \to X\) the closed immersion. Consider a coherent sheaf \(\mathcal{G}\) as in (4). By Lemma 01YE there exists a sheaf of ideals \(\mathcal{I}\) on \(Z_0\) and a short exact sequence \[0 \to i_*\mathcal{I}^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0\] where the support of \(\mathcal{Q}\) is strictly contained in \(Z_0\). In particular \(r > 0\) and \(\mathcal{I}\) is nonzero because the support of \(\mathcal{G}\) is equal to \(Z_0\). Let \(\mathcal{I}' \subset \mathcal{I}\) be any nonzero quasi-coherent sheaf of ideals on \(Z_0\) contained in \(\mathcal{I}\). Then we also get a short exact sequence \[0 \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{G} \to \mathcal{Q}' \to 0\] where \(\mathcal{Q}'\) has support properly contained in \(Z_0\). Let \(\mathcal{J} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals cutting out the support of \(\mathcal{Q}'\) (for example the ideal corresponding to the reduced induced closed subscheme structure on the support of \(\mathcal{Q}'\)). Then \(\mathcal{J}_\xi = \mathcal{O}_{X, \xi}\). By Lemma 01Y9 we see that \(\mathcal{J}^n\mathcal{Q}' = 0\) for some \(n\). Hence \(\mathcal{J}^n\mathcal{G} \subset i_*(\mathcal{I}')^{\oplus r}\). By assumption (4)(c) of the lemma we see there exists a quasi-coherent subsheaf \(\mathcal{G}' \subset \mathcal{J}^n\mathcal{G}\) with \(\mathcal{G}'_\xi = \mathcal{G}_\xi\) for which property \(\mathcal{P}\) holds. Hence we get a short exact sequence \[0 \to \mathcal{G}' \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{Q}'' \to 0\] where \(\mathcal{Q}''\) has support properly contained in \(Z_0\). Thus by our initial remarks and property (1) of the lemma we conclude that \(i_*(\mathcal{I}')^{\oplus r}\) satisfies \(\mathcal{P}\). Hence we see that \(i_*\mathcal{I}'\) satisfies \(\mathcal{P}\) by (2). Finally, for an arbitrary quasi-coherent sheaf of ideals \(\mathcal{I}'' \subset \mathcal{O}_{Z_0}\) we can set \(\mathcal{I}' = \mathcal{I}'' \cap \mathcal{I}\) and we get a short exact sequence \[0 \to i_*(\mathcal{I}') \to i_*(\mathcal{I}'') \to \mathcal{Q}''' \to 0\] where \(\mathcal{Q}'''\) has support properly contained in \(Z_0\). Hence we conclude that property \(\mathcal{P}\) holds for \(i_*\mathcal{I}''\) as well.

The final step of the proof is to note that any coherent sheaf \(\mathcal{F}\) on \(X\) whose support is contained in \(Z_0\) has a filtration (see Lemma 01YF again) whose subquotients all have property \(\mathcal{P}\) by what we just said.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{P}\) be a property of coherent sheaves on \(X\) such that

1. For any short exact sequence of coherent sheaves \[0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0\] if \(\mathcal{F}_i\), \(i = 1, 2\) have property \(\mathcal{P}\) then so does \(\mathcal{F}\).

2. If \(\mathcal{P}\) holds for \(\mathcal{F}^{\oplus r}\) for some \(r \geq 1\), then it holds for \(\mathcal{F}\).

3. For every integral closed subscheme \(Z \subset X\) with generic point \(\xi\) there exists some coherent sheaf \(\mathcal{G}\) such that

   1. \(\text{Supp}(\mathcal{G}) = Z\),

   2. \(\mathcal{G}_\xi\) is annihilated by \(\mathfrak m_\xi\), and

   3. for every quasi-coherent sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_X\) such that \(\mathcal{J}_\xi = \mathcal{O}_{X, \xi}\) there exists a quasi-coherent subsheaf \(\mathcal{G}' \subset \mathcal{J}\mathcal{G}\) with \(\mathcal{G}'_\xi = \mathcal{G}_\xi\) and such that \(\mathcal{P}\) holds for \(\mathcal{G}'\).

Then property \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Proof

Follows from Lemma 01YL in exactly the same way that Lemma 01YI follows from Lemma 01YH.

Finite morphisms and affines

In this section we use the results of the preceding sections to show that the image of a Noetherian affine scheme under a finite morphism is affine. We will see later that this result holds more generally (see Limits, Lemma 01ZT and Proposition 05YU).

Lemma

Let \(f : Y \to X\) be a morphism of schemes. Assume \(f\) is finite, surjective and \(X\) locally Noetherian. Let \(Z \subset X\) be an integral closed subscheme with generic point \(\xi\). Then there exists a coherent sheaf \(\mathcal{F}\) on \(Y\) such that the support of \(f_*\mathcal{F}\) is equal to \(Z\) and \((f_*\mathcal{F})_\xi\) is annihilated by \(\mathfrak m_\xi\).

Proof

Note that \(Y\) is locally Noetherian by Morphisms, Lemma 01T6. Because \(f\) is surjective the fibre \(Y_\xi\) is not empty. Pick \(\xi' \in Y\) mapping to \(\xi\). Let \(Z' = \overline{\{\xi'\}}\). We may think of \(Z' \subset Y\) as a reduced closed subscheme, see Schemes, Lemma 01J3. Hence the sheaf \(\mathcal{F} = (Z' \to Y)_*\mathcal{O}_{Z'}\) is a coherent sheaf on \(Y\) (see Lemma 01Y6). Look at the commutative diagram \[\xymatrix{ Z' \ar[r]_{i'} \ar[d]_{f'} & Y \ar[d]^f \\ Z \ar[r]^i & X }\] We see that \(f_*\mathcal{F} = i_*f'_*\mathcal{O}_{Z'}\). Hence the stalk of \(f_*\mathcal{F}\) at \(\xi\) is the stalk of \(f'_*\mathcal{O}_{Z'}\) at \(\xi\). Note that since \(Z'\) is integral with generic point \(\xi'\) we have that \(\xi'\) is the only point of \(Z'\) lying over \(\xi\), see Algebra, Lemmas 00GK and 00GT. Hence the stalk of \(f'_*\mathcal{O}_{Z'}\) at \(\xi\) equal \(\mathcal{O}_{Z', \xi'} = \kappa(\xi')\). In particular the stalk of \(f_*\mathcal{F}\) at \(\xi\) is not zero. This combined with the fact that \(f_*\mathcal{F}\) is of the form \(i_*f'_*(\text{something})\) implies the lemma.

Lemma

Let \(f : Y \to X\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(Y\). Let \(\mathcal{I}\) be a quasi-coherent sheaf of ideals on \(X\). If the morphism \(f\) is affine then \(\mathcal{I}f_*\mathcal{F} = f_*(f^{-1}\mathcal{I}\mathcal{F})\).

Proof

The notation means the following. Since \(f^{-1}\) is an exact functor we see that \(f^{-1}\mathcal{I}\) is a sheaf of ideals of \(f^{-1}\mathcal{O}_X\). Via the map \(f^\sharp : f^{-1}\mathcal{O}_X \to \mathcal{O}_Y\) this acts on \(\mathcal{F}\). Then \(f^{-1}\mathcal{I}\mathcal{F}\) is the subsheaf generated by sums of local sections of the form \(as\) where \(a\) is a local section of \(f^{-1}\mathcal{I}\) and \(s\) is a local section of \(\mathcal{F}\). It is a quasi-coherent \(\mathcal{O}_Y\)-submodule of \(\mathcal{F}\) because it is also the image of a natural map \(f^*\mathcal{I} \otimes_{\mathcal{O}_Y} \mathcal{F} \to \mathcal{F}\).

Having said this the proof is straightforward. Namely, the question is local and hence we may assume \(X\) is affine. Since \(f\) is affine we see that \(Y\) is affine too. Thus we may write \(Y = \Spec(B)\), \(X = \Spec(A)\), \(\mathcal{F} = \widetilde{M}\), and \(\mathcal{I} = \widetilde{I}\). The assertion of the lemma in this case boils down to the statement that \[I(M_A) = ((IB)M)_A\] where \(M_A\) indicates the \(A\)-module associated to the \(B\)-module \(M\).

Lemma

Let \(f : Y \to X\) be a morphism of schemes. Assume

1. \(f\) finite,

2. \(f\) surjective,

3. \(Y\) affine, and

4. \(X\) Noetherian.

Then \(X\) is affine.

Proof

We will prove that under the assumptions of the lemma for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we have \(H^1(X, \mathcal{F}) = 0\). This will in particular imply that \(H^1(X, \mathcal{I}) = 0\) for every quasi-coherent sheaf of ideals of \(\mathcal{O}_X\). Then it follows that \(X\) is affine from either Lemma 01XF or Lemma 01XG.

Let \(\mathcal{P}\) be the property of coherent sheaves \(\mathcal{F}\) on \(X\) defined by the rule \[\mathcal{P}(\mathcal{F}) \Leftrightarrow H^1(X, \mathcal{F}) = 0.\] We are going to apply Lemma 01YM. Thus we have to verify (1), (2) and (3) of that lemma for \(\mathcal{P}\). Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves. Property (2) follows since \(H^1(X, -)\) is an additive functor. To see (3) let \(Z \subset X\) be an integral closed subscheme with generic point \(\xi\). Let \(\mathcal{F}\) be a coherent sheaf on \(Y\) such that the support of \(f_*\mathcal{F}\) is equal to \(Z\) and \((f_*\mathcal{F})_\xi\) is annihilated by \(\mathfrak m_\xi\), see Lemma 01YO. We claim that taking \(\mathcal{G} = f_*\mathcal{F}\) works. We only have to verify part (3)(c) of Lemma 01YM. Hence assume that \(\mathcal{J} \subset \mathcal{O}_X\) is a quasi-coherent sheaf of ideals such that \(\mathcal{J}_\xi = \mathcal{O}_{X, \xi}\). A finite morphism is affine hence by Lemma 01YP we see that \(\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})\). Also, as pointed out in the proof of Lemma 01YP the sheaf \(f^{-1}\mathcal{J}\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_Y\)-module. Since \(Y\) is affine we see that \(H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0\), see Lemma 01XB. Since \(f\) is finite, hence affine, we see that \[H^1(X, \mathcal{J}\mathcal{G}) = H^1(X, f_*(f^{-1}\mathcal{J}\mathcal{F})) = H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0\] by Lemma 089W. Hence the quasi-coherent subsheaf \(\mathcal{G}' = \mathcal{J}\mathcal{G}\) satisfies \(\mathcal{P}\). This verifies property (3)(c) of Lemma 01YM as desired.

Coherent sheaves on Proj, I

In this section we discuss coherent sheaves on \(\text{Proj}(A)\) where \(A\) is a Noetherian graded ring generated by \(A_1\) over \(A_0\). In the next section we discuss what happens if \(A\) is not generated by degree \(1\) elements. First, we formulate an all-in-one result for projective space over a Noetherian ring.

Lemma

Let \(R\) be a Noetherian ring. Let \(n \geq 0\) be an integer. For every coherent sheaf \(\mathcal{F}\) on \(\mathbf{P}^n_R\) we have the following:

1. There exists an \(r \geq 0\) and \(d_1, \ldots, d_r \in \mathbf{Z}\) and a surjection \[\bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j) \longrightarrow \mathcal{F}.\]

2. We have \(H^i(\mathbf{P}^n_R, \mathcal{F}) = 0\) unless \(0 \leq i \leq n\).

3. For any \(i\) the cohomology group \(H^i(\mathbf{P}^n_R, \mathcal{F})\) is a finite \(R\)-module.

4. If \(i > 0\), then \(H^i(\mathbf{P}^n_R, \mathcal{F}(d)) = 0\) for all \(d\) large enough.

5. For any \(k \in \mathbf{Z}\) the graded \(R[T_0, \ldots, T_n]\)-module \[\bigoplus\nolimits_{d \geq k} H^0(\mathbf{P}^n_R, \mathcal{F}(d))\] is a finite \(R[T_0, \ldots, T_n]\)-module.

Proof

We will use that \(\mathcal{O}_{\mathbf{P}^n_R}(1)\) is an ample invertible sheaf on the scheme \(\mathbf{P}^n_R\). This follows directly from the definition since \(\mathbf{P}^n_R\) covered by the standard affine opens \(D_{+}(T_i)\). Hence by Properties, Proposition 01Q3 every finite type quasi-coherent \(\mathcal{O}_{\mathbf{P}^n_R}\)-module is a quotient of a finite direct sum of tensor powers of \(\mathcal{O}_{\mathbf{P}^n_R}(1)\). On the other hand coherent sheaves and finite type quasi-coherent sheaves are the same thing on projective space over \(R\) by Lemma 01XZ. Thus we see (1).

Projective \(n\)-space \(\mathbf{P}^n_R\) is covered by \(n + 1\) affines, namely the standard opens \(D_{+}(T_i)\), \(i = 0, \ldots, n\), see Constructions, Lemma 01NG. Hence we see that for any quasi-coherent sheaf \(\mathcal{F}\) on \(\mathbf{P}^n_R\) we have \(H^i(\mathbf{P}^n_R, \mathcal{F}) = 0\) for \(i \geq n + 1\), see Lemma 01XI. Hence (2) holds.

Let us prove (3) and (4) simultaneously for all coherent sheaves on \(\mathbf{P}^n_R\) by descending induction on \(i\). Clearly the result holds for \(i \geq n + 1\) by (2). Suppose we know the result for \(i + 1\) and we want to show the result for \(i\). (If \(i = 0\), then part (4) is vacuous.) Let \(\mathcal{F}\) be a coherent sheaf on \(\mathbf{P}^n_R\). Choose a surjection as in (1) and denote \(\mathcal{G}\) the kernel so that we have a short exact sequence \[0 \to \mathcal{G} \to \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j) \to \mathcal{F} \to 0\] By Lemma 01Y0 we see that \(\mathcal{G}\) is coherent. The long exact cohomology sequence gives an exact sequence \[H^i(\mathbf{P}^n_R, \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j)) \to H^i(\mathbf{P}^n_R, \mathcal{F}) \to H^{i + 1}(\mathbf{P}^n_R, \mathcal{G}).\] By induction assumption the right \(R\)-module is finite and by Lemma 01XT the left \(R\)-module is finite. Since \(R\) is Noetherian it follows immediately that \(H^i(\mathbf{P}^n_R, \mathcal{F})\) is a finite \(R\)-module. This proves the induction step for assertion (3). Since \(\mathcal{O}_{\mathbf{P}^n_R}(d)\) is invertible we see that twisting on \(\mathbf{P}^n_R\) is an exact functor (since you get it by tensoring with an invertible sheaf, see Constructions, Definition 01MN). This means that for all \(d \in \mathbf{Z}\) the sequence \[0 \to \mathcal{G}(d) \to \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j + d) \to \mathcal{F}(d) \to 0\] is short exact. The resulting cohomology sequence is \[H^i(\mathbf{P}^n_R, \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j + d)) \to H^i(\mathbf{P}^n_R, \mathcal{F}(d)) \to H^{i + 1}(\mathbf{P}^n_R, \mathcal{G}(d)).\] By induction assumption we see the module on the right is zero for \(d \gg 0\) and by the computation in Lemma 01XT the module on the left is zero as soon as \(d \geq -\min\{d_j\}\) and \(i \geq 1\). Hence the induction step for assertion (4). This concludes the proof of (3) and (4).

In order to prove (5) note that for all sufficiently large \(d\) the map \[H^0(\mathbf{P}^n_R, \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j + d)) \to H^0(\mathbf{P}^n_R, \mathcal{F}(d))\] is surjective by the vanishing of \(H^1(\mathbf{P}^n_R, \mathcal{G}(d))\) we just proved. In other words, the module \[M_k = \bigoplus\nolimits_{d \geq k} H^0(\mathbf{P}^n_R, \mathcal{F}(d))\] is for \(k\) large enough a quotient of the corresponding module \[N_k = \bigoplus\nolimits_{d \geq k} H^0(\mathbf{P}^n_R, \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_R}(d_j + d) )\] When \(k\) is sufficiently small (e.g. \(k < -d_j\) for all \(j\)) then \[N_k = \bigoplus\nolimits_{j = 1, \ldots, r} R[T_0, \ldots, T_n](d_j)\] by our computations in Section 01XS. In particular it is finitely generated. Suppose \(k \in \mathbf{Z}\) is arbitrary. Choose \(k_{-} \ll k \ll k_{+}\). Consider the diagram \[\xymatrix{ N_{k_{-}} & N_{k_{+}} \ar[d] \ar[l] \\ M_k & M_{k_{+}} \ar[l] }\] where the vertical arrow is the surjective map above and the horizontal arrows are the obvious inclusion maps. By what was said above we see that \(N_{k_{-}}\) is a finitely generated \(R[T_0, \ldots, T_n]\)-module. Hence \(N_{k_{+}}\) is a finitely generated \(R[T_0, \ldots, T_n]\)-module because it is a submodule of a finitely generated module and the ring \(R[T_0, \ldots, T_n]\) is Noetherian. Since the vertical arrow is surjective we conclude that \(M_{k_{+}}\) is a finitely generated \(R[T_0, \ldots, T_n]\)-module. The quotient \(M_k/M_{k_{+}}\) is finite as an \(R\)-module since it is a finite direct sum of the finite \(R\)-modules \(H^0(\mathbf{P}^n_R, \mathcal{F}(d))\) for \(k \leq d < k_{+}\). Note that we use part (3) for \(i = 0\) here. Hence \(M_k/M_{k_{+}}\) is a fortiori a finite \(R[T_0, \ldots, T_n]\)-module. In other words, we have sandwiched \(M_k\) between two finite \(R[T_0, \ldots, T_n]\)-modules and we win.

Lemma

Let \(A\) be a graded ring such that \(A_0\) is Noetherian and \(A\) is generated by finitely many elements of \(A_1\) over \(A_0\). Set \(X = \text{Proj}(A)\). Then \(X\) is a Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module.

1. There exists an \(r \geq 0\) and \(d_1, \ldots, d_r \in \mathbf{Z}\) and a surjection \[\bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_X(d_j) \longrightarrow \mathcal{F}.\]

2. For any \(p\) the cohomology group \(H^p(X, \mathcal{F})\) is a finite \(A_0\)-module.

3. If \(p > 0\), then \(H^p(X, \mathcal{F}(d)) = 0\) for all \(d\) large enough.

4. For any \(k \in \mathbf{Z}\) the graded \(A\)-module \[\bigoplus\nolimits_{d \geq k} H^0(X, \mathcal{F}(d))\] is a finite \(A\)-module.

Proof

By assumption there exists a surjection of graded \(A_0\)-algebras \[A_0[T_0, \ldots, T_n] \longrightarrow A\] where \(\deg(T_j) = 1\) for \(j = 0, \ldots, n\). By Constructions, Lemma 01N1 this defines a closed immersion \(i : X \to \mathbf{P}^n_{A_0}\) such that \(i^*\mathcal{O}_{\mathbf{P}^n_{A_0}}(1) = \mathcal{O}_X(1)\). In particular, \(X\) is Noetherian as a closed subscheme of the Noetherian scheme \(\mathbf{P}^n_{A_0}\). We claim that the results of the lemma for \(\mathcal{F}\) follow from the corresponding results of Lemma 01YS for the coherent sheaf \(i_*\mathcal{F}\) (Lemma 087T) on \(\mathbf{P}^n_{A_0}\). For example, by this lemma there exists a surjection \[\bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_{A_0}}(d_j) \longrightarrow i_*\mathcal{F}.\] By adjunction this corresponds to a map \(\bigoplus_{j = 1, \ldots, r} \mathcal{O}_X(d_j) \longrightarrow \mathcal{F}\) which is surjective as well. The statements on cohomology follow from the fact that \(H^p(X, \mathcal{F}(d)) = H^p(\mathbf{P}^n_{A_0}, i_*\mathcal{F}(d))\) by Lemma 089W.

Lemma

Let \(A\) be a graded ring such that \(A_0\) is Noetherian and \(A\) is generated by finitely many elements of \(A_1\) over \(A_0\). Let \(M\) be a finite graded \(A\)-module. Set \(X = \text{Proj}(A)\) and let \(\widetilde{M}\) be the quasi-coherent \(\mathcal{O}_X\)-module on \(X\) associated to \(M\). The maps \[M_n \longrightarrow \Gamma(X, \widetilde{M}(n))\] from Constructions, Lemma 01MT are isomorphisms for all sufficiently large \(n\).

Proof

Because \(M\) is a finite \(A\)-module we see that \(\widetilde{M}\) is a finite type \(\mathcal{O}_X\)-module, i.e., a coherent \(\mathcal{O}_X\)-module. Set \(N = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \widetilde{M}(n))\). We have to show that the map \(M \to N\) of graded \(A\)-modules is an isomorphism in all sufficiently large degrees. By Properties, Lemma 0AG5 we have a canonical isomorphism \(\widetilde{N} \to \widetilde{M}\) such that the induced maps \(N_n \to N_n = \Gamma(X, \widetilde{M}(n))\) are the identity maps. Thus we have maps \(\widetilde{M} \to \widetilde{N} \to \widetilde{M}\) such that for all \(n\) the diagram \[\xymatrix{ M_n \ar[d] \ar[r] & N_n \ar[d] \ar@{=}[rd] \\ \Gamma(X, \widetilde{M}(n)) \ar[r] & \Gamma(X, \widetilde{N}(n)) \ar[r]^{\cong} & \Gamma(X, \widetilde{M}(n)) }\] is commutative. This means that the composition \[M_n \to \Gamma(X, \widetilde{M}(n)) \to \Gamma(X, \widetilde{N}(n)) \to \Gamma(X, \widetilde{M}(n))\] is equal to the canonical map \(M_n \to \Gamma(X, \widetilde{M}(n))\). Clearly this implies that the composition \(\widetilde{M} \to \widetilde{N} \to \widetilde{M}\) is the identity. Hence \(\widetilde{M} \to \widetilde{N}\) is an isomorphism. Let \(K = \Ker(M \to N)\) and \(Q = \Coker(M \to N)\). Recall that the functor \(M \mapsto \widetilde{M}\) is exact, see Constructions, Lemma 01M7. Hence we see that \(\widetilde{K} = 0\) and \(\widetilde{Q} = 0\). Recall that \(A\) is a Noetherian ring, \(M\) is a finitely generated \(A\)-module, and \(N\) is a graded \(A\)-module such that \(N' = \bigoplus_{n \geq 0} N_n\) is finitely generated by the last part of Lemma 0AG6. Hence \(K' = \bigoplus_{n \geq 0} K_n\) and \(Q' = \bigoplus_{n \geq 0} Q_n\) are finite \(A\)-modules. Observe that \(\widetilde{Q} = \widetilde{Q'}\) and \(\widetilde{K} = \widetilde{K'}\). Thus to finish the proof it suffices to show that a finite \(A\)-module \(K\) with \(\widetilde{K} = 0\) has only finitely many nonzero homogeneous parts \(K_d\) with \(d \geq 0\). To do this, let \(x_1, \ldots, x_r \in K\) be homogeneous generators say sitting in degrees \(d_1, \ldots, d_r\). Let \(f_1, \ldots, f_n \in A_1\) be elements generating \(A\) over \(A_0\). For each \(i\) and \(j\) there exists an \(n_{ij} \geq 0\) such that \(f_i^{n_{ij}} x_j = 0\) in \(K_{d_j + n_{ij}}\): if not then \(x_i/f_i^{d_i} \in K_{(f_i)}\) would not be zero, i.e., \(\widetilde{K}\) would not be zero. Then we see that \(K_d\) is zero for \(d > \max_j(d_j + \sum_i n_{ij})\) as every element of \(K_d\) is a sum of terms where each term is a monomials in the \(f_i\) times one of the \(x_j\) of total degree \(d\).

Let \(A\) be a graded ring such that \(A_0\) is Noetherian and \(A\) is generated by finitely many elements of \(A_1\) over \(A_0\). Recall that \(A_+ = \bigoplus_{n > 0} A_n\) is the irrelevant ideal. Let \(M\) be a graded \(A\)-module. Recall that \(M\) is an \(A_+\)-power torsion module if for all \(x \in M\) there is an \(n \geq 1\) such that \((A_+)^n x = 0\), see More on Algebra, Definition 05E6. If \(M\) is finitely generated, then we see that this is equivalent to \(M_n = 0\) for \(n \gg 0\). Sometimes \(A_+\)-power torsion modules are called torsion modules. Sometimes a graded \(A\)-module \(M\) is called torsion free if \(x \in M\) with \((A_+)^n x = 0\), \(n > 0\) implies \(x = 0\). Denote \(\text{Mod}_A\) the category of graded \(A\)-modules, \(\text{Mod}^{fg}_A\) the full subcategory of finitely generated ones, and \(\text{Mod}^{fg}_{A, torsion}\) the full subcategory of modules \(M\) such that \(M_n = 0\) for \(n \gg 0\).

Proposition

Let \(A\) be a graded ring such that \(A_0\) is Noetherian and \(A\) is generated by finitely many elements of \(A_1\) over \(A_0\). Set \(X = \text{Proj}(A)\). The functor \(M \mapsto \widetilde M\) induces an equivalence \[\text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, torsion} \longrightarrow \textit{Coh}(\mathcal{O}_X)\] whose quasi-inverse is given by \(\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))\).

Proof

The subcategory \(\text{Mod}^{fg}_{A, torsion}\) is a Serre subcategory of \(\text{Mod}^{fg}_A\), see Homology, Definition 02MO. This is clear from the description of objects given above but it also follows from More on Algebra, Lemma 0A6K. Hence the quotient category on the left of the arrow is defined in Homology, Lemma 02MS. To define the functor of the proposition, it suffices to show that the functor \(M \mapsto \widetilde M\) sends torsion modules to \(0\). This is clear because for any \(f \in A_+\) homogeneous the module \(M_f\) is zero and hence the value \(M_{(f)}\) of \(\widetilde M\) on \(D_+(f)\) is zero too.

By Lemma 0AG6 the proposed quasi-inverse makes sense. Namely, the lemma shows that \(\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))\) is a functor \(\textit{Coh}(\mathcal{O}_X) \to \text{Mod}^{fg}_A\) which we can compose with the quotient functor \(\text{Mod}^{fg}_A \to \text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, torsion}\).

By Lemma 0AG7 the composite left to right to left is isomorphic to the identity functor.

Finally, let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Set \(M = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F}(n))\) viewed as a graded \(A\)-module, so that our functor sends \(\mathcal{F}\) to \(M_{\geq 0} = \bigoplus_{n \geq 0} M_n\). By Properties, Lemma 0AG5 the canonical map \(\widetilde M \to \mathcal{F}\) is an isomorphism. Since the inclusion map \(M_{\geq 0} \to M\) defines an isomorphism \(\widetilde{M_{\geq 0}} \to \widetilde M\) we conclude that the composite right to left to right is isomorphic to the identity functor as well.

Lemma

Let \(K\) be a field, let \(r\geq1\), put \(S=K[t_0,\ldots,t_r]\), and set \(X=\mathbf P^r_K\). Let \(M\) be a finite graded \(S\)-module and let \(\mathcal F=\widetilde M\). Denote the canonical maps by \[\alpha_n:M_n\longrightarrow H^0(X,\mathcal F(n)).\] Then:

1. \(M\) has projective dimension at most \(r\) if and only if \(\alpha_n\) is injective for every \(n\in\mathbf Z\).

2. If \(1\leq k\leq r\), then \(M\) has projective dimension at most \(r-k\) if and only if every \(\alpha_n\) is an isomorphism and \[H^q(X,\mathcal F(n))=0\] for every \(n\in\mathbf Z\) and every \(0<q<k\).

Proof

Put \(\Omega=S(-r-1)\). On the standard affine covering, Constructions, Lemma 01M7 and Lemma 01XD identify the module \(B^q(M)\) of More on Algebra, Lemma more-algebra-lemma-graded-local-duality-polynomial-ring with \(\bigoplus_n H^q(X,\mathcal F(n))\). For \(q=0\), its canonical map \(M\to B^0(M)\) is the sum of the maps \(\alpha_n\). Every homogeneous piece in sight is finite dimensional over \(K\) by Lemma 01YS.

Algebra, Lemmas algebra-lemma-graded-ext-properties and algebra-lemma-projective-dimension-ext-finite-module show that a projective-dimension bound is equivalent to the corresponding vanishing of \(\text{GrExt}^i_S(M,\Omega)\). Graded local duality identifies these Ext modules with \(B^{r-i}(M)^\vee\) for \(0\leq i<r\) and gives the exact sequence \[0\to\text{GrExt}^r_S(M,\Omega)\to B^0(M)^\vee \xrightarrow{\alpha^\vee}M^\vee\to \text{GrExt}^{r+1}_S(M,\Omega)\to0.\] Because all homogeneous pieces are finite dimensional, the last Ext module vanishes exactly when every \(\alpha_n\) is injective. This proves (1). Vanishing of both exceptional Ext modules says that every \(\alpha_n\) is an isomorphism, while vanishing in degrees \(r-k<i<r\) says precisely that \(B^q(M)=0\) for \(0<q<k\). This proves (2).

Lemma

Let \(k\) be a field, let \(r\geq1\), and set \(X=\mathbf{P}^r_k\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module and let \(0\leq q\leq r\). The following are equivalent:

1. \(H^q(X,\mathcal{F}(-n))=0\) for all sufficiently large \(n\),

2. \[\Ext^{r-q}_{\mathcal{O}_{X,x}} (\mathcal{F}_x,\mathcal{O}_{X,x})=0\] for every \(x\in X\).

Proof

Put \(S=k[T_0,\ldots,T_r]\) with its standard grading and \(\Omega=S(-r-1)\). By Proposition 0BXD, choose a finite graded \(S\)-module \(M\) and an isomorphism \(\widetilde M\cong\mathcal{F}\). Use the notation \(B^q(M)\) and \(T^q(M)=B^q(M)^\vee\) of More on Algebra, Lemma more-algebra-lemma-graded-local-duality-polynomial-ring.

On the standard affine covering, Constructions, Lemma 01M7 identifies the degree \(-n\) part of the positive localization complex of \(M\) with the alternating Čech complex of \(\mathcal{F}(-n)\). Lemma 01XD therefore gives \[H^q(X,\mathcal{F}(-n))=B^q(M)_{-n}.\] By the definition of the graded \(k\)-dual, condition (1) is equivalent to \(T^q(M)_n=0\) for all sufficiently large \(n\).

Set \[E=\text{GrExt}^{r-q}_S(M,\Omega).\] If \(q>0\), More on Algebra, Lemma more-algebra-lemma-graded-local-duality-polynomial-ring identifies \(E\) with \(T^q(M)\). If \(q=0\), the five-term sequence in that lemma and the equality \(M^\vee_n=0\) for \(n\gg0\) identify \(E_n\) with \(T^0(M)_n\) for all sufficiently large \(n\). Thus condition (1) is equivalent to \(E_n=0\) for \(n\gg0\).

The graded module \(E\) is finite by Algebra, Lemma algebra-lemma-graded-ext-properties. For a finite graded module, \(E_n=0\) for \(n\gg0\) is equivalent to \(\widetilde E=0\). Indeed, one direction follows from Lemma 0AG7; for the other, high-degree vanishing and finite generation give \(S_+^aE=0\) for some \(a\), so every homogeneous localization defining \(\widetilde E\) is zero.

It remains to test \(\widetilde E\) on stalks. If \(x\) corresponds to the homogeneous prime \(\mathfrak p\), Constructions, Lemma constructions-lemma-proj-internal-hom-ext gives \[(\widetilde E)_x = \Ext^{r-q}_{S_{(\mathfrak p)}} \left(M_{(\mathfrak p)},\Omega_{(\mathfrak p)}\right).\] Here \(S_{(\mathfrak p)}=\mathcal{O}_{X,x}\), \(M_{(\mathfrak p)}=\mathcal{F}_x\), and \(\widetilde\Omega=\mathcal{O}_X(-r-1)\) is invertible. After choosing a local basis of this invertible module, the last display is the Ext module in condition (2). Hence \(\widetilde E=0\) if and only if condition (2) holds at every point.

Lemma

Let \(k\) be a field, let \(r\geq1\), and set \(X=\mathbf{P}^r_k\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module and let \(0\leq p\leq r\). The following are equivalent:

1. \(H^q(X,\mathcal{F}(-n))=0\) for all sufficiently large \(n\) and every \(0\leq q<p\),

2. for every \(x\in X\), the \(\mathcal{O}_{X,x}\)-module \(\mathcal{F}_x\) has projective dimension at most \(r-p\).

Proof

Every point \(x\) lies in a standard affine chart of \(X\), whose coordinate ring is a polynomial ring in \(r\) variables over \(k\). Algebra, Proposition 00OQ and Lemma 00O8 show that \(\mathcal{O}_{X,x}\) has global dimension at most \(r\). Thus the finite module \(\mathcal{F}_x\) has finite projective dimension at most \(r\).

By Algebra, Lemma algebra-lemma-projective-dimension-ext-local-ring, condition (2) is equivalent to \[\Ext^i_{\mathcal{O}_{X,x}} (\mathcal{F}_x,\mathcal{O}_{X,x})=0 \quad\text{for }r-p<i\leq r\] at every point \(x\). Writing \(i=r-q\), this is exactly the collection of local conditions for \(0\leq q<p\) in Lemma coherent-lemma-negative-twists-local-ext, and the result follows.

Lemma

Let \(k\) be a field, let \(r\geq1\), and set \(X=\mathbf{P}^r_k\). Let \(0\leq c\leq r\) and let \(i:Z\to X\) be a closed immersion such that, for every \(z\in Z\), the kernel of \[\mathcal{O}_{X,i(z)}\longrightarrow\mathcal{O}_{Z,z}\] can be generated by a regular sequence of length \(c\). If \(\mathcal{E}\) is a finite locally free \(\mathcal{O}_Z\)-module, then \[H^q\left(Z,\mathcal{E}\otimes_{\mathcal{O}_Z} i^*\mathcal{O}_X(-n)\right)=0\] for all sufficiently large \(n\) and every \(0\leq q<r-c\).

Proof

Set \(\mathcal{G}=i_*\mathcal{E}\). A closed immersion is finite by Morphisms, Lemma 035C, and \(X\) is Noetherian. Hence \(\mathcal{G}\) is coherent by Lemma 01Y6.

We claim that every stalk of \(\mathcal{G}\) has projective dimension at most \(c\) over the corresponding local ring of \(X\). This is immediate away from \(i(Z)\). If \(x=i(z)\), put \[R=\mathcal{O}_{X,x},\qquad A=\mathcal{O}_{Z,z},\qquad I=\Ker(R\longrightarrow A).\] Choose a regular sequence \(f_1,\ldots,f_c\) generating \(I\). By More on Algebra, Definition 0623 and Lemma 062F, the augmented Koszul complex on this sequence is a finite free \(R\)-resolution of \(A\) of length \(c\). Since \(\mathcal{E}_z\) is a finite free \(A\)-module, a finite direct sum of this complex resolves \(\mathcal{G}_x=\mathcal{E}_z\). This proves the claim.

Apply Lemma coherent-lemma-negative-twists-projective-dimension with \(p=r-c\). We obtain \[H^q(X,\mathcal{G}(-n))=0\] for all sufficiently large \(n\) and every \(0\leq q<r-c\). The projection formula of Cohomology, Lemma 01E8 identifies \[\mathcal{G}(-n)= i_*\left(\mathcal{E}\otimes_{\mathcal{O}_Z}i^*\mathcal{O}_X(-n)\right).\] Finally, Lemma 01Y6 and Cohomology, Lemma 01F4 identify the cohomology of the right hand side on \(X\) with the cohomology of its argument on \(Z\). This gives the asserted vanishing.

Lemma

Let \(K\) be an algebraically closed field, let \(r\geq1\), and let \(1\leq k\leq r\). Put \(S=K[t_0,\ldots,t_r]\). Let \(X\subset\mathbf P^r_K\) be a connected reduced closed subscheme, let \(I\) be its defining saturated homogeneous ideal, and put \(A=S/I\). Assume that

1. \(H^q(X,\mathcal O_X)=0\) for every \(0<q<k\), and

2. for every \(x\in X\), the \(\mathcal O_{\mathbf P^r_K,x}\)-module \(\mathcal O_{X,x}\) has projective dimension at most \(r-k\).

For \(d\geq1\), put \[P_d=\operatorname{Sym}_K(S_d),\qquad N_d=\dim_K(S_d)-1,\] where \(S_d\) is placed in degree one. The natural graded map \(P_d\to A^{(d)}\) is surjective. For every sufficiently large \(d\), \[\operatorname{pd}_{P_d}(A^{(d)})\leq N_d-k.\]

Proof

The scheme \(X\) is proper over \(K\). Since it is connected and reduced, Varieties, Lemma 0BUG shows that \(H^0(X,\mathcal O_X)\) is a finite field extension of \(K\), hence is \(K\).

Let \(Y\) be an irreducible component of dimension \(p\), and let \(\eta\) be its generic point. The ambient local ring \(\mathcal O_{\mathbf P^r_K,\eta}\) is regular of dimension \(r-p\), while \(\mathcal O_{X,\eta}\) is a field. The Auslander–Buchsbaum formula and hypothesis (2) therefore give \(r-p\leq r-k\), so every component has dimension at least \(k\). In particular, \(\operatorname{Ass}(\mathcal O_X)\) contains no closed point. Varieties, Lemma 0FD7, applied to the ample invertible sheaf \(\mathcal O_X(1)\), gives an integer \(d_0\) such that \[H^0(X,\mathcal O_X(-m))=0\quad(m\geq d_0).\]

Let \(i:X\to\mathbf P^r_K\) be the closed immersion. Lemma 0AG7 gives an integer \(d_1\) such that \[A_m\longrightarrow H^0(X,\mathcal O_X(m))\] is an isomorphism for \(m\geq d_1\). Lemma 01YS, applied to \(i_*\mathcal O_X\), gives an integer \(d_2\) such that \[H^q(X,\mathcal O_X(m))=0\] for \(m\geq d_2\) and \(0<q<k\). Finally, Lemma coherent-lemma-negative-twists-projective-dimension and hypothesis (2) give an integer \(d_3\) such that the same vanishing holds for \(m\leq-d_3\) and \(0<q<k\).

Choose \(d\geq\max(d_0,d_1,d_2,d_3)\) large enough that \(N_d\geq k\). Constructions, Lemma 0B5J identifies \(\operatorname{Proj}(A^{(d)})\) with \(X\) and its twist \(n\) with \(\mathcal O_X(nd)\). The canonical map for the finite graded \(P_d\)-module \(A^{(d)}\) is an isomorphism in every degree: this follows from the choice of \(d_1\) for \(n>0\), from \(H^0(X,\mathcal O_X)=K\) for \(n=0\), and from the choice of \(d_0\) for \(n<0\). Its cohomology in degrees \(0<q<k\) vanishes for every twist by the choices of \(d_2,d_3\) and hypothesis (1) at \(n=0\). Lemma coherent-lemma-graded-projective-dimension-cohomology, applied in \(\mathbf P^{N_d}_K\), proves the assertion.

Lemma

Let \(K\) be an algebraically closed field, let \(k\geq1\), and let \(X\) be a smooth projective \(K\)-scheme of pure dimension \(p\geq k\). The following are equivalent:

1. \(X\) is connected and \(H^q(X,\mathcal O_X)=0\) for every \(0<q<k\),

2. there is a closed immersion \(X\to\mathbf P^N_K\) whose homogeneous coordinate ring \(B\) satisfies \(\operatorname{pd}_{K[x_0,\ldots,x_N]}(B)\leq N-k\).

Proof

If (2) holds, Lemma coherent-lemma-graded-projective-dimension-cohomology gives the asserted cohomology vanishing at twist zero and identifies \(K=B_0\) with \(H^0(X,\mathcal O_X)\). The latter equality implies that \(X\) is connected, since a decomposition would give a nontrivial idempotent in its ring of global functions.

Conversely, choose a projective embedding \(X\to\mathbf P^r_K\). Both schemes are smooth over \(K\), so Divisors, Lemma 067U makes this a regular immersion of codimension \(r-p\). The Koszul resolution gives projective dimension \(r-p\leq r-k\) at every stalk. Lemma coherent-lemma-veronese-projective-dimension now gives (2) after any sufficiently high Veronese embedding.

Lemma

Let \(K\) be a field, let \(r\geq1\), and let \(0\leq c\leq r-1\). Set \[S=K[t_0,\ldots,t_r],\qquad p=r-c.\] Let \(f_1,\ldots,f_c\in S\) be a homogeneous regular sequence of positive degrees \(m_1,\ldots,m_c\), and put \[A=S/(f_1,\ldots,f_c),\qquad X=\text{Proj}(A),\qquad N=m_1+\ldots+m_c-r-1.\] Then for every \(n\in\mathbf Z\) the following hold:

1. the canonical map \[A_n\longrightarrow H^0(X,\mathcal O_X(n))\] is an isomorphism;

2. \(H^q(X,\mathcal O_X(n))=0\) for \(0<q<p\);

3. after fixing the displayed order of the equations, there are isomorphisms \[H^p(X,\mathcal O_X(n))\cong \Hom_K(A_{N-n},K)\cong \Hom_K(H^0(X,\mathcal O_X(N-n)),K).\]

Moreover, if \(H^p(X,\mathcal O_X)=0\), then \(N<0\).

Proof

The Koszul resolution in More on Algebra, Lemma more-algebra-lemma-graded-ext-complete-intersection shows that \(A\) has projective dimension \(c=r-p\) over \(S\). Since \(p\geq1\), Lemma coherent-lemma-graded-projective-dimension-cohomology, applied with \(k=p\), proves (1) and (2).

Put \(\Omega=S(-r-1)\). The same Koszul–Ext lemma gives \[\text{GrExt}^c_S(A,\Omega)\cong A(N).\] Use the notation \(B^q(A)\) and \(T^q(A)=B^q(A)^\vee\) of More on Algebra, Lemma more-algebra-lemma-graded-local-duality-polynomial-ring. Because \(c<r\), that lemma identifies \[T^p(A)\cong\text{GrExt}^{r-p}_S(A,\Omega)\cong A(N).\] The degree \(n\) part of \(B^p(A)\) is \(H^p(X,\mathcal O_X(n))\): this follows from the standard affine covering and Lemma 01XD. Hence the degree \(-n\) part of the last display is \[\Hom_K(H^p(X,\mathcal O_X(n)),K)\cong A_{N-n}.\] Both vector spaces are finite dimensional by Lemma 01YS. Dualizing and using (1) with twist \(N-n\) proves (3).

Finally, if \(H^p(X,\mathcal O_X)=0\), then (3) gives \(A_N=0\). This is impossible for \(N=0\) because \(A_0=K\). If \(N>0\), generation of \(A\) by \(A_1\) would make \(A_+\) nilpotent. On the other hand, applying Algebra, Lemma 00KW successively to the \(c\) equations gives \(\dim(A)\geq r+1-c=p+1>0\), whereas a nilpotent \(A_+\) would give \(\dim(A)=\dim(A/A_+)=0\). Thus \(N<0\).

Remark

Keep the notation above but take \(c=r\), so that \(X\) has dimension zero. The canonical section map need not be an isomorphism in negative degree. More on Algebra, Lemmas more-algebra-lemma-graded-ext-complete-intersection and more-algebra-lemma-graded-local-duality-polynomial-ring instead give, for every \(n\in\mathbf Z\), an exact sequence \[0\longrightarrow A_n\longrightarrow H^0(X,\mathcal O_X(n)) \longrightarrow\Hom_K(A_{N-n},K)\longrightarrow0.\] Thus this sequence, rather than an all-degree identification of its first two terms, is the graded-local-duality statement in dimension zero.

The top-duality assertion itself remains valid in this case. Indeed, \(X\to\mathbf P^r_K\) is a regular immersion with normal determinant \(\mathcal O_X(m_1+\ldots+m_r)\). Duality, Lemmas 0BR0, 0A9W, 0ATX, and 0B6U identify the dualizing module of \(X\) with \(\mathcal O_X(N)\). Duality, Lemma 0FVZ, applied to \(\mathcal O_X(n)\), therefore gives \[H^0(X,\mathcal O_X(n))\cong \Hom_K(H^0(X,\mathcal O_X(N-n)),K).\] Thus part (c) of the source survives in dimension zero even though part (a) does not.

Coherent sheaves on Proj, II

In this section we discuss coherent sheaves on \(\text{Proj}(A)\) where \(A\) is a Noetherian graded ring. Most of the results will be deduced by sleight of hand from the corresponding result in the previous section where we discussed what happens if \(A\) is generated by degree \(1\) elements.

Lemma

Let \(A\) be a Noetherian graded ring. Set \(X = \text{Proj}(A)\). Then \(X\) is a Noetherian scheme. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module.

1. There exists an \(r \geq 0\) and \(d_1, \ldots, d_r \in \mathbf{Z}\) and a surjection \[\bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_X(d_j) \longrightarrow \mathcal{F}.\]

2. For any \(p\) the cohomology group \(H^p(X, \mathcal{F})\) is a finite \(A_0\)-module.

3. If \(p > 0\), then \(H^p(X, \mathcal{F}(d)) = 0\) for all \(d\) large enough.

4. For any \(k \in \mathbf{Z}\) the graded \(A\)-module \[\bigoplus\nolimits_{d \geq k} H^0(X, \mathcal{F}(d))\] is a finite \(A\)-module.

Proof

We will prove this by reducing the statement to Lemma 0AG6. By Algebra, Lemmas 00JW and 07Z4 the ring \(A_0\) is Noetherian and \(A\) is generated over \(A_0\) by finitely many elements \(f_1, \ldots, f_r\) homogeneous of positive degree. Let \(d\) be a sufficiently divisible integer. Set \(A' = A^{(d)}\) with notation as in Algebra, Section 00JL. Then \(A'\) is generated over \(A'_0 = A_0\) by elements of degree \(1\), see Algebra, Lemma 0EGH. Thus Lemma 0AG6 applies to \(X' = \text{Proj}(A')\).

By Constructions, Lemma 0B5J there exist an isomorphism of schemes \(i : X \to X'\) and isomorphisms \(\mathcal{O}_X(nd) \to i^*\mathcal{O}_{X'}(n)\) compatible with the map \(A' \to A\) and the maps \(A_n \to H^0(X, \mathcal{O}_X(n))\) and \(A'_n \to H^0(X', \mathcal{O}_{X'}(n))\). Thus Lemma 0AG6 implies \(X\) is Noetherian and that (1) and (2) hold. To see (3) and (4) we can use that for any fixed \(k\), \(p\), and \(q\) we have \[\bigoplus\nolimits_{dn + q \geq k} H^p(X, \mathcal{F}(dn + q)) = \bigoplus\nolimits_{dn + q \geq k} H^p(X', (i_*\mathcal{F}(q))(n)\] by the compatibilities above. If \(p > 0\), we have the vanishing of the right hand side for \(k\) depending on \(q\) large enough by Lemma 0AG6. Since there are only a finite number of congruence classes of integers modulo \(d\), we see that (3) holds for \(\mathcal{F}\) on \(X\). If \(p = 0\), then we have that the right hand side is a finite \(A'\)-module by Lemma 0AG6. Using the finiteness of congruence classes once more, we find that \(\bigoplus_{n \geq k} H^0(X, \mathcal{F}(n))\) is a finite \(A'\)-module too. Since the \(A'\)-module structure comes from the \(A\)-module structure (by the compatibilities mentioned above), we conclude it is finite as an \(A\)-module as well.

Lemma

Let \(A\) be a Noetherian graded ring and let \(d\) be the lcm of generators of \(A\) over \(A_0\). Let \(M\) be a finite graded \(A\)-module. Set \(X = \text{Proj}(A)\) and let \(\widetilde{M}\) be the quasi-coherent \(\mathcal{O}_X\)-module on \(X\) associated to \(M\). Let \(k \in \mathbf{Z}\).

1. \(N' = \bigoplus_{n \geq k} H^0(X, \widetilde{M(n)})\) is a finite \(A\)-module,

2. \(N = \bigoplus_{n \geq k} H^0(X, \widetilde{M}(n))\) is a finite \(A\)-module,

3. there is a canonical map \(N \to N'\),

4. if \(k\) is small enough there is a canonical map \(M \to N'\),

5. the map \(M_n \to N'_n\) is an isomorphism for \(n \gg 0\),

6. \(N_n \to N'_n\) is an isomorphism for \(d | n\).

Proof

The map \(N \to N'\) in (3) comes from Constructions, Equation (01MQ) by taking global sections.

By Constructions, Equation (01MR) there is a map of graded \(A\)-modules \(M \to \bigoplus_{n \in \mathbf{Z}} H^0(X, \widetilde{M(n)})\). If the generators of \(M\) sit in degrees \(\geq k\), then the image is contained in the submodule \(N' \subset \bigoplus_{n \in \mathbf{Z}} H^0(X, \widetilde{M(n)})\) and we get the map in (4).

By Algebra, Lemmas 00JW and 07Z4 the ring \(A_0\) is Noetherian and \(A\) is generated over \(A_0\) by finitely many elements \(f_1, \ldots, f_r\) homogeneous of positive degree. Let \(d = \text{lcm}(\deg(f_i))\). Then we see that (6) holds for example by Constructions, Lemma 01MU.

Because \(M\) is a finite \(A\)-module we see that \(\widetilde{M}\) is a finite type \(\mathcal{O}_X\)-module, i.e., a coherent \(\mathcal{O}_X\)-module. Thus part (2) follows from Lemma 0B5Q.

We will deduce (1) from (2) using a trick. For \(q \in \{0, \ldots, d - 1\}\) write \[{}^qN = \bigoplus\nolimits_{n + q \geq k} H^0(X, \widetilde{M(q)}(n))\] By part (2) these are finite \(A\)-modules. The Noetherian ring \(A\) is finite over \(A^{(d)} = \bigoplus_{n \geq 0} A_{dn}\), because it is generated by \(f_i\) over \(A^{(d)}\) and \(f_i^d \in A^{(d)}\). Hence \({}^qN\) is a finite \(A^{(d)}\)-module. Moreover, \(A^{(d)}\) is Noetherian (follows from Algebra, Lemma 07Z2). It follows that the \(A^{(d)}\)-submodule \({}^qN^{(d)} = \bigoplus_{n \in \mathbf{Z}} {}^qN_{dn}\) is a finite module over \(A^{(d)}\). Using the isomorphisms \(\widetilde{M(dn + q)} = \widetilde{M(q)}(dn)\) we can write \[N' = \bigoplus\nolimits_{q \in \{0, \ldots, d - 1\}} \bigoplus\nolimits_{dn + q \geq k} H^0(X, \widetilde{M(q)}(dn)) = \bigoplus\nolimits_{q \in \{0, \ldots, d - 1\}} {}^qN^{(d)}\] Thus \(N'\) is finite over \(A^{(d)}\) and a fortiori finite over \(A\). Thus (1) is true.

Let \(K\) be a finite \(A\)-module such that \(\widetilde{K} = 0\). We claim that \(K_n = 0\) for \(d|n\) and \(n \gg 0\). Arguing as above we see that \(K^{(d)}\) is a finite \(A^{(d)}\)-module. Let \(x_1, \ldots, x_m \in K\) be homogeneous generators of \(K^{(d)}\) over \(A^{(d)}\), say sitting in degrees \(d_1, \ldots, d_m\) with \(d | d_j\). For each \(i\) and \(j\) there exists an \(n_{ij} \geq 0\) such that \(f_i^{n_{ij}} x_j = 0\) in \(K_{d_j + n_{ij}}\): if not then \(x_j/f_i^{d_i/\deg(f_i)} \in K_{(f_i)}\) would not be zero, i.e., \(\widetilde{K}\) would not be zero. Here we use that \(\deg(f_i) | d | d_j\) for all \(i, j\). We conclude that \(K_n\) is zero for \(n\) with \(d | n\) and \(n > \max_j (d_j + \sum_i n_{ij} \deg(f_i))\) as every element of \(K_n\) is a sum of terms where each term is a monomials in the \(f_i\) times one of the \(x_j\) of total degree \(n\).

To finish the proof, we have to show that \(M \to N'\) is an isomorphism in all sufficiently large degrees. The map \(N \to N'\) induces an isomorphism \(\widetilde{N} \to \widetilde{N'}\) because on the affine opens \(D_+(f_i) = D_+(f_i^d)\) the corresponding modules are isomorphic: \(N_{(f_i)} \cong N_{(f_i^d)} \cong N'_{(f_i^d)} \cong N'_{(f_i)}\) by property (6). By Properties, Lemma 0AG5 we have a canonical isomorphism \(\widetilde{N} \to \widetilde{M}\). The composition \(\widetilde{N} \to \widetilde{M} \to \widetilde{N'}\) is the isomorphism above (proof omitted; hint: look on standard affine opens to check this). Thus the map \(M \to N'\) induces an isomorphism \(\widetilde{M} \to \widetilde{N'}\). Let \(K = \Ker(M \to N')\) and \(Q = \Coker(M \to N')\). Recall that the functor \(M \mapsto \widetilde{M}\) is exact, see Constructions, Lemma 01M7. Hence we see that \(\widetilde{K} = 0\) and \(\widetilde{Q} = 0\). By the result of the previous paragraph we see that \(K_n = 0\) and \(Q_n = 0\) for \(d | n\) and \(n \gg 0\). At this point we finally see the advantage of using \(N'\) over \(N\): the functor \(M \leadsto N'\) is compatible with shifts (immediate from the construction). Thus, repeating the whole argument with \(M\) replaced by \(M(q)\) we find that \(K_n = 0\) and \(Q_n = 0\) for \(n \equiv q \bmod d\) and \(n \gg 0\). Since there are only finitely many congruence classes modulo \(n\) the proof is finished.

Let \(A\) be a Noetherian graded ring. Recall that \(A_+ = \bigoplus_{n > 0} A_n\) is the irrelevant ideal. By Algebra, Lemmas 00JW and 07Z4 the ring \(A_0\) is Noetherian and \(A\) is generated over \(A_0\) by finitely many elements \(f_1, \ldots, f_r\) homogeneous of positive degree. Let \(d = \text{lcm}(\deg(f_i))\). Let \(M\) be a graded \(A\)-module. In this situation we say a homogeneous element \(x \in M\) is irrelevant² if \[(A_+ x)_{nd} = 0\text{ for all }n \gg 0\] If \(x \in M\) is homogeneous and irrelevant and \(f \in A\) is homogeneous, then \(fx\) is irrelevant too. Hence the set of irrelevant elements generate a graded submodule \(M_{irrelevant} \subset M\). We will say \(M\) is irrelevant if every homogeneous element of \(M\) is irrelevant, i.e., if \(M_{irrelevant} = M\). If \(M\) is finitely generated, then we see that this is equivalent to \(M_{nd} = 0\) for \(n \gg 0\). Denote \(\text{Mod}_A\) the category of graded \(A\)-modules, \(\text{Mod}^{fg}_A\) the full subcategory of finitely generated ones, and \(\text{Mod}^{fg}_{A, irrelevant}\) the full subcategory of irrelevant modules.

Proposition

Let \(A\) be a Noetherian graded ring. Set \(X = \text{Proj}(A)\). The functor \(M \mapsto \widetilde M\) induces an equivalence \[\text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, irrelevant} \longrightarrow \textit{Coh}(\mathcal{O}_X)\] whose quasi-inverse is given by \(\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))\).

Proof

We urge the reader to read the proof in the case where \(A\) is generated in degree \(1\) first, see Proposition 0BXD. Let \(f_1, \ldots, f_r \in A\) be homogeneous elements of positive degree which generate \(A\) over \(A_0\). Let \(d\) be the lcm of the degrees \(d_i\) of \(f_i\). Let \(M\) be a finite \(A\)-module. Let us show that \(\widetilde{M}\) is zero if and only if \(M\) is an irrelevant graded \(A\)-module (as defined above the statement of the proposition). Namely, let \(x \in M\) be a homogeneous element. Choose \(k \in \mathbf{Z}\) sufficiently small and let \(N \to N'\) and \(M \to N'\) be as in Lemma 0B5R. We may also pick \(l\) sufficiently large such that \(M_n \to N_n\) is an isomorphism for \(n \geq l\). If \(\widetilde{M}\) is zero, then \(N = 0\). Thus for any \(f \in A_+\) homogeneous with \(\deg(f) + \deg(x) = nd\) and \(nd > l\) we see that \(fx\) is zero because \(N_{nd} \to N'_{nd}\) and \(M_{nd} \to N'_{nd}\) are isomorphisms. Hence \(x\) is irrelevant. Conversely, assume \(M\) is irrelevant. Then \(M_{nd}\) is zero for \(n \gg 0\) (see discussion above proposition). Clearly this implies that \(M_{(f_i)} = M_{(f_i^{d/\deg(f_i)})} = 0\), whence \(\widetilde{M} = 0\) by construction.

It follows that the subcategory \(\text{Mod}^{fg}_{A, irrelevant}\) is a Serre subcategory of \(\text{Mod}^{fg}_A\) as the kernel of the exact functor \(M \mapsto \widetilde M\), see Homology, Lemma 02MQ and Constructions, Lemma 01M7. Hence the quotient category on the left of the arrow is defined in Homology, Lemma 02MS. To define the functor of the proposition, it suffices to show that the functor \(M \mapsto \widetilde M\) sends irrelevant modules to \(0\) which we have shown above.

By Lemma 0B5Q the proposed quasi-inverse makes sense. Namely, the lemma shows that \(\mathcal{F} \longmapsto \bigoplus_{n \geq 0} \Gamma(X, \mathcal{F}(n))\) is a functor \(\textit{Coh}(\mathcal{O}_X) \to \text{Mod}^{fg}_A\) which we can compose with the quotient functor \(\text{Mod}^{fg}_A \to \text{Mod}^{fg}_A/\text{Mod}^{fg}_{A, irrelevant}\).

By Lemma 0B5R the composite left to right to left is isomorphic to the identity functor. Namely, let \(M\) be a finite graded \(A\)-module and let \(k \in \mathbf{Z}\) sufficiently small and let \(N \to N'\) and \(M \to N'\) be as in Lemma 0B5R. Then the kernel and cokernel of \(M \to N'\) are nonzero in only finitely many degrees, hence are irrelevant. Moreover, the kernel and cokernel of the map \(N \to N'\) are zero in all sufficiently large degrees divisible by \(d\), hence these are irrelevant modules too. Thus \(M \to N'\) and \(N \to N'\) are both isomorphisms in the quotient category, as desired.

Finally, let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Set \(M = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F}(n))\) viewed as a graded \(A\)-module, so that our functor sends \(\mathcal{F}\) to \(M_{\geq 0} = \bigoplus_{n \geq 0} M_n\). By Properties, Lemma 0AG5 the canonical map \(\widetilde M \to \mathcal{F}\) is an isomorphism. Since the inclusion map \(M_{\geq 0} \to M\) defines an isomorphism \(\widetilde{M_{\geq 0}} \to \widetilde M\) we conclude that the composite right to left to right is isomorphic to the identity functor as well.

Higher direct images along projective morphisms

We first state and prove a result for when the base is affine and then we deduce some results for projective morphisms.

Lemma

Let \(R\) be a Noetherian ring. Let \(X \to \Spec(R)\) be a proper morphism. Let \(\mathcal{L}\) be an ample invertible sheaf on \(X\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module.

1. The graded ring \(A = \bigoplus_{d \geq 0} H^0(X, \mathcal{L}^{\otimes d})\) is a finitely generated \(R\)-algebra.

2. There exists an \(r \geq 0\) and \(d_1, \ldots, d_r \in \mathbf{Z}\) and a surjection \[\bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{L}^{\otimes d_j} \longrightarrow \mathcal{F}.\]

3. For any \(p\) the cohomology group \(H^p(X, \mathcal{F})\) is a finite \(R\)-module.

4. If \(p > 0\), then \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\) for all \(d\) large enough.

5. For any \(k \in \mathbf{Z}\) the graded \(A\)-module \[\bigoplus\nolimits_{d \geq k} H^0(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\] is a finite \(A\)-module.

Proof

By Morphisms, Lemma 01VT there exists a \(d > 0\) and an immersion \(i : X \to \mathbf{P}^n_R\) such that \(\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^n_R}(1)\). Since \(X\) is proper over \(R\) the morphism \(i\) is a closed immersion (Morphisms, Lemma 01W6). Thus we have \(H^i(X, \mathcal{G}) = H^i(\mathbf{P}^n_R, i_*\mathcal{G})\) for any quasi-coherent sheaf \(\mathcal{G}\) on \(X\) (by Lemma 089W and the fact that closed immersions are affine, see Morphisms, Lemma 01SE). Moreover, if \(\mathcal{G}\) is coherent, then \(i_*\mathcal{G}\) is coherent as well (Lemma 087T). We will use these facts without further mention.

Proof of (1). Set \(S = R[T_0, \ldots, T_n]\) so that \(\mathbf{P}^n_R = \text{Proj}(S)\). Observe that \(A\) is an \(S\)-algebra (but the ring map \(S \to A\) is not a homomorphism of graded rings because \(S_n\) maps into \(A_{dn}\)). By the projection formula (Cohomology, Lemma 01E8) we have \[i_*(\mathcal{L}^{\otimes nd + q}) = i_*(\mathcal{L}^{\otimes q}) \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{O}_{\mathbf{P}^n_R}(n)\] for all \(n \in \mathbf{Z}\). We conclude that \(\bigoplus_{n \geq 0} A_{nd + q}\) is a finite graded \(S\)-module by Lemma 01YS. Since \(A = \bigoplus_{q \in \{0, \ldots, d - 1} \bigoplus_{n \geq 0} A_{nd + q}\) we see that \(A\) is finite as an \(S\)-algebra, hence (1) is true.

Proof of (2). This follows from Properties, Proposition 01Q3.

Proof of (3). Apply Lemma 01YS and use \(H^p(X, \mathcal{F}) = H^p(\mathbf{P}^n_R, i_*\mathcal{F})\).

Proof of (4). Fix \(p > 0\). By the projection formula we have \[i_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes nd + q}) = i_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes q}) \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{O}_{\mathbf{P}^n_R}(n)\] for all \(n \in \mathbf{Z}\). By Lemma 01YS we conclude that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{nd + q}) = 0\) for \(n \gg 0\). Since there are only finitely many congruence classes of integers modulo \(d\) this proves (4).

Proof of (5). Fix an integer \(k\). Set \(M = \bigoplus_{n \geq k} H^0(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})\). Arguing as above we conclude that \(\bigoplus_{nd + q \geq k} M_{nd + q}\) is a finite graded \(S\)-module. Since \(M = \bigoplus_{q \in \{0, \ldots, d - 1\}} \bigoplus_{nd + q \geq k} M_{nd + q}\) we see that \(M\) is finite as an \(S\)-module. Since the \(S\)-module structure factors through the ring map \(S \to A\), we conclude that \(M\) is finite as an \(A\)-module.

Lemma

Let \(f : X \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{L}\) be an invertible sheaf on \(X\). Assume that

1. \(S\) is Noetherian,

2. \(f\) is proper,

3. \(\mathcal{F}\) is coherent, and

4. \(\mathcal{L}\) is relatively ample on \(X/S\).

Then there exists an \(n_0\) such that for all \(n \geq n_0\) we have \[R^pf_*\left(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\right) = 0\] for all \(p > 0\).

Proof

Choose a finite affine open covering \(S = \bigcup V_j\) and set \(X_j = f^{-1}(V_j)\). Clearly, if we solve the question for each of the finitely many systems \((X_j \to V_j, \mathcal{L}|_{X_j}, \mathcal{F}|_{V_j})\) then the result follows. Thus we may assume \(S\) is affine. In this case the vanishing of \(R^pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes n})\) is equivalent to the vanishing of \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})\), see Lemma 01XK. Thus the required vanishing follows from Lemma 0B5T (which applies because \(\mathcal{L}\) is ample on \(X\) by Morphisms, Lemma 01VT).

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a locally projective morphism. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Then \(R^if_*\mathcal{F}\) is a coherent \(\mathcal{O}_S\)-module for all \(i \geq 0\).

Proof

We first remark that a locally projective morphism is proper (Morphisms, Lemma 01WC) and hence of finite type. In particular \(X\) is locally Noetherian (Morphisms, Lemma 01T6) and hence the statement makes sense. Moreover, by Lemma 01XJ the sheaves \(R^pf_*\mathcal{F}\) are quasi-coherent.

Having said this the statement is local on \(S\) (for example by Cohomology, Lemma 01E5). Hence we may assume \(S = \Spec(R)\) is the spectrum of a Noetherian ring, and \(X\) is a closed subscheme of \(\mathbf{P}^n_R\) for some \(n\), see Morphisms, Lemma 01WB. In this case, the sheaves \(R^pf_*\mathcal{F}\) are the quasi-coherent sheaves associated to the \(R\)-modules \(H^p(X, \mathcal{F})\), see Lemma 01XK. Hence it suffices to show that \(R\)-modules \(H^p(X, \mathcal{F})\) are finite \(R\)-modules (Lemma 01XZ). This follows from Lemma 0B5T (because the restriction of \(\mathcal{O}_{\mathbf{P}^n_R}(1)\) to \(X\) is ample on \(X\)).

Ample invertible sheaves and cohomology

Here is a criterion for ampleness on proper schemes over affine bases in terms of vanishing of cohomology after twisting.

Lemma

Let \(R\) be a Noetherian ring. Let \(f : X \to \Spec(R)\) be a proper morphism. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{L}\) is ample on \(X\) (this is equivalent to many other things, see Properties, Proposition 01Q3 and Morphisms, Lemma 01VT),

2. for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exists an \(n_0 \geq 0\) such that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0\) for all \(n \geq n_0\) and \(p > 0\), and

3. for every quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\), there exists an \(n \geq 1\) such that \(H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n}) = 0\).

Proof

The implication (1) \(\Rightarrow\) (2) follows from Lemma 0B5T. The implication (2) \(\Rightarrow\) (3) is trivial. The implication (3) \(\Rightarrow\) (1) is Lemma 0B5P.

Lemma

Let \(R\) be a Noetherian ring. Let \(f : Y \to X\) be a morphism of schemes proper over \(R\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume \(f\) is finite and surjective. Then \(\mathcal{L}\) is ample if and only if \(f^*\mathcal{L}\) is ample.

Proof

The pullback of an ample invertible sheaf by a quasi-affine morphism is ample, see Morphisms, Lemma 0892. This proves one of the implications as a finite morphism is affine by definition.

Assume that \(f^*\mathcal{L}\) is ample. Let \(P\) be the following property on coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\): there exists an \(n_0\) such that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0\) for all \(n \geq n_0\) and \(p > 0\). We will prove that \(P\) holds for any coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), which implies \(\mathcal{L}\) is ample by Lemma 0B5U. We are going to apply Lemma 01YM. Thus we have to verify (1), (2) and (3) of that lemma for \(P\). Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves and the fact that tensoring with an invertible sheaf is an exact functor. Property (2) follows since \(H^p(X, -)\) is an additive functor. To see (3) let \(Z \subset X\) be an integral closed subscheme with generic point \(\xi\). Let \(\mathcal{F}\) be a coherent sheaf on \(Y\) such that the support of \(f_*\mathcal{F}\) is equal to \(Z\) and \((f_*\mathcal{F})_\xi\) is annihilated by \(\mathfrak m_\xi\), see Lemma 01YO. We claim that taking \(\mathcal{G} = f_*\mathcal{F}\) works. We only have to verify part (3)(c) of Lemma 01YM. Hence assume that \(\mathcal{J} \subset \mathcal{O}_X\) is a quasi-coherent sheaf of ideals such that \(\mathcal{J}_\xi = \mathcal{O}_{X, \xi}\). A finite morphism is affine hence by Lemma 01YP we see that \(\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})\). Also, as pointed out in the proof of Lemma 01YP the sheaf \(f^{-1}\mathcal{J}\mathcal{F}\) is a coherent \(\mathcal{O}_Y\)-module. As \(\mathcal{L}\) is ample we see from Lemma 0B5U that there exists an \(n_0\) such that \[H^p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0,\] for \(n \geq n_0\) and \(p > 0\). Since \(f\) is finite, hence affine, we see that \[\begin{align*} H^p(X, \mathcal{J}\mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) & = H^p(X, f_*(f^{-1}\mathcal{J}\mathcal{F}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \\ & = H^p(X, f_*(f^{-1}\mathcal{J}\mathcal{F} \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n})) \\ & = H^p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*}\] Here we have used the projection formula (Cohomology, Lemma 01E8) and Lemma 089W. Hence the quasi-coherent subsheaf \(\mathcal{G}' = \mathcal{J}\mathcal{G}\) satisfies \(P\). This verifies property (3)(c) of Lemma 01YM as desired.

Cohomology is functorial. In particular, given a ringed space \(X\), an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\), a section \(s \in \Gamma(X, \mathcal{L})\) we get maps \[H^p(X, \mathcal{F}) \longrightarrow H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}), \quad \xi \longmapsto s\xi\] induced by the map \(\mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}\) which is multiplication by \(s\). We set \(\Gamma_*(X, \mathcal{L}) = \bigoplus_{n \geq 0} \Gamma(X, \mathcal{L}^{\otimes n})\) as a graded ring, see Modules, Definition 01CV. Given a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) and an integer \(p \geq 0\) we set \[H^p_*(X, \mathcal{L}, \mathcal{F}) = \bigoplus\nolimits_{n \in \mathbf{Z}} H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\] This is a graded \(\Gamma_*(X, \mathcal{L})\)-module by the multiplication defined above. Warning: the notation \(H^p_*(X, \mathcal{L}, \mathcal{F})\) is nonstandard.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible sheaf on \(X\). Let \(s \in \Gamma(X, \mathcal{L})\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(X\) is quasi-compact and quasi-separated, the canonical map \[H^p_*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^p(X_s, \mathcal{F})\] which maps \(\xi/s^n\) to \(s^{-n}\xi\) is an isomorphism.

Proof

Note that for \(p = 0\) this is Properties, Lemma 01PW. We will prove the statement using the induction principle (Lemma 08DR) where for \(U \subset X\) quasi-compact open we let \(P(U)\) be the property: for all \(p \geq 0\) the map \[H^p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^p(U_s, \mathcal{F})\] is an isomorphism.

If \(U\) is affine, then both sides of the arrow displayed above are zero for \(p > 0\) by Lemma 01XB and Properties, Lemma 01PV and the statement is true. If \(P\) is true for \(U\), \(V\), and \(U \cap V\), then we can use the Mayer-Vietoris sequences (Cohomology, Lemma 01EB) to obtain a map of long exact sequences \[\xymatrix{ H^{p - 1}_*(U \cap V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^p_*(U \cup V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \oplus H^p_*(V, \mathcal{L}, \mathcal{F})_{(s)} \ar[d] \\ H^{p - 1}(U_s \cap V_s, \mathcal{F}) \ar[r]& H^p(U_s \cup V_s, \mathcal{F}) \ar[r] & H^p(U_s, \mathcal{F}) \oplus H^p(V_s, \mathcal{F}) }\] (only a snippet shown). Observe that \(U_s \cap V_s = (U \cap V)_s\) and that \(U_s \cup V_s = (U \cup V)_s\). Thus the left and right vertical maps are isomorphisms (as well as one more to the right and one more to the left which are not shown in the diagram). We conclude that \(P(U \cup V)\) holds by the 5-lemma (Homology, Lemma 05QB). This finishes the proof.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\) be a section. Assume that

1. \(X\) is quasi-compact and quasi-separated, and

2. \(X_s\) is affine.

Then for every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) and every \(p > 0\) and all \(\xi \in H^p(X, \mathcal{F})\) there exists an \(n \geq 0\) such that \(s^n\xi = 0\) in \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})\).

Proof

Recall that \(H^p(X_s, \mathcal{G})\) is zero for every quasi-coherent module \(\mathcal{G}\) by Lemma 01XB. Hence the lemma follows from Lemma 09MR.

For a more general version of the following lemma see Limits, Lemma 09MW.

Lemma

Let \(i : Z \to X\) be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Let \(\mathcal{L}\) be an invertible sheaf on \(X\). Then \(i^*\mathcal{L}\) is ample on \(Z\), if and only if \(\mathcal{L}\) is ample on \(X\).

Proof

If \(\mathcal{L}\) is ample, then \(i^*\mathcal{L}\) is ample for example by Morphisms, Lemma 0892. Assume \(i^*\mathcal{L}\) is ample. We have to show that \(\mathcal{L}\) is ample on \(X\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be the coherent sheaf of ideals cutting out the closed subscheme \(Z\). Since \(i(Z) = X\) set theoretically we see that \(\mathcal{I}^n = 0\) for some \(n\) by Lemma 01Y9. Consider the sequence \[X = Z_n \supset Z_{n - 1} \supset Z_{n - 2} \supset \ldots \supset Z_1 = Z\] of closed subschemes cut out by \(0 = \mathcal{I}^n \subset \mathcal{I}^{n - 1} \subset \ldots \subset \mathcal{I}\). Then each of the closed immersions \(Z_{i - 1} \to Z_{i}\) is defined by a coherent sheaf of ideals of square zero. In this way we reduce to the case that \(\mathcal{I}^2 = 0\).

Consider the short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0\] of quasi-coherent \(\mathcal{O}_X\)-modules. Tensoring with \(\mathcal{L}^{\otimes n}\) we obtain short exact sequences [0B8T]\[\begin{equation} 0 \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} \to \mathcal{L}^{\otimes n} \to i_*i^*\mathcal{L}^{\otimes n} \to 0 \end{equation}\] As \(\mathcal{I}^2 = 0\), we can use Morphisms, Lemma 01QY to think of \(\mathcal{I}\) as a quasi-coherent \(\mathcal{O}_Z\)-module and then \(\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n} = \mathcal{I} \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes n}\) with obvious abuse of notation. Moreover, the cohomology of this sheaf over \(Z\) is canonically the same as the cohomology of this sheaf over \(X\) (as \(i\) is a homeomorphism).

Let \(x \in X\) be a point and denote \(z \in Z\) the corresponding point. Because \(i^*\mathcal{L}\) is ample there exists an \(n\) and a section \(s \in \Gamma(Z, i^*\mathcal{L}^{\otimes n})\) with \(z \in Z_s\) and with \(Z_s\) affine. The obstruction to lifting \(s\) to a section of \(\mathcal{L}^{\otimes n}\) over \(X\) is the boundary \[\xi = \partial s \in H^1(X, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) = H^1(Z, \mathcal{I} \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes n})\] coming from the short exact sequence of sheaves (0B8T). If we replace \(s\) by \(s^{e + 1}\) then \(\xi\) is replaced by \(\partial(s^{e + 1}) = (e + 1) s^e \xi\) in \(H^1(Z, \mathcal{I} \otimes_{\mathcal{O}_Z} i^*\mathcal{L}^{\otimes (e + 1)n})\) because the boundary map for \[0 \to \bigoplus\nolimits_{m \geq 0} \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes m} \to \bigoplus\nolimits_{m \geq 0} \mathcal{L}^{\otimes m} \to \bigoplus\nolimits_{m \geq 0} i_*i^*\mathcal{L}^{\otimes m} \to 0\] is a derivation by Cohomology, Lemma 0B8S. By Lemma 01XR we see that \(s^e \xi\) is zero for \(e\) large enough. Hence, after replacing \(s\) by a power, we can assume \(s\) is the image of a section \(s' \in \Gamma(X, \mathcal{L}^{\otimes n})\). Then \(X_{s'}\) is an open subscheme and \(Z_s \to X_{s'}\) is a surjective closed immersion of Noetherian schemes with \(Z_s\) affine. Hence \(X_s\) is affine by Lemma 01YQ and we conclude that \(\mathcal{L}\) is ample.

For a more general version of the following lemma see Limits, Lemma 0B7L.

Lemma

Let \(i : Z \to X\) be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Then \(X\) is quasi-affine if and only if \(Z\) is quasi-affine.

Proof

Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma 01QE. Hence if \(Z\) is quasi-affine, then \(\mathcal{O}_Z\) is ample, hence \(\mathcal{O}_X\) is ample by Lemma 09MS, hence \(X\) is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards.

Lemma

Let \(X\) be a scheme. Let \(\mathcal{L}\) be an ample invertible \(\mathcal{O}_X\)-module. Let \(n_0\) be an integer. If \(H^p(X, \mathcal{L}^{\otimes -n}) = 0\) for \(n \geq n_0\) and \(p > 0\), then \(X\) is affine.

Proof

We claim \(H^p(X, \mathcal{F}) = 0\) for every quasi-coherent \(\mathcal{O}_X\)-module and \(p > 0\). Since \(X\) is quasi-compact by Properties, Definition 01PS the claim finishes the proof by Lemma 01XF. The scheme \(X\) is separated by Properties, Lemma 09MP. Say \(X\) is covered by \(e + 1\) affine opens. Then \(H^p(X, \mathcal{F}) = 0\) for \(p > e\), see Lemma 01XI. Thus we may use descending induction on \(p\) to prove the claim. Writing \(\mathcal{F}\) as a filtered colimit of finite type quasi-coherent modules (Properties, Lemma 01PG) and using Cohomology, Lemma 01FF we may assume \(\mathcal{F}\) is of finite type. Then we can choose \(n > n_0\) such that \(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}\) is globally generated, see Properties, Proposition 01Q3. This means there is a short exact sequence \[0 \to \mathcal{F}' \to \bigoplus\nolimits_{i \in I} \mathcal{L}^{\otimes -n} \to \mathcal{F} \to 0\] for some set \(I\) (in fact we can choose \(I\) finite). By induction hypothesis we have \(H^{p + 1}(X, \mathcal{F}') = 0\) and by assumption (combined with the already used commutation of cohomology with colimits) we have \(H^p(X, \bigoplus_{i \in I} \mathcal{L}^{\otimes -n}) = 0\). From the long exact cohomology sequence we conclude that \(H^p(X, \mathcal{F}) = 0\) as desired.

Lemma

Let \(X\) be a quasi-affine scheme. If \(H^p(X, \mathcal{O}_X) = 0\) for \(p > 0\), then \(X\) is affine.

Proof

Since \(\mathcal{O}_X\) is ample by Properties, Lemma 01QE this follows from Lemma 0EBD.

Chow’s Lemma

In this section we prove Chow’s lemma in the Noetherian case (Lemma 0200). In Limits, Section 01ZZ we prove some variants for the non-Noetherian case.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a separated morphism of finite type. Then there exist an \(n \geq 0\) and a diagram \[\xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & }\] where \(X' \to \mathbf{P}^n_S\) is an immersion, and \(\pi : X' \to X\) is proper and surjective. Moreover, we may arrange it such that there exists a dense open subscheme \(U \subset X\) such that \(\pi^{-1}(U) \to U\) is an isomorphism.

Proof

All of the schemes we will encounter during the rest of the proof are going to be of finite type over the Noetherian scheme \(S\) and hence Noetherian (see Morphisms, Lemma 01T6). All morphisms between them will automatically be quasi-compact, locally of finite type and quasi-separated, see Morphisms, Lemma 01T8 and Properties, Lemmas 01OY and 01P0.

The scheme \(X\) has only finitely many irreducible components (Properties, Lemma 0BA8). Say \(X = X_1 \cup \ldots \cup X_r\) is the decomposition of \(X\) into irreducible components. Let \(\eta_i \in X_i\) be the generic point. For every point \(x \in X\) there exists an affine open \(U_x \subset X\) which contains \(x\) and each of the generic points \(\eta_i\). See Properties, Lemma 01ZX. Since \(X\) is quasi-compact, we can find a finite affine open covering \(X = U_1 \cup \ldots \cup U_m\) such that each \(U_i\) contains \(\eta_1, \ldots, \eta_r\). In particular we conclude that the open \(U = U_1 \cap \ldots \cap U_m \subset X\) is a dense open. This and the fact that the \(U_i\) are affine opens covering \(X\) are all that we will use below.

Let \(X^* \subset X\) be the scheme theoretic closure of \(U \to X\), see Morphisms, Definition 01R7. Let \(U_i^* = X^* \cap U_i\). Note that \(U_i^*\) is a closed subscheme of \(U_i\). Hence \(U_i^*\) is affine. Since \(U\) is dense in \(X\) the morphism \(X^* \to X\) is a surjective closed immersion. It is an isomorphism over \(U\). Hence we may replace \(X\) by \(X^*\) and \(U_i\) by \(U_i^*\) and assume that \(U\) is scheme theoretically dense in \(X\), see Morphisms, Definition 01RB.

By Morphisms, Lemma 01VS we can find an immersion \(j_i : U_i \to \mathbf{P}_S^{n_i}\) for each \(i\). By Morphisms, Lemma 01RG we can find closed subschemes \(Z_i \subset \mathbf{P}_S^{n_i}\) such that \(j_i : U_i \to Z_i\) is a scheme theoretically dense open immersion. Note that \(Z_i \to S\) is proper, see Morphisms, Lemma 01WC. Consider the morphism \[j = (j_1|_U, \ldots, j_m|_U) : U \longrightarrow \mathbf{P}_S^{n_1} \times_S \ldots \times_S \mathbf{P}_S^{n_m}.\] By the lemma cited above we can find a closed subscheme \(Z\) of \(\mathbf{P}_S^{n_1} \times_S \ldots \times_S \mathbf{P}_S^{n_m}\) such that \(j : U \to Z\) is an open immersion and such that \(U\) is scheme theoretically dense in \(Z\). The morphism \(Z \to S\) is proper. Consider the \(i\)th projection \[\text{pr}_i|_Z : Z \longrightarrow \mathbf{P}^{n_i}_S.\] This morphism factors through \(Z_i\) (see Morphisms, Lemma 01R9). Denote \(p_i : Z \to Z_i\) the induced morphism. This is a proper morphism, see Morphisms, Lemma 01W6 for example. At this point we have that \(U \subset U_i \subset Z_i\) are scheme theoretically dense open immersions. Moreover, we can think of \(Z\) as the scheme theoretic image of the “diagonal” morphism \(U \to Z_1 \times_S \ldots \times_S Z_m\).

Set \(V_i = p_i^{-1}(U_i)\). Note that \(p_i|_{V_i} : V_i \to U_i\) is proper. Set \(X' = V_1 \cup \ldots \cup V_m\). By construction \(X'\) has an immersion into the scheme \(\mathbf{P}^{n_1}_S \times_S \ldots \times_S \mathbf{P}^{n_m}_S\). Thus by the Segre embedding (see Constructions, Lemma 01WD) we see that \(X'\) has an immersion into a projective space over \(S\).

We claim that the morphisms \(p_i|_{V_i}: V_i \to U_i\) glue to a morphism \(X' \to X\). Namely, it is clear that \(p_i|_U\) is the identity map from \(U\) to \(U\). Since \(U \subset X'\) is scheme theoretically dense by construction, it is also scheme theoretically dense in the open subscheme \(V_i \cap V_j\). Thus we see that \(p_i|_{V_i \cap V_j} = p_j|_{V_i \cap V_j}\) as morphisms into the separated \(S\)-scheme \(X\), see Morphisms, Lemma 01RH. We denote the resulting morphism \(\pi : X' \to X\).

We claim that \(\pi^{-1}(U_i) = V_i\). Since \(\pi|_{V_i} = p_i|_{V_i}\) it follows that \(V_i \subset \pi^{-1}(U_i)\). Consider the diagram \[\xymatrix{ V_i \ar[r] \ar[rd]_{p_i|_{V_i}} & \pi^{-1}(U_i) \ar[d] \\ & U_i }\] Since \(V_i \to U_i\) is proper we see that the image of the horizontal arrow is closed, see Morphisms, Lemma 01W6. Since \(V_i \subset \pi^{-1}(U_i)\) is scheme theoretically dense (as it contains \(U\)) we conclude that \(V_i = \pi^{-1}(U_i)\) as claimed.

This shows that \(\pi^{-1}(U_i) \to U_i\) is identified with the proper morphism \(p_i|_{V_i} : V_i \to U_i\). Hence we see that \(X\) has a finite affine covering \(X = \bigcup U_i\) such that the restriction of \(\pi\) is proper on each member of the covering. Thus by Morphisms, Lemma 01W2 we see that \(\pi\) is proper.

Finally we have to show that \(\pi^{-1}(U) = U\). To see this we argue in the same way as above using the diagram \[\xymatrix{ U \ar[r] \ar[rd] & \pi^{-1}(U) \ar[d] \\ & U }\] and using that \(\text{id}_U : U \to U\) is proper and that \(U\) is scheme theoretically dense in \(\pi^{-1}(U)\).

Remark

In the situation of Chow’s Lemma 0200:

1. The morphism \(\pi\) is actually H-projective (hence projective, see Morphisms, Lemma 01W9) since the morphism \(X' \to \mathbf{P}^n_S \times_S X = \mathbf{P}^n_X\) is a closed immersion (use the fact that \(\pi\) is proper, see Morphisms, Lemma 01W6).

2. We may assume that \(\pi^{-1}(U)\) is scheme theoretically dense in \(X'\). Namely, we can simply replace \(X'\) by the scheme theoretic closure of \(\pi^{-1}(U)\). In this case we can think of \(U\) as a scheme theoretically dense open subscheme of \(X'\). See Morphisms, Section 01R5.

3. If \(X\) is reduced then we may choose \(X'\) reduced. This is clear from (2).

Higher direct images of coherent sheaves

In this section we prove the fundamental fact that the higher direct images of a coherent sheaf under a proper morphism are coherent.

Proposition

Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a proper morphism. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Then \(R^if_*\mathcal{F}\) is a coherent \(\mathcal{O}_S\)-module for all \(i \geq 0\).

Proof

Since the problem is local on \(S\) we may assume that \(S\) is a Noetherian scheme. Since a proper morphism is of finite type we see that in this case \(X\) is a Noetherian scheme also. Consider the property \(\mathcal{P}\) of coherent sheaves on \(X\) defined by the rule \[\mathcal{P}(\mathcal{F}) \Leftrightarrow R^pf_*\mathcal{F}\text{ is coherent for all }p \geq 0\] We are going to use the result of Lemma 01YI to prove that \(\mathcal{P}\) holds for every coherent sheaf on \(X\).

Let \[0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\] be a short exact sequence of coherent sheaves on \(X\). Consider the long exact sequence of higher direct images \[R^{p - 1}f_*\mathcal{F}_3 \to R^pf_*\mathcal{F}_1 \to R^pf_*\mathcal{F}_2 \to R^pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1\] Then it is clear that if 2-out-of-3 of the sheaves \(\mathcal{F}_i\) have property \(\mathcal{P}\), then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemma 01Y0 and 01Y1. Hence property \(\mathcal{P}\) holds for the third as well.

Let \(Z \subset X\) be an integral closed subscheme. We have to find a coherent sheaf \(\mathcal{F}\) on \(X\) whose support is contained in \(Z\), whose stalk at the generic point \(\xi\) of \(Z\) is a \(1\)-dimensional vector space over \(\kappa(\xi)\) such that \(\mathcal{P}\) holds for \(\mathcal{F}\). Denote \(g = f|_Z : Z \to S\) the restriction of \(f\). Suppose we can find a coherent sheaf \(\mathcal{G}\) on \(Z\) such that (a) \(\mathcal{G}_\xi\) is a \(1\)-dimensional vector space over \(\kappa(\xi)\), (b) \(R^pg_*\mathcal{G} = 0\) for \(p > 0\), and (c) \(g_*\mathcal{G}\) is coherent. Then we can consider \(\mathcal{F} = (Z \to X)_*\mathcal{G}\). As \(Z \to X\) is a closed immersion we see that \((Z \to X)_*\mathcal{G}\) is coherent on \(X\) and \(R^p(Z \to X)_*\mathcal{G} = 0\) for \(p > 0\) (Lemma 01Y6). Hence by the relative Leray spectral sequence (Cohomology, Lemma 01F6) we will have \(R^pf_*\mathcal{F} = R^pg_*\mathcal{G} = 0\) for \(p > 0\) and \(f_*\mathcal{F} = g_*\mathcal{G}\) is coherent. Finally \(\mathcal{F}_\xi = ((Z \to X)_*\mathcal{G})_\xi = \mathcal{G}_\xi\) which verifies the condition on the stalk at \(\xi\). Hence everything depends on finding a coherent sheaf \(\mathcal{G}\) on \(Z\) which has properties (a), (b), and (c).

We can apply Chow’s Lemma 0200 to the morphism \(Z \to S\). Thus we get a diagram \[\xymatrix{ Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^m_S \ar[dl] \\ & S & }\] as in the statement of Chow’s lemma. Also, let \(U \subset Z\) be the dense open subscheme such that \(\pi^{-1}(U) \to U\) is an isomorphism. By the discussion in Remark 0201 we see that \(i' = (i, \pi) : Z' \to \mathbf{P}^m_Z\) is a closed immersion. Hence \[\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^m_S}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^m_Z}(1)\] is \(g'\)-relatively ample and \(\pi\)-relatively ample (for example by Morphisms, Lemma 02NR). Hence by Lemma 02O1 there exists an \(n \geq 0\) such that both \(R^p\pi_*\mathcal{L}^{\otimes n} = 0\) for all \(p > 0\) and \(R^p(g')_*\mathcal{L}^{\otimes n} = 0\) for all \(p > 0\). Set \(\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}\). Property (a) holds because \(\pi_*\mathcal{L}^{\otimes n}|_U\) is an invertible sheaf (as \(\pi^{-1}(U) \to U\) is an isomorphism). Properties (b) and (c) hold because by the relative Leray spectral sequence (Cohomology, Lemma 01F6) we have \[E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes n} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes n}\] and by choice of \(n\) the only nonzero terms in \(E_2^{p, q}\) are those with \(q = 0\) and the only nonzero terms of \(R^{p + q}(g')_*\mathcal{L}^{\otimes n}\) are those with \(p = q = 0\). This implies that \(R^pg_*\mathcal{G} = 0\) for \(p > 0\) and that \(g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}\). Finally, applying the previous Lemma 02O4 we see that \(g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}\) is coherent as desired.

Lemma

Let \(S = \Spec(A)\) with \(A\) a Noetherian ring. Let \(f : X \to S\) be a proper morphism. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Then \(H^i(X, \mathcal{F})\) is a finite \(A\)-module for all \(i \geq 0\).

Proof

This is just the affine case of Proposition 02O5. Namely, by Lemmas 01XJ and 01XK we know that \(R^if_*\mathcal{F}\) is the quasi-coherent sheaf associated to the \(A\)-module \(H^i(X, \mathcal{F})\) and by Lemma 01XZ this is a coherent sheaf if and only if \(H^i(X, \mathcal{F})\) is an \(A\)-module of finite type.

Lemma

Let \(A\) be a Noetherian ring. Let \(B\) be a finitely generated graded \(A\)-algebra. Let \(f : X \to \Spec(A)\) be a proper morphism. Set \(\mathcal{B} = f^*\widetilde B\). Let \(\mathcal{F}\) be a quasi-coherent graded \(\mathcal{B}\)-module of finite type.

1. For every \(p \geq 0\) the graded \(B\)-module \(H^p(X, \mathcal{F})\) is a finite \(B\)-module.

2. If \(\mathcal{L}\) is an ample invertible \(\mathcal{O}_X\)-module, then there exists an integer \(d_0\) such that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0\) for all \(p > 0\) and \(d \geq d_0\).

Proof

To prove this we consider the fibre product diagram \[\xymatrix{ X' = \Spec(B) \times_{\Spec(A)} X \ar[r]_-\pi \ar[d]_{f'} & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) }\] Note that \(f'\) is a proper morphism, see Morphisms, Lemma 01W4. Also, \(B\) is a finitely generated \(A\)-algebra, and hence Noetherian (Algebra, Lemma 00FN). This implies that \(X'\) is a Noetherian scheme (Morphisms, Lemma 01T6). Note that \(X'\) is the relative spectrum of the quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{B}\) by Constructions, Lemma 01LX. Since \(\mathcal{F}\) is a quasi-coherent \(\mathcal{B}\)-module we see that there is a unique quasi-coherent \(\mathcal{O}_{X'}\)-module \(\mathcal{F}'\) such that \(\pi_*\mathcal{F}' = \mathcal{F}\), see Morphisms, Lemma 01SB Since \(\mathcal{F}\) is finite type as a \(\mathcal{B}\)-module we conclude that \(\mathcal{F}'\) is a finite type \(\mathcal{O}_{X'}\)-module (details omitted). In other words, \(\mathcal{F}'\) is a coherent \(\mathcal{O}_{X'}\)-module (Lemma 01XZ). Since the morphism \(\pi : X' \to X\) is affine we have \[H^p(X, \mathcal{F}) = H^p(X', \mathcal{F}')\] by Lemma 089W. Thus (1) follows from Lemma 02O6. Given \(\mathcal{L}\) as in (2) we set \(\mathcal{L}' = \pi^*\mathcal{L}\). Note that \(\mathcal{L}'\) is ample on \(X'\) by Morphisms, Lemma 0892. By the projection formula (Cohomology, Lemma 01E8) we have \(\pi_*(\mathcal{F}' \otimes \mathcal{L}') = \mathcal{F} \otimes \mathcal{L}\). Thus part (2) follows by the same reasoning as above from Lemma 02O1.

The theorem on formal functions

In this section we study the behaviour of cohomology of sequences of sheaves either of the form \(\{I^n\mathcal{F}\}_{n \geq 0}\) or of the form \(\{\mathcal{F}/I^n\mathcal{F}\}_{n \geq 0}\) as \(n\) varies.

Here and below we use the following notation. Given a morphism of schemes \(f : X \to Y\), a quasi-coherent sheaf \(\mathcal{F}\) on \(X\), and a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Y\) we denote \(\mathcal{I}^n\mathcal{F}\) the quasi-coherent subsheaf generated by products of local sections of \(f^{-1}(\mathcal{I}^n)\) and \(\mathcal{F}\). In a formula \[\mathcal{I}^n\mathcal{F} = \Im\left( f^*(\mathcal{I}^n) \otimes_{\mathcal{O}_X} \mathcal{F} \longrightarrow \mathcal{F} \right).\] Note that there are natural maps \[f^{-1}(\mathcal{I}^n) \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{I}^m\mathcal{F} \longrightarrow f^*(\mathcal{I}^n) \otimes_{\mathcal{O}_X} \mathcal{I}^m\mathcal{F} \longrightarrow \mathcal{I}^{n + m}\mathcal{F}\] Hence a section of \(\mathcal{I}^n\) will give rise to a map \(R^pf_*(\mathcal{I}^m\mathcal{F}) \to R^pf_*(\mathcal{I}^{n + m}\mathcal{F})\) by functoriality of higher direct images. Localizing and then sheafifying we see that there are \(\mathcal{O}_Y\)-module maps \[\mathcal{I}^n \otimes_{\mathcal{O}_Y} R^pf_*(\mathcal{I}^m\mathcal{F}) \longrightarrow R^pf_*(\mathcal{I}^{n + m}\mathcal{F}).\] In other words we see that \(\bigoplus_{n \geq 0} R^pf_*(\mathcal{I}^n\mathcal{F})\) is a graded \(\bigoplus_{n \geq 0} \mathcal{I}^n\)-module.

If \(Y = \Spec(A)\) and \(\mathcal{I} = \widetilde{I}\) we denote \(\mathcal{I}^n\mathcal{F}\) simply \(I^n\mathcal{F}\). The maps introduced above give \(M = \bigoplus H^p(X, I^n\mathcal{F})\) the structure of a graded \(S = \bigoplus I^n\)-module. If \(f\) is proper, \(A\) is Noetherian and \(\mathcal{F}\) is coherent, then this turns out to be a module of finite type.

Lemma

Let \(A\) be a Noetherian ring. Let \(I \subset A\) be an ideal. Set \(B = \bigoplus_{n \geq 0} I^n\). Let \(f : X \to \Spec(A)\) be a proper morphism. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Then for every \(p \geq 0\) the graded \(B\)-module \(\bigoplus_{n \geq 0} H^p(X, I^n\mathcal{F})\) is a finite \(B\)-module.

Proof

Let \(\mathcal{B} = f^*\widetilde{B}\). Since \(\mathcal{B}\) surjects onto \(\bigoplus I^n\mathcal{O}_X\) it is clear that \(\bigoplus I^n\mathcal{F}\) is a finite type graded \(\mathcal{B}\)-module. Hence the result follows from Lemma 0897 part (1).

Lemma

Given a morphism of schemes \(f : X \to Y\), a quasi-coherent sheaf \(\mathcal{F}\) on \(X\), and a quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_Y\). Assume \(Y\) locally Noetherian, \(f\) proper, and \(\mathcal{F}\) coherent. Then \[\mathcal{M} = \bigoplus\nolimits_{n \geq 0} R^pf_*(\mathcal{I}^n\mathcal{F})\] is a graded \(\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{I}^n\)-module which is quasi-coherent and of finite type.

Proof

The statement is local on \(Y\), hence this reduces to the case where \(Y\) is affine. In the affine case the result follows from Lemma 02O8. Details omitted.

Lemma

Let \(A\) be a Noetherian ring. Let \(I \subset A\) be an ideal. Let \(f : X \to \Spec(A)\) be a proper morphism. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Then for every \(p \geq 0\) there exists an integer \(c \geq 0\) such that

1. the multiplication map \(I^{n - c} \otimes H^p(X, I^c\mathcal{F}) \to H^p(X, I^n\mathcal{F})\) is surjective for all \(n \geq c\),

2. the image of \(H^p(X, I^{n + m}\mathcal{F}) \to H^p(X, I^n\mathcal{F})\) is contained in the submodule \(I^{m - e} H^p(X, I^n\mathcal{F})\) where \(e = \max(0, c - n)\) for \(n + m \geq c\), \(n, m \geq 0\),

3. we have \[\Ker(H^p(X, I^n\mathcal{F}) \to H^p(X, \mathcal{F})) = \Ker(H^p(X, I^n\mathcal{F}) \to H^p(X, I^{n - c}\mathcal{F}))\] for \(n \geq c\),

4. there are maps \(I^nH^p(X, \mathcal{F}) \to H^p(X, I^{n - c}\mathcal{F})\) for \(n \geq c\) such that the compositions \[H^p(X, I^n\mathcal{F}) \to I^{n - c}H^p(X, \mathcal{F}) \to H^p(X, I^{n - 2c}\mathcal{F})\] and \[I^nH^p(X, \mathcal{F}) \to H^p(X, I^{n - c}\mathcal{F}) \to I^{n - 2c}H^p(X, \mathcal{F})\] for \(n \geq 2c\) are the canonical ones, and

5. the inverse systems \((H^p(X, I^n\mathcal{F}))\) and \((I^nH^p(X, \mathcal{F}))\) are pro-isomorphic.

Proof

Write \(M_n = H^p(X, I^n\mathcal{F})\) for \(n \geq 1\) and \(M_0 = H^p(X, \mathcal{F})\) so that we have maps \(\ldots \to M_3 \to M_2 \to M_1 \to M_0\). Setting \(B = \bigoplus_{n \geq 0} I^n\), then \(M = \bigoplus_{n \geq 0} M_n\) is a finite graded \(B\)-module, see Lemma 02O8. Observe that the products \(B_n \otimes M_m \to M_{m + n}\), \(a \otimes m \mapsto a \cdot m\) are compatible with the maps in our inverse system in the sense that the diagrams \[\xymatrix{ B_n \otimes_A M_m \ar[r] \ar[d] & M_{n + m} \ar[d] \\ B_n \otimes_A M_{m'} \ar[r] & M_{n + m'} }\] commute for \(n, m' \geq 0\) and \(m \geq m'\).

Proof of (1). Choose \(d_1, \ldots, d_t \geq 0\) and \(x_i \in M_{d_i}\) such that \(M\) is generated by \(x_1, \ldots, x_t\) over \(B\). For any \(c \geq \max\{d_i\}\) we conclude that \(B_{n - c} \cdot M_c = M_n\) for \(n \geq c\) and we conclude (1) is true.

Proof of (2). Let \(c\) be as in the proof of (1). Let \(n + m \geq c\). We have \(M_{n + m} = B_{n + m - c} \cdot M_c\). If \(c > n\) then we use \(M_c \to M_n\) and the compatibility of products with transition maps pointed out above to conclude that the image of \(M_{n + m} \to M_n\) is contained in \(I^{n + m - c}M_n\). If \(c \leq n\), then we write \(M_{n + m} = B_m \cdot B_{n - c} \cdot M_c = B_m \cdot M_n\) to see that the image is contained in \(I^m M_n\). This proves (2).

Let \(K_n \subset M_n\) be the kernel of the map \(M_n \to M_0\). The compatibility of products with transition maps pointed out above shows that \(K = \bigoplus K_n \subset M\) is a graded \(B\)-submodule. As \(B\) is Noetherian and \(M\) is a finitely generated graded \(B\)-module, this shows that \(K\) is a finitely generated graded \(B\)-module. Choose \(d'_1, \ldots, d'_{t'} \geq 0\) and \(y_i \in K_{d'_i}\) such that \(K\) is generated by \(y_1, \ldots, y_{t'}\) over \(B\). Set \(c = \max(d'_i, d'_j)\). Since \(y_i \in \Ker(M_{d'_i} \to M_0)\) we see that \(B_n \cdot y_i \subset \Ker(M_{n + d'_i} \to M_n)\). In this way we see that \(K_n = \Ker(M_n \to M_{n - c})\) for \(n \geq c\). This proves (3).

Consider the following commutative solid diagram \[\xymatrix{ I^n \otimes_A M_0 \ar[r] \ar[d] & I^nM_0 \ar[r] \ar@{..>}[d] & M_0 \ar[d] \\ M_n \ar[r] & M_{n - c} \ar[r] & M_0 }\] Since the kernel of the surjective arrow \(I^n \otimes_A M_0 \to I^nM_0\) maps into \(K_n\) by the above we obtain the dotted arrow and the composition \(I^nM_0 \to M_{n - c} \to M_0\) is the canonical map. Then clearly the composition \(I^nM_0 \to M_{n - c} \to I^{n - 2c}M_0\) is the canonical map for \(n \geq 2c\). Consider the composition \(M_n \to I^{n - c}M_0 \to M_{n - 2c}\). The first map sends an element of the form \(a \cdot m\) with \(a \in I^{n - c}\) and \(m \in M_c\) to \(a m'\) where \(m'\) is the image of \(m\) in \(M_0\). Then the second map sends this to \(a \cdot m'\) in \(M_{n - 2c}\) and we see (4) is true.

Part (5) is an immediate consequence of (4) and the definition of morphisms of pro-objects.

In the situation of Lemmas 02O8 and 02OA consider the inverse system \[\mathcal{F}/I\mathcal{F} \leftarrow \mathcal{F}/I^2\mathcal{F} \leftarrow \mathcal{F}/I^3\mathcal{F} \leftarrow \ldots\] We would like to know what happens to the cohomology groups. Here is a first result.

Lemma

Let \(A\) be a Noetherian ring. Let \(I \subset A\) be an ideal. Let \(f : X \to \Spec(A)\) be a proper morphism. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Fix \(p \geq 0\). There exists a \(c \geq 0\) such that

1. for all \(n \geq c\) we have \[\Ker(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) \subset I^{n - c}H^p(X, \mathcal{F}).\]

2. the inverse system \[\left(H^p(X, \mathcal{F}/I^n\mathcal{F})\right)_{n \in \mathbf{N}}\] satisfies the Mittag-Leffler condition (see Homology, Definition 02N0), and

3. we have \[\Im(H^p(X, \mathcal{F}/I^k\mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) = \Im(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}))\] for all \(k \geq n + c\).

Proof

Let \(c = \max\{c_p, c_{p + 1}\}\), where \(c_p, c_{p + 1}\) are the integers found in Lemma 02OA for \(H^p\) and \(H^{p + 1}\).

Let us prove part (1). Consider the short exact sequence \[0 \to I^n\mathcal{F} \to \mathcal{F} \to \mathcal{F}/I^n\mathcal{F} \to 0\] From the long exact cohomology sequence we see that \[\Ker( H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}) ) = \Im( H^p(X, I^n\mathcal{F}) \to H^p(X, \mathcal{F}) )\] Hence by Lemma 02OA part (2) we see that this is contained in \(I^{n - c}H^p(X, \mathcal{F})\) for \(n \geq c\).

Note that part (3) implies part (2) by definition of the Mittag-Leffler systems.

Let us prove part (3). Fix an \(n\). Consider the commutative diagram \[\xymatrix{ 0 \ar[r] & I^n\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}/I^n\mathcal{F} \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F} \ar[r] \ar[u] & \mathcal{F} \ar[r] \ar[u] & \mathcal{F}/I^{n + m}\mathcal{F} \ar[r] \ar[u] & 0 }\] This gives rise to the following commutative diagram \[\xymatrix{ H^p(X, \mathcal{F}) \ar[r] & H^p(X, \mathcal{F}/I^n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^n\mathcal{F}) \ar[r] & H^{p + 1}(X, \mathcal{F}) \\ H^p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u]^\gamma & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^\alpha \ar[r]^-\beta & H^{p + 1}(X, \mathcal{F}) \ar[u]_1 }\] with exact rows. By Lemma 02OA part (4) the kernel of \(\beta\) is equal to the kernel of \(\alpha\) for \(m \geq c\). By a diagram chase this shows that the image of \(\gamma\) is contained in the kernel of \(\delta\) which shows that part (3) is true (set \(k = n + m\) to get it).

Theorem

Let \(A\) be a Noetherian ring. Let \(I \subset A\) be an ideal. Let \(f : X \to \Spec(A)\) be a proper morphism. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Fix \(p \geq 0\). The system of maps \[H^p(X, \mathcal{F})/I^nH^p(X, \mathcal{F}) \longrightarrow H^p(X, \mathcal{F}/I^n\mathcal{F})\] define an isomorphism of limits \[H^p(X, \mathcal{F})^\wedge \longrightarrow \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F})\] where the left hand side is the completion of the \(A\)-module \(H^p(X, \mathcal{F})\) with respect to the ideal \(I\), see Algebra, Section 00M9. Moreover, this is in fact a homeomorphism for the limit topologies.

Proof

This follows from Lemma 02OB as follows. Set \(M = H^p(X, \mathcal{F})\), \(M_n = H^p(X, \mathcal{F}/I^n\mathcal{F})\), and denote \(N_n = \Im(M \to M_n)\). By Lemma 02OB parts (2) and (3) we see that \((M_n)\) is a Mittag-Leffler system with \(N_n \subset M_n\) equal to the image of \(M_k\) for all \(k \gg n\). It follows that \(\lim M_n = \lim N_n\) as topological modules (with limit topologies). On the other hand, the \(N_n\) form an inverse system of quotients of the module \(M\) and hence \(\lim N_n\) is the completion of \(M\) with respect to the topology given by the kernels \(K_n = \Ker(M \to N_n)\). By Lemma 02OB part (1) we have \(K_n \subset I^{n - c}M\) and since \(N_n \subset M_n\) is annihilated by \(I^n\) we have \(I^n M \subset K_n\). Thus the topology defined using the submodules \(K_n\) as a fundamental system of open neighbourhoods of \(0\) is the same as the \(I\)-adic topology and we find that the induced map \(M^\wedge = \lim M/I^nM \to \lim N_n = \lim M_n\) is an isomorphism of topological modules³.

Lemma

Let \(A\) be a ring. Let \(I \subset A\) be an ideal. Assume \(A\) is Noetherian and complete with respect to \(I\). Let \(f : X \to \Spec(A)\) be a proper morphism. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Then \[H^p(X, \mathcal{F}) = \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F})\] for all \(p \geq 0\).

Proof

This is a reformulation of the theorem on formal functions (Theorem 02OC) in the case of a complete Noetherian base ring. Namely, in this case the \(A\)-module \(H^p(X, \mathcal{F})\) is finite (Lemma 02O6) hence \(I\)-adically complete (Algebra, Lemma 00MA) and we see that completion on the left hand side is not necessary.

Lemma

Given a morphism of schemes \(f : X \to Y\) and a quasi-coherent sheaf \(\mathcal{F}\) on \(X\). Assume

1. \(Y\) locally Noetherian,

2. \(f\) proper, and

3. \(\mathcal{F}\) coherent.

Let \(y \in Y\) be a point. Consider the infinitesimal neighbourhoods \[\xymatrix{ X_n = \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n) \times_Y X \ar[r]_-{i_n} \ar[d]_{f_n} & X \ar[d]^f \\ \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n) \ar[r]^-{c_n} & Y }\] of the fibre \(X_1 = X_y\) and set \(\mathcal{F}_n = i_n^*\mathcal{F}\). Then we have \[\left(R^pf_*\mathcal{F}\right)_y^\wedge \cong \lim_n H^p(X_n, \mathcal{F}_n)\] as \(\mathcal{O}_{Y, y}^\wedge\)-modules.

Proof

This is just a reformulation of a special case of the theorem on formal functions, Theorem 02OC. Let us spell it out. Note that \(\mathcal{O}_{Y, y}\) is a Noetherian local ring. Consider the canonical morphism \(c : \Spec(\mathcal{O}_{Y, y}) \to Y\), see Schemes, Equation (02NA). This is a flat morphism as it identifies local rings. Denote momentarily \(f' : X' \to \Spec(\mathcal{O}_{Y, y})\) the base change of \(f\) to this local ring. We see that \(c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'\) by Lemma 02KH. Moreover, the infinitesimal neighbourhoods of the fibre \(X_y\) and \(X'_y\) are identified (verification omitted; hint: the morphisms \(c_n\) factor through \(c\)).

Hence we may assume that \(Y = \Spec(A)\) is the spectrum of a Noetherian local ring \(A\) with maximal ideal \(\mathfrak m\) and that \(y \in Y\) corresponds to the closed point (i.e., to \(\mathfrak m\)). In particular it follows that \[\left(R^pf_*\mathcal{F}\right)_y = \Gamma(Y, R^pf_*\mathcal{F}) = H^p(X, \mathcal{F}).\]

In this case also, the morphisms \(c_n\) are each closed immersions. Hence their base changes \(i_n\) are closed immersions as well. Note that \(i_{n, *}\mathcal{F}_n = i_{n, *}i_n^*\mathcal{F} = \mathcal{F}/\mathfrak m^n\mathcal{F}\). By the Leray spectral sequence for \(i_n\), and Lemma 01Y6 we see that \[H^p(X_n, \mathcal{F}_n) = H^p(X, i_{n, *}\mathcal{F}_n) = H^p(X, \mathcal{F}/\mathfrak m^n\mathcal{F})\] Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win.

Here is a lemma which we will generalize later to fibres of dimension \(> 0\), namely the next lemma.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(y \in Y\). Assume

1. \(Y\) locally Noetherian,

2. \(f\) is proper, and

3. \(f^{-1}(\{y\})\) is finite.

Then for any coherent sheaf \(\mathcal{F}\) on \(X\) we have \((R^pf_*\mathcal{F})_y = 0\) for all \(p > 0\).

Proof

The fibre \(X_y\) is finite, and by Morphisms, Lemma 02NG it is a finite discrete space. Moreover, the underlying topological space of each infinitesimal neighbourhood \(X_n\) is the same. Hence each of the schemes \(X_n\) is affine according to Schemes, Lemma 02O0. Hence it follows that \(H^p(X_n, \mathcal{F}_n) = 0\) for all \(p > 0\). Hence we see that \((R^pf_*\mathcal{F})_y^\wedge = 0\) by Lemma 02OD. Note that \(R^pf_*\mathcal{F}\) is coherent by Proposition 02O5 and hence \(R^pf_*\mathcal{F}_y\) is a finite \(\mathcal{O}_{Y, y}\)-module. By Nakayama’s lemma (Algebra, Lemma 00DV) if the completion of a finite module over a local ring is zero, then the module is zero. Whence \((R^pf_*\mathcal{F})_y = 0\).

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Let \(y \in Y\). Assume

1. \(Y\) locally Noetherian,

2. \(f\) is proper, and

3. \(\dim(X_y) = d\).

Then for any coherent sheaf \(\mathcal{F}\) on \(X\) we have \((R^pf_*\mathcal{F})_y = 0\) for all \(p > d\).

Proof

The fibre \(X_y\) is of finite type over \(\Spec(\kappa(y))\). Hence \(X_y\) is a Noetherian scheme by Morphisms, Lemma 01T6. Hence the underlying topological space of \(X_y\) is Noetherian, see Properties, Lemma 01OZ. Moreover, the underlying topological space of each infinitesimal neighbourhood \(X_n\) is the same as that of \(X_y\). Hence \(H^p(X_n, \mathcal{F}_n) = 0\) for all \(p > d\) by Cohomology, Proposition 02UZ. Hence we see that \((R^pf_*\mathcal{F})_y^\wedge = 0\) by Lemma 02OD for \(p > d\). Note that \(R^pf_*\mathcal{F}\) is coherent by Proposition 02O5 and hence \(R^pf_*\mathcal{F}_y\) is a finite \(\mathcal{O}_{Y, y}\)-module. By Nakayama’s lemma (Algebra, Lemma 00DV) if the completion of a finite module over a local ring is zero, then the module is zero. Whence \((R^pf_*\mathcal{F})_y = 0\).

Applications of the theorem on formal functions

We will add more here as needed. For the moment we need the following characterization of finite morphisms in the Noetherian case.

Lemma

(For a more general version see More on Morphisms, Lemma 02LS.) Let \(f : X \to S\) be a morphism of schemes. Assume \(S\) is locally Noetherian. The following are equivalent

1. \(f\) is finite, and

2. \(f\) is proper with finite fibres.

Proof

A finite morphism is proper according to Morphisms, Lemma 01WN. A finite morphism is quasi-finite according to Morphisms, Lemma 02NU. A quasi-finite morphism has finite fibres, see Morphisms, Lemma 02NH. Hence a finite morphism is proper and has finite fibres.

Assume \(f\) is proper with finite fibres. We want to show \(f\) is finite. In fact it suffices to prove \(f\) is affine. Namely, if \(f\) is affine, then it follows that \(f\) is integral by Morphisms, Lemma 01WM whereupon it follows from Morphisms, Lemma 01WJ that \(f\) is finite.

To show that \(f\) is affine we may assume that \(S\) is affine, and our goal is to show that \(X\) is affine too. Since \(f\) is proper we see that \(X\) is separated and quasi-compact. Hence we may use the criterion of Lemma 01XG to prove that \(X\) is affine. To see this let \(\mathcal{I} \subset \mathcal{O}_X\) be a finite type ideal sheaf. In particular \(\mathcal{I}\) is a coherent sheaf on \(X\). By Lemma 02OE we conclude that \(R^1f_*\mathcal{I}_s = 0\) for all \(s \in S\). In other words, \(R^1f_*\mathcal{I} = 0\). Hence we see from the Leray Spectral Sequence for \(f\) that \(H^1(X , \mathcal{I}) = H^1(S, f_*\mathcal{I})\). Since \(S\) is affine, and \(f_*\mathcal{I}\) is quasi-coherent (Schemes, Lemma 01LC) we conclude \(H^1(S, f_*\mathcal{I}) = 0\) from Lemma 01XB as desired. Hence \(H^1(X, \mathcal{I}) = 0\) as desired.

As a consequence we have the following useful result.

Lemma

(For a more general version see More on Morphisms, Lemma 02UP.) Let \(f : X \to S\) be a morphism of schemes. Let \(s \in S\). Assume

1. \(S\) is locally Noetherian,

2. \(f\) is proper, and

3. \(f^{-1}(\{s\})\) is a finite set.

Then there exists an open neighbourhood \(V \subset S\) of \(s\) such that \(f|_{f^{-1}(V)} : f^{-1}(V) \to V\) is finite.

Proof

The morphism \(f\) is quasi-finite at all the points of \(f^{-1}(\{s\})\) by Morphisms, Lemma 02NG. By Morphisms, Lemma 01TI the set of points at which \(f\) is quasi-finite is an open \(U \subset X\). Let \(Z = X \setminus U\). Then \(s \not \in f(Z)\). Since \(f\) is proper the set \(f(Z) \subset S\) is closed. Choose any open neighbourhood \(V \subset S\) of \(s\) with \(Z \cap V = \emptyset\). Then \(f^{-1}(V) \to V\) is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 01TJ), hence has finite fibres (Morphisms, Lemma 02NH), hence is finite by Lemma 02OG.

Lemma

Let \(f : X \to Y\) be a proper morphism of schemes with \(Y\) Noetherian. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Let \(y \in Y\) be a point such that \(\mathcal{L}_y\) is ample on \(X_y\). Then there exists a \(d_0\) such that for all \(d \geq d_0\) we have \[R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = 0 \text{ for }p > 0\] and the map \[f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y \longrightarrow H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})\] is surjective.

Proof

Note that \(\mathcal{O}_{Y, y}\) is a Noetherian local ring. Consider the canonical morphism \(c : \Spec(\mathcal{O}_{Y, y}) \to Y\), see Schemes, Equation (02NA). This is a flat morphism as it identifies local rings. Denote momentarily \(f' : X' \to \Spec(\mathcal{O}_{Y, y})\) the base change of \(f\) to this local ring. We see that \(c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'\) by Lemma 02KH. Moreover, the fibres \(X_y\) and \(X'_y\) are identified. Hence we may assume that \(Y = \Spec(A)\) is the spectrum of a Noetherian local ring \((A, \mathfrak m, \kappa)\) and \(y \in Y\) corresponds to \(\mathfrak m\). In this case \(R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) for all \(p \geq 0\). Denote \(f_y : X_y \to \Spec(\kappa)\) the projection.

Let \(B = \text{Gr}_\mathfrak m(A) = \bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}\). Consider the sheaf \(\mathcal{B} = f_y^*\widetilde{B}\) of quasi-coherent graded \(\mathcal{O}_{X_y}\)-algebras. We will use notation as in Section 02O7 with \(I\) replaced by \(\mathfrak m\). Since \(X_y\) is the closed subscheme of \(X\) cut out by \(\mathfrak m\mathcal{O}_X\) we may think of \(\mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}\) as a coherent \(\mathcal{O}_{X_y}\)-module, see Lemma 087T. Then \(\bigoplus_{n \geq 0} \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}\) is a quasi-coherent graded \(\mathcal{B}\)-module of finite type because it is generated in degree zero over \(\mathcal{B}\) and because the degree zero part is \(\mathcal{F}_y = \mathcal{F}/\mathfrak m \mathcal{F}\) which is a coherent \(\mathcal{O}_{X_y}\)-module. Hence by Lemma 0897 part (2) we see that \[H^p(X_y, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d}) = 0\] for all \(p > 0\), \(d \geq d_0\), and \(n \geq 0\). By Lemma 089W this is the same as the statement that \(H^p(X, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\) for all \(p > 0\), \(d \geq d_0\), and \(n \geq 0\).

Consider the short exact sequences \[0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \to 0\] of coherent \(\mathcal{O}_X\)-modules. Tensoring with \(\mathcal{L}^{\otimes d}\) is an exact functor and we obtain short exact sequences \[0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to 0\] Using the long exact cohomology sequence and the vanishing above we conclude (using induction) that

1. \(H^p(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\) for all \(p > 0\), \(d \geq d_0\), and \(n \geq 0\), and

2. \(H^0(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})\) is surjective for all \(d \geq d_0\) and \(n \geq 1\).

By the theorem on formal functions (Theorem 02OC) we find that the \(\mathfrak m\)-adic completion of \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) is zero for all \(d \geq d_0\) and \(p > 0\). Since \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) is a finite \(A\)-module by Lemma 02O6 it follows from Nakayama’s lemma (Algebra, Lemma 00DV) that \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) is zero for all \(d \geq d_0\) and \(p > 0\). For \(p = 0\) we deduce from Lemma 02OB part (3) that \(H^0(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})\) is surjective, which gives the final statement of the lemma.

Lemma

(For a more general version see More on Morphisms, Lemma 0D2S.) Let \(f : X \to Y\) be a proper morphism of schemes with \(Y\) Noetherian. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(y \in Y\) be a point such that \(\mathcal{L}_y\) is ample on \(X_y\). Then there is an open neighbourhood \(V \subset Y\) of \(y\) such that \(\mathcal{L}|_{f^{-1}(V)}\) is ample on \(f^{-1}(V)/V\).

Proof

Pick \(d_0\) as in Lemma 0D2M for \(\mathcal{F} = \mathcal{O}_X\). Pick \(d \geq d_0\) so that we can find \(r \geq 0\) and sections \(s_{y, 0}, \ldots, s_{y, r} \in H^0(X_y, \mathcal{L}_y^{\otimes d})\) which define a closed immersion \[\varphi_y = \varphi_{\mathcal{L}_y^{\otimes d}, (s_{y, 0}, \ldots, s_{y, r})} : X_y \to \mathbf{P}^r_{\kappa(y)}.\] This is possible by Morphisms, Lemma 01VT but we also use Morphisms, Lemma 01W6 to see that \(\varphi_y\) is a closed immersion and Constructions, Section 01ND for the description of morphisms into projective space in terms of invertible sheaves and sections. By our choice of \(d_0\), after replacing \(Y\) by an open neighbourhood of \(y\), we can choose \(s_0, \ldots, s_r \in H^0(X, \mathcal{L}^{\otimes d})\) mapping to \(s_{y, 0}, \ldots, s_{y, r}\). Let \(X_{s_i} \subset X\) be the open subset where \(s_i\) is a generator of \(\mathcal{L}^{\otimes d}\). Since the \(s_{y, i}\) generate \(\mathcal{L}_y^{\otimes d}\) we see that \(X_y \subset U = \bigcup X_{s_i}\). Since \(X \to Y\) is closed, we see that there is an open neighbourhood \(y \in V \subset Y\) such that \(f^{-1}(V) \subset U\). After replacing \(Y\) by \(V\) we may assume that the \(s_i\) generate \(\mathcal{L}^{\otimes d}\). Thus we obtain a morphism \[\varphi = \varphi_{\mathcal{L}^{\otimes d}, (s_0, \ldots, s_r)} : X \longrightarrow \mathbf{P}^r_Y\] with \(\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)\) whose base change to \(y\) gives \(\varphi_y\).

We will finish the proof by a sleight of hand; the “correct” proof proceeds by directly showing that \(\varphi\) is a closed immersion after base changing to an open neighbourhood of \(y\). Namely, by Lemma 02OH we see that \(\varphi\) is a finite over an open neighbourhood of the fibre \(\mathbf{P}^r_{\kappa(y)}\) of \(\mathbf{P}^r_Y \to Y\) above \(y\). Using that \(\mathbf{P}^r_Y \to Y\) is closed, after shrinking \(Y\) we may assume that \(\varphi\) is finite. Then \(\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)\) is ample by the very general Morphisms, Lemma 0892.

Cohomology and base change, III

In this section we prove the simplest case of a very general phenomenon that will be discussed in Derived Categories of Schemes, Section 08ET. Please see Remark 07VL for a translation of the following lemma into algebra.

Lemma

Let \(A\) be a Noetherian ring and set \(S = \Spec(A)\). Let \(f : X \to S\) be a proper morphism of schemes. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module flat over \(S\). Then

1. \(R\Gamma(X, \mathcal{F})\) is a perfect object of \(D(A)\), and

2. for any ring map \(A \to A'\) the base change map \[R\Gamma(X, \mathcal{F}) \otimes_A^{\mathbf{L}} A' \longrightarrow R\Gamma(X_{A'}, \mathcal{F}_{A'})\] is an isomorphism.

Proof

Choose a finite affine open covering \(X = \bigcup_{i = 1, \ldots, n} U_i\). By Lemmas 01XL and 01XM the Čech complex \(K^\bullet = {\check C}^\bullet(\mathcal{U}, \mathcal{F})\) satisfies \[K^\bullet \otimes_A A' = R\Gamma(X_{A'}, \mathcal{F}_{A'})\] for all ring maps \(A \to A'\). Let \(K_{alt}^\bullet = {\check C}_{alt}^\bullet(\mathcal{U}, \mathcal{F})\) be the alternating Čech complex. By Cohomology, Lemma 01FM there is a homotopy equivalence \(K_{alt}^\bullet \to K^\bullet\) of \(A\)-modules. In particular, we have \[K_{alt}^\bullet \otimes_A A' = R\Gamma(X_{A'}, \mathcal{F}_{A'})\] as well. Since \(\mathcal{F}\) is flat over \(A\) we see that each \(K_{alt}^n\) is flat over \(A\) (see Morphisms, Lemma 01U4). Since moreover \(K_{alt}^\bullet\) is bounded above (this is why we switched to the alternating Čech complex) \(K_{alt}^\bullet \otimes_A A' = K_{alt}^\bullet \otimes_A^{\mathbf{L}} A'\) by the definition of derived tensor products (see More on Algebra, Section 06XY). By Lemma 02O6 the cohomology groups \(H^i(K_{alt}^\bullet)\) are finite \(A\)-modules. As \(K_{alt}^\bullet\) is bounded, we conclude that \(K_{alt}^\bullet\) is pseudo-coherent, see More on Algebra, Lemma 066E. Given any \(A\)-module \(M\) set \(A' = A \oplus M\) where \(M\) is a square zero ideal, i.e., \((a, m) \cdot (a', m') = (aa', am' + a'm)\). By the above we see that \(K_{alt}^\bullet \otimes_A^\mathbf{L} A'\) has cohomology in degrees \(0, \ldots, n\). Hence \(K_{alt}^\bullet \otimes_A^\mathbf{L} M\) has cohomology in degrees \(0, \ldots, n\). Hence \(K_{alt}^\bullet\) has finite Tor dimension, see More on Algebra, Definition 0652. We win by More on Algebra, Lemma 0658.

Remark

A consequence of Lemma 07VK is that there exists a finite complex of finite projective \(A\)-modules \(M^\bullet\) such that we have \[H^i(X_{A'}, \mathcal{F}_{A'}) = H^i(M^\bullet \otimes_A A')\] functorially in \(A'\). The condition that \(\mathcal{F}\) is flat over \(A\) is essential, see [Hartshorne].

Coherent formal modules

As we do not yet have the theory of formal schemes to our disposal, we develop a bit of language that replaces the notion of a “coherent module on a Noetherian adic formal scheme”.

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. We will consider inverse systems \((\mathcal{F}_n)\) of coherent \(\mathcal{O}_X\)-modules such that

1. \(\mathcal{F}_n\) is annihilated by \(\mathcal{I}^n\), and

2. the transition maps induce isomorphisms \(\mathcal{F}_{n + 1}/\mathcal{I}^n\mathcal{F}_{n + 1} \to \mathcal{F}_n\).

A morphism of such inverse systems is defined as usual. Let us denote the category of these inverse systems with \(\textit{Coh}(X, \mathcal{I})\). We are going to proceed by proving a bunch of lemmas about objects in this category. In fact, most of the lemmas that follow are straightforward consequences of the following description of the category in the affine case.

Lemma

If \(X = \Spec(A)\) is the spectrum of a Noetherian ring and \(\mathcal{I}\) is the quasi-coherent sheaf of ideals associated to the ideal \(I \subset A\), then \(\textit{Coh}(X, \mathcal{I})\) is equivalent to the category of finite \(A^\wedge\)-modules where \(A^\wedge\) is the completion of \(A\) with respect to \(I\).

Proof

Let \(\text{Mod}^{fg}_{A, I}\) be the category of inverse systems \((M_n)\) of finite \(A\)-modules satisfying: (1) \(M_n\) is annihilated by \(I^n\) and (2) \(M_{n + 1}/I^nM_{n + 1} = M_n\). By the correspondence between coherent sheaves on \(X\) and finite \(A\)-modules (Lemma 01XZ) it suffices to show \(\text{Mod}^{fg}_{A, I}\) is equivalent to the category of finite \(A^\wedge\)-modules. To see this it suffices to prove that given an object \((M_n)\) of \(\text{Mod}^{fg}_{A, I}\) the module \[M = \lim M_n\] is a finite \(A^\wedge\)-module and that \(M/I^nM = M_n\). As the transition maps are surjective, we see that \(M \to M_1\) is surjective. Pick \(x_1, \ldots, x_t \in M\) which map to generators of \(M_1\). This induces a map of systems \((A/I^n)^{\oplus t} \to M_n\). By Nakayama’s lemma (Algebra, Lemma 00DV) these maps are surjective. Let \(K_n \subset (A/I^n)^{\oplus t}\) be the kernel. Property (2) implies that \(K_{n + 1} \to K_n\) is surjective, in particular the system \((K_n)\) satisfies the Mittag-Leffler condition. By Homology, Lemma 02N1 we obtain an exact sequence \(0 \to K \to (A^\wedge)^{\oplus t} \to M \to 0\) with \(K = \lim K_n\). Hence \(M\) is a finite \(A^\wedge\)-module. As \(K \to K_n\) is surjective it follows that \[M/I^nM = \Coker(K \to (A/I^n)^{\oplus t}) = (A/I^n)^{\oplus t}/K_n = M_n\] as desired.

Lemma

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals.

1. The category \(\textit{Coh}(X, \mathcal{I})\) is abelian.

2. For \(U \subset X\) open the restriction functor \(\textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(U, \mathcal{I}|_U)\) is exact.

3. Exactness in \(\textit{Coh}(X, \mathcal{I})\) may be checked by restricting to the members of an open covering of \(X\).

Proof

Let \(\alpha =(\alpha_n) : (\mathcal{F}_n) \to (\mathcal{G}_n)\) be a morphism of \(\textit{Coh}(X, \mathcal{I})\). The cokernel of \(\alpha\) is the inverse system \((\Coker(\alpha_n))\) (details omitted). To describe the kernel let \[\mathcal{K}'_{l, m} = \Im(\Ker(\alpha_l) \to \mathcal{F}_m)\] for \(l \geq m\). We claim:

1. the inverse system \((\mathcal{K}'_{l, m})_{l \geq m}\) is eventually constant, say with value \(\mathcal{K}'_m\),

2. the system \((\mathcal{K}'_m/\mathcal{I}^n\mathcal{K}'_m)_{m \geq n}\) is eventually constant, say with value \(\mathcal{K}_n\),

3. the system \((\mathcal{K}_n)\) forms an object of \(\textit{Coh}(X, \mathcal{I})\), and

4. this object is the kernel of \(\alpha\).

To see (a), (b), and (c) we may work affine locally, say \(X = \Spec(A)\) and \(\mathcal{I}\) corresponds to the ideal \(I \subset A\). By Lemma 087W \(\alpha\) corresponds to a map \(f : M \to N\) of finite \(A^\wedge\)-modules. Denote \(K = \Ker(f)\). Note that \(A^\wedge\) is a Noetherian ring (Algebra, Lemma 0316). Choose an integer \(c \geq 0\) such that \(K \cap I^n M \subset I^{n - c}K\) for \(n \geq c\) (Algebra, Lemma 00IN) and which satisfies Algebra, Lemma 00IO for the map \(f\) and the ideal \(I^\wedge = IA^\wedge\). Then \(\mathcal{K}'_{l, m}\) corresponds to the \(A\)-module \[K'_{l, m} = \frac{a^{-1}(I^lN) + I^mM}{I^mM} = \frac{K + I^{l - c}f^{-1}(I^cN) + I^mM}{I^mM} = \frac{K + I^mM}{I^mM}\] where the last equality holds if \(l \geq m + c\). So \(\mathcal{K}'_m\) corresponds to the \(A\)-module \(K/K \cap I^mM\) and \(\mathcal{K}'_m/\mathcal{I}^n\mathcal{K}'_m\) corresponds to \[\frac{K}{K \cap I^mM + I^nK} = \frac{K}{I^nK}\] for \(m \geq n + c\) by our choice of \(c\) above. Hence \(\mathcal{K}_n\) corresponds to \(K/I^nK\).

We prove (d). It is clear from the description on affines above that the composition \((\mathcal{K}_n) \to (\mathcal{F}_n) \to (\mathcal{G}_n)\) is zero. Let \(\beta : (\mathcal{H}_n) \to (\mathcal{F}_n)\) be a morphism such that \(\alpha \circ \beta = 0\). Then \(\mathcal{H}_l \to \mathcal{F}_l\) maps into \(\Ker(\alpha_l)\). Since \(\mathcal{H}_m = \mathcal{H}_l/\mathcal{I}^m\mathcal{H}_l\) for \(l \geq m\) we obtain a system of maps \(\mathcal{H}_m \to \mathcal{K}'_{l, m}\). Thus a map \(\mathcal{H}_m \to \mathcal{K}_m'\). Since \(\mathcal{H}_n = \mathcal{H}_m/\mathcal{I}^n\mathcal{H}_m\) we obtain a system of maps \(\mathcal{H}_n \to \mathcal{K}'_m/\mathcal{I}^n\mathcal{K}'_m\) and hence a map \(\mathcal{H}_n \to \mathcal{K}_n\) as desired.

To finish the proof of (1) we still have to show that \(\Coim = \Im\) in \(\textit{Coh}(X, \mathcal{I})\). We have seen above that taking kernels and cokernels commutes, over affines, with the description of \(\textit{Coh}(X, \mathcal{I})\) as a category of modules. Since \(\Im = \Coim\) holds in the category of modules this gives \(\Coim = \Im\) in \(\textit{Coh}(X, \mathcal{I})\). Parts (2) and (3) of the lemma are immediate from our construction of kernels and cokernels.

Lemma

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. A map \((\mathcal{F}_n) \to (\mathcal{G}_n)\) is surjective in \(\textit{Coh}(X, \mathcal{I})\) if and only if \(\mathcal{F}_1 \to \mathcal{G}_1\) is surjective.

Proof

Omitted. Hint: Look on affine opens, use Lemma 087W, and use Algebra, Lemma 00DV.

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. There is a functor [0880]\[\begin{equation} \textit{Coh}(\mathcal{O}_X) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \end{equation}\] which associates to the coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the object \(\mathcal{F}^\wedge = (\mathcal{F}/\mathcal{I}^n\mathcal{F})\) of \(\textit{Coh}(X, \mathcal{I})\).

Lemma

The functor (0880) is exact.

Proof

It suffices to check this locally on \(X\). Hence we may assume \(X\) is affine, i.e., we have a situation as in Lemma 087W. The functor is the functor \(\text{Mod}^{fg}_A \to \text{Mod}^{fg}_{A^\wedge}\) which associates to a finite \(A\)-module \(M\) the completion \(M^\wedge\). Thus the result follows from Algebra, Lemma 00MB.

Lemma

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Set \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})\). Then \[\lim H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge, \mathcal{F}^\wedge).\]

Proof

To prove this we may work affine locally on \(X\). Hence we may assume \(X = \Spec(A)\) and \(\mathcal{F}\), \(\mathcal{G}\) given by finite \(A\)-module \(M\) and \(N\). Then \(\mathcal{H}\) corresponds to the finite \(A\)-module \(H = \Hom_A(M, N)\). The statement of the lemma becomes the statement \[H^\wedge = \Hom_{A^\wedge}(M^\wedge, N^\wedge)\] via the equivalence of Lemma 087W. By Algebra, Lemma 00MB (used 3 times) we have \[H^\wedge = \Hom_A(M, N) \otimes_A A^\wedge = \Hom_{A^\wedge}(M \otimes_A A^\wedge, N \otimes_A A^\wedge) = \Hom_{A^\wedge}(M^\wedge, N^\wedge)\] where the second equality uses that \(A^\wedge\) is flat over \(A\) (see More on Algebra, Lemma 087R). The lemma follows.

Let \(X\) be a Noetherian scheme. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. We say an object \((\mathcal{F}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) is \(\mathcal{I}\)-power torsion or is annihilated by a power of \(\mathcal{I}\) if there exists a \(c \geq 1\) such that \(\mathcal{F}_n = \mathcal{F}_c\) for all \(n \geq c\). If this is the case we will say that \((\mathcal{F}_n)\) is annihilated by \(\mathcal{I}^c\). If \(X = \Spec(A)\) is affine, then, via the equivalence of Lemma 087W, these objects corresponds exactly to the finite \(A\)-modules annihilated by a power of \(I\) or by \(I^c\).

Lemma

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{G}\) be a coherent \(\mathcal{O}_X\)-module. Let \((\mathcal{F}_n)\) an object of \(\textit{Coh}(X, \mathcal{I})\).

1. If \(\alpha : (\mathcal{F}_n) \to \mathcal{G}^\wedge\) is a map whose kernel and cokernel are annihilated by a power of \(\mathcal{I}\), then there exists a unique (up to unique isomorphism) triple \((\mathcal{F}, a, \beta)\) where

   1. \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module,

   2. \(a : \mathcal{F} \to \mathcal{G}\) is an \(\mathcal{O}_X\)-module map whose kernel and cokernel are annihilated by a power of \(\mathcal{I}\),

   3. \(\beta : (\mathcal{F}_n) \to \mathcal{F}^\wedge\) is an isomorphism, and

   4. \(\alpha = a^\wedge \circ \beta\).

2. If \(\alpha : \mathcal{G}^\wedge \to (\mathcal{F}_n)\) is a map whose kernel and cokernel are annihilated by a power of \(\mathcal{I}\), then there exists a unique (up to unique isomorphism) triple \((\mathcal{F}, a, \beta)\) where

   1. \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module,

   2. \(a : \mathcal{G} \to \mathcal{F}\) is an \(\mathcal{O}_X\)-module map whose kernel and cokernel are annihilated by a power of \(\mathcal{I}\),

   3. \(\beta : \mathcal{F}^\wedge \to (\mathcal{F}_n)\) is an isomorphism, and

   4. \(\alpha = \beta \circ a^\wedge\).

Proof

Proof of (1). The uniqueness implies it suffices to construct \((\mathcal{F}, a, \beta)\) Zariski locally on \(X\). Thus we may assume \(X = \Spec(A)\) and \(\mathcal{I}\) corresponds to the ideal \(I \subset A\). In this situation Lemma 087W applies. Let \(M'\) be the finite \(A^\wedge\)-module corresponding to \((\mathcal{F}_n)\). Let \(N\) be the finite \(A\)-module corresponding to \(\mathcal{G}\). Then \(\alpha\) corresponds to a map \[\varphi : M' \longrightarrow N^\wedge\] whose kernel and cokernel are annihilated by \(I^t\) for some \(t\). Recall that \(N^\wedge = N \otimes_A A^\wedge\) (Algebra, Lemma 00MA). By More on Algebra, Lemma 0ALK there is an \(A\)-module map \(\psi : M \to N\) whose kernel and cokernel are \(I\)-power torsion and an isomorphism \(M \otimes_A A^\wedge = M'\) compatible with \(\varphi\). As \(N\) and \(M'\) are finite modules, we conclude that \(M\) is a finite \(A\)-module, see More on Algebra, Remark 05EU. Hence \(M \otimes_A A^\wedge = M^\wedge\). We omit the verification that the triple \((M, N \to M, M^\wedge \to M')\) so obtained is unique up to unique isomorphism.

The proof of (2) is exactly the same and we omit it.

Lemma

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Any object of \(\textit{Coh}(X, \mathcal{I})\) which is annihilated by a power of \(\mathcal{I}\) is in the essential image of (0880). Moreover, if \(\mathcal{F}\), \(\mathcal{G}\) are in \(\textit{Coh}(\mathcal{O}_X)\) and either \(\mathcal{F}\) or \(\mathcal{G}\) is annihilated by a power of \(\mathcal{I}\), then the maps \[\xymatrix{ \Hom_X(\mathcal{F}, \mathcal{G}) \ar[d] & \Ext_X(\mathcal{F}, \mathcal{G}) \ar[d] \\ \Hom_{\textit{Coh}(X, \mathcal{I})}(\mathcal{F}^\wedge, \mathcal{G}^\wedge) & \Ext_{\textit{Coh}(X, \mathcal{I})}(\mathcal{F}^\wedge, \mathcal{G}^\wedge) }\] are isomorphisms.

Proof

Suppose \((\mathcal{F}_n)\) is an object of \(\textit{Coh}(X, \mathcal{I})\) which is annihilated by \(\mathcal{I}^c\) for some \(c \geq 1\). Then \(\mathcal{F}_n \to \mathcal{F}_c\) is an isomorphism for \(n \geq c\). Hence if we set \(\mathcal{F} = \mathcal{F}_c\), then we see that \(\mathcal{F}^\wedge \cong (\mathcal{F}_n)\). This proves the first assertion.

Let \(\mathcal{F}\), \(\mathcal{G}\) be objects of \(\textit{Coh}(\mathcal{O}_X)\) such that either \(\mathcal{F}\) or \(\mathcal{G}\) is annihilated by \(\mathcal{I}^c\) for some \(c \geq 1\). Then \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})\) is a coherent \(\mathcal{O}_X\)-module annihilated by \(\mathcal{I}^c\). Hence we see that \[\Hom_X(\mathcal{G}, \mathcal{F}) = H^0(X, \mathcal{H}) = \lim H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge, \mathcal{F}^\wedge).\] see Lemma 0882. This proves the statement on homomorphisms.

The notation \(\Ext\) refers to extensions as defined in Homology, Section 010I. The injectivity of the map on \(\Ext\)’s follows immediately from the bijectivity of the map on \(\Hom\)’s. For surjectivity, assume \(\mathcal{F}\) is annihilated by a power of \(I\). Then part (1) of Lemma 0889 shows that given an extension \[0 \to \mathcal{G}^\wedge \to (\mathcal{E}_n) \to \mathcal{F}^\wedge \to 0\] in \(\textit{Coh}(U, I\mathcal{O}_U)\) the morphism \(\mathcal{G}^\wedge \to (\mathcal{E}_n)\) is isomorphic to \(\mathcal{G} \to \mathcal{E}^\wedge\) for some \(\mathcal{G} \to \mathcal{E}\) in \(\textit{Coh}(\mathcal{O}_U)\). Similarly in the other case.

Lemma

Let \(X\) be a Noetherian scheme and let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. If \((\mathcal{F}_n)\) is an object of \(\textit{Coh}(X, \mathcal{I})\) then \(\bigoplus \Ker(\mathcal{F}_{n + 1} \to \mathcal{F}_n)\) is a finite type, graded, quasi-coherent \(\bigoplus \mathcal{I}^n/\mathcal{I}^{n + 1}\)-module.

Proof

The question is local on \(X\) hence we may assume \(X\) is affine, i.e., we have a situation as in Lemma 087W. In this case, if \((\mathcal{F}_n)\) corresponds to the finite \(A^\wedge\) module \(M\), then \(\bigoplus \Ker(\mathcal{F}_{n + 1} \to \mathcal{F}_n)\) corresponds to \(\bigoplus I^nM/I^{n + 1}M\) which is clearly a finite module over \(\bigoplus I^n/I^{n + 1}\).

Lemma

Let \(f : X \to Y\) be a morphism of Noetherian schemes. Let \(\mathcal{J} \subset \mathcal{O}_Y\) be a quasi-coherent sheaf of ideals and set \(\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_X\). Then there is a right exact functor \[f^* : \textit{Coh}(Y, \mathcal{J}) \longrightarrow \textit{Coh}(X, \mathcal{I})\] which sends \((\mathcal{G}_n)\) to \((f^*\mathcal{G}_n)\). If \(f\) is flat, then \(f^*\) is an exact functor.

Proof

Since \(f^* : \textit{Coh}(\mathcal{O}_Y) \to \textit{Coh}(\mathcal{O}_X)\) is right exact we have \[f^*\mathcal{G}_n = f^*(\mathcal{G}_{n + 1}/\mathcal{I}^n\mathcal{G}_{n + 1}) = f^*\mathcal{G}_{n + 1}/f^{-1}\mathcal{I}^nf^*\mathcal{G}_{n + 1} = f^*\mathcal{G}_{n + 1}/\mathcal{J}^nf^*\mathcal{G}_{n + 1}\] hence the pullback of a system is a system. The construction of cokernels in the proof of Lemma 087X shows that \(f^* : \textit{Coh}(Y, \mathcal{J}) \to \textit{Coh}(X, \mathcal{I})\) is always right exact. If \(f\) is flat, then \(f^* : \textit{Coh}(\mathcal{O}_Y) \to \textit{Coh}(\mathcal{O}_X)\) is an exact functor. It follows from the construction of kernels in the proof of Lemma 087X that in this case \(f^* : \textit{Coh}(Y, \mathcal{J}) \to \textit{Coh}(X, \mathcal{I})\) also transforms kernels into kernels.

Lemma

Let \(f : X' \to X\) be a morphism of Noetherian schemes. Let \(Z \subset X\) be a closed subscheme and denote \(Z' = f^{-1}Z\) the scheme theoretic inverse image. Let \(\mathcal{I} \subset \mathcal{O}_X\), \(\mathcal{I}' \subset \mathcal{O}_{X'}\) be the corresponding quasi-coherent sheaves of ideals. If \(f\) is flat and the induced morphism \(Z' \to Z\) is an isomorphism, then the pullback functor \(f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', \mathcal{I}')\) (Lemma 0887) is an equivalence.

Proof

If \(X\) and \(X'\) are affine, then this follows immediately from More on Algebra, Lemma 05EC. To prove it in general we let \(Z_n \subset X\), \(Z'_n \subset X'\) be the \(n\)th infinitesimal neighbourhoods of \(Z\), \(Z'\). The induced morphism \(Z_n \to Z'_n\) is a homeomorphism on underlying topological spaces. On the other hand, if \(z' \in Z'\) maps to \(z \in Z\), then the ring map \(\mathcal{O}_{X, z} \to \mathcal{O}_{X', z'}\) is flat and induces an isomorphism \(\mathcal{O}_{X, z}/\mathcal{I}_z \to \mathcal{O}_{X', z'}/\mathcal{I}'_{z'}\). Hence it induces an isomorphism \(\mathcal{O}_{X, z}/\mathcal{I}_z^n \to \mathcal{O}_{X', z'}/(\mathcal{I}'_{z'})^n\) for all \(n \geq 1\) for example by More on Algebra, Lemma 05E9. Thus \(Z'_n \to Z_n\) is an isomorphism of schemes. Thus \(f^*\) induces an equivalence between the category of coherent \(\mathcal{O}_X\)-modules annihilated by \(\mathcal{I}^n\) and the category of coherent \(\mathcal{O}_{X'}\)-modules annihilated by \((\mathcal{I}')^n\), see Lemma 087T. This clearly implies the lemma.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{I}, \mathcal{J} \subset \mathcal{O}_X\) be quasi-coherent sheaves of ideals. If \(V(\mathcal{I}) = V(\mathcal{J})\) is the same closed subset of \(X\), then \(\textit{Coh}(X, \mathcal{I})\) and \(\textit{Coh}(X, \mathcal{J})\) are equivalent.

Proof

First, assume \(X = \Spec(A)\) is affine. Let \(I, J \subset A\) be the ideals corresponding to \(\mathcal{I}, \mathcal{J}\). Then \(V(I) = V(J)\) implies we have \(I^c \subset J\) and \(J^d \subset I\) for some \(c, d \geq 1\) by elementary properties of the Zariski topology (see Algebra, Section 00DY and Lemma 00IM). Hence the \(I\)-adic and \(J\)-adic completions of \(A\) agree, see Algebra, Lemma 0319. Thus the equivalence follows from Lemma 087W in this case.

In general, using what we said above and the fact that \(X\) is quasi-compact, to choose \(c, d \geq 1\) such that \(\mathcal{I}^c \subset \mathcal{J}\) and \(\mathcal{J}^d \subset \mathcal{I}\). Then given an object \((\mathcal{F}_n)\) in \(\textit{Coh}(X, \mathcal{I})\) we claim that the inverse system \[(\mathcal{F}_{cn}/\mathcal{J}^n\mathcal{F}_{cn})\] is in \(\textit{Coh}(X, \mathcal{J})\). This may be checked on the members of an affine covering; we omit the details. In the same manner we can construct an object of \(\textit{Coh}(X, \mathcal{I})\) starting with an object of \(\textit{Coh}(X, \mathcal{J})\). We omit the verification that these constructions define mutually quasi-inverse functors.

Grothendieck’s existence theorem, I

In this section we discuss Grothendieck’s existence theorem for the projective case. We will use the notion of coherent formal modules developed in Section 0EHN. The reader who is familiar with formal schemes is encouraged to read the statement and proof of the theorem in [EGA].

Lemma

Let \(A\) be Noetherian ring complete with respect to an ideal \(I\). Let \(f : X \to \Spec(A)\) be a proper morphism. Let \(\mathcal{I} = I\mathcal{O}_X\). Then the functor (0880) is fully faithful.

Proof

Let \(\mathcal{F}\), \(\mathcal{G}\) be coherent \(\mathcal{O}_X\)-modules. Then \(\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})\) is a coherent \(\mathcal{O}_X\)-module, see Modules, Lemma 01CQ. By Lemma 0882 the map \[\lim_n H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) \to \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge, \mathcal{F}^\wedge)\] is bijective. Hence fully faithfulness of (0880) follows from the theorem on formal functions (Lemma 087U) for the coherent sheaf \(\mathcal{H}\).

Lemma

Let \(A\) be Noetherian ring and \(I \subset A\) an ideal. Let \(f : X \to \Spec(A)\) be a proper morphism and let \(\mathcal{L}\) be an \(f\)-ample invertible sheaf. Let \(\mathcal{I} = I\mathcal{O}_X\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(X, \mathcal{I})\). Then there exists an integer \(d_0\) such that \[H^1(X, \Ker(\mathcal{F}_{n + 1} \to \mathcal{F}_n) \otimes \mathcal{L}^{\otimes d} ) = 0\] for all \(n \geq 0\) and all \(d \geq d_0\).

Proof

Set \(B = \bigoplus I^n/I^{n + 1}\) and \(\mathcal{B} = \bigoplus \mathcal{I}^n/\mathcal{I}^{n + 1} = f^*\widetilde{B}\). By Lemma 087Z the graded quasi-coherent \(\mathcal{B}\)-module \(\mathcal{G} = \bigoplus \Ker(\mathcal{F}_{n + 1} \to \mathcal{F}_n)\) is of finite type. Hence the lemma follows from Lemma 0897 part (2).

Lemma

Let \(A\) be Noetherian ring complete with respect to an ideal \(I\). Let \(f : X \to \Spec(A)\) be a projective morphism. Let \(\mathcal{I} = I\mathcal{O}_X\). Then the functor (0880) is an equivalence.

Proof

We have already seen that (0880) is fully faithful in Lemma 0883. Thus it suffices to show that the functor is essentially surjective.

We first show that every object \((\mathcal{F}_n)\) of \(\textit{Coh}(X, \mathcal{I})\) is the quotient of an object in the image of (0880). Let \(\mathcal{L}\) be an \(f\)-ample invertible sheaf on \(X\). Choose \(d_0\) as in Lemma 0884. Choose a \(d \geq d_0\) such that \(\mathcal{F}_1 \otimes \mathcal{L}^{\otimes d}\) is globally generated by some sections \(s_{1, 1}, \ldots, s_{t, 1}\). Since the transition maps of the system \[H^0(X, \mathcal{F}_{n + 1} \otimes \mathcal{L}^{\otimes d}) \longrightarrow H^0(X, \mathcal{F}_n \otimes \mathcal{L}^{\otimes d})\] are surjective by the vanishing of \(H^1\) we can lift \(s_{1, 1}, \ldots, s_{t, 1}\) to a compatible system of global sections \(s_{1, n}, \ldots, s_{t, n}\) of \(\mathcal{F}_n \otimes \mathcal{L}^{\otimes d}\). These determine a compatible system of maps \[(s_{1, n}, \ldots, s_{t, n}) : (\mathcal{L}^{\otimes -d})^{\oplus t} \longrightarrow \mathcal{F}_n\] Using Lemma 087Y we deduce that we have a surjective map \[\left((\mathcal{L}^{\otimes -d})^{\oplus t}\right)^\wedge \longrightarrow (\mathcal{F}_n)\] as desired.

The result of the previous paragraph and the fact that \(\textit{Coh}(X, \mathcal{I})\) is abelian (Lemma 087X) implies that every object of \(\textit{Coh}(X, \mathcal{I})\) is a cokernel of a map between objects coming from \(\textit{Coh}(\mathcal{O}_X)\). As (0880) is fully faithful and exact by Lemmas 0883 and 0881 we conclude.

Grothendieck’s existence theorem, II

In this section we discuss Grothendieck’s existence theorem in the proper case. Before we give the statement and proof, we need to develop a bit more theory regarding the categories \(\textit{Coh}(X, \mathcal{I})\) of coherent formal modules introduced in Section 0EHN.

Remark

Let \(X\) be a Noetherian scheme and let \(\mathcal{I}, \mathcal{K} \subset \mathcal{O}_X\) be quasi-coherent sheaves of ideals. Let \(\alpha : (\mathcal{F}_n) \to (\mathcal{G}_n)\) be a morphism of \(\textit{Coh}(X, \mathcal{I})\). Given an affine open \(\Spec(A) = U \subset X\) with \(\mathcal{I}|_U, \mathcal{K}|_U\) corresponding to ideals \(I, K \subset A\) denote \(\alpha_U : M \to N\) of finite \(A^\wedge\)-modules which corresponds to \(\alpha|_U\) via Lemma 087W. We claim the following are equivalent

1. there exists an integer \(t \geq 1\) such that \(\Ker(\alpha_n)\) and \(\Coker(\alpha_n)\) are annihilated by \(\mathcal{K}^t\) for all \(n \geq 1\),

2. for any affine open \(\Spec(A) = U \subset X\) as above the modules \(\Ker(\alpha_U)\) and \(\Coker(\alpha_U)\) are annihilated by \(K^t\) for some integer \(t \geq 1\), and

3. there exists a finite affine open covering \(X = \bigcup U_i\) such that the conclusion of (2) holds for \(\alpha_{U_i}\).

If these equivalent conditions hold we will say that \(\alpha\) is a map whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\). To see the equivalence we use the following commutative algebra fact: suppose given an exact sequence \[0 \to T \to M \to N \to Q \to 0\] of \(A\)-modules with \(T\) and \(Q\) annihilated by \(K^t\) for some ideal \(K \subset A\). Then for every \(f, g \in K^t\) there exists a canonical map \("fg": N \to M\) such that \(M \to N \to M\) is equal to multiplication by \(fg\). Namely, for \(y \in N\) we can pick \(x \in M\) mapping to \(fy\) in \(N\) and then we can set \("fg"(y) = gx\). Thus it is clear that \(\Ker(M/JM \to N/JN)\) and \(\Coker(M/JM \to N/JN)\) are annihilated by \(K^{2t}\) for any ideal \(J \subset A\).

Applying the commutative algebra fact to \(\alpha_{U_i}\) and \(J = I^n\) we see that (3) implies (1). Conversely, suppose (1) holds and \(M \to N\) is equal to \(\alpha_U\). Then there is a \(t \geq 1\) such that \(\Ker(M/I^nM \to N/I^nN)\) and \(\Coker(M/I^nM \to N/I^nN)\) are annihilated by \(K^t\) for all \(n\). We obtain maps \("fg" : N/I^nN \to M/I^nM\) which in the limit induce a map \(N \to M\) as \(N\) and \(M\) are \(I\)-adically complete. Since the composition with \(N \to M \to N\) is multiplication by \(fg\) we conclude that \(fg\) annihilates \(T\) and \(Q\). In other words \(T\) and \(Q\) are annihilated by \(K^{2t}\) as desired.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{I}, \mathcal{K} \subset \mathcal{O}_X\) be quasi-coherent sheaves of ideals. Let \(X_e \subset X\) be the closed subscheme cut out by \(\mathcal{K}^e\). Let \(\mathcal{I}_e = \mathcal{I}\mathcal{O}_{X_e}\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(X, \mathcal{I})\). Assume

1. the functor \(\textit{Coh}(\mathcal{O}_{X_e}) \to \textit{Coh}(X_e, \mathcal{I}_e)\) is an equivalence for all \(e \geq 1\), and

2. there exists a coherent sheaf \(\mathcal{H}\) on \(X\) and a map \(\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge\) whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\).

Then \((\mathcal{F}_n)\) is in the essential image of (0880).

Proof

During this proof we will use without further mention that for a closed immersion \(i : Z \to X\) the functor \(i_*\) gives an equivalence between the category of coherent modules on \(Z\) and coherent modules on \(X\) annihilated by the ideal sheaf of \(Z\), see Lemma 087T. In particular we may identify \(\textit{Coh}(\mathcal{O}_{X_e})\) with the category of coherent \(\mathcal{O}_X\)-modules annihilated by \(\mathcal{K}^e\) and \(\textit{Coh}(X_e, \mathcal{I}_e)\) as the full subcategory of \(\textit{Coh}(X, \mathcal{I})\) of objects annihilated by \(\mathcal{K}^e\). Moreover (1) tells us these two categories are equivalent under the completion functor (0880).

Applying this equivalence we get a coherent \(\mathcal{O}_X\)-module \(\mathcal{G}_e\) annihilated by \(\mathcal{K}^e\) corresponding to the system \((\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n)\) of \(\textit{Coh}(X, \mathcal{I})\). The maps \(\mathcal{F}_n/\mathcal{K}^{e + 1}\mathcal{F}_n \to \mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n\) correspond to canonical maps \(\mathcal{G}_{e + 1} \to \mathcal{G}_e\) which induce isomorphisms \(\mathcal{G}_{e + 1}/\mathcal{K}^e\mathcal{G}_{e + 1} \to \mathcal{G}_e\). Hence \((\mathcal{G}_e)\) is an object of \(\textit{Coh}(X, \mathcal{K})\). The map \(\alpha\) induces a system of maps \[\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n \longrightarrow \mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H}\] whence maps \(\mathcal{G}_e \to \mathcal{H}/\mathcal{K}^e\mathcal{H}\) (by the equivalence of categories again). Let \(t \geq 1\) be an integer, which exists by assumption (2), such that \(\mathcal{K}^t\) annihilates the kernel and cokernel of all the maps \(\mathcal{F}_n \to \mathcal{H}/\mathcal{I}^n\mathcal{H}\). Then \(\mathcal{K}^{2t}\) annihilates the kernel and cokernel of the maps \(\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n \to \mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H}\), see Remark 0888. Whereupon we conclude that \(\mathcal{K}^{4t}\) annihilates the kernel and the cokernel of the maps \[\mathcal{G}_e \longrightarrow \mathcal{H}/\mathcal{K}^e\mathcal{H},\] see Remark 0888. We apply Lemma 0889 to obtain a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), a map \(a : \mathcal{F} \to \mathcal{H}\) and an isomorphism \(\beta : (\mathcal{G}_e) \to (\mathcal{F}/\mathcal{K}^e\mathcal{F})\) in \(\textit{Coh}(X, \mathcal{K})\). Working backwards, for a given \(n\) the triple \((\mathcal{F}/\mathcal{I}^n\mathcal{F}, a \bmod \mathcal{I}^n, \beta \bmod \mathcal{I}^n)\) is a triple as in the lemma for the morphism \(\alpha_n \bmod \mathcal{K}^e : (\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n) \to (\mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H})\) of \(\textit{Coh}(X, \mathcal{K})\). Thus the uniqueness in Lemma 0889 gives a canonical isomorphism \(\mathcal{F}/\mathcal{I}^n\mathcal{F} \to \mathcal{F}_n\) compatible with all the morphisms in sight. This finishes the proof of the lemma.

Lemma

Let \(Y\) be a Noetherian scheme. Let \(\mathcal{J}, \mathcal{K} \subset \mathcal{O}_Y\) be quasi-coherent sheaves of ideals. Let \(f : X \to Y\) be a proper morphism which is an isomorphism over \(V = Y \setminus V(\mathcal{K})\). Set \(\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_X\). Let \((\mathcal{G}_n)\) be an object of \(\textit{Coh}(Y, \mathcal{J})\), let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module, and let \(\beta : (f^*\mathcal{G}_n) \to \mathcal{F}^\wedge\) be an isomorphism in \(\textit{Coh}(X, \mathcal{I})\). Then there exists a map \[\alpha : (\mathcal{G}_n) \longrightarrow (f_*\mathcal{F})^\wedge\] in \(\textit{Coh}(Y, \mathcal{J})\) whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\).

Proof

Since \(f\) is a proper morphism we see that \(f_*\mathcal{F}\) is a coherent \(\mathcal{O}_Y\)-module (Proposition 02O5). Thus the statement of the lemma makes sense. Consider the compositions \[\gamma_n : \mathcal{G}_n \to f_*f^*\mathcal{G}_n \to f_*(\mathcal{F}/\mathcal{I}^n\mathcal{F}).\] Here the first map is the adjunction map and the second is \(f_*\beta_n\). We claim that there exists a unique \(\alpha\) as in the lemma such that the compositions \[\mathcal{G}_n \xrightarrow{\alpha_n} f_*\mathcal{F}/\mathcal{J}^nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^n\mathcal{F})\] equal \(\gamma_n\) for all \(n\). Because of the uniqueness we may assume that \(Y = \Spec(B)\) is affine. Let \(J \subset B\) corresponds to the ideal \(\mathcal{J}\). Set \[M_n = H^0(X, \mathcal{F}/\mathcal{I}^n\mathcal{F}) \quad\text{and}\quad M = H^0(X, \mathcal{F})\] By Lemma 02OB and Theorem 02OC the inverse limit of the modules \(M_n\) equals the completion \(M^\wedge = \lim M/J^nM\). Set \(N_n = H^0(Y, \mathcal{G}_n)\) and \(N = \lim N_n\). Via the equivalence of categories of Lemma 087W the finite \(B^\wedge\) modules \(N\) and \(M^\wedge\) correspond to \((\mathcal{G}_n)\) and \(f_*\mathcal{F}^\wedge\). It follows from this that \(\alpha\) has to be the morphism of \(\textit{Coh}(Y, \mathcal{J})\) corresponding to the homomorphism \[\lim \gamma_n : N = \lim_n N_n \longrightarrow \lim M_n = M^\wedge\] of finite \(B^\wedge\)-modules.

We still have to show that the kernel and cokernel of \(\alpha\) are annihilated by a power of \(\mathcal{K}\). Set \(Y' = \Spec(B^\wedge)\) and \(X' = Y' \times_Y X\). Let \(\mathcal{K}'\), \(\mathcal{J}'\), \(\mathcal{G}'_n\) and \(\mathcal{I}'\), \(\mathcal{F}'\) be the pullback of \(\mathcal{K}\), \(\mathcal{J}\), \(\mathcal{G}_n\) and \(\mathcal{I}\), \(\mathcal{F}\), to \(Y'\) and \(X'\). The projection morphism \(f' : X' \to Y'\) is the base change of \(f\) by \(Y' \to Y\). Note that \(Y' \to Y\) is a flat morphism of schemes as \(B \to B^\wedge\) is flat by Algebra, Lemma 00MB. Hence \(f'_*\mathcal{F}'\), resp. \(f'_*(f')^*\mathcal{G}_n'\) is the pullback of \(f_*\mathcal{F}\), resp. \(f_*f^*\mathcal{G}_n\) to \(Y'\) by Lemma 02KH. The uniqueness of our construction shows the pullback of \(\alpha\) to \(Y'\) is the corresponding map \(\alpha'\) constructed for the situation on \(Y'\). Moreover, to check that the kernel and cokernel of \(\alpha\) are annihilated by \(\mathcal{K}^t\) it suffices to check that the kernel and cokernel of \(\alpha'\) are annihilated by \((\mathcal{K}')^t\). Namely, to see this we need to check this for kernels and cokernels of the maps \(\alpha_n\) and \(\alpha'_n\) (see Remark 0888) and the ring map \(B \to B^\wedge\) induces an equivalence of categories between modules annihilated by \(J^n\) and \((J')^n\), see More on Algebra, Lemma 05EC. Thus we may assume \(B\) is complete with respect to \(J\).

Assume \(Y = \Spec(B)\) is affine, \(\mathcal{J}\) corresponds to the ideal \(J \subset B\), and \(B\) is complete with respect to \(J\). In this case \((\mathcal{G}_n)\) is in the essential image of the functor \(\textit{Coh}(\mathcal{O}_Y) \to \textit{Coh}(Y, \mathcal{J})\). Say \(\mathcal{G}\) is a coherent \(\mathcal{O}_Y\)-module such that \((\mathcal{G}_n) = \mathcal{G}^\wedge\). Note that \(f^*(\mathcal{G}^\wedge) = (f^*\mathcal{G})^\wedge\). Hence Lemma 0883 tells us that \(\beta\) comes from an isomorphism \(b : f^*\mathcal{G} \to \mathcal{F}\) and \(\alpha\) is the completion functor applied to \[\mathcal{G} \to f_*f^*\mathcal{G} \cong f_*\mathcal{F}\] Hence we are trying to verify that the kernel and cokernel of the adjunction map \(c : \mathcal{G} \to f_*f^*\mathcal{G}\) are annihilated by a power of \(\mathcal{K}\). However, since the restriction \(f|_{f^{-1}(V)} : f^{-1}(V) \to V\) is an isomorphism we see that \(c|_V\) is an isomorphism. Thus the coherent sheaves \(\Ker(c)\) and \(\Coker(c)\) are supported on \(V(\mathcal{K})\) hence are annihilated by a power of \(\mathcal{K}\) (Lemma 01Y9) as desired.

The following proposition is the form of Grothendieck’s existence theorem which is most often used in practice.

Proposition

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Let \(f : X \to \Spec(A)\) be a proper morphism of schemes. Set \(\mathcal{I} = I\mathcal{O}_X\). Then the functor (0880) is an equivalence.

Proof

We have already seen that (0880) is fully faithful in Lemma 0883. Thus it suffices to show that the functor is essentially surjective.

Consider the collection \(\Xi\) of quasi-coherent sheaves of ideals \(\mathcal{K} \subset \mathcal{O}_X\) such that every object \((\mathcal{F}_n)\) annihilated by \(\mathcal{K}\) is in the essential image. We want to show \((0)\) is in \(\Xi\). If not, then since \(X\) is Noetherian there exists a maximal quasi-coherent sheaf of ideals \(\mathcal{K}\) not in \(\Xi\), see Lemma 01Y8. After replacing \(X\) by the closed subscheme of \(X\) corresponding to \(\mathcal{K}\) we may assume that every nonzero \(\mathcal{K}\) is in \(\Xi\). (This uses the correspondence by coherent modules annihilated by \(\mathcal{K}\) and coherent modules on the closed subscheme corresponding to \(\mathcal{K}\), see Lemma 087T.) Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(X, \mathcal{I})\). We will show that this object is in the essential image of the functor (0880), thereby completion the proof of the proposition.

Apply Chow’s lemma (Lemma 0200) to find a proper surjective morphism \(f : X' \to X\) which is an isomorphism over a dense open \(U \subset X\) such that \(X'\) is projective over \(A\). Let \(\mathcal{K}\) be the quasi-coherent sheaf of ideals cutting out the reduced complement \(X \setminus U\). By the projective case of Grothendieck’s existence theorem (Lemma 0885) there exists a coherent module \(\mathcal{F}'\) on \(X'\) such that \((\mathcal{F}')^\wedge \cong (f^*\mathcal{F}_n)\). By Proposition 02O5 the \(\mathcal{O}_X\)-module \(\mathcal{H} = f_*\mathcal{F}'\) is coherent and by Lemma 088B there exists a morphism \((\mathcal{F}_n) \to \mathcal{H}^\wedge\) of \(\textit{Coh}(X, \mathcal{I})\) whose kernel and cokernel are annihilated by a power of \(\mathcal{K}\). The powers \(\mathcal{K}^e\) are all in \(\Xi\) so that (0880) is an equivalence for the closed subschemes \(X_e = V(\mathcal{K}^e)\). We conclude by Lemma 088A.

Being proper over a base

This is just a short section to point out some useful features of closed subsets proper over a base and finite type, quasi-coherent modules with support proper over a base.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(Z \subset X\) be a closed subset. The following are equivalent

1. the morphism \(Z \to S\) is proper if \(Z\) is endowed with the reduced induced closed subscheme structure (Schemes, Definition 01J4),

2. for some closed subscheme structure on \(Z\) the morphism \(Z \to S\) is proper,

3. for any closed subscheme structure on \(Z\) the morphism \(Z \to S\) is proper.

Proof

The implications (3) \(\Rightarrow\) (1) and (1) \(\Rightarrow\) (2) are immediate. Thus it suffices to prove that (2) implies (3). We urge the reader to find their own proof of this fact. Let \(Z'\) and \(Z''\) be closed subscheme structures on \(Z\) such that \(Z' \to S\) is proper. We have to show that \(Z'' \to S\) is proper. Let \(Z''' = Z' \cup Z''\) be the scheme theoretic union, see Morphisms, Definition 0C4H. Then \(Z'''\) is another closed subscheme structure on \(Z\). This follows for example from the description of scheme theoretic unions in Morphisms, Lemma 0C4J. Since \(Z'' \to Z'''\) is a closed immersion it suffices to prove that \(Z''' \to S\) is proper (see Morphisms, Lemmas 01W5 and 01W3). The morphism \(Z' \to Z'''\) is a bijective closed immersion and in particular surjective and universally closed. Then the fact that \(Z' \to S\) is separated implies that \(Z''' \to S\) is separated, see Morphisms, Lemma 09MQ. Moreover \(Z''' \to S\) is locally of finite type as \(X \to S\) is locally of finite type (Morphisms, Lemmas 01T5 and 01T3). Since \(Z' \to S\) is quasi-compact and \(Z' \to Z'''\) is a homeomorphism we see that \(Z''' \to S\) is quasi-compact. Finally, since \(Z' \to S\) is universally closed, we see that the same thing is true for \(Z''' \to S\) by Morphisms, Lemma 03GN. This finishes the proof.

Definition

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(Z \subset X\) be a closed subset. We say \(Z\) is proper over \(S\) if the equivalent conditions of Lemma 0CYL are satisfied.

The lemma used in the definition above is false if the morphism \(f : X \to S\) is not locally of finite type. Therefore we urge the reader not to use this terminology if \(f\) is not locally of finite type.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(Y \subset Z \subset X\) be closed subsets. If \(Z\) is proper over \(S\), then the same is true for \(Y\).

Proof

Omitted.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] with \(f\) locally of finite type. If \(Z\) is a closed subset of \(X\) proper over \(S\), then \((g')^{-1}(Z)\) is a closed subset of \(X'\) proper over \(S'\).

Proof

Observe that the statement makes sense as \(f'\) is locally of finite type by Morphisms, Lemma 01T4. Endow \(Z\) with the reduced induced closed subscheme structure. Denote \(Z' = (g')^{-1}(Z)\) the scheme theoretic inverse image (Schemes, Definition 01JV). Then \(Z' = X' \times_X Z = (S' \times_S X) \times_X Z = S' \times_S Z\) is proper over \(S'\) as a base change of \(Z\) over \(S\) (Morphisms, Lemma 01W4).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of schemes which are locally of finite type over \(S\).

1. If \(Y\) is separated over \(S\) and \(Z \subset X\) is a closed subset proper over \(S\), then \(f(Z)\) is a closed subset of \(Y\) proper over \(S\).

2. If \(f\) is universally closed and \(Z \subset X\) is a closed subset proper over \(S\), then \(f(Z)\) is a closed subset of \(Y\) proper over \(S\).

3. If \(f\) is proper and \(Z \subset Y\) is a closed subset proper over \(S\), then \(f^{-1}(Z)\) is a closed subset of \(X\) proper over \(S\).

Proof

Proof of (1). Assume \(Y\) is separated over \(S\) and \(Z \subset X\) is a closed subset proper over \(S\). Endow \(Z\) with the reduced induced closed subscheme structure and apply Morphisms, Lemma 0AH6 to \(Z \to Y\) over \(S\) to conclude.

Proof of (2). Assume \(f\) is universally closed and \(Z \subset X\) is a closed subset proper over \(S\). Endow \(Z\) and \(Z' = f(Z)\) with their reduced induced closed subscheme structures. We obtain an induced morphism \(Z \to Z'\). Denote \(Z'' = f^{-1}(Z')\) the scheme theoretic inverse image (Schemes, Definition 01JV). Then \(Z'' \to Z'\) is universally closed as a base change of \(f\) (Morphisms, Lemma 01W4). Hence \(Z \to Z'\) is universally closed as a composition of the closed immersion \(Z \to Z''\) and \(Z'' \to Z'\) (Morphisms, Lemmas 01W5 and 01W3). We conclude that \(Z' \to S\) is separated by Morphisms, Lemma 09MQ. Since \(Z \to S\) is quasi-compact and \(Z \to Z'\) is surjective we see that \(Z' \to S\) is quasi-compact. Since \(Z' \to S\) is the composition of \(Z' \to Y\) and \(Y \to S\) we see that \(Z' \to S\) is locally of finite type (Morphisms, Lemmas 01T5 and 01T3). Finally, since \(Z \to S\) is universally closed, we see that the same thing is true for \(Z' \to S\) by Morphisms, Lemma 03GN. This finishes the proof.

Proof of (3). Assume \(f\) is proper and \(Z \subset Y\) is a closed subset proper over \(S\). Endow \(Z\) with the reduced induced closed subscheme structure. Denote \(Z' = f^{-1}(Z)\) the scheme theoretic inverse image (Schemes, Definition 01JV). Then \(Z' \to Z\) is proper as a base change of \(f\) (Morphisms, Lemma 01W4). Whence \(Z' \to S\) is proper as the composition of \(Z' \to Z\) and \(Z \to S\) (Morphisms, Lemma 01W3). This finishes the proof.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(Z_i \subset X\), \(i = 1, \ldots, n\) be closed subsets. If \(Z_i\), \(i = 1, \ldots, n\) are proper over \(S\), then the same is true for \(Z_1 \cup \ldots \cup Z_n\).

Proof

Endow \(Z_i\) with their reduced induced closed subscheme structures. The morphism \[Z_1 \amalg \ldots \amalg Z_n \longrightarrow X\] is finite by Morphisms, Lemmas 035C and 0CYI. As finite morphisms are universally closed (Morphisms, Lemma 01WN) and since \(Z_1 \amalg \ldots \amalg Z_n\) is proper over \(S\) we conclude by Lemma 0CYQ part (2) that the image \(Z_1 \cup \ldots \cup Z_n\) is proper over \(S\).

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. Then the support \(\text{Supp}(\mathcal{F})\) of \(\mathcal{F}\) is a closed subset of \(X\), see Morphisms, Lemma 056J. Hence it makes sense to say “the support of \(\mathcal{F}\) is proper over \(S\)”.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. The following are equivalent

1. the support of \(\mathcal{F}\) is proper over \(S\),

2. the scheme theoretic support of \(\mathcal{F}\) (Morphisms, Definition 05JV) is proper over \(S\), and

3. there exists a closed subscheme \(Z \subset X\) and a finite type, quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) such that (a) \(Z \to S\) is proper, and (b) \((Z \to X)_*\mathcal{G} = \mathcal{F}\).

Proof

The support \(\text{Supp}(\mathcal{F})\) of \(\mathcal{F}\) is a closed subset of \(X\), see Morphisms, Lemma 056J. Hence we can apply Definition 0CYM. Since the scheme theoretic support of \(\mathcal{F}\) is a closed subscheme whose underlying closed subset is \(\text{Supp}(\mathcal{F})\) we see that (1) and (2) are equivalent by Definition 0CYM. It is clear that (2) implies (3). Conversely, if (3) is true, then \(\text{Supp}(\mathcal{F}) \subset Z\) (an inclusion of closed subsets of \(X\)) and hence \(\text{Supp}(\mathcal{F})\) is proper over \(S\) for example by Lemma 0CYN.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] with \(f\) locally of finite type. Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. If the support of \(\mathcal{F}\) is proper over \(S\), then the support of \((g')^*\mathcal{F}\) is proper over \(S'\).

Proof

Observe that the statement makes sense because \((g')*\mathcal{F}\) is of finite type by Modules, Lemma 01B6. We have \(\text{Supp}((g')^*\mathcal{F}) = (g')^{-1}(\text{Supp}(\mathcal{F}))\) by Morphisms, Lemma 056J. Thus the lemma follows from Lemma 0CYP.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\), \(\mathcal{G}\) be finite type, quasi-coherent \(\mathcal{O}_X\)-module.

1. If the supports of \(\mathcal{F}\), \(\mathcal{G}\) are proper over \(S\), then the same is true for \(\mathcal{F} \oplus \mathcal{G}\), for any extension of \(\mathcal{G}\) by \(\mathcal{F}\), for \(\Im(u)\) and \(\Coker(u)\) given any \(\mathcal{O}_X\)-module map \(u : \mathcal{F} \to \mathcal{G}\), and for any quasi-coherent quotient of \(\mathcal{F}\) or \(\mathcal{G}\).

2. If \(S\) is locally Noetherian, then the category of coherent \(\mathcal{O}_X\)-modules with support proper over \(S\) is a Serre subcategory (Homology, Definition 02MO) of the abelian category of coherent \(\mathcal{O}_X\)-modules.

Proof

Proof of (1). Let \(Z\), \(Z'\) be the support of \(\mathcal{F}\) and \(\mathcal{G}\). Then all the sheaves mentioned in (1) have support contained in \(Z \cup Z'\). Thus the assertion itself is clear from Lemmas 0CYN and 0CYR provided we check that these sheaves are finite type and quasi-coherent. For quasi-coherence we refer the reader to Schemes, Section 01LA. For “finite type” we suggest the reader take a look at Modules, Section 01B4.

Proof of (2). The proof is the same as the proof of (1). Note that the assertions make sense as \(X\) is locally Noetherian by Morphisms, Lemma 01T6 and by the description of the category of coherent modules in Section 01XY.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module with support proper over \(S\). Then \(R^pf_*\mathcal{F}\) is a coherent \(\mathcal{O}_S\)-module for all \(p \geq 0\).

Proof

By Lemma 0CYS there exists a closed immersion \(i : Z \to X\) and a finite type, quasi-coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) such that (a) \(g = f \circ i : Z \to S\) is proper, and (b) \(i_*\mathcal{G} = \mathcal{F}\). We see that \(R^pg_*\mathcal{G}\) is coherent on \(S\) by Proposition 02O5. On the other hand, \(R^qi_*\mathcal{G} = 0\) for \(q > 0\) (Lemma 01Y6). By Cohomology, Lemma 01F6 we get \(R^pf_*\mathcal{F} = R^pg_*\mathcal{G}\) which concludes the proof.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a finite type morphism. Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. The following are Serre subcategories of \(\textit{Coh}(X, \mathcal{I})\)

1. the full subcategory of \(\textit{Coh}(X, \mathcal{I})\) consisting of those objects \((\mathcal{F}_n)\) such that the support of \(\mathcal{F}_1\) is proper over \(S\),

2. the full subcategory of \(\textit{Coh}(X, \mathcal{I})\) consisting of those objects \((\mathcal{F}_n)\) such that there exists a closed subscheme \(Z \subset X\) proper over \(S\) with \(\mathcal{I}_Z \mathcal{F}_n = 0\) for all \(n \geq 1\).

Proof

We will use the criterion of Homology, Lemma 02MP. Moreover, we will use that if \(0 \to (\mathcal{G}_n) \to (\mathcal{F}_n) \to (\mathcal{H}_n) \to 0\) is a short exact sequence of \(\textit{Coh}(X, \mathcal{I})\), then (a) \(\mathcal{G}_n \to \mathcal{F}_n \to \mathcal{H}_n \to 0\) is exact for all \(n \geq 1\) and (b) \(\mathcal{G}_n\) is a quotient of \(\Ker(\mathcal{F}_m \to \mathcal{H}_m)\) for some \(m \geq n\). See proof of Lemma 087X.

Proof of (1). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}(X, \mathcal{I})\). Then \(\text{Supp}(\mathcal{F}_n) = \text{Supp}(\mathcal{F}_1)\) for all \(n \geq 1\). Hence by remarks (a) and (b) above we see that for any short exact sequence \(0 \to (\mathcal{G}_n) \to (\mathcal{F}_n) \to (\mathcal{H}_n) \to 0\) of \(\textit{Coh}(X, \mathcal{I})\) we have \(\text{Supp}(\mathcal{G}_1) \cup \text{Supp}(\mathcal{H}_1) = \text{Supp}(\mathcal{F}_1)\). This proves that the category defined in (1) is a Serre subcategory of \(\textit{Coh}(X, \mathcal{I})\).

Proof of (2). Here we argue the same way. Let \(0 \to (\mathcal{G}_n) \to (\mathcal{F}_n) \to (\mathcal{H}_n) \to 0\) be a short exact sequence of \(\textit{Coh}(X, \mathcal{I})\). If \(Z \subset X\) is a closed subscheme and \(\mathcal{I}_Z\) annihilates \(\mathcal{F}_n\) for all \(n\), then \(\mathcal{I}_Z\) annihilates \(\mathcal{G}_n\) and \(\mathcal{H}_n\) for all \(n\) by (a) and (b) above. Hence if \(Z \to S\) is proper, then we conclude that the category defined in (2) is closed under taking sub and quotient objects inside of \(\textit{Coh}(X, \mathcal{I})\). Finally, suppose that \(Z \subset X\) and \(Y \subset X\) are closed subschemes proper over \(S\) such that \(\mathcal{I}_Z \mathcal{G}_n = 0\) and \(\mathcal{I}_Y \mathcal{H}_n = 0\) for all \(n \geq 1\). Then it follows from (a) above that \(\mathcal{I}_{Z \cup Y} = \mathcal{I}_Z \cdot \mathcal{I}_Y\) annihilates \(\mathcal{F}_n\) for all \(n\). By Lemma 0CYR (and via Definition 0CYM which tells us we may choose an arbitrary scheme structure used on the union) we see that \(Z \cup Y \to S\) is proper and the proof is complete.

Grothendieck’s existence theorem, III

To state the general version of Grothendieck’s existence theorem we introduce a bit more notation. Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Let \(f : X \to \Spec(A)\) be a separated finite type morphism of schemes. Set \(\mathcal{I} = I\mathcal{O}_X\). In this situation we let \[\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X)\] be the full subcategory of \(\textit{Coh}(\mathcal{O}_X)\) consisting of those coherent \(\mathcal{O}_X\)-modules whose support is proper over \(\Spec(A)\). This is a Serre subcategory of \(\textit{Coh}(\mathcal{O}_X)\), see Lemma 0CYU. Similarly, we let \[\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})\] be the full subcategory of \(\textit{Coh}(X, \mathcal{I})\) consisting of those objects \((\mathcal{F}_n)\) such that the support of \(\mathcal{F}_1\) is proper over \(\Spec(A)\). This is a Serre subcategory of \(\textit{Coh}(X, \mathcal{I})\) by Lemma 0CYV part (1). Since the support of a quotient module is contained in the support of the module, it follows that (0880) induces a functor [088D]\[\begin{equation} \textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X) \longrightarrow \textit{Coh}_{\text{support proper over }A}(X, \mathcal{I}) \end{equation}\] We are now ready to state the main theorem of this section.

Theorem

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Let \(X\) be a separated, finite type scheme over \(A\). Then the functor (088D) \[\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X) \longrightarrow \textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\] is an equivalence.

Proof

We will use the equivalence of categories of Lemma 087T without further mention. For a closed subscheme \(Z \subset X\) proper over \(A\) in this proof we will say a coherent module on \(X\) is “supported on \(Z\)” if it is annihilated by the ideal sheaf of \(Z\) or equivalently if it is the pushforward of a coherent module on \(Z\). By Proposition 088C we know that the result is true for the functor between coherent modules and systems of coherent modules supported on \(Z\). Hence it suffices to show that every object of \(\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X)\) and every object of \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\) is supported on a closed subscheme \(Z \subset X\) proper over \(A\). This holds by definition for objects of \(\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_X)\). We will prove this statement for objects of \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\) using the method of proof of Proposition 088C. We urge the reader to read that proof first.

Consider the collection \(\Xi\) of quasi-coherent sheaves of ideals \(\mathcal{K} \subset \mathcal{O}_X\) such that the statement holds for every object \((\mathcal{F}_n)\) of \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\) annihilated by \(\mathcal{K}\). We want to show \((0)\) is in \(\Xi\). If not, then since \(X\) is Noetherian there exists a maximal quasi-coherent sheaf of ideals \(\mathcal{K}\) not in \(\Xi\), see Lemma 01Y8. After replacing \(X\) by the closed subscheme of \(X\) corresponding to \(\mathcal{K}\) we may assume that every nonzero \(\mathcal{K}\) is in \(\Xi\). Let \((\mathcal{F}_n)\) be an object of \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\). We will show that this object is supported on a closed subscheme \(Z \subset X\) proper over \(A\), thereby completing the proof of the theorem.

Apply Chow’s lemma (Lemma 0200) to find a proper surjective morphism \(f : Y \to X\) which is an isomorphism over a dense open \(U \subset X\) such that \(Y\) is H-quasi-projective over \(A\). Choose an open immersion \(j : Y \to Y'\) with \(Y'\) projective over \(A\), see Morphisms, Lemma 01WA. Observe that \[\text{Supp}(f^*\mathcal{F}_n) = f^{-1}\text{Supp}(\mathcal{F}_n) = f^{-1}\text{Supp}(\mathcal{F}_1)\] The first equality by Morphisms, Lemma 056J. By assumption and Lemma 0CYQ part (3) we see that \(f^{-1}\text{Supp}(\mathcal{F}_1)\) is proper over \(A\). Hence the image of \(f^{-1}\text{Supp}(\mathcal{F}_1)\) under \(j\) is closed in \(Y'\) by Lemma 0CYQ part (1). Thus \(\mathcal{F}'_n = j_*f^*\mathcal{F}_n\) is coherent on \(Y'\) by Lemma 0CYJ. It follows that \((\mathcal{F}_n')\) is an object of \(\textit{Coh}(Y', I\mathcal{O}_{Y'})\). By the projective case of Grothendieck’s existence theorem (Lemma 0885) there exists a coherent \(\mathcal{O}_{Y'}\)-module \(\mathcal{F}'\) and an isomorphism \((\mathcal{F}')^\wedge \cong (\mathcal{F}'_n)\) in \(\textit{Coh}(Y', I\mathcal{O}_{Y'})\). Since \(\mathcal{F}'/I\mathcal{F}' = \mathcal{F}'_1\) we see that \[\text{Supp}(\mathcal{F}') \cap V(I\mathcal{O}_{Y'}) = \text{Supp}(\mathcal{F}'_1) = j(f^{-1}\text{Supp}(\mathcal{F}_1))\] The structure morphism \(p' : Y' \to \Spec(A)\) is proper, hence \(p'(\text{Supp}(\mathcal{F}') \setminus j(Y))\) is closed in \(\Spec(A)\). A nonempty closed subset of \(\Spec(A)\) contains a point of \(V(I)\) as \(I\) is contained in the Jacobson radical of \(A\) by Algebra, Lemma 05GI. The displayed equation shows that \(\text{Supp}(\mathcal{F}') \cap (p')^{-1}V(I) \subset j(Y)\) hence we conclude that \(\text{Supp}(\mathcal{F}') \subset j(Y)\). Thus \(\mathcal{F}'|_Y = j^*\mathcal{F}'\) is supported on a closed subscheme \(Z'\) of \(Y\) proper over \(A\) and \((\mathcal{F}'|_Y)^\wedge = (f^*\mathcal{F}_n)\).

Let \(\mathcal{K}\) be the quasi-coherent sheaf of ideals cutting out the reduced complement \(X \setminus U\). By Proposition 02O5 the \(\mathcal{O}_X\)-module \(\mathcal{H} = f_*(\mathcal{F}'|_Y)\) is coherent and by Lemma 088B there exists a morphism \(\alpha : (\mathcal{F}_n) \to \mathcal{H}^\wedge\) of \(\textit{Coh}(X, \mathcal{I})\) whose kernel and cokernel are annihilated by a power \(\mathcal{K}^t\) of \(\mathcal{K}\). We obtain an exact sequence \[0 \to \Ker(\alpha) \to (\mathcal{F}_n) \to \mathcal{H}^\wedge \to \Coker(\alpha) \to 0\] in \(\textit{Coh}(X, \mathcal{I})\). If \(Z_0 \subset X\) is the scheme theoretic support of \(\mathcal{H}\), then it is clear that \(Z_0 \subset f(Z')\) set-theoretically. Hence \(Z_0\) is proper over \(A\) by Lemma 0CYN and Lemma 0CYQ part (2). Hence \(\mathcal{H}^\wedge\) is in the subcategory defined in Lemma 0CYV part (2) and a fortiori in \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\). We conclude that \(\Ker(\alpha)\) and \(\Coker(\alpha)\) are in \(\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})\) by Lemma 0CYV part (1). By induction hypothesis, more precisely because \(\mathcal{K}^t\) is in \(\Xi\), we see that \(\Ker(\alpha)\) and \(\Coker(\alpha)\) are in the subcategory defined in Lemma 0CYV part (2). Since this is a Serre subcategory by the lemma, we conclude that the same is true for \((\mathcal{F}_n)\) which is what we wanted to show.

Remark

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X \to S\) be a separated morphism of finite type. For \(n \geq 1\) we set \(X_n = X \times_S S_n\). Picture: \[\xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots & X \ar[d] \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots & S }\] In this situation we consider systems \((\mathcal{F}_n, \varphi_n)\) where

1. \(\mathcal{F}_n\) is a coherent \(\mathcal{O}_{X_n}\)-module,

2. \(\varphi_n : i_n^*\mathcal{F}_{n + 1} \to \mathcal{F}_n\) is an isomorphism, and

3. \(\text{Supp}(\mathcal{F}_1)\) is proper over \(S_1\).

Theorem 088E says that the completion functor \[\begin{matrix} \text{coherent }\mathcal{O}_X\text{-modules }\mathcal{F} \\ \text{with support proper over }A \end{matrix} \quad \longrightarrow \quad \begin{matrix} \text{systems }(\mathcal{F}_n) \\ \text{as above} \end{matrix}\] is an equivalence of categories. In the special case that \(X\) is proper over \(A\) we can omit the conditions on the supports.

Grothendieck’s algebraization theorem

Our first result is a translation of Grothendieck’s existence theorem in terms of closed subschemes and finite morphisms.

Lemma

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X \to S\) be a separated morphism of finite type. For \(n \geq 1\) we set \(X_n = X \times_S S_n\). Suppose given a commutative diagram \[\xymatrix{ Z_1 \ar[r] \ar[d] & Z_2 \ar[r] \ar[d] & Z_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots }\] of schemes with cartesian squares. Assume that

1. \(Z_1 \to X_1\) is a closed immersion, and

2. \(Z_1 \to S_1\) is proper.

Then there exists a closed immersion of schemes \(Z \to X\) such that \(Z_n = Z \times_S S_n\). Moreover, \(Z\) is proper over \(S\).

Proof

Let’s write \(j_n : Z_n \to X_n\) for the vertical morphisms. As the squares in the statement are cartesian we see that the base change of \(j_n\) to \(X_1\) is \(j_1\). Thus Morphisms, Lemma 0896 shows that \(j_n\) is a closed immersion. Set \(\mathcal{F}_n = j_{n, *}\mathcal{O}_{Z_n}\), so that \(j_n^\sharp\) is a surjection \(\mathcal{O}_{X_n} \to \mathcal{F}_n\). Again using that the squares are cartesian we see that the pullback of \(\mathcal{F}_{n + 1}\) to \(X_n\) is \(\mathcal{F}_n\). Hence Grothendieck’s existence theorem, as reformulated in Remark 088F, tells us there exists a map \(\mathcal{O}_X \to \mathcal{F}\) of coherent \(\mathcal{O}_X\)-modules whose restriction to \(X_n\) recovers \(\mathcal{O}_{X_n} \to \mathcal{F}_n\). Moreover, the support of \(\mathcal{F}\) is proper over \(S\). As the completion functor is exact (Lemma 0881) we see that the cokernel \(\mathcal{Q}\) of \(\mathcal{O}_X \to \mathcal{F}\) has vanishing completion. Since \(\mathcal{F}\) has support proper over \(S\) and so does \(\mathcal{Q}\) this implies that \(\mathcal{Q} = 0\) for example because the functor (088D) is an equivalence by Grothendieck’s existence theorem. Thus \(\mathcal{F} = \mathcal{O}_X/\mathcal{J}\) for some quasi-coherent sheaf of ideals \(\mathcal{J}\). Setting \(Z = V(\mathcal{J})\) finishes the proof.

In the following lemma it is actually enough to assume that \(Y_1 \to X_1\) is finite as it will imply that \(Y_n \to X_n\) is finite too (see More on Morphisms, Lemma 09ZW).

Lemma

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X \to S\) be a separated morphism of finite type. For \(n \geq 1\) we set \(X_n = X \times_S S_n\). Suppose given a commutative diagram \[\xymatrix{ Y_1 \ar[r] \ar[d] & Y_2 \ar[r] \ar[d] & Y_3 \ar[r] \ar[d] & \ldots \\ X_1 \ar[r]^{i_1} & X_2 \ar[r]^{i_2} & X_3 \ar[r] & \ldots }\] of schemes with cartesian squares. Assume that

1. \(Y_n \to X_n\) is a finite morphism, and

2. \(Y_1 \to S_1\) is proper.

Then there exists a finite morphism of schemes \(Y \to X\) such that \(Y_n = Y \times_S S_n\). Moreover, \(Y\) is proper over \(S\).

Proof

Let’s write \(f_n : Y_n \to X_n\) for the vertical morphisms. Set \(\mathcal{F}_n = f_{n, *}\mathcal{O}_{Y_n}\). This is a coherent \(\mathcal{O}_{X_n}\)-module as \(f_n\) is finite (Lemma 01Y6). Using that the squares are cartesian we see that the pullback of \(\mathcal{F}_{n + 1}\) to \(X_n\) is \(\mathcal{F}_n\). Hence Grothendieck’s existence theorem, as reformulated in Remark 088F, tells us there exists a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) whose restriction to \(X_n\) recovers \(\mathcal{F}_n\). Moreover, the support of \(\mathcal{F}\) is proper over \(S\). As the completion functor is fully faithful (Theorem 088E) we see that the multiplication maps \(\mathcal{F}_n \otimes_{\mathcal{O}_{X_n}} \mathcal{F}_n \to \mathcal{F}_n\) fit together to give an algebra structure on \(\mathcal{F}\). Setting \(Y = \underline{\Spec}_X(\mathcal{F})\) finishes the proof.

Lemma

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Write \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Let \(X\), \(Y\) be schemes over \(S\). For \(n \geq 1\) we set \(X_n = X \times_S S_n\) and \(Y_n = Y \times_S S_n\). Suppose given a compatible system of commutative diagrams \[\xymatrix{ & & X_{n + 1} \ar[rd] \ar[rr]_{g_{n + 1}} & & Y_{n + 1} \ar[ld] \\ X_n \ar[rru] \ar[rd] \ar[rr]_{g_n} & & Y_n \ar[rru] \ar[ld] & S_{n + 1} \\ & S_n \ar[rru] }\] Assume that

1. \(X \to S\) is proper, and

2. \(Y \to S\) is separated of finite type.

Then there exists a unique morphism of schemes \(g : X \to Y\) over \(S\) such that \(g_n\) is the base change of \(g\) to \(S_n\).

Proof

The morphisms \((1, g_n) : X_n \to X_n \times_S Y_n\) are closed immersions because \(Y_n \to S_n\) is separated (Schemes, Lemma 01KT). Thus by Lemma 0899 there exists a closed subscheme \(Z \subset X \times_S Y\) proper over \(S\) whose base change to \(S_n\) recovers \(X_n \subset X_n \times_S Y_n\). The first projection \(p : Z \to X\) is a proper morphism (as \(Z\) is proper over \(S\), see Morphisms, Lemma 01W6) whose base change to \(S_n\) is an isomorphism for all \(n\). In particular, \(p : Z \to X\) is finite over an open neighbourhood of \(X_0\) by Lemma 02OH. As \(X\) is proper over \(S\) this open neighbourhood is all of \(X\) and we conclude \(p : Z \to X\) is finite. Applying the equivalence of Proposition 088C we see that \(p_*\mathcal{O}_Z = \mathcal{O}_X\) as this is true modulo \(I^n\) for all \(n\). Hence \(p\) is an isomorphism and we obtain the morphism \(g\) as the composition \(X \cong Z \to Y\). We omit the proof of uniqueness.

In order to prove an “abstract” algebraization theorem we need to assume we have an ample invertible sheaf, as the result is false without such an assumption.

Theorem

Let \(A\) be a Noetherian ring complete with respect to an ideal \(I\). Set \(S = \Spec(A)\) and \(S_n = \Spec(A/I^n)\). Consider a commutative diagram \[\xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots }\] of schemes with cartesian squares. Suppose given \((\mathcal{L}_n, \varphi_n)\) where each \(\mathcal{L}_n\) is an invertible sheaf on \(X_n\) and \(\varphi_n : i_n^*\mathcal{L}_{n + 1} \to \mathcal{L}_n\) is an isomorphism. If

1. \(X_1 \to S_1\) is proper, and

2. \(\mathcal{L}_1\) is ample on \(X_1\)

then there exists a proper morphism of schemes \(X \to S\) and an ample invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) and isomorphisms \(X_n \cong X \times_S S_n\) and \(\mathcal{L}_n \cong \mathcal{L}|_{X_n}\) compatible with the morphisms \(i_n\) and \(\varphi_n\).

Proof

Since the squares in the diagram are cartesian and since the morphisms \(S_n \to S_{n + 1}\) are closed immersions, we see that the morphisms \(i_n\) are closed immersions too. In particular we may think of \(X_m\) as a closed subscheme of \(X_n\) for \(m < n\). In fact \(X_m\) is the closed subscheme cut out by the quasi-coherent sheaf of ideals \(I^m\mathcal{O}_{X_n}\). Moreover, the underlying topological spaces of the schemes \(X_1, X_2, X_3, \ldots\) are all identified, hence we may (and do) think of sheaves \(\mathcal{O}_{X_n}\) as living on the same underlying topological space; similarly for coherent \(\mathcal{O}_{X_n}\)-modules. Set \[\mathcal{F}_n = \Ker(\mathcal{O}_{X_{n + 1}} \to \mathcal{O}_{X_n})\] so that we obtain short exact sequences \[0 \to \mathcal{F}_n \to \mathcal{O}_{X_{n + 1}} \to \mathcal{O}_{X_n} \to 0\] By the above we have \(\mathcal{F}_n = I^n\mathcal{O}_{X_{n + 1}}\). It follows \(\mathcal{F}_n\) is a coherent sheaf on \(X_{n + 1}\) annihilated by \(I\), hence we may (and do) think of it as a coherent module \(\mathcal{O}_{X_1}\)-module. Observe that for \(m > n\) the sheaf \[I^n\mathcal{O}_{X_m}/I^{n + 1}\mathcal{O}_{X_m}\] maps isomorphically to \(\mathcal{F}_n\) under the map \(\mathcal{O}_{X_m} \to \mathcal{O}_{X_{n + 1}}\). Hence given \(n_1, n_2 \geq 0\) we can pick an \(m > n_1 + n_2\) and consider the multiplication map \[I^{n_1}\mathcal{O}_{X_m} \times I^{n_2}\mathcal{O}_{X_m} \longrightarrow I^{n_1 + n_2}\mathcal{O}_{X_m} \to \mathcal{F}_{n_1 + n_2}\] This induces an \(\mathcal{O}_{X_1}\)-bilinear map \[\mathcal{F}_{n_1} \times \mathcal{F}_{n_2} \longrightarrow \mathcal{F}_{n_1 + n_2}\] which in turn defines the structure of a graded \(\mathcal{O}_{X_1}\)-algebra on \(\mathcal{F} = \bigoplus_{n \geq 0} \mathcal{F}_n\).

Set \(B = \bigoplus I^n/I^{n + 1}\); this is a finitely generated graded \(A/I\)-algebra. Set \(\mathcal{B} = (X_1 \to S_1)^*\widetilde{B}\). The discussion above provides us with a canonical surjection \[\mathcal{B} \longrightarrow \mathcal{F}\] of graded \(\mathcal{O}_{X_1}\)-algebras. In particular we see that \(\mathcal{F}\) is a finite type quasi-coherent graded \(\mathcal{B}\)-module. By Lemma 0897 we can find an integer \(d_0\) such that \(H^1(X_1, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0\) for all \(d \geq d_0\). Pick a \(d \geq d_0\) such that there exist sections \(s_{0, 1}, \ldots, s_{N, 1} \in \Gamma(X_1, \mathcal{L}_1^{\otimes d})\) which induce an immersion \[\psi_1 : X_1 \to \mathbf{P}^N_{S_1}\] over \(S_1\), see Morphisms, Lemma 01VT. As \(X_1\) is proper over \(S_1\) we see that \(\psi_1\) is a closed immersion, see Morphisms, Lemma 01W6 and Schemes, Lemma 01IQ. We are going to “lift” \(\psi_1\) to a compatible system of closed immersions of \(X_n\) into \(\mathbf{P}^N\).

Upon tensoring the short exact sequences of the first paragraph of the proof by \(\mathcal{L}_{n + 1}^{\otimes d}\) we obtain short exact sequences \[0 \to \mathcal{F}_n \otimes \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to 0\] Using the isomorphisms \(\varphi_n\) we obtain isomorphisms \(\mathcal{L}_{n + 1} \otimes \mathcal{O}_{X_l} = \mathcal{L}_l\) for \(l \leq n\). Whence the sequence above becomes \[0 \to \mathcal{F}_n \otimes \mathcal{L}_1^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_n^{\otimes d} \to 0\] The vanishing of \(H^1(X, \mathcal{F}_n \otimes \mathcal{L}_1^{\otimes d})\) implies we can inductively lift \(s_{0, 1}, \ldots, s_{N, 1} \in \Gamma(X_1, \mathcal{L}_1^{\otimes d})\) to sections \(s_{0, n}, \ldots, s_{N, n} \in \Gamma(X_n, \mathcal{L}_n^{\otimes d})\). Thus we obtain a commutative diagram \[\xymatrix{ X_1 \ar[r]_{i_1} \ar[d]_{\psi_1} & X_2 \ar[r]_{i_2} \ar[d]_{\psi_2} & X_3 \ar[r] \ar[d]_{\psi_3} & \ldots \\ \mathbf{P}^N_{S_1} \ar[r] & \mathbf{P}^N_{S_2} \ar[r] & \mathbf{P}^N_{S_3} \ar[r] & \ldots }\] where \(\psi_n = \varphi_{(\mathcal{L}_n, (s_{0, n}, \ldots, s_{N, n}))}\) in the notation of Constructions, Section 01ND. As the squares in the statement of the theorem are cartesian we see that the squares in the above diagram are cartesian. We win by applying Lemma 0899.

Lemma

Let \((A, \mathfrak m, k)\) be a complete Noetherian local ring. Set \(S = \Spec(A)\) and \(S_n = \Spec(A/\mathfrak m^n)\). Consider a commutative diagram \[\xymatrix{ X_1 \ar[r] \ar[d] & X_2 \ar[r] \ar[d] & X_3 \ar[r] \ar[d] & \ldots \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots }\] with cartesian squares. Assume that

1. \(X_n \to S_n\) is flat for every \(n \geq 1\),

2. \(X_1\) is projective over \(k\), and

3. \(H^2(X_1, \mathcal{O}_{X_1}) = 0\).

Then there exists a projective flat morphism \(X \to S\) and compatible isomorphisms \(X_n \cong X \times_S S_n\).

Proof

Choose an ample invertible sheaf \(\mathcal{L}_1\) on \(X_1\). Suppose \(\mathcal{L}_n\) has been constructed on \(X_n\). Let \(\mathcal{I}_n\) be the ideal sheaf of \(X_n\) in \(X_{n + 1}\). By More on Morphisms, Lemma 063Y, flatness gives \[\mathcal{I}_n = \mathcal{O}_{X_n} \otimes_{A/\mathfrak m^n} \mathfrak m^n/\mathfrak m^{n + 1}.\] This is the pushforward from \(X_1\) of the tensor product of \(\mathcal{O}_{X_1}\) with the finite dimensional \(k\)-vector space \(\mathfrak m^n/\mathfrak m^{n + 1}\). Hence \[H^2(X_n, \mathcal{I}_n) = H^2(X_1, \mathcal{O}_{X_1}) \otimes_k \mathfrak m^n/\mathfrak m^{n + 1} = 0.\] The exact sequence of More on Morphisms, Lemma 0C6R shows that \(\mathcal{L}_n\) lifts to an invertible sheaf \(\mathcal{L}_{n + 1}\) on \(X_{n + 1}\). This constructs a compatible system \((\mathcal{L}_n)\).

Theorem 089A gives a proper morphism \(X \to S\), an ample invertible sheaf on \(X\), and compatible isomorphisms of its reductions with the \(X_n\). Thus \(X \to S\) is projective by Morphisms, Lemma 01VT.

It remains to prove flatness. Let \(x\) be a point of the closed fibre and choose an affine open neighbourhood \(\Spec(B) \subset X\) of \(x\). The ring \(B\) is Noetherian and \(B/\mathfrak m^nB\) is flat over \(A/\mathfrak m^n\) for every \(n\). Algebra, Lemma 0523 shows that \(X \to S\) is flat at \(x\). Thus the flat locus is an open neighbourhood of the closed fibre by More on Morphisms, Theorem 0399. Its closed complement has closed image in \(S\) because \(X\) is proper. A nonempty closed subset of the spectrum of a local ring contains the closed point. The complement must therefore be empty, and \(X\) is flat over \(A\).