Derived Categories of Schemes

Introduction

In this chapter we discuss derived categories of modules on schemes. Most of the material discussed here can be found in [TT], [Bokstedt-Neeman], [BvdB], and [LN]. Of course there are many other references.

Conventions

If \(\mathcal{A}\) is an abelian category and \(M\) is an object of \(\mathcal{A}\) then we also denote \(M\) the object of \(K(\mathcal{A})\) and/or \(D(\mathcal{A})\) corresponding to the complex which has \(M\) in degree \(0\) and is zero in all other degrees.

If we have a ring \(A\), then \(K(A)\) denotes the homotopy category of complexes of \(A\)-modules and \(D(A)\) the associated derived category. Similarly, if we have a ringed space \((X, \mathcal{O}_X)\) the symbol \(K(\mathcal{O}_X)\) denotes the homotopy category of complexes of \(\mathcal{O}_X\)-modules and \(D(\mathcal{O}_X)\) the associated derived category.

Derived category of quasi-coherent modules

In this section we discuss the relationship between quasi-coherent modules and all modules on a scheme \(X\). A reference is [TT, Appendix B]. By the discussion in Schemes, Section 01LA the embedding \(\QCoh(\mathcal{O}_X) \subset \textit{Mod}(\mathcal{O}_X)\) exhibits \(\QCoh(\mathcal{O}_X)\) as a weak Serre subcategory of the category of \(\mathcal{O}_X\)-modules. Denote \[D_\QCoh(\mathcal{O}_X) \subset D(\mathcal{O}_X)\] the subcategory of complexes whose cohomology sheaves are quasi-coherent, see Derived Categories, Section 06UP. Thus we obtain a canonical functor [06VT]\[\begin{equation} D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) \end{equation}\] see Derived Categories, Equation (06UR).

Lemma

Let \(X\) be a scheme. Then \(D_\QCoh(\mathcal{O}_X)\) has direct sums.

Proof

By Injectives, Lemma 07D9 the derived category \(D(\mathcal{O}_X)\) has direct sums and they are computed by taking termwise direct sums of any representatives. Thus it is clear that the cohomology sheaf of a direct sum is the direct sum of the cohomology sheaves as taking direct sums is an exact functor (in any Grothendieck abelian category). The lemma follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see Schemes, Section 01LA.

We will need some information on derived limits. We warn the reader that in the lemma below the derived limit will typically not be an object of \(D_\QCoh\).

Lemma

Let \(X\) be a scheme. Let \((K_n)\) be an inverse system of \(D_\QCoh(\mathcal{O}_X)\) with derived limit \(K = R\lim K_n\) in \(D(\mathcal{O}_X)\). Assume \(H^q(K_{n + 1}) \to H^q(K_n)\) is surjective for all \(q \in \mathbf{Z}\) and \(n \geq 1\). Then

1. \(H^q(K) = \lim H^q(K_n)\),

2. \(R\lim H^q(K_n) = \lim H^q(K_n)\), and

3. for every affine open \(U \subset X\) we have \(H^p(U, \lim H^q(K_n)) = 0\) for \(p > 0\).

Proof

Let \(\mathcal{B}\) be the set of affine opens of \(X\). Since \(H^q(K_n)\) is quasi-coherent we have \(H^p(U, H^q(K_n)) = 0\) for \(U \in \mathcal{B}\) by Cohomology of Schemes, Lemma 01XB. Moreover, the maps \(H^0(U, H^q(K_{n + 1})) \to H^0(U, H^q(K_n))\) are surjective for \(U \in \mathcal{B}\) by Schemes, Lemma 01IB. Part (1) follows from Cohomology, Lemma 0BKU whose conditions we have just verified. Parts (2) and (3) follow from Cohomology, Lemma 0BKS.

The following lemma will help us to “compute” a right derived functor on an object of \(D_\QCoh(\mathcal{O}_X)\).

Lemma

Let \(X\) be a scheme. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Then the map \(E \to R\lim \tau_{\geq -n}E\) of Derived Categories, Remark 0H72 is an isomorphism¹.

Proof

Denote \(\mathcal{H}^i = H^i(E)\) the \(i\)th cohomology sheaf of \(E\). Let \(\mathcal{B}\) be the set of affine open subsets of \(X\). Then \(H^p(U, \mathcal{H}^i) = 0\) for all \(p > 0\), all \(i \in \mathbf{Z}\), and all \(U \in \mathcal{B}\), see Cohomology of Schemes, Lemma 01XB. Thus the lemma follows from Cohomology, Lemma 0D64.

Lemma

Let \(X\) be a scheme. Let \(F : \textit{Mod}(\mathcal{O}_X) \to \textit{Ab}\) be an additive functor and \(N \geq 0\) an integer. Assume that

1. \(F\) commutes with countable direct products,

2. \(R^pF(\mathcal{F}) = 0\) for all \(p \geq N\) and \(\mathcal{F}\) quasi-coherent.

Then for \(E \in D_\QCoh(\mathcal{O}_X)\)

1. \(H^i(RF(\tau_{\leq a}E)) \to H^i(RF(E))\) is an isomorphism for \(i \leq a\),

2. \(H^i(RF(E)) \to H^i(RF(\tau_{\geq b - N + 1}E))\) is an isomorphism for \(i \geq b\),

3. if \(H^i(E) = 0\) for \(i \not \in [a, b]\) for some \(-\infty \leq a \leq b \leq \infty\), then \(H^i(RF(E)) = 0\) for \(i \not \in [a, b + N - 1]\).

Proof

Statement (1) is Derived Categories, Lemma 05TC.

Proof of statement (2). Write \(E_n = \tau_{\geq -n}E\). We have \(E = R\lim E_n\), see Lemma 08D3. Thus \(RF(E) = R\lim RF(E_n)\) in \(D(\textit{Ab})\) by Injectives, Lemma 08U1. Thus for every \(i \in \mathbf{Z}\) we have a short exact sequence \[0 \to R^1\lim H^{i - 1}(RF(E_n)) \to H^i(RF(E)) \to \lim H^i(RF(E_n)) \to 0\] see More on Algebra, Remark 08U5. To prove (2) we will show that the term on the left is zero and that the term on the right equals \(H^i(RF(E_{-b + N - 1}))\) for any \(b\) with \(i \geq b\).

For every \(n\) we have a distinguished triangle \[H^{-n}(E)[n] \to E_n \to E_{n - 1} \to H^{-n}(E)[n + 1]\] (Derived Categories, Remark 08J5) in \(D(\mathcal{O}_X)\). Since \(H^{-n}(E)\) is quasi-coherent we have \[H^i(RF(H^{-n}(E)[n])) = R^{i + n}F(H^{-n}(E)) = 0\] for \(i + n \geq N\) and \[H^i(RF(H^{-n}(E)[n + 1])) = R^{i + n + 1}F(H^{-n}(E)) = 0\] for \(i + n + 1 \geq N\). We conclude that \[H^i(RF(E_n)) \to H^i(RF(E_{n - 1}))\] is an isomorphism for \(n \geq N - i\). Thus the systems \(H^i(RF(E_n))\) all satisfy the ML condition and the \(R^1\lim\) term in our short exact sequence is zero (see discussion in More on Algebra, Section 07KV). Moreover, the system \(H^i(RF(E_n))\) is constant starting with \(n = N - i - 1\) as desired.

Proof of (3). Under the assumption on \(E\) we have \(\tau_{\leq a - 1}E = 0\) and we get the vanishing of \(H^i(RF(E))\) for \(i \leq a - 1\) from (1). Similarly, we have \(\tau_{\geq b + 1}E = 0\) and hence we get the vanishing of \(H^i(RF(E))\) for \(i \geq b + N\) from part (2).

The following lemma is the key ingredient to many of the results in this chapter.

Lemma

Let \(X = \Spec(A)\) be an affine scheme. All the functors in the diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[rr]_{(\href{perfect.html#perfect-equation-compare}{06VT})} & & D_\QCoh(\mathcal{O}_X) \ar[ld]^{R\Gamma(X, -)} \\ & D(A) \ar[lu]^{\widetilde{\ \ }} }\] are equivalences of triangulated categories. Moreover, for \(E\) in \(D_\QCoh(\mathcal{O}_X)\) we have \(H^0(X, E) = H^0(X, H^0(E))\).

Proof

The functor \(R\Gamma(X, -)\) gives a functor \(D(\mathcal{O}_X) \to D(A)\) and hence by restriction a functor [06VU]\[\begin{equation} R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \longrightarrow D(A). \end{equation}\] We will show this functor is quasi-inverse to (06VT) via the equivalence between quasi-coherent modules on \(X\) and the category of \(A\)-modules.

Elucidation. Denote \((Y, \mathcal{O}_Y)\) the one point space with sheaf of rings given by \(A\). Denote \(\pi : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)\) the obvious morphism of ringed spaces. Then \(R\Gamma(X, -)\) can be identified with \(R\pi_*\) and the functor (06VT) via the equivalence \(\textit{Mod}(\mathcal{O}_Y) = \text{Mod}_A = \QCoh(\mathcal{O}_X)\) can be identified with \(L\pi^* = \pi^* = \widetilde{\ }\) (see Modules, Lemma 01BH and Schemes, Lemmas 01I7 and 01IB). Thus the functors \[\xymatrix{ D(A) \ar@<1ex>[r] & D(\mathcal{O}_X) \ar@<1ex>[l] }\] are adjoint (by Cohomology, Lemma 079W). In particular we obtain canonical adjunction mappings \[a : \widetilde{R\Gamma(X, E)} \longrightarrow E\] for \(E\) in \(D(\mathcal{O}_X)\) and \[b : M^\bullet \longrightarrow R\Gamma(X, \widetilde{M^\bullet})\] for \(M^\bullet\) a complex of \(A\)-modules.

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). We may apply Lemma 08D4 to the functor \(F(-) = \Gamma(X, -)\) with \(N = 1\) by Cohomology of Schemes, Lemma 01XB. Hence \[H^0(R\Gamma(X, E)) = H^0(R\Gamma(X, \tau_{\geq 0}E)) = \Gamma(X, H^0(E))\] (the last equality by definition of the canonical truncation). Using this we will show that the adjunction mappings \(a\) and \(b\) induce isomorphisms \(H^0(a)\) and \(H^0(b)\). Thus \(a\) and \(b\) are quasi-isomorphisms (as the statement is invariant under shifts) and the lemma is proved.

In both cases we use that \(\widetilde{\ }\) is an exact functor (Schemes, Lemma 01HV). Namely, this implies that \[H^0\left(\widetilde{R\Gamma(X, E)}\right) = \widetilde{H^0(R\Gamma(X, E))} = \widetilde{\Gamma(X, H^0(E))}\] which is equal to \(H^0(E)\) because \(H^0(E)\) is quasi-coherent. Thus \(H^0(a)\) is an isomorphism. For the other direction we have \[H^0(R\Gamma(X, \widetilde{M^\bullet})) = \Gamma(X, H^0(\widetilde{M^\bullet})) = \Gamma(X, \widetilde{H^0(M^\bullet)}) = H^0(M^\bullet)\] which proves that \(H^0(b)\) is an isomorphism.

Lemma

Let \(X = \Spec(A)\) be an affine scheme. If \(K^\bullet\) is a K-flat complex of \(A\)-modules, then \(\widetilde{K^\bullet}\) is a K-flat complex of \(\mathcal{O}_X\)-modules.

Proof

By More on Algebra, Lemma 06Y1 we see that \(K^\bullet \otimes_A A_\mathfrak p\) is a K-flat complex of \(A_\mathfrak p\)-modules for every \(\mathfrak p \in \Spec(A)\). Hence we conclude from Cohomology, Lemma 06YB (and Schemes, Lemma 01HV) that \(\widetilde{K^\bullet}\) is K-flat.

Lemma

If \(f : X \to Y\) is a morphism of affine schemes given by the ring map \(A \to B\), then the diagram \[\xymatrix{ D(B) \ar[d] \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(A) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] commutes.

Proof

Follows from Lemma 06Z0 using that \(R\Gamma(Y, Rf_*K) = R\Gamma(X, K)\) by Cohomology, Lemma 0D5W.

Lemma

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

1. The functor \(Lf^*\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\).

2. If \(X\) and \(Y\) are affine and \(f\) is given by the ring map \(A \to B\), then the diagram \[\xymatrix{ D(B) \ar[r] & D_\QCoh(\mathcal{O}_Y) \\ D(A) \ar[r] \ar[u]^{- \otimes_A^\mathbf{L} B} & D_\QCoh(\mathcal{O}_X) \ar[u]_{Lf^*} }\] commutes.

Proof

We first prove the diagram \[\xymatrix{ D(B) \ar[r] & D(\mathcal{O}_Y) \\ D(A) \ar[r] \ar[u]^{- \otimes_A^\mathbf{L} B} & D(\mathcal{O}_X) \ar[u]_{Lf^*} }\] commutes. This is clear from Lemma 08DV and the constructions of the functors in question. To see (1) let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). To see that \(Lf^*E\) has quasi-coherent cohomology sheaves we may work locally on \(X\). Note that \(Lf^*\) is compatible with restricting to open subschemes. Hence we can assume that \(f\) is a morphism of affine schemes as in (2). Then we can apply Lemma 06Z0 to see that \(E\) comes from a complex of \(A\)-modules. By the commutativity of the first diagram of the proof the same holds for \(Lf^*E\) and we conclude (1) is true.

Lemma

Let \(X\) be a scheme.

1. For objects \(K, L\) of \(D_\QCoh(\mathcal{O}_X)\) the derived tensor product \(K \otimes^\mathbf{L}_{\mathcal{O}_X} L\) is in \(D_\QCoh(\mathcal{O}_X)\).

2. If \(X = \Spec(A)\) is affine then \[\widetilde{M^\bullet} \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{K^\bullet} = \widetilde{M^\bullet \otimes_A^\mathbf{L} K^\bullet}\] for any pair of complexes of \(A\)-modules \(K^\bullet\), \(M^\bullet\).

Proof

The equality of (2) follows immediately from Lemma 08DV and the construction of the derived tensor product. To see (1) let \(K, L\) be objects of \(D_\QCoh(\mathcal{O}_X)\). To check that \(K \otimes^\mathbf{L} L\) is in \(D_\QCoh(\mathcal{O}_X)\) we may work locally on \(X\), hence we may assume \(X = \Spec(A)\) is affine. By Lemma 06Z0 we may represent \(K\) and \(L\) by complexes of \(A\)-modules. Then part (2) implies the result.

Total direct image

The following lemma is the analogue of Cohomology of Schemes, Lemma 01XJ.

Lemma

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

1. The functor \(Rf_*\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_S)\).

2. If \(S\) is quasi-compact, there exists an integer \(N = N(X, S, f)\) such that for an object \(E\) of \(D_\QCoh(\mathcal{O}_X)\) with \(H^m(E) = 0\) for \(m > 0\) we have \(H^m(Rf_*E) = 0\) for \(m \geq N\).

3. In fact, if \(S\) is quasi-compact we can find \(N = N(X, S, f)\) such that for every morphism of schemes \(S' \to S\) the same conclusion holds for the functor \(R(f')_*\) where \(f' : X' \to S'\) is the base change of \(f\).

Proof

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). To prove (1) we have to show that \(Rf_*E\) has quasi-coherent cohomology sheaves. The question is local on \(S\), hence we may assume \(S\) is quasi-compact. Pick \(N = N(X, S, f)\) as in Cohomology of Schemes, Lemma 01XJ. Thus \(R^pf_*\mathcal{F} = 0\) for all quasi-coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\) and all \(p \geq N\) and the same remains true after base change.

First, assume \(E\) is bounded below. We will show (1) and (2) and (3) hold for such \(E\) with our choice of \(N\). In this case we can for example use the spectral sequence \[R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E\] (Derived Categories, Lemma 015J), the quasi-coherence of \(R^pf_*H^q(E)\), and the vanishing of \(R^pf_*H^q(E)\) for \(p \geq N\) to see that (1), (2), and (3) hold in this case.

Next we prove (2) and (3). Say \(H^m(E) = 0\) for \(m > 0\). Let \(U \subset S\) be affine open. By Cohomology of Schemes, Lemma 01XK and our choice of \(N\) we have \(H^p(f^{-1}(U), \mathcal{F}) = 0\) for \(p \geq N\) and any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\). Hence we may apply Lemma 08D4 to the functor \(\Gamma(f^{-1}(U), -)\) to see that \[R\Gamma(U, Rf_*E) = R\Gamma(f^{-1}(U), E)\] has vanishing cohomology in degrees \(\geq N\). Since this holds for all \(U \subset S\) affine open we conclude that \(H^m(Rf_*E) = 0\) for \(m \geq N\).

Next, we prove (1) in the general case. Recall that there is a distinguished triangle \[\tau_{\leq -n - 1}E \to E \to \tau_{\geq -n}E \to (\tau_{\leq -n - 1}E)[1]\] in \(D(\mathcal{O}_X)\), see Derived Categories, Remark 08J5. By (2) we see that \(Rf_*\tau_{\leq -n - 1}E\) has vanishing cohomology sheaves in degrees \(\geq -n + N\). Thus, given an integer \(q\) we see that \(R^qf_*E\) is equal to \(R^qf_*\tau_{\geq -n}E\) for some \(n\) and the result above applies.

Lemma

Let \(f : X \to S\) be a quasi-separated and quasi-compact morphism of schemes. Let \(\mathcal{F}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules each of which is right acyclic for \(f_*\). Then \(f_*\mathcal{F}^\bullet\) represents \(Rf_*\mathcal{F}^\bullet\) in \(D(\mathcal{O}_S)\).

Proof

There is always a canonical map \(f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet\). Our task is to show that this is an isomorphism on cohomology sheaves. As the statement is invariant under shifts it suffices to show that \(H^0(f_*(\mathcal{F}^\bullet)) \to R^0f_*\mathcal{F}^\bullet\) is an isomorphism. The statement is local on \(S\) hence we may assume \(S\) affine. By Lemma 08D5 we have \(R^0f_*\mathcal{F}^\bullet = R^0f_*\tau_{\geq -n}\mathcal{F}^\bullet\) for all sufficiently large \(n\). Thus we may assume \(\mathcal{F}^\bullet\) bounded below. As each \(\mathcal{F}^n\) is right \(f_*\)-acyclic by assumption we see that \(f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet\) is a quasi-isomorphism by Leray’s acyclicity lemma (Derived Categories, Lemma 015E).

Lemma

Let \(X\) be a quasi-separated and quasi-compact scheme. Let \(\mathcal{F}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules each of which is right acyclic for \(\Gamma(X, -)\). Then \(\Gamma(X, \mathcal{F}^\bullet)\) represents \(R\Gamma(X, \mathcal{F}^\bullet)\) in \(D(\Gamma(X, \mathcal{O}_X)\).

Proof

Apply Lemma 0G9N to the canonical morphism \(X \to \Spec(\Gamma(X, \mathcal{O}_X))\). Some details omitted.

Lemma

Let \(X\) be a quasi-separated and quasi-compact scheme. For any object \(K\) of \(D_\QCoh(\mathcal{O}_X)\) the spectral sequence \[E_2^{i, j} = H^i(X, H^j(K)) \Rightarrow H^{i + j}(X, K)\] of Cohomology, Example 0BKM is bounded and converges.

Proof

By the construction of the spectral sequence via Cohomology, Lemma 0BKK using the filtration given by \(\tau_{\leq -p}K\), we see that suffices to show that given \(n \in \mathbf{Z}\) we have \[H^n(X, \tau_{\leq -p}K) = 0 \text{ for } p \gg 0\] and \[H^n(X, K) = H^n(X, \tau_{\leq -p}K) \text{ for } p \ll 0\] The first follows from Lemma 08D4 applied with \(F = \Gamma(X, -)\) and the bound in Cohomology of Schemes, Lemma 01XJ. The second holds whenever \(-p \leq n\) for any ringed space \((X, \mathcal{O}_X)\) and any \(K \in D(\mathcal{O}_X)\).

Lemma

Let \(f : X \to S\) be a quasi-separated and quasi-compact morphism of schemes. Then \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)\) commutes with direct sums.

Proof

Let \(E_i\) be a family of objects of \(D_\QCoh(\mathcal{O}_X)\) and set \(E = \bigoplus E_i\). We want to show that the map \[\bigoplus Rf_*E_i \longrightarrow Rf_*E\] is an isomorphism. We will show it induces an isomorphism on cohomology sheaves in degree \(0\) which will imply the lemma. To prove this we may work locally on \(S\), hence we may and do assume that \(S\) is quasi-compact. Choose an integer \(N\) as in Lemma 08D5. Then \(R^0f_*E = R^0f_*\tau_{\geq -N}E\) and \(R^0f_*E_i = R^0f_*\tau_{\geq -N}E_i\) by the lemma cited. Observe that \(\tau_{\geq -N}E = \bigoplus \tau_{\geq -N}E_i\). Thus we may assume all of the \(E_i\) have vanishing cohomology sheaves in degrees \(< -N\). Next we use the spectral sequences \[R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E \quad\text{and}\quad R^pf_*H^q(E_i) \Rightarrow R^{p + q}f_*E_i\] (Derived Categories, Lemma 015J) to reduce to the case of a direct sum of quasi-coherent sheaves. This case is handled by Cohomology of Schemes, Lemma 07TB.

Affine morphisms

In this section we collect some information about pushforward along an affine morphism of schemes.

Lemma

Let \(f : X \to S\) be an affine morphism of schemes. Let \(\mathcal{F}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules. Then \(f_*\mathcal{F}^\bullet = Rf_*\mathcal{F}^\bullet\).

Proof

Combine Lemma 0G9N with Cohomology of Schemes, Lemma 01XC. An alternative proof is to work affine locally on \(S\) and use Lemma 0DJK.

Lemma

Let \(f : X \to S\) be an affine morphism of schemes. Then \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)\) reflects isomorphisms.

Proof

The statement means that a morphism \(\alpha : E \to F\) of \(D_\QCoh(\mathcal{O}_X)\) is an isomorphism if \(Rf_*\alpha\) is an isomorphism. We may check this on cohomology sheaves. In particular, the question is local on \(S\). Hence we may assume \(S\) and therefore \(X\) is affine. In this case the statement is clear from the description of the derived categories \(D_\QCoh(\mathcal{O}_X)\) and \(D_\QCoh(\mathcal{O}_S)\) given in Lemma 06Z0. Some details omitted.

Lemma

Let \(f : X \to S\) be an affine morphism of schemes. For \(E\) in \(D_\QCoh(\mathcal{O}_S)\) we have \(Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_S} f_*\mathcal{O}_X\).

Proof

Since \(f\) is affine the map \(f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X\) is an isomorphism (Cohomology of Schemes, Lemma 01XC). There is a canonical map \(E \otimes^\mathbf{L} f_*\mathcal{O}_X = E \otimes^\mathbf{L} Rf_*\mathcal{O}_X \to Rf_* Lf^* E\) adjoint to the map \[Lf^*(E \otimes^\mathbf{L} Rf_*\mathcal{O}_X) = Lf^*E \otimes^\mathbf{L} Lf^*Rf_*\mathcal{O}_X \longrightarrow Lf^* E \otimes^\mathbf{L} \mathcal{O}_X = Lf^* E\] coming from \(1 : Lf^*E \to Lf^*E\) and the canonical map \(Lf^*Rf_*\mathcal{O}_X \to \mathcal{O}_X\). To check the map so constructed is an isomorphism we may work locally on \(S\). Hence we may assume \(S\) and therefore \(X\) is affine. In this case the statement is clear from the description of the derived categories \(D_\QCoh(\mathcal{O}_X)\) and \(D_\QCoh(\mathcal{O}_S)\) and the functor \(Lf^*\) given in Lemmas 06Z0 and 08DW. Some details omitted.

Let \(Y\) be a scheme. Let \(\mathcal{A}\) be a sheaf of \(\mathcal{O}_Y\)-algebras. We will denote \(D_\QCoh(\mathcal{A})\) the inverse image of \(D_\QCoh(\mathcal{O}_X)\) under the restriction functor \(D(\mathcal{A}) \to D(\mathcal{O}_X)\). In other words, \(K \in D(\mathcal{A})\) is in \(D_\QCoh(\mathcal{A})\) if and only if its cohomology sheaves are quasi-coherent as \(\mathcal{O}_X\)-modules. If \(\mathcal{A}\) is quasi-coherent itself this is the same as asking the cohomology sheaves to be quasi-coherent as \(\mathcal{A}\)-modules, see Morphisms, Lemma 01SB.

Lemma

Let \(f : X \to Y\) be an affine morphism of schemes. Then \(f_*\) induces an equivalence \[\Phi : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(f_*\mathcal{O}_X)\] whose composition with \(D_\QCoh(f_*\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) is \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\).

Proof

Recall that \(Rf_*\) is computed on an object \(K \in D_\QCoh(\mathcal{O}_X)\) by choosing a K-injective complex \(\mathcal{I}^\bullet\) of \(\mathcal{O}_X\)-modules representing \(K\) and taking \(f_*\mathcal{I}^\bullet\). Thus we let \(\Phi(K)\) be the complex \(f_*\mathcal{I}^\bullet\) viewed as a complex of \(f_*\mathcal{O}_X\)-modules. Denote \(g : (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)\) the obvious morphism of ringed spaces. Then \(g\) is a flat morphism of ringed spaces (see below for a description of the stalks) and \(\Phi\) is the restriction of \(Rg_*\) to \(D_\QCoh(\mathcal{O}_X)\). We claim that \(Lg^*\) is a quasi-inverse. First, observe that \(Lg^*\) sends \(D_\QCoh(f_*\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_X)\) because \(g^*\) transforms quasi-coherent modules into quasi-coherent modules (Modules, Lemma 01BG). To finish the proof it suffices to show that the adjunction mappings \[Lg^*\Phi(K) = Lg^*Rg_*K \to K \quad\text{and}\quad M \to Rg_*Lg^*M = \Phi(Lg^*M)\] are isomorphisms for \(K \in D_\QCoh(\mathcal{O}_X)\) and \(M \in D_\QCoh(f_*\mathcal{O}_X)\). This is a local question, hence we may assume \(Y\) and therefore \(X\) are affine.

Assume \(Y = \Spec(B)\) and \(X = \Spec(A)\). Let \(\mathfrak p = x \in \Spec(A) = X\) be a point mapping to \(\mathfrak q = y \in \Spec(B) = Y\). Then \((f_*\mathcal{O}_X)_y = A_\mathfrak q\) and \(\mathcal{O}_{X, x} = A_\mathfrak p\) hence \(g\) is flat. Hence \(g^*\) is exact and \(H^i(Lg^*M) = g^*H^i(M)\) for any \(M\) in \(D(f_*\mathcal{O}_X)\). For \(K \in D_\QCoh(\mathcal{O}_X)\) we see that \[H^i(\Phi(K)) = H^i(Rf_*K) = f_*H^i(K)\] by the vanishing of higher direct images (Cohomology of Schemes, Lemma 01XC) and Lemma 08D4 (small detail omitted). Thus it suffice to show that \[g^*g_*\mathcal{F} \to \mathcal{F} \quad\text{and}\quad \mathcal{G} \to g_*g^*\mathcal{G}\] are isomorphisms where \(\mathcal{F}\) is a quasi-coherent \(\mathcal{O}_X\)-module and \(\mathcal{G}\) is a quasi-coherent \(f_*\mathcal{O}_X\)-module. This follows from Morphisms, Lemma 01SB.

Cohomology with support in a closed subset

We elaborate on the material in Cohomology, Sections 0A39 and 0G6Y for schemes and quasi-coherent modules.

Definition

Let \(X\) be a scheme. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(T \subset X\) be a closed subset. We say \(E\) is supported on \(T\) if the cohomology sheaves \(H^i(E)\) are supported on \(T\).

We repeat some of the discussion from Cohomology, Section 0G6Y in the situation of the definition. Let \(X\) be a scheme. Let \(T \subset X\) be a closed subset. The category of \(\mathcal{O}_X\)-modules whose support is contained in \(T\) is a Serre subcategory of the category of all \(\mathcal{O}_X\)-modules, see Homology, Definition 02MO and Modules, Lemma 01AU. Thus we may apply the discussion in Derived Categories, Section 06UP and we obtain a strictly full, saturated triangulated subcategory of \(D(\mathcal{O}_X)\) consisting of objects supported on \(T\); we will denote this category \(D_T(\mathcal{O}_X)\) in the following.

In the situation of Definition 08DA denote \(i : T \to X\) the inclusion map. Recall from Cohomology, Section 0G6Y that in this situation we have a functor \(R\mathcal{H}_T : D(\mathcal{O}_X) \to D(i^{-1}\mathcal{O}_X)\) which is right adjoint to \(i_* : D(i^{-1}\mathcal{O}_X) \to D(\mathcal{O}_X)\).

Lemma

Let \(X\) be a scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is a retrocompact open of \(X\). Let \(i : T \to X\) be the inclusion.

1. For \(E\) in \(D_\QCoh(\mathcal{O}_X)\) we have \(i_*R\mathcal{H}_T(E)\) in \(D_{\QCoh, T}(\mathcal{O}_X)\).

2. The functor \(i_* \circ R\mathcal{H}_T : D_\QCoh(\mathcal{O}_X) \to D_{\QCoh, T}(\mathcal{O}_X)\) is right adjoint to the inclusion functor \(D_{\QCoh, T}(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_X)\).

Proof

Set \(U = X \setminus T\) and denote \(j : U \to X\) the inclusion. By Cohomology, Lemma 0G72 there is a distinguished triangle \[i_*R\mathcal{H}_T(E) \to E \to Rj_*(E|_U) \to i_*R\mathcal{H}_Z(E)[1]\] in \(D(\mathcal{O}_X)\). By Lemma 08D5 the complex \(Rj_*(E|_U)\) has quasi-coherent cohomology sheaves (this is where we use that \(U\) is retrocompact in \(X\)). Thus we see that (1) is true. Part (2) follows from this and the adjointness of functors in Cohomology, Lemma 0AEF.

Lemma

Let \(X\) be a scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is a retrocompact open of \(X\). Then for a family of objects \(E_i\), \(i \in I\) of \(D_\QCoh(\mathcal{O}_X)\) we have \(R\mathcal{H}_T(\bigoplus E_i) = \bigoplus R\mathcal{H}_T(E_i)\).

Proof

Set \(U = X \setminus T\) and denote \(j : U \to X\) the inclusion. By Cohomology, Lemma 0G72 there is a distinguished triangle \[i_*R\mathcal{H}_T(E) \to E \to Rj_*(E|_U) \to i_*R\mathcal{H}_Z(E)[1]\] in \(D(\mathcal{O}_X)\) for any \(E\) in \(D(\mathcal{O}_X)\). The functor \(E \mapsto Rj_*(E|_U)\) commutes with direct sums on \(D_\QCoh(\mathcal{O}_X)\) by Lemma 08DZ. It follows that the same is true for the functor \(i_* \circ R\mathcal{H}_T\) (details omitted). Since \(i_* : D(i^{-1}\mathcal{O}_X) \to D_T(\mathcal{O}_X)\) is an equivalence (Cohomology, Lemma 0AEF) we conclude.

Remark

Let \(X\) be a scheme. Let \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\). Denote \(Z \subset X\) the closed subscheme cut out by \(f_1, \ldots, f_c\). For \(0 \leq p < c\) and \(1 \leq i_0 < \ldots < i_p \leq c\) we denote \(U_{i_0 \ldots i_p} \subset X\) the open subscheme where \(f_{i_0} \ldots f_{i_p}\) is invertible. For any \(\mathcal{O}_X\)-module \(\mathcal{F}\) we set \[\mathcal{F}_{i_0 \ldots i_p} = (U_{i_0 \ldots i_p} \to X)_*(\mathcal{F}|_{U_{i_0 \ldots i_p}})\] In this situation the extended alternating Čech complex is the complex of \(\mathcal{O}_X\)-modules [0G7J]\[\begin{equation} 0 \to \mathcal{F} \to \bigoplus\nolimits_{i_0} \mathcal{F}_{i_0} \to \ldots \to \bigoplus\nolimits_{i_0 < \ldots < i_p} \mathcal{F}_{i_0 \ldots i_p} \to \ldots \to \mathcal{F}_{1 \ldots c} \to 0 \end{equation}\] where \(\mathcal{F}\) is put in degree \(0\). The maps are constructed as follows. Given \(1 \leq i_0 < \ldots < i_{p + 1} \leq c\) and \(0 \leq j \leq p + 1\) we have the canonical map \[\mathcal{F}_{i_0 \ldots \hat i_j \ldots i_{p + 1}} \to \mathcal{F}_{i_0 \ldots i_p}\] coming from the inclusion \(U_{i_0 \ldots i_p} \subset U_{i_0 \ldots \hat i_j \ldots i_{p + 1}}\). The differentials in the extended alternating complex use these canonical maps with sign \((-1)^j\).

Lemma

With \(X\), \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\), and \(\mathcal{F}\) as in Remark 0G7I the complex (0G7J) restricts to an acyclic complex over \(X \setminus Z\).

We remark that this lemma holds more generally for any extended alternating Čech complex defined as in Remark 0G7I starting with a finite open covering \(X \setminus Z = U_1 \cup \ldots \cup U_c\).

Proof

Let \(W \subset X \setminus Z\) be an open subset. Evaluating the complex of sheaves (0G7J) on \(W\) we obtain the complex \[\mathcal{F}(W) \to \bigoplus\nolimits_{i_0} \mathcal{F}(U_{i_0} \cap W) \to \bigoplus\nolimits_{i_0 < i_1} \mathcal{F}(U_{i_0i_1} \cap W) \to \ldots\] In other words, we obtain the extended ordered Čech complex for the covering \(W = \bigcup U_i \cap W\) and the standard ordering on \(\{1, \ldots, c\}\), see Cohomology, Section 01FG. By Cohomology, Lemma 0G6T this complex is homotopic to zero as soon as \(W\) is contained in \(V(f_i)\) for some \(1 \leq i \leq c\). This finishes the proof.

Remark

Let \(X\), \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\), and \(\mathcal{F}\) be as in Remark 0G7I. Denote \(\mathcal{F}^\bullet\) the complex (0G7J). By Lemma 0G7K the cohomology sheaves of \(\mathcal{F}^\bullet\) are supported on \(Z\) hence \(\mathcal{F}^\bullet\) is an object of \(D_Z(\mathcal{O}_X)\). On the other hand, the equality \(\mathcal{F}^0 = \mathcal{F}\) determines a canonical map \(\mathcal{F}^\bullet \to \mathcal{F}\) in \(D(\mathcal{O}_X)\). As \(i_* \circ R\mathcal{H}_Z\) is a right adjoint to the inclusion functor \(D_Z(\mathcal{O}_X) \to D(\mathcal{O}_X)\), see Cohomology, Lemma 0AEF, we obtain a canonical commutative diagram \[\xymatrix{ \mathcal{F}^\bullet \ar[rd] \ar[rr] & & \mathcal{F} \\ & i_*R\mathcal{H}_Z(\mathcal{F}) \ar[ru] }\] in \(D(\mathcal{O}_X)\) functorial in the \(\mathcal{O}_X\)-module \(\mathcal{F}\).

Lemma

With \(X\), \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\), and \(\mathcal{F}\) as in Remark 0G7I. If \(\mathcal{F}\) is quasi-coherent, then the complex (0G7J) represents \(i_* R\mathcal{H}_Z(\mathcal{F})\) in \(D_Z(\mathcal{O}_X)\).

Proof

Let us denote \(\mathcal{F}^\bullet\) the complex (0G7J). The statement of the lemma means that the map \(\mathcal{F}^\bullet \to i_*R\mathcal{H}_Z(\mathcal{F})\) of Remark 0G7L is an isomorphism. Since \(\mathcal{F}^\bullet\) is in \(D_Z(\mathcal{O}_X)\) (see remark cited), we see that \(i_*R\mathcal{H}_Z(\mathcal{F}^\bullet) = \mathcal{F}^\bullet\) by Cohomology, Lemma 0AEF. The morphism \(U_{i_0 \ldots i_p} \to X\) is affine as it is given over affine opens of \(X\) by inverting the function \(f_{i_0} \ldots f_{i_p}\). Thus we see that \[\mathcal{F}_{i_0 \ldots i_p} = (U_{i_0 \ldots i_p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_p}} = R(U_{i_0 \ldots i_p} \to X)_*\mathcal{F}|_{U_{i_0 \ldots i_p}}\] by Cohomology of Schemes, Lemma 01XC and the assumption that \(\mathcal{F}\) is quasi-coherent. We conclude that \(R\mathcal{H}_Z(\mathcal{F}_{i_0 \ldots i_p}) = 0\) by Cohomology, Lemma 0G73. Thus \(i_*R\mathcal{H}_Z(\mathcal{F}^p) = 0\) for \(p > 0\). Putting everything together we obtain \[\mathcal{F}^\bullet = i_*R\mathcal{H}_Z(\mathcal{F}^\bullet) = i_*R\mathcal{H}_Z(\mathcal{F})\] as desired.

Lemma

Let \(X\) be a scheme. Let \(T \subset X\) be a closed subset which can locally be cut out by at most \(c\) elements of the structure sheaf. Then \(\mathcal{H}^i_Z(\mathcal{F}) = 0\) for \(i > c\) and any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\).

Proof

This follows immediately from the local description of \(R\mathcal{H}_T(\mathcal{F})\) given in Lemma 0G7M.

Lemma

Let \(X\) be a scheme. Let \(Z \subset X\) be a closed subset which can locally be cut out by a Koszul regular sequence having \(c\) elements. Then \(\mathcal{H}^i_Z(\mathcal{F}) = 0\) for \(i \not = c\) for every flat, quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\).

Proof

By the description of \(R\mathcal{H}_Z(\mathcal{F})\) given in Lemma 0G7M this boils down to the following algebra statement: given a ring \(R\), a Koszul regular sequence \(f_1, \ldots, f_c \in R\), and a flat \(R\)-module \(M\), the extended alternating Čech complex \(M \to \bigoplus\nolimits_{i_0} M_{f_{i_0}} \to \bigoplus\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \to \ldots \to M_{f_1 \ldots f_c}\) from More on Algebra, Section 0G6F only has cohomology in degree \(c\). By More on Algebra, Lemma 0G6L we obtain the desired vanishing for the extended alternating Čech complex of \(R\). Since the complex for \(M\) is obtained by tensoring this with the flat \(R\)-module \(M\) (More on Algebra, Lemma 0G6H) we conclude.

Remark

With \(X\), \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\), and \(\mathcal{F}\) as in Remark 0G7I. There is a canonical \(\mathcal{O}_X|_Z\)-linear map \[c_{f_1, \ldots, f_c} : i^*\mathcal{F} \longrightarrow \mathcal{H}^c_Z(\mathcal{F})\] functorial in \(\mathcal{F}\). Namely, denoting \(\mathcal{F}^\bullet\) the extended alternating Čech complex (0G7J) we have the canonical map \(\mathcal{F}^\bullet \to i_*R\mathcal{H}_Z(\mathcal{F})\) of Remark 0G7L. This determines a canonical map \[\Coker\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^c_Z(\mathcal{F})\] on cohomology sheaves in degree \(c\). Given a local section \(s\) of \(\mathcal{F}\) we can consider the local section \[\frac{s}{f_1 \ldots f_c}\] of \(\mathcal{F}_{1 \ldots c}\). The class of this section in the cokernel displayed above depends only on \(s\) modulo the image of \((f_1, \ldots, f_c) : \mathcal{F}^{\oplus c} \to \mathcal{F}\). Since \(i_*i^*\mathcal{F}\) is equal to the cokernel of \((f_1, \ldots, f_c) : \mathcal{F}^{\oplus c} \to \mathcal{F}\) we see that we get an \(\mathcal{O}_X\)-module map \(i_*i^*\mathcal{F} \to i_*\mathcal{H}_Z^c(\mathcal{F})\). As \(i_*\) is fully faithful we get the map \(c_{f_1, \ldots, f_c}\).

Example

Let \(X = \Spec(A)\) be affine, \(f_1, \ldots, f_c \in A\), and let \(\mathcal{F} = \widetilde{M}\) for some \(A\)-module \(M\). The map \(c_{f_1, \ldots, f_c}\) of Remark 0G7Q can be described as the map \[M/(f_1, \ldots, f_c)M \longrightarrow \Coker\left( \bigoplus M_{f_1 \ldots \hat f_i \ldots f_c} \to M_{f_1 \ldots f_c} \right)\] sending the class of \(s \in M\) to the class of \(s/f_1 \ldots f_c\) in the cokernel.

Lemma

With \(X\), \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\), and \(\mathcal{F}\) as in Remark 0G7I. Let \(a_{ji} \in \Gamma(X, \mathcal{O}_X)\) for \(1 \leq i, j \leq c\) and set \(g_j = \sum_{i = 1, \ldots, c} a_{ji}f_i\). Assume \(g_1, \ldots, g_c\) scheme theoretically cut out \(Z\). If \(\mathcal{F}\) is quasi-coherent, then \[c_{f_1, \ldots, f_c} = \det(a_{ji}) c_{g_1, \ldots, g_c}\] where \(c_{f_1, \ldots, f_c}\) and \(c_{g_1, \ldots, g_c}\) are as in Remark 0G7Q.

Proof

We will prove that \(c_{f_1, \ldots, f_c}(s) = \det(a_{ij}) c_{g_1, \ldots, g_c}(s)\) as global sections of \(\mathcal{H}_Z(\mathcal{F})\) for any \(s \in \mathcal{F}(X)\). This is sufficient since we then obtain the same result for section over any open subscheme of \(X\). To do this, for \(1 \leq i_0 < \ldots < i_p \leq c\) and \(1 \leq j_0 < \ldots < j_q \leq c\) we denote \(U_{i_0 \ldots i_p} \subset X\), \(V_{j_0 \ldots j_q} \subset X\), and \(W_{i_0 \ldots i_p, j_0 \ldots j_q} \subset X\) the open subscheme where \(f_{i_0} \ldots f_{i_p}\) is invertible, \(g_{j_0} \ldots g_{j_q}\) is invertible, and where \(f_{i_0} \ldots f_{i_p}g_{j_0} \ldots g_{j_q}\) is invertible. We denote \(\mathcal{F}_{i_0 \ldots i_p}\), resp. \(\mathcal{F}'_{j_0 \ldots j_q}\) \(\mathcal{F}''_{i_0 \ldots i_p, j_0 \ldots j_q}\) the pushforward to \(X\) of the restriction of \(\mathcal{F}\) to \(U_{i_0 \ldots i_p}\), resp. \(V_{j_0 \ldots j_q}\), resp. \(W_{i_0 \ldots i_p, j_0 \ldots j_q}\). Then we obtain three extended alternating Čech complexes \[\mathcal{F}^\bullet : \mathcal{F} \to \bigoplus\nolimits_{i_0} \mathcal{F}_{i_0} \to \bigoplus\nolimits_{i_0 < i_1} \mathcal{F}_{i_0i_1} \to \ldots\] and \[(\mathcal{F}')^\bullet : \mathcal{F} \to \bigoplus\nolimits_{j_0} \mathcal{F}'_{j_0} \to \bigoplus\nolimits_{j_0 < j_1} \mathcal{F}'_{j_0j_1} \to \ldots\] and \[(\mathcal{F}'')^\bullet : \mathcal{F} \to \bigoplus\nolimits_{i_0} \mathcal{F}_{i_0} \oplus \bigoplus\nolimits_{j_0} \mathcal{F}'_{j_0} \to \bigoplus\nolimits_{i_0 < i_1} \mathcal{F}_{i_0i_1} \oplus \bigoplus\nolimits_{i_0, j_0} \mathcal{F}''_{i_0, j_0} \oplus \bigoplus\nolimits_{j_0 < j_1} \mathcal{F}'_{j_0j_1} \to \ldots\] whose differentials are those used in defining (0G7J). There are maps of complexes \[(\mathcal{F}'')^\bullet \to \mathcal{F}^\bullet \quad\text{and}\quad (\mathcal{F}'')^\bullet \to (\mathcal{F}')^\bullet\] given by the projection maps on the terms (and hence inducing the identity map in degree \(0\)). Observe that by Lemma 0G7M each of these complexes represents \(i_*R\mathcal{H}_Z(\mathcal{F})\) and these maps represent the identity on this object. Thus it suffices to find an element \[\sigma \in H^c((\mathcal{F}'')^\bullet(X))\] mapping to \(c_{f_1, \ldots, f_c}(s)\) and \(\det(a_{ji})c_{g_1, \ldots, g_c}(s)\) by these two maps. It turns out we can explicitly give a cocycle for \(\sigma\). Namely, we take \[\sigma_{1 \ldots c} = \frac{s}{f_1 \ldots f_c} \in \mathcal{F}_{1 \ldots c}(X) \quad\text{and}\quad \sigma'_{1 \ldots c} = \frac{\det(a_{ji})s}{g_1 \ldots g_c} \in \mathcal{F}'_{1 \ldots c}(X)\] and we take \[\sigma_{i_0 \ldots i_p, j_0 \ldots j_{c - p - 2}} = \frac{\lambda(i_0 \ldots i_p, j_0 \ldots j_{c - p - 2})s} {f_{i_0} \ldots f_{i_p}g_{j_0} \ldots g_{j_{c - p - 2}}} \in \mathcal{F}''_{i_0 \ldots i_p, j_0 \ldots j_{c - p - 2}}(X)\] where \(\lambda(i_0 \ldots i_p, j_0 \ldots j_{c - p - 2})\) is the coefficient of \(e_1 \wedge \ldots \wedge e_c\) in the formal expression \[e_{i_0} \wedge \ldots \wedge e_{i_p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_c) \wedge \ldots \wedge (a_{j_{c - p - 2}1} e_1 + \ldots + a_{j_{c - p - 2}c}e_c)\] To verify that \(\sigma\) is a cocycle, we have to show for \(1 \leq i_0 < \ldots < i_p \leq c\) and \(1 \leq j_0 < \ldots < j_{c - p - 1} \leq c\) that we have \[\begin{align*} 0 & = \sum\nolimits_{a = 0, \ldots, p} (-1)^a f_{i_a} \lambda(i_0 \ldots \hat i_a \ldots i_p, j_0 \ldots j_{c - p - 1}) \\ & + \sum\nolimits_{b = 0, \ldots, c - p - 1} (-1)^{p + b + 1}g_{j_b} \lambda(i_0 \ldots i_p, j_0 \ldots \hat j_b \ldots j_{c - p - 1}) \end{align*}\] The easiest way to see this is perhaps to argue that the formal expression \[\xi = e_{i_0} \wedge \ldots \wedge e_{i_p} \wedge (a_{j_01} e_1 + \ldots + a_{j_0c}e_c) \wedge \ldots \wedge (a_{j_{c - p - 1}1} e_1 + \ldots + a_{j_{c - p - 1}c}e_c)\] is \(0\) as it is an element of the \((c + 1)\)st wedge power of the free module on \(e_1, \ldots, e_c\) and that the expression above is the image of \(\xi\) under the Koszul differential sending \(e_i \to f_i\). Some details omitted.

Lemma

Let \(X\) be a scheme. Let \(Z \to X\) be a closed immersion of finite presentation whose conormal sheaf \(\mathcal{C}_{Z/X}\) is locally free of rank \(c\). Then there is a canonical map \[c : \wedge^c(\mathcal{C}_{Z/X})^\vee \otimes_{\mathcal{O}_Z} i^*\mathcal{F} \longrightarrow \mathcal{H}_Z^c(\mathcal{F})\] functorial in the quasi-coherent module \(\mathcal{F}\).

Proof

Follows from the construction in Remark 0G7Q and the independence of the choice of generators of the ideal sheaf shown in Lemma 0G7S. Some details omitted.

Remark

Let \(g : X' \to X\) be a morphism of schemes. Let \(f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)\). Set \(f'_i = g^\sharp(f_i) \in \Gamma(X', \mathcal{O}_{X'})\). Denote \(Z \subset X\), resp. \(Z' \subset X'\) the closed subscheme cut out by \(f_1, \ldots, f_c\), resp. \(f'_1, \ldots, f'_c\). Then \(Z' = Z \times_X X'\). Denote \(h : Z' \to Z\) the induced morphism of schemes. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Set \(\mathcal{F}' = g^*\mathcal{F}\). In this setting, if \(\mathcal{F}\) is quasi-coherent, then the diagram \[\xymatrix{ (i')^{-1}\mathcal{O}_{X'} \otimes_{h^{-1}i^{-1}\mathcal{O}_X} h^{-1}\mathcal{H}^c_Z(\mathcal{F}) \ar[r] & \mathcal{H}_{Z'}^c(\mathcal{F}') \\ h^*i^*\mathcal{F} \ar[r] \ar[u]_-{c_{f_1, \ldots, f_c}} & (i')^*\mathcal{F}' \ar[u]^-{c_{f'_1, \ldots, f'_c}} }\] is commutative where the top horizontal arrow is the map of Cohomology, Remark 0G78 on cohomology sheaves in degree \(c\). Namely, denote \(\mathcal{F}^\bullet\), resp. \((\mathcal{F}')^\bullet\) the extended alternating Čech complex constructed in Remark 0G7I using \(\mathcal{F}, f_1, \ldots, f_c\), resp. \(\mathcal{F}', f'_1, \ldots, f'_c\). Note that \((\mathcal{F}')^\bullet = g^*\mathcal{F}^\bullet\). Then, without assuming \(\mathcal{F}\) is quasi-coherent, the diagram \[\xymatrix{ i'_* L(g|_{Z'})^* R\mathcal{H}_Z(\mathcal{F}) \ar[r] \ar@{=}[d] & i'_*R\mathcal{H}_{Z'}(Lg^*\mathcal{F}) \ar[d] \\ Lg^*i_*R\mathcal{H}_Z(\mathcal{F}) & i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \\ Lg^*(\mathcal{F}^\bullet) \ar[u] \ar[r] & (\mathcal{F}')^\bullet \ar[u] }\] is commutative where \(g|_{Z'} : (Z', (i')^{-1}\mathcal{O}_{X'}) \to (Z, i^{-1}\mathcal{O}_X)\) is the induced morphism of ringed spaces. Here the top horizontal arrow is given in Cohomology, Remark 0G78 as is the explanation for the equal sign. The arrows pointing up are from Remark 0G7L. The lower horizontal arrow is the map \(Lg^*\mathcal{F}^\bullet \to g^*\mathcal{F}^\bullet = (\mathcal{F}')^\bullet\) and the arrow pointing down is induced by \(Lg^*\mathcal{F} \to g^*\mathcal{F} = \mathcal{F}'\). The diagram commutes because going around the diagram both ways we obtain two arrows \(Lg^*\mathcal{F}^\bullet \to i'_*R\mathcal{H}_{Z'}(\mathcal{F}')\) whose composition with \(i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \to \mathcal{F}'\) is the canonical map \(Lg^*\mathcal{F}^\bullet \to \mathcal{F}'\). Some details omitted. Now the commutativity of the first diagram follows by looking at this diagram on cohomology sheaves in degree \(c\) and using that the construction of the map \(i^*\mathcal{F} \to \Coker(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c})\) used in Remark 0G7Q is compatible with pullbacks.

The coherator

Let \(X\) be a scheme. The coherator is a functor \[Q_X : \textit{Mod}(\mathcal{O}_X) \longrightarrow \QCoh(\mathcal{O}_X)\] which is right adjoint to the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\). It exists for any scheme \(X\) and moreover the adjunction mapping \(Q_X(\mathcal{F}) \to \mathcal{F}\) is an isomorphism for every quasi-coherent module \(\mathcal{F}\), see Properties, Proposition 077P. Since \(Q_X\) is left exact (as a right adjoint) we can consider its right derived extension \[RQ_X : D(\mathcal{O}_X) \longrightarrow D(\QCoh(\mathcal{O}_X)).\] Since \(Q_X\) is right adjoint to the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) we see that \(RQ_X\) is right adjoint to the canonical functor \(D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)\) by Derived Categories, Lemma 09T5.

In this section we will study the functor \(RQ_X\). In Section 0CQZ we will study the (closely related) right adjoint to the inclusion functor \(D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)\) (when it exists).

Lemma

Let \(f : X \to Y\) be an affine morphism of schemes. Then \(f_*\) defines a derived functor \(f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\). This functor has the property that \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{f_*} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] commutes.

Proof

The functor \(f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)\) is exact, see Cohomology of Schemes, Lemma 01XC. Hence \(f_*\) defines a derived functor \(f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\) by simply applying \(f_*\) to any representative complex, see Derived Categories, Lemma 015F. The diagram commutes by Lemma 0G9R.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(f\) is quasi-compact, quasi-separated, and flat. Then, denoting \[\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\] the right derived functor of \(f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)\) we have \(RQ_Y \circ Rf_* = \Phi \circ RQ_X\).

Proof

We will prove this by showing that \(RQ_Y \circ Rf_*\) and \(\Phi \circ RQ_X\) are right adjoint to the same functor \(D(\QCoh(\mathcal{O}_Y)) \to D(\mathcal{O}_X)\).

Since \(f\) is quasi-compact and quasi-separated, we see that \(f_*\) preserves quasi-coherence, see Schemes, Lemma 01LC. Recall that \(\QCoh(\mathcal{O}_X)\) is a Grothendieck abelian category (Properties, Proposition 077P). Hence any \(K\) in \(D(\QCoh(\mathcal{O}_X))\) can be represented by a K-injective complex \(\mathcal{I}^\bullet\) of \(\QCoh(\mathcal{O}_X)\), see Injectives, Theorem 079P. Then we can define \(\Phi(K) = f_*\mathcal{I}^\bullet\).

Since \(f\) is flat, the functor \(f^*\) is exact. Hence \(f^*\) defines \(f^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)\) and also \(f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))\). The functor \(f^* = Lf^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)\) is left adjoint to \(Rf_* : D(\mathcal{O}_X) \to D(\mathcal{O}_Y)\), see Cohomology, Lemma 079W. Similarly, the functor \(f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))\) is left adjoint to \(\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\) by Derived Categories, Lemma 09T5.

Let \(A\) be an object of \(D(\QCoh(\mathcal{O}_Y))\) and \(E\) an object of \(D(\mathcal{O}_X)\). Then \[\begin{align*} \Hom_{D(\QCoh(\mathcal{O}_Y))}(A, RQ_Y(Rf_*E)) & = \Hom_{D(\mathcal{O}_Y)}(A, Rf_*E) \\ & = \Hom_{D(\mathcal{O}_X)}(f^*A, E) \\ & = \Hom_{D(\QCoh(\mathcal{O}_X))}(f^*A, RQ_X(E)) \\ & = \Hom_{D(\QCoh(\mathcal{O}_Y))}(A, \Phi(RQ_X(E))) \end{align*}\] This implies what we want.

Lemma

Let \(X = \Spec(A)\) be an affine scheme. Then

1. \(Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)\) is the functor which sends \(\mathcal{F}\) to the quasi-coherent \(\mathcal{O}_X\)-module associated to the \(A\)-module \(\Gamma(X, \mathcal{F})\),

2. \(RQ_X : D(\mathcal{O}_X) \to D(\QCoh(\mathcal{O}_X))\) is the functor which sends \(E\) to the complex of quasi-coherent \(\mathcal{O}_X\)-modules associated to the object \(R\Gamma(X, E)\) of \(D(A)\),

3. restricted to \(D_\QCoh(\mathcal{O}_X)\) the functor \(RQ_X\) defines a quasi-inverse to (06VT).

Proof

The functor \(Q_X\) is the functor \[\mathcal{F} \mapsto \widetilde{\Gamma(X, \mathcal{F})}\] by Schemes, Lemma 01I7. This immediately implies (1) and (2). The third assertion follows from (the proof of) Lemma 06Z0.

At this point we are ready to prove a criterion for when the functor \(D(\QCoh(\mathcal{O}_X)) \to D_\QCoh(\mathcal{O}_X)\) is an equivalence.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Suppose that for every affine open \(U \subset X\) the right derived functor \[\Phi : D(\QCoh(\mathcal{O}_U)) \to D(\QCoh(\mathcal{O}_X))\] of the left exact functor \(j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)\) fits into a commutative diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_U)) \ar[d]_\Phi \ar[r]_{i_U} & D_\QCoh(\mathcal{O}_U) \ar[d]^{Rj_*} \\ D(\QCoh(\mathcal{O}_X)) \ar[r]^{i_X} & D_\QCoh(\mathcal{O}_X) }\] Then the functor (06VT) \[D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an equivalence with quasi-inverse given by \(RQ_X\).

Proof

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\) and let \(A\) be an object of \(D(\QCoh(\mathcal{O}_X))\). We have to show that the adjunction maps \[A \to RQ_X(i_X(A)) \quad\text{and}\quad i_X(RQ_X(E)) \to E\] are isomorphisms. Consider the hypothesis \(H_n\): the adjunction maps above are isomorphisms whenever \(E\) and \(i_X(A)\) are supported (Definition 08DA) on a closed subset of \(X\) which is contained in the union of \(n\) affine opens of \(X\). We will prove \(H_n\) by induction on \(n\).

Base case: \(n = 0\). In this case \(E = 0\), hence the map \(i_X(RQ_X(E)) \to E\) is an isomorphism. Similarly \(i_X(A) = 0\). Thus the cohomology sheaves of \(i_X(A)\) are zero. Since the inclusion functor \(\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) is fully faithful and exact, we conclude that the cohomology objects of \(A\) are zero, i.e., \(A = 0\) and \(A \to RQ_X(i_X(A))\) is an isomorphism as well.

Induction step. Suppose that \(E\) and \(i_X(A)\) are supported on a closed subset \(T\) of \(X\) contained in \(U_1 \cup \ldots \cup U_n\) with \(U_i \subset X\) affine open. Set \(U = U_n\). Consider the distinguished triangles \[A \to \Phi(A|_U) \to A' \to A[1] \quad\text{and}\quad E \to Rj_*(E|_U) \to E' \to E[1]\] where \(\Phi\) is as in the statement of the lemma. Note that \(E \to Rj_*(E|_U)\) is a quasi-isomorphism over \(U = U_n\). Since \(i_X \circ \Phi = Rj_* \circ i_U\) by assumption and since \(i_X(A)|_U = i_U(A|_U)\) we see that \(i_X(A) \to i_X(\Phi(A|_U))\) is a quasi-isomorphism over \(U\). Hence \(i_X(A')\) and \(E'\) are supported on the closed subset \(T \setminus U\) of \(X\) which is contained in \(U_1 \cup \ldots \cup U_{n - 1}\). By induction hypothesis the statement is true for \(A'\) and \(E'\). By Derived Categories, Lemma 014A it suffices to prove the maps \[\Phi(A|_U) \to RQ_X(i_X(\Phi(A|_U))) \quad\text{and}\quad i_X(RQ_X(Rj_*E|_U)) \to Rj_*(E|_U)\] are isomorphisms. By assumption and by Lemma 08D8 (the inclusion morphism \(j : U \to X\) is flat, quasi-compact, and quasi-separated) we have \[RQ_X(i_X(\Phi(A|_U))) = RQ_X(Rj_*(i_U(A|_U))) = \Phi(RQ_U(i_U(A|_U)))\] and \[i_X(RQ_X(Rj_*(E|_U))) = i_X(\Phi(RQ_U(E|_U))) = Rj_*(i_U(RQ_U(E|_U)))\] Finally, the maps \[A|_U \to RQ_U(i_U(A|_U)) \quad\text{and}\quad i_U(RQ_U(E|_U)) \to E|_U\] are isomorphisms by Lemma 08D9. The result follows.

Proposition

Let \(X\) be a quasi-compact scheme with affine diagonal. Then the functor (06VT) \[D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an equivalence with quasi-inverse given by \(RQ_X\).

Proof

Let \(U \subset X\) be an affine open. Then the morphism \(U \to X\) is affine by Morphisms, Lemma 01SG. Thus the assumption of Lemma 09T6 holds by Lemma 08D7 and we win.

Lemma

Let \(f : X \to Y\) be a morphism of schemes. Assume \(X\) and \(Y\) are quasi-compact and have affine diagonal. Then, denoting \[\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\] the right derived functor of \(f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)\) the diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_\Phi \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] is commutative.

Proof

Observe that the horizontal arrows in the diagram are equivalences of categories by Proposition 08DB. Hence we can identify these categories (and similarly for other quasi-compact schemes with affine diagonal). The statement of the lemma is that the canonical map \(\Phi(K) \to Rf_*(K)\) is an isomorphism for all \(K\) in \(D(\QCoh(\mathcal{O}_X))\). Note that if \(K_1 \to K_2 \to K_3 \to K_1[1]\) is a distinguished triangle in \(D(\QCoh(\mathcal{O}_X))\) and the statement is true for two-out-of-three, then it is true for the third.

Let \(U \subset X\) be an affine open. Since the diagonal of \(X\) is affine, the inclusion morphism \(j : U \to X\) is affine (Morphisms, Lemma 01SG). Similarly, the composition \(g = f \circ j : U \to Y\) is affine. Let \(\mathcal{I}^\bullet\) be a K-injective complex in \(\QCoh(\mathcal{O}_U)\). Since \(j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)\) has an exact left adjoint \(j^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)\) we see that \(j_*\mathcal{I}^\bullet\) is a K-injective complex in \(\QCoh(\mathcal{O}_X)\), see Derived Categories, Lemma 08BJ. It follows that \[\Phi(j_*\mathcal{I}^\bullet) = f_*j_*\mathcal{I}^\bullet = g_*\mathcal{I}^\bullet\] By Lemma 08D7 we see that \(j_*\mathcal{I}^\bullet\) represents \(Rj_*\mathcal{I}^\bullet\) and \(g_*\mathcal{I}^\bullet\) represents \(Rg_*\mathcal{I}^\bullet\). On the other hand, we have \(Rf_* \circ Rj_* = Rg_*\). Hence \(f_*j_*\mathcal{I}^\bullet\) represents \(Rf_*(j_*\mathcal{I}^\bullet)\). We conclude that the lemma is true for any complex of the form \(j_*\mathcal{G}^\bullet\) with \(\mathcal{G}^\bullet\) a complex of quasi-coherent modules on \(U\). (Note that if \(\mathcal{G}^\bullet \to \mathcal{I}^\bullet\) is a quasi-isomorphism, then \(j_*\mathcal{G}^\bullet \to j_*\mathcal{I}^\bullet\) is a quasi-isomorphism as well since \(j_*\) is an exact functor on quasi-coherent modules.)

Let \(\mathcal{F}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules. Let \(T \subset X\) be a closed subset such that the support of \(\mathcal{F}^p\) is contained in \(T\) for all \(p\). We will use induction on the minimal number \(n\) of affine opens \(U_1, \ldots, U_n\) such that \(T \subset U_1 \cup \ldots \cup U_n\). The base case \(n = 0\) is trivial. If \(n \geq 1\), then set \(U = U_1\) and denote \(j : U \to X\) the open immersion as above. We consider the map of complexes \(c : \mathcal{F}^\bullet \to j_*j^*\mathcal{F}^\bullet\). We obtain two short exact sequences of complexes: \[0 \to \Ker(c) \to \mathcal{F}^\bullet \to \Im(c) \to 0\] and \[0 \to \Im(c) \to j_*j^*\mathcal{F}^\bullet \to \Coker(c) \to 0\] The complexes \(\Ker(c)\) and \(\Coker(c)\) are supported on \(T \setminus U \subset U_2 \cup \ldots \cup U_n\) and the result holds for them by induction. The result holds for \(j_*j^*\mathcal{F}^\bullet\) by the discussion in the preceding paragraph. We conclude by looking at the distinguished triangles associated to the short exact sequences and using the initial remark of the proof.

Remark

Let \(X\) be a quasi-compact scheme with affine diagonal. Even though we know that \(D(\QCoh(\mathcal{O}_X)) = D_\QCoh(\mathcal{O}_X)\) by Proposition 08DB strange things can happen and it is easy to make mistakes with this material. One pitfall is to carelessly assume that this equality means derived functors are the same. For example, suppose we have a quasi-compact open \(U \subset X\). Then we can consider the higher right derived functors \[R^i(\QCoh)\Gamma(U, -) : \QCoh(\mathcal{O}_X) \to \textit{Ab}\] of the left exact functor \(\Gamma(U, -)\). Since this is a universal \(\delta\)-functor, and since the functors \(H^i(U, -)\) (defined for all abelian sheaves on \(X\)) restricted to \(\QCoh(\mathcal{O}_X)\) form a \(\delta\)-functor, we obtain canonical transformations \[t^i : R^i(\QCoh)\Gamma(U, -) \to H^i(U, -).\] These transformations aren’t in general isomorphisms even if \(X = \Spec(A)\) is affine! Namely, we have \(R^1(\QCoh)\Gamma(U, \widetilde{I}) = 0\) if \(I\) an injective \(A\)-module by construction of right derived functors and the equivalence of \(\QCoh(\mathcal{O}_X)\) and \(\text{Mod}_A\). But Examples, Lemma 0CRZ shows there exists \(A\), \(I\), and \(U\) such that \(H^1(U, \widetilde{I}) \not = 0\).

The coherator for Noetherian schemes

In the case of Noetherian schemes we can use the following lemma.

Lemma

Let \(X\) be a Noetherian scheme. Let \(\mathcal{J}\) be an injective object of \(\QCoh(\mathcal{O}_X)\). Then \(\mathcal{J}\) is a flasque sheaf of \(\mathcal{O}_X\)-modules.

Proof

Let \(U \subset X\) be an open subset and let \(s \in \mathcal{J}(U)\) be a section. Let \(\mathcal{I} \subset \mathcal{O}_X\) be the quasi-coherent sheaf of ideals defining the reduced induced scheme structure on \(X \setminus U\) (see Schemes, Definition 01J4). By Cohomology of Schemes, Lemma 01YB the section \(s\) corresponds to a map \(\sigma : \mathcal{I}^n \to \mathcal{J}\) for some \(n\). As \(\mathcal{J}\) is an injective object of \(\QCoh(\mathcal{O}_X)\) we can extend \(\sigma\) to a map \(\tilde s : \mathcal{O}_X \to \mathcal{J}\). Then \(\tilde s\) corresponds to a global section of \(\mathcal{J}\) restricting to \(s\).

Lemma

Let \(f : X \to Y\) be a morphism of Noetherian schemes. Then \(f_*\) on quasi-coherent sheaves has a right derived extension \(\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))\) such that the diagram \[\xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{\Phi} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) }\] commutes.

Proof

Since \(X\) and \(Y\) are Noetherian schemes the morphism is quasi-compact and quasi-separated (see Properties, Lemma 01OY and Schemes, Remark 0816). Thus \(f_*\) preserve quasi-coherence, see Schemes, Lemma 01LC. Next, let \(K\) be an object of \(D(\QCoh(\mathcal{O}_X))\). Since \(\QCoh(\mathcal{O}_X)\) is a Grothendieck abelian category (Properties, Proposition 077P), we can represent \(K\) by a K-injective complex \(\mathcal{I}^\bullet\) such that each \(\mathcal{I}^n\) is an injective object of \(\QCoh(\mathcal{O}_X)\), see Injectives, Theorem 079P. Thus we see that the functor \(\Phi\) is defined by setting \[\Phi(K) = f_*\mathcal{I}^\bullet\] where the right hand side is viewed as an object of \(D(\QCoh(\mathcal{O}_Y))\). To finish the proof of the lemma it suffices to show that the canonical map \[f_*\mathcal{I}^\bullet \longrightarrow Rf_*\mathcal{I}^\bullet\] is an isomorphism in \(D(\mathcal{O}_Y)\). To see this by Lemma 0G9N it suffices to show that \(\mathcal{I}^n\) is right \(f_*\)-acyclic for all \(n \in \mathbf{Z}\). This is true because \(\mathcal{I}^n\) is flasque by Lemma 09T2 and flasque modules are right \(f_*\)-acyclic by Cohomology, Lemma 09T0.

Proposition

Let \(X\) be a Noetherian scheme. Then the functor (06VT) \[D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an equivalence with quasi-inverse given by \(RQ_X\).

Proof

This follows from Lemma 09T6 and Lemma 09T3.

Koszul complexes

Let \(A\) be a ring and let \(f_1, \ldots, f_r\) be a sequence of elements of \(A\). We have defined the Koszul complex \(K_\bullet(f_1, \ldots, f_r)\) in More on Algebra, Definition 0623. It is a chain complex sitting in degrees \(r, \ldots, 0\). We turn this into a cochain complex \(K^\bullet(f_1, \ldots, f_r)\) by setting \(K^{-n}(f_1, \ldots, f_r) = K_n(f_1, \ldots, f_r)\) and using the same differentials. In the rest of this section all the complexes will be cochain complexes.

We define a complex \(I^\bullet(f_1, \ldots, f_r)\) such that we have a distinguished triangle \[I^\bullet(f_1, \ldots, f_r) \to A \to K^\bullet(f_1, \ldots, f_r) \to I^\bullet(f_1, \ldots, f_r)[1]\] in \(K(A)\). In other words, we set \[I^i(f_1, \ldots, f_r) = \left\{ \begin{matrix} K^{i - 1}(f_1, \ldots, f_r) & \text{if } i \leq 0 \\ 0 & \text{else} \end{matrix} \right.\] and we use the negative of the differential on \(K^\bullet(f_1, \ldots, f_r)\). The maps in the distinguished triangle are the obvious ones. Note that \(I^0(f_1, \ldots, f_r) = A^{\oplus r} \to A\) is given by multiplication by \(f_i\) on the \(i\)th factor. Hence \(I^\bullet(f_1, \ldots, f_r) \to A\) factors as \[I^\bullet(f_1, \ldots, f_r) \to I \to A\] where \(I = (f_1, \ldots, f_r)\). In fact, there is a short exact sequence \[0 \to H^{-1}(K^\bullet(f_1, \ldots, f_s)) \to H^0(I^\bullet(f_1, \ldots, f_s)) \to I \to 0\] and for every \(i < 0\) we have \(H^i(I^\bullet(f_1, \ldots, f_r)) = H^{i - 1}(K^\bullet(f_1, \ldots, f_r))\). Observe that given a second sequence \(g_1, \ldots, g_r\) of elements of \(A\) there are canonical maps \[I^\bullet(f_1g_1, \ldots, f_rg_r) \to I^\bullet(f_1, \ldots, f_r) \quad\text{and}\quad K^\bullet(f_1g_1, \ldots, f_rg_r) \to K^\bullet(f_1, \ldots, f_r)\] compatible with the maps described above. The first of these maps is given by multiplication by \(g_i\) on the \(i\)th summand of \(I^0(f_1g_1, \ldots, f_rg_r) = A^{\oplus r}\). In particular, given \(f_1, \ldots, f_r\) we obtain an inverse system of complexes [08CY]\[\begin{equation} I^\bullet(f_1, \ldots, f_r) \leftarrow I^\bullet(f_1^2, \ldots, f_r^2) \leftarrow I^\bullet(f_1^3, \ldots, f_r^3) \leftarrow \ldots \end{equation}\] which will play an important role in that which is to follow. To easily formulate the following lemmas we fix some notation.

Situation

Here \(A\) is a ring and \(f_1, \ldots, f_r\) is a sequence of elements of \(A\). We set \(X = \Spec(A)\) and \(U = D(f_1) \cup \ldots \cup D(f_r) \subset X\). We denote \(\mathcal{U} : U = \bigcup_{i = 1, \ldots, r} D(f_i)\) the given open covering of \(U\).

Our first lemma is that the complexes above can be used to compute the cohomology of quasi-coherent sheaves on \(U\).

Lemma

In Situation 08CZ. Let \(M\) be an \(A\)-module and denote \(\mathcal{F}\) the associated \(\mathcal{O}_X\)-module. Then there is a canonical isomorphism of complexes \[\Psi : \colim_e \Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M) \longrightarrow \check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})\] functorial in \(M\) where the differentials on the \(\Hom\)-complex are the contragredients of the differentials on \(I^\bullet(f_1^e, \ldots, f_r^e)\).

Proof

Recall that the alternating Čech complex is the subcomplex of the usual Čech complex given by alternating cochains, see Cohomology, Section 01FG. As usual we view a \(p\)-cochain in \(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})\) as an alternating function \(s\) on \(\{1, \ldots, r\}^{p + 1}\) whose value \(s_{i_0\ldots i_p}\) at \((i_0, \ldots, i_p)\) lies in \(M_{f_{i_0}\ldots f_{i_p}} = \mathcal{F}(U_{i_0\ldots i_p})\). On the other hand, a \(p\)-cochain \(t\) in \(\Hom^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M)\) is a map \(t : \wedge^{p + 1}(A^{\oplus r}) \to M\). Write \([i] \in A^{\oplus r}\) for the \(i\)th basis element and write \[[i_0, \ldots, i_p] = [i_0] \wedge \ldots \wedge [i_p] \in \wedge^{p + 1}(A^{\oplus r})\] For \(t\) as above we set \[\Psi(t)_{i_0 \ldots i_p} = (-1)^p \frac{t([i_0, \ldots, i_p])}{f_{i_0}^e\ldots f_{i_p}^e}\] It is clear that \(\Psi(t)\) is an alternating cochain. The rule above is compatible with the transition maps of the system as the transition map \[I^\bullet(f_1^e, \ldots, f_r^e) \leftarrow I^\bullet(f_1^{e + 1}, \ldots, f_r^{e + 1}),\] of (08CY) sends \([i_0, \ldots, i_p]\) to \(f_{i_0}\ldots f_{i_p}[i_0, \ldots, i_p]\). It is clear from the description of the localizations \(M_{f_{i_0} \ldots f_{i_p}}\) in Algebra, Lemma 00CR that the rule \(\Psi\) defines an isomorphism of cochain modules in degree \(p\) in the colimit. To finish the proof we have to show that the map is compatible with differentials. To see this, for \(t\) as above we compute \[\begin{align*} d(\Psi(t))_{i_0 \ldots i_{p + 1}} & = \sum\nolimits_{j = 0}^{p + 1} (-1)^j \Psi(t)_{i_0\ldots \hat i_j \ldots i_{p + 1}} \\ & = (-1)^p \sum\nolimits_{j = 0}^{p + 1} (-1)^j t([i_0 \ldots \hat i_j \ldots i_{p + 1}]) (f_{i_0} \ldots \hat f_{i_j} \ldots f_{i_p})^{-e} \end{align*}\] Recall that the differentials on \(I^\bullet(f_1^e, \ldots, f_r^e)\) are the negative of the differentials on \(K^\bullet(f_1, \ldots, f_r)\). Thus \[\begin{align*} \Psi(d(t))_{i_0 \ldots i_{p + 1}} & = (-1)^{p + 1} d(t)([i_0, \ldots, i_{p + 1}]) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = (-1)^{p + 1} t(d([i_0, \ldots, i_{p + 1}])) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = (-1)^{p + 1} t(-\sum\nolimits_{j = 0}^{p + 1} (-1)^j f_{i_j}^e [i_0, \ldots, \hat i_j, \ldots i_{p + 1}]) (f_{i_0} \ldots f_{i_{p + 1}})^{-e} \\ & = -(-1)^{p + 1} \sum\nolimits_{j = 0}^{p + 1} (-1)^j t([i_0, \ldots, \hat i_j, \ldots i_{p + 1}]) (f_{i_0} \ldots \hat f_{i_j} \ldots f_{i_p})^{-e} \end{align*}\] The two formulas agree concluding the proof.

Suppose given a finite complex \(I^\bullet\) of \(A\)-modules and a complex of \(A\)-modules \(M^\bullet\). Then we have the corresponding Hom complex \(\Hom^\bullet(I^\bullet, M^\bullet)\). This is a complex with degree \(n\) term given by \[\bigoplus\nolimits_{p + q = n} \Hom_A(I^{-q}, M^p)\] and differential as described in More on Algebra, Section 0A8H. As our complex \(I^\bullet\) has only finitely many nonzero terms, the direct sum displayed above is finite. The conventions for taking the total complex associated to a Čech complex of a complex are as in Cohomology, Section 01FP.

Lemma

In Situation 08CZ. Let \(M^\bullet\) be a complex of \(A\)-modules and denote \(\mathcal{F}^\bullet\) the associated complex of \(\mathcal{O}_X\)-modules. Then there is a canonical isomorphism of complexes \[\colim_e \Hom^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))\] functorial in \(M^\bullet\).

Proof

Consider the double complex \(F^{\bullet, \bullet}\) with terms \(F^{p, q} = \mathcal{C}_{alt}^p(\mathcal{U}, \mathcal{F}^q)\) discussed in Cohomology, Section 01FP. Consider the double complex \(G^{\bullet, \bullet}\) with terms \(G^{p, q} = \colim_e \Hom_A(I^{-p}(f_1^e, \ldots, f_r^e), M^q)\) and differentials given by functoriality (without the intervention of signs). The maps \(\psi^{p, q} : G^{p, q} \to F^{p, q}\) constructed in the proof of Lemma 08D0 are isomorphisms and compatible with the differentials \(d_1\) (by the lemma) and \(d_2\) (this is clear). However, the differentials \(d\) on the complexes on the left and right hand side of the arrow in the lemma have different signs. Namely, for \(g \in G^{p, q}\) is given by \[d(g) = d_2(g) - (-1)^{p + q} d_1(g)\] (see More on Algebra, Section 0A8H) and the differential for \(f \in F^{p, q}\) is given by \[d(f) = d_1(f) + (-1)^p d_2(f)\] Thus we can fix the signs by multiplying \(\psi^{p, q}\) by \((-1)^{pq + p(p - 1)/2}\).

Lemma

In Situation 08CZ. Let \(\mathcal{F}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules. Then there is a canonical isomorphism \[\text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(U, \mathcal{F}^\bullet)\] in \(D(A)\) functorial in \(\mathcal{F}^\bullet\).

Proof

Let \(\mathcal{B}\) be the set of affine opens of \(U\). Since the higher cohomology groups of a quasi-coherent module on an affine scheme are zero (Cohomology of Schemes, Lemma 01XB) this is a special case of Cohomology, Lemma 08C2.

In Situation 08CZ denote \(I_e\) the object of \(D(\mathcal{O}_X)\) corresponding to the complex of \(A\)-modules \(I^\bullet(f_1^e, \ldots, f_r^e)\) via the equivalence of Lemma 06Z0. The maps (08CY) give a system \[I_1 \leftarrow I_2 \leftarrow I_3 \leftarrow \ldots\] Moreover, there is a compatible system of maps \(I_e \to \mathcal{O}_X\) which become isomorphisms when restricted to \(U\). Thus we see that for every object \(E\) of \(D(\mathcal{O}_X)\) there is a canonical map [08DC]\[\begin{equation} \colim_e \Hom_{D(\mathcal{O}_X)}(I_e, E) \longrightarrow H^0(U, E) \end{equation}\] constructed by sending a map \(I_e \to E\) to its restriction to \(U\) and using that \(\Hom_{D(\mathcal{O}_U)}(\mathcal{O}_U, E|_U) = H^0(U, E)\).

Proposition

In Situation 08CZ. For every object \(E\) of \(D_\QCoh(\mathcal{O}_X)\) the map (08DC) is an isomorphism.

Proof

By Lemma 06Z0 we may assume that \(E\) is given by a complex of quasi-coherent sheaves \(\mathcal{F}^\bullet\). Let \(M^\bullet = \Gamma(X, \mathcal{F}^\bullet)\) be the corresponding complex of \(A\)-modules. By Lemmas 08D1 and 08D2 we have quasi-isomorphisms \[\colim_e \Hom^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(U, \mathcal{F}^\bullet)\] By More on Algebra, Lemma 0A66 and Equation (0A64) taking \(H^0\) of the complex \(\Hom^\bullet(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet)\) computes \(\Hom\) in \(D(A)\). Thus taking \(H^0\) on both sides we obtain \[\colim_e \Hom_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) = H^0(U, E)\] Since \(\Hom_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) = \Hom_{D(\mathcal{O}_X)}(I_e, E)\) by Lemma 06Z0 the lemma follows.

In Situation 08CZ denote \(K_e\) the object of \(D(\mathcal{O}_X)\) corresponding to the complex of \(A\)-modules \(K^\bullet(f_1^e, \ldots, f_r^e)\) via the equivalence of Lemma 06Z0. Thus we have distinguished triangles \[I_e \to \mathcal{O}_X \to K_e \to I_e[1]\] and a system \[K_1 \leftarrow K_2 \leftarrow K_3 \leftarrow \ldots\] compatible with the system \((I_e)\). Moreover, there is a compatible system of maps \[K_e \to H^0(K_e) = \mathcal{O}_X/(f_1^e, \ldots, f_r^e)\]

Lemma

In Situation 08CZ. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Assume that \(H^i(E)|_U = 0\) for \(i = - r + 1, \ldots, 0\). Then given \(s \in H^0(X, E)\) there exists an \(e \geq 0\) and a morphism \(K_e \to E\) such that \(s\) is in the image of \(H^0(X, K_e) \to H^0(X, E)\).

Proof

Since \(U\) is covered by \(r\) affine opens we have \(H^j(U, \mathcal{F}) = 0\) for \(j \geq r\) and any quasi-coherent module (Cohomology of Schemes, Lemma 01XI). By Lemma 08D4 we see that \(H^0(U, E)\) is equal to \(H^0(U, \tau_{\geq -r + 1}E)\). There is a spectral sequence \[H^j(U, H^i(\tau_{\geq -r + 1}E)) \Rightarrow H^{i + j}(U, \tau_{\geq -N}E)\] see Derived Categories, Lemma 015J. Hence \(H^0(U, E) = 0\) by our assumed vanishing of cohomology sheaves of \(E\). We conclude that \(s|_U = 0\). Think of \(s\) as a morphism \(\mathcal{O}_X \to E\) in \(D(\mathcal{O}_X)\). By Proposition 08DD the composition \(I_e \to \mathcal{O}_X \to E\) is zero for some \(e\). By the distinguished triangle \(I_e \to \mathcal{O}_X \to K_e \to I_e[1]\) we obtain a morphism \(K_e \to E\) such that \(s\) is the composition \(\mathcal{O}_X \to K_e \to E\).

Pseudo-coherent and perfect complexes

In this section we make the connection between the general notions defined in Cohomology, Sections 08C3, 08CA, 08CF, and 08CL and the corresponding notions for complexes of modules in More on Algebra, Sections 064N, 0651, and 0656.

Lemma

Let \(X\) be a scheme. If \(E\) is an \(m\)-pseudo-coherent object of \(D(\mathcal{O}_X)\), then \(H^i(E)\) is a quasi-coherent \(\mathcal{O}_X\)-module for \(i > m\) and \(H^m(E)\) is a quotient of a quasi-coherent \(\mathcal{O}_X\)-module. If \(E\) is pseudo-coherent, then \(E\) is an object of \(D_\QCoh(\mathcal{O}_X)\).

Proof

Locally on \(X\) there exists a strictly perfect complex \(\mathcal{E}^\bullet\) such that \(H^i(E)\) is isomorphic to \(H^i(\mathcal{E}^\bullet)\) for \(i > m\) and \(H^m(E)\) is a quotient of \(H^m(\mathcal{E}^\bullet)\). The sheaves \(\mathcal{E}^i\) are direct summands of finite free modules, hence quasi-coherent. The lemma follows.

Lemma

Let \(X = \Spec(A)\) be an affine scheme. Let \(M^\bullet\) be a complex of \(A\)-modules and let \(E\) be the corresponding object of \(D(\mathcal{O}_X)\). Then \(E\) is an \(m\)-pseudo-coherent (resp. pseudo-coherent) as an object of \(D(\mathcal{O}_X)\) if and only if \(M^\bullet\) is \(m\)-pseudo-coherent (resp. pseudo-coherent) as a complex of \(A\)-modules.

Proof

It is immediate from the definitions that if \(M^\bullet\) is \(m\)-pseudo-coherent, so is \(E\). To prove the converse, assume \(E\) is \(m\)-pseudo-coherent. As \(X = \Spec(A)\) is quasi-compact with a basis for the topology given by standard opens, we can find a standard open covering \(X = D(f_1) \cup \ldots \cup D(f_n)\) and strictly perfect complexes \(\mathcal{E}_i^\bullet\) on \(D(f_i)\) and maps \(\alpha_i : \mathcal{E}_i^\bullet \to E|_{U_i}\) inducing isomorphisms on \(H^j\) for \(j > m\) and surjections on \(H^m\). By Cohomology, Lemma 08C9 after refining the open covering we may assume \(\alpha_i\) is given by a map of complexes \(\mathcal{E}_i^\bullet \to \widetilde{M^\bullet}|_{U_i}\) for each \(i\). By Modules, Lemma 0BCI the terms \(\mathcal{E}_i^n\) are finite locally free modules. Hence after refining the open covering we may assume each \(\mathcal{E}_i^n\) is a finite free \(\mathcal{O}_{U_i}\)-module. From the definition it follows that \(M^\bullet_{f_i}\) is an \(m\)-pseudo-coherent complex of \(A_{f_i}\)-modules. We conclude by applying More on Algebra, Lemma 066D.

The case “pseudo-coherent” follows from the fact that \(E\) is pseudo-coherent if and only if \(E\) is \(m\)-pseudo-coherent for all \(m\) (by definition) and the same is true for \(M^\bullet\) by More on Algebra, Lemma 064U.

Lemma

Let \(X\) be a Noetherian scheme. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). For \(m \in \mathbf{Z}\) the following are equivalent

1. \(H^i(E)\) is coherent for \(i \geq m\) and zero for \(i \gg 0\), and

2. \(E\) is \(m\)-pseudo-coherent.

In particular, \(E\) is pseudo-coherent if and only if \(E\) is an object of \(D^-_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

As \(X\) is quasi-compact we see that in both (1) and (2) the object \(E\) is bounded above. Thus the question is local on \(X\) and we may assume \(X\) is affine. Say \(X = \Spec(A)\) for some Noetherian ring \(A\). In this case \(E\) corresponds to a complex of \(A\)-modules \(M^\bullet\) by Lemma 06Z0. By Lemma 08E7 we see that \(E\) is \(m\)-pseudo-coherent if and only if \(M^\bullet\) is \(m\)-pseudo-coherent. On the other hand, \(H^i(E)\) is coherent if and only if \(H^i(M^\bullet)\) is a finite \(A\)-module (Properties, Lemma 01PB). Thus the result follows from More on Algebra, Lemma 066E.

Lemma

Let \(X = \Spec(A)\) be an affine scheme. Let \(M^\bullet\) be a complex of \(A\)-modules and let \(E\) be the corresponding object of \(D(\mathcal{O}_X)\). Then

1. \(E\) has tor amplitude in \([a, b]\) if and only if \(M^\bullet\) has tor amplitude in \([a, b]\).

2. \(E\) has finite tor dimension if and only if \(M^\bullet\) has finite tor dimension.

Proof

Part (2) follows trivially from part (1). In the proof of (1) we will use the equivalence \(D(A) = D_\QCoh(X)\) of Lemma 06Z0 without further mention. Assume \(M^\bullet\) has tor amplitude in \([a, b]\). Then \(K^\bullet\) is isomorphic in \(D(A)\) to a complex \(K^\bullet\) of flat \(A\)-modules with \(K^i = 0\) for \(i \not \in [a, b]\), see More on Algebra, Lemma 0654. Then \(E\) is isomorphic to \(\widetilde{K^\bullet}\). Since each \(\widetilde{K^i}\) is a flat \(\mathcal{O}_X\)-module, we see that \(E\) has tor amplitude in \([a, b]\) by Cohomology, Lemma 08CI.

Assume that \(E\) has tor amplitude in \([a, b]\). Then \(E\) is bounded whence \(M^\bullet\) is in \(K^-(A)\). Thus we may replace \(M^\bullet\) by a bounded above complex of \(A\)-modules. We may even choose a projective resolution and assume that \(M^\bullet\) is a bounded above complex of free \(A\)-modules. Then for any \(A\)-module \(N\) we have \[E \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{N} \cong \widetilde{M^\bullet} \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{N} \cong \widetilde{M^\bullet \otimes_A N}\] in \(D(\mathcal{O}_X)\). Thus the vanishing of cohomology sheaves of the left hand side implies \(M^\bullet\) has tor amplitude in \([a, b]\).

Lemma

Let \(f : X \to S\) be a morphism of affine schemes corresponding to the ring map \(R \to A\). Let \(M^\bullet\) be a complex of \(A\)-modules and let \(E\) be the corresponding object of \(D(\mathcal{O}_X)\). Then

1. \(E\) as an object of \(D(f^{-1}\mathcal{O}_S)\) has tor amplitude in \([a, b]\) if and only if \(M^\bullet\) has tor amplitude in \([a, b]\) as an object of \(D(R)\).

2. \(E\) locally has finite tor dimension as an object of \(D(f^{-1}\mathcal{O}_S)\) if and only if \(M^\bullet\) has finite tor dimension as an object of \(D(R)\).

Proof

Consider a prime \(\mathfrak q \subset A\) lying over \(\mathfrak p \subset R\). Let \(x \in X\) and \(s = f(x) \in S\) be the corresponding points. Then \((f^{-1}\mathcal{O}_S)_x = \mathcal{O}_{S, s} = R_\mathfrak p\) and \(E_x = M^\bullet_\mathfrak q\). Keeping this in mind we can see the equivalence as follows.

If \(M^\bullet\) has tor amplitude in \([a, b]\) as a complex of \(R\)-modules, then the same is true for the localization of \(M^\bullet\) at any prime of \(A\). Then we conclude by Cohomology, Lemma 09U9 that \(E\) has tor amplitude in \([a, b]\) as a complex of sheaves of \(f^{-1}\mathcal{O}_S\)-modules. Conversely, assume that \(E\) has tor amplitude in \([a, b]\) as an object of \(D(f^{-1}\mathcal{O}_S)\). We conclude (using the last cited lemma) that \(M^\bullet_\mathfrak q\) has tor amplitude in \([a, b]\) as a complex of \(R_\mathfrak p\)-modules for every prime \(\mathfrak q \subset A\) lying over \(\mathfrak p \subset R\). By More on Algebra, Lemma 0B67 we find that \(M^\bullet\) has tor amplitude in \([a, b]\) as a complex of \(R\)-modules. This finishes the proof of (1).

Since \(X\) is quasi-compact, if \(E\) locally has finite tor dimension as a complex of \(f^{-1}\mathcal{O}_S\)-modules, then actually \(E\) has tor amplitude in \([a, b]\) for some \(a, b\) as a complex of \(f^{-1}\mathcal{O}_S\)-modules. Thus (2) follows from (1).

Lemma

Let \(X\) be a quasi-separated scheme. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Let \(a \leq b\). The following are equivalent

1. \(E\) has tor amplitude in \([a, b]\), and

2. for all \(\mathcal{F}\) in \(\QCoh(\mathcal{O}_X)\) we have \(H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}) = 0\) for \(i \not \in [a, b]\).

Proof

It is clear that (1) implies (2). Assume (2). Let \(U \subset X\) be an affine open. As \(X\) is quasi-separated the morphism \(j : U \to X\) is quasi-compact and separated, hence \(j_*\) transforms quasi-coherent modules into quasi-coherent modules (Schemes, Lemma 01LC). Thus the functor \(\QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)\) is essentially surjective. It follows that condition (2) implies the vanishing of \(H^i(E|_U \otimes_{\mathcal{O}_U}^\mathbf{L} \mathcal{G})\) for \(i \not \in [a, b]\) for all quasi-coherent \(\mathcal{O}_U\)-modules \(\mathcal{G}\). Write \(U = \Spec(A)\) and let \(M^\bullet\) be the complex of \(A\)-modules corresponding to \(E|_U\) by Lemma 06Z0. We have just shown that \(M^\bullet \otimes_A^\mathbf{L} N\) has vanishing cohomology groups outside the range \([a, b]\), in other words \(M^\bullet\) has tor amplitude in \([a, b]\). By Lemma 08E9 we conclude that \(E|_U\) has tor amplitude in \([a, b]\). This proves the lemma.

Lemma

Let \(X = \Spec(A)\) be an affine scheme. Let \(M^\bullet\) be a complex of \(A\)-modules and let \(E\) be the corresponding object of \(D(\mathcal{O}_X)\). Then \(E\) is a perfect object of \(D(\mathcal{O}_X)\) if and only if \(M^\bullet\) is perfect as an object of \(D(A)\).

Proof

This is a logical consequence of Lemmas 08E7 and 08E9, Cohomology, Lemma 08CQ, and More on Algebra, Lemma 0658.

As a consequence of our description of pseudo-coherent complexes on schemes we can prove certain internal homs are quasi-coherent.

Lemma

Let \(X\) be a scheme.

1. If \(L\) is in \(D^+_\QCoh(\mathcal{O}_X)\) and \(K\) in \(D(\mathcal{O}_X)\) is pseudo-coherent, then \(R\SheafHom(K, L)\) is in \(D_\QCoh(\mathcal{O}_X)\) and locally bounded below.

2. If \(L\) is in \(D_\QCoh(\mathcal{O}_X)\) and \(K\) in \(D(\mathcal{O}_X)\) is perfect, then \(R\SheafHom(K, L)\) is in \(D_\QCoh(\mathcal{O}_X)\).

3. If \(X = \Spec(A)\) is affine and \(K, L \in D(A)\) then \[R\SheafHom(\widetilde{K}, \widetilde{L}) = \widetilde{R\Hom_A(K, L)}\] in the following two cases

   1. \(K\) is pseudo-coherent and \(L\) is bounded below,

   2. \(K\) is perfect and \(L\) arbitrary.

4. If \(X = \Spec(A)\) and \(K, L\) are in \(D(A)\), then the \(n\)th cohomology sheaf of \(R\SheafHom(\widetilde{K}, \widetilde{L})\) is the sheaf associated to the presheaf \[X \supset D(f) \longmapsto \Ext^n_{A_f}(K \otimes_A A_f, L \otimes_A A_f)\] for \(f \in A\).

Proof

The construction of the internal hom in the derived category of \(\mathcal{O}_X\) commutes with localization (see Cohomology, Section 08DH). Hence to prove (1) and (2) we may replace \(X\) by an affine open. By Lemmas 06Z0, 08E7, and 08EB in order to prove (1) and (2) it suffices to prove (3).

Part (3) follows from the computation of the internal hom of Cohomology, Lemma 08I5 by representing \(K\) by a bounded above (resp. finite) complex of finite projective \(A\)-modules and \(L\) by a bounded below (resp. arbitrary) complex of \(A\)-modules.

To prove (4) recall that on any ringed space the \(n\)th cohomology sheaf of \(R\SheafHom(A, B)\) is the sheaf associated to the presheaf \[U \mapsto \Hom_{D(U)}(A|_U, B|_U[n]) = \Ext^n_{D(\mathcal{O}_U)}(A|_U, B|_U)\] See Cohomology, Section 08DH. On the other hand, the restriction of \(\widetilde{K}\) to a principal open \(D(f)\) is the image of \(K \otimes_A A_f\) and similarly for \(L\). Hence (4) follows from the equivalence of categories of Lemma 06Z0.

Lemma

Let \(X\) be a scheme. Let \(K, L, M\) be objects of \(D_\QCoh(\mathcal{O}_X)\). The map \[K \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] of Cohomology, Lemma 0BYS is an isomorphism in the following cases

1. \(M\) perfect, or

2. \(K\) is perfect, or

3. \(M\) is pseudo-coherent, \(L \in D^+(\mathcal{O}_X)\), and \(K\) has finite tor dimension.

Proof

Lemma 0A6H reduces cases (1) and (3) to the affine case which is treated in More on Algebra, Lemma 0ATK. (You also have to use Lemmas 08E7, 08EB, and 08E9 to do the translation into algebra.) If \(K\) is perfect but no other assumptions are made, then we do not know that either side of the arrow is in \(D_\QCoh(\mathcal{O}_X)\) but the result is still true because we can work locally and reduce to the case that \(K\) is a finite complex of finite free modules in which case it is clear.

Derived category of coherent modules

Let \(X\) be a locally Noetherian scheme. In this case the category \(\textit{Coh}(\mathcal{O}_X) \subset \textit{Mod}(\mathcal{O}_X)\) of coherent \(\mathcal{O}_X\)-modules is a weak Serre subcategory, see Homology, Section 02MN and Cohomology of Schemes, Lemma 01Y0. Denote \[D_{\textit{Coh}}(\mathcal{O}_X) \subset D(\mathcal{O}_X)\] the subcategory of complexes whose cohomology sheaves are coherent, see Derived Categories, Section 06UP. Thus we obtain a canonical functor [08E1]\[\begin{equation} D(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_X) \end{equation}\] see Derived Categories, Equation (06UR).

Lemma

Let \(X\) be a Noetherian scheme. Then the functor \[D^-(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^-_{\textit{Coh}(\mathcal{O}_X)}(\QCoh(\mathcal{O}_X))\] is an equivalence.

Proof

Observe that \(\textit{Coh}(\mathcal{O}_X) \subset \QCoh(\mathcal{O}_X)\) is a Serre subcategory, see Homology, Definition 02MO and Lemma 02MP and Cohomology of Schemes, Lemmas 01Y0 and 01Y1. On the other hand, if \(\mathcal{G} \to \mathcal{F}\) is a surjection from a quasi-coherent \(\mathcal{O}_X\)-module to a coherent \(\mathcal{O}_X\)-module, then there exists a coherent submodule \(\mathcal{G}' \subset \mathcal{G}\) which surjects onto \(\mathcal{F}\). Namely, we can write \(\mathcal{G}\) as the filtered union of its coherent submodules by Properties, Lemma 01PG and then one of these will do the job. Thus the lemma follows from Derived Categories, Lemma 0FCL.

Proposition

Let \(X\) be a Noetherian scheme. Then the functors \[D^-(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^-_{\textit{Coh}}(\mathcal{O}_X) \quad\text{and}\quad D^b(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_X)\] are equivalences.

Proof

Consider the commutative diagram \[\xymatrix{ D^-(\textit{Coh}(\mathcal{O}_X)) \ar[r] \ar[d] & D^-_{\textit{Coh}}(\mathcal{O}_X) \ar[d] \\ D^-(\QCoh(\mathcal{O}_X)) \ar[r] & D^-_\QCoh(\mathcal{O}_X) }\] By Lemma 0FDA the left vertical arrow is fully faithful. By Proposition 09T4 the bottom arrow is an equivalence. By construction the right vertical arrow is fully faithful. We conclude that the top horizontal arrow is fully faithful. If \(K\) is an object of \(D^-_{\textit{Coh}}(\mathcal{O}_X)\) then the object \(K'\) of \(D^-(\QCoh(\mathcal{O}_X))\) which corresponds to it by Proposition 09T4 will have coherent cohomology sheaves. Hence \(K'\) is in the essential image of the left vertical arrow by Lemma 0FDA and we find that the top horizontal arrow is essentially surjective. This finishes the proof for the bounded above case. The bounded case follows immediately from the bounded above case.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) such that the support of \(H^i(E)\) is proper over \(S\) for all \(i\). Then \(Rf_*E\) is an object of \(D^b_{\textit{Coh}}(\mathcal{O}_S)\).

Proof

Consider the spectral sequence \[R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E\] see Derived Categories, Lemma 015J. By assumption and Cohomology of Schemes, Lemma 08DS the sheaves \(R^pf_*H^q(E)\) are coherent. Hence \(R^{p + q}f_*E\) is coherent, i.e., \(Rf_*E \in D_{\textit{Coh}}(\mathcal{O}_S)\). Boundedness from below is trivial. Boundedness from above follows from Cohomology of Schemes, Lemma 01XJ or from Lemma 08D5.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E\) be an object of \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) such that the support of \(H^i(E)\) is proper over \(S\) for all \(i\). Then \(Rf_*E\) is an object of \(D^+_{\textit{Coh}}(\mathcal{O}_S)\).

Proof

The proof is the same as the proof of Lemma 08E2. You can also deduce it from Lemma 08E2 by considering what the exact functor \(Rf_*\) does to the distinguished triangles \(\tau_{\leq a}E \to E \to \tau_{\geq a + 1}E \to \tau_{\leq a}E[1]\).

Lemma

Let \(X\) be a locally Noetherian scheme. If \(L\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) and \(K\) in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\), then \(R\SheafHom(K, L)\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

It suffices to prove this when \(X\) is the spectrum of a Noetherian ring \(A\). By Lemma 08E8 we see that \(K\) is pseudo-coherent. Then we can use Lemma 0A6H to translate the problem into the following algebra problem: for \(L \in D^+_{\textit{Coh}}(A)\) and \(K\) in \(D^-_{\textit{Coh}}(A)\), then \(R\Hom_A(K, L)\) is in \(D^+_{\textit{Coh}}(A)\). Since \(L\) is bounded below and \(K\) is bounded above there is a convergent spectral sequence \[\Ext^p_A(K, H^q(L)) \Rightarrow \text{Ext}^{p + q}_A(K, L)\] and there are convergent spectral sequences \[\Ext^i_A(H^{-j}(K), H^q(L)) \Rightarrow \text{Ext}^{i + j}_A(K, H^q(L))\] See Injectives, Remarks 0G1Y and 0G20. This finishes the proof as the modules \(\Ext^p_A(M, N)\) are finite for finite \(A\)-modules \(M\), \(N\) by Algebra, Lemma 08YR.

Lemma

Let \(X\) be a Noetherian scheme. Let \(E\) in \(D(\mathcal{O}_X)\) be perfect. Then

1. \(E\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\),

2. if \(L\) is in \(D_{\textit{Coh}}(\mathcal{O}_X)\) then \(E \otimes_{\mathcal{O}_X}^\mathbf{L} L\) and \(R\SheafHom_{\mathcal{O}_X}(E, L)\) are in \(D_{\textit{Coh}}(\mathcal{O}_X)\),

3. if \(L\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) then \(E \otimes_{\mathcal{O}_X}^\mathbf{L} L\) and \(R\SheafHom_{\mathcal{O}_X}(E, L)\) are in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\),

4. if \(L\) is in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) then \(E \otimes_{\mathcal{O}_X}^\mathbf{L} L\) and \(R\SheafHom_{\mathcal{O}_X}(E, L)\) are in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\),

5. if \(L\) is in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\) then \(E \otimes_{\mathcal{O}_X}^\mathbf{L} L\) and \(R\SheafHom_{\mathcal{O}_X}(E, L)\) are in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

Since \(X\) is quasi-compact, each of these statements can be checked over the members of any open covering of \(X\). Thus we may assume \(E\) is represented by a bounded complex \(\mathcal{E}^\bullet\) of finite free modules, see Cohomology, Lemma 0BCJ. In this case each of the statements is clear as both \(R\SheafHom_{\mathcal{O}_X}(E, L)\) and \(E \otimes_{\mathcal{O}_X}^\mathbf{L} L\) can be computed on the level of complexes using \(\mathcal{E}^\bullet\), see Cohomology, Lemmas 08DM and 06YD. Some details omitted.

Lemma

Let \(A\) be a Noetherian ring. Let \(X\) be a proper scheme over \(A\). For \(L\) in \(D^+_{\textit{Coh}}(\mathcal{O}_X)\) and \(K\) in \(D^-_{\textit{Coh}}(\mathcal{O}_X)\), the \(A\)-modules \(\Ext_{\mathcal{O}_X}^n(K, L)\) are finite.

Proof

Recall that \[\Ext_{\mathcal{O}_X}^n(K, L) = H^n(X, R\SheafHom_{\mathcal{O}_X}(K, L)) = H^n(\Spec(A), Rf_*R\SheafHom_{\mathcal{O}_X}(K, L))\] see Cohomology, Lemma 08DK and Cohomology, Section 01EY. Thus the result follows from Lemmas 0D0C and 0D0B.

Lemma

Let \(X\) be a regular scheme. Then every object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\) is perfect. If \(X\) is quasi-compact, i.e., Noetherian and regular, then conversely every perfect object of \(D(\mathcal{O}_X)\) is in \(D^b_{\textit{Coh}}(\mathcal{O}_X)\).

Proof

Let \(K\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_X)\). To check that \(K\) is perfect, we may work affine locally on \(X\) (see Cohomology, Section 08CL). Then \(K\) is perfect by Lemma 08EB and More on Algebra, Lemma 066Z. The converse is Lemma 0FXU.

Descent finiteness properties of complexes

This section is the analogue of Descent, Section 05AY for objects of the derived category of a scheme. The easiest such result is probably the following.

Lemma

Let \(f : X \to Y\) be a surjective flat morphism of schemes (or more generally locally ringed spaces). Let \(E \in D(\mathcal{O}_Y)\). Let \(a, b \in \mathbf{Z}\). Then \(E\) has tor-amplitude in \([a, b]\) if and only if \(Lf^*E\) has tor-amplitude in \([a, b]\).

Proof

Pullback always preserves tor-amplitude, see Cohomology, Lemma 09U8. We may check tor-amplitude in \([a, b]\) on stalks, see Cohomology, Lemma 09U9. A flat local ring homomorphism is faithfully flat by Algebra, Lemma 00HR. Thus the result follows from More on Algebra, Lemma 068S.

Lemma

Let \(\{f_i : X_i \to X\}\) be an fpqc covering of schemes. Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Then \(E\) is \(m\)-pseudo-coherent if and only if each \(Lf_i^*E\) is \(m\)-pseudo-coherent.

Proof

Pullback always preserves \(m\)-pseudo-coherence, see Cohomology, Lemma 09U7. Conversely, assume that \(Lf_i^*E\) is \(m\)-pseudo-coherent for all \(i\). Let \(U \subset X\) be an affine open. It suffices to prove that \(E|_U\) is \(m\)-pseudo-coherent. Since \(\{f_i : X_i \to X\}\) is an fpqc covering, we can find finitely many affine open \(V_j \subset X_{a(j)}\) such that \(f_{a(j)}(V_j) \subset U\) and \(U = \bigcup f_{a(j)}(V_j)\). Set \(V = \coprod V_i\). Thus we may replace \(X\) by \(U\) and \(\{f_i : X_i \to X\}\) by \(\{V \to U\}\) and assume that \(X\) is affine and our covering is given by a single surjective flat morphism \(\{f : Y \to X\}\) of affine schemes. In this case the result follows from More on Algebra, Lemma 068R via Lemmas 06Z0 and 08E7.

Lemma

Let \(\{f_i : X_i \to X\}\) be an fppf covering of schemes. Let \(E \in D(\mathcal{O}_X)\). Let \(m \in \mathbf{Z}\). Then \(E\) is \(m\)-pseudo-coherent if and only if each \(Lf_i^*E\) is \(m\)-pseudo-coherent.

Proof

Pullback always preserves \(m\)-pseudo-coherence, see Cohomology, Lemma 09U7. Conversely, assume that \(Lf_i^*E\) is \(m\)-pseudo-coherent for all \(i\). Let \(U \subset X\) be an affine open. It suffices to prove that \(E|_U\) is \(m\)-pseudo-coherent. Since \(\{f_i : X_i \to X\}\) is an fppf covering, we can find finitely many affine open \(V_j \subset X_{a(j)}\) such that \(f_{a(j)}(V_j) \subset U\) and \(U = \bigcup f_{a(j)}(V_j)\). Set \(V = \coprod V_i\). Thus we may replace \(X\) by \(U\) and \(\{f_i : X_i \to X\}\) by \(\{V \to U\}\) and assume that \(X\) is affine and our covering is given by a single surjective flat morphism \(\{f : Y \to X\}\) of finite presentation.

Since \(f\) is flat the derived functor \(Lf^*\) is just given by \(f^*\) and \(f^*\) is exact. Hence \(H^i(Lf^*E) = f^*H^i(E)\). Since \(Lf^*E\) is \(m\)-pseudo-coherent, we see that \(Lf^*E \in D^-(\mathcal{O}_Y)\). Since \(f\) is surjective and flat, we see that \(E \in D^-(\mathcal{O}_X)\). Let \(i \in \mathbf{Z}\) be the largest integer such that \(H^i(E)\) is nonzero. If \(i < m\), then we are done. Otherwise, \(f^*H^i(E)\) is a finite type \(\mathcal{O}_Y\)-module by Cohomology, Lemma 08DN. Then by Descent, Lemma 09UB the \(\mathcal{O}_X\)-module \(H^i(E)\) is of finite type. Thus, after replacing \(X\) by the members of a finite affine open covering, we may assume there exists a map \[\alpha : \mathcal{O}_X^{\oplus n}[-i] \longrightarrow E\] such that \(H^i(\alpha)\) is a surjection. Let \(C\) be the cone of \(\alpha\) in \(D(\mathcal{O}_X)\). Pulling back to \(Y\) and using Cohomology, Lemma 08CD we find that \(Lf^*C\) is \(m\)-pseudo-coherent. Moreover \(H^j(C) = 0\) for \(j \geq i\). Thus by induction on \(i\) we see that \(C\) is \(m\)-pseudo-coherent. Using Cohomology, Lemma 08CD again we conclude.

Lemma

Let \(\{f_i : X_i \to X\}\) be an fpqc covering of schemes. Let \(E \in D(\mathcal{O}_X)\). Then \(E\) is perfect if and only if each \(Lf_i^*E\) is perfect.

Proof

Pullback always preserves perfect complexes, see Cohomology, Lemma 09UA. Conversely, assume that \(Lf_i^*E\) is perfect for all \(i\). Then the cohomology sheaves of each \(Lf_i^*E\) are quasi-coherent, see Lemma 08E5 and Cohomology, Lemma 08CQ. Since the morphisms \(f_i\) is flat we see that \(H^p(Lf_i^*E) = f_i^*H^p(E)\). Thus the cohomology sheaves of \(E\) are quasi-coherent by Descent, Proposition 023T. Having said this the lemma follows formally from Cohomology, Lemma 08CQ and Lemmas 09UD and 09UE.

Lemma

Let \(i : Z \to X\) be a morphism of ringed spaces such that \(i\) is a closed immersion of underlying topological spaces and such that \(i_*\mathcal{O}_Z\) is pseudo-coherent as an \(\mathcal{O}_X\)-module. Let \(E \in D(\mathcal{O}_Z)\). Then \(E\) is \(m\)-pseudo-coherent if and only if \(Ri_*E\) is \(m\)-pseudo-coherent.

Proof

Throughout this proof we will use that \(i_*\) is an exact functor, and hence that \(Ri_* = i_*\), see Modules, Lemma 01AX.

Assume \(E\) is \(m\)-pseudo-coherent. Let \(x \in X\). We will find a neighbourhood of \(x\) such that \(i_*E\) is \(m\)-pseudo-coherent on it. If \(x \not \in Z\) then this is clear. Thus we may assume \(x \in Z\). We will use that \(U \cap Z\) for \(x \in U \subset X\) open form a fundamental system of neighbourhoods of \(x\) in \(Z\). After shrinking \(X\) we may assume \(E\) is bounded above. We will argue by induction on the largest integer \(p\) such that \(H^p(E)\) is nonzero. If \(p < m\), then there is nothing to prove. If \(p \geq m\), then \(H^p(E)\) is an \(\mathcal{O}_Z\)-module of finite type, see Cohomology, Lemma 08DN. Thus we may choose, after shrinking \(X\), a map \(\mathcal{O}_Z^{\oplus n}[-p] \to E\) which induces a surjection \(\mathcal{O}_Z^{\oplus n} \to H^p(E)\). Choose a distinguished triangle \[\mathcal{O}_Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_Z^{\oplus n}[-p + 1]\] We see that \(H^j(C) = 0\) for \(j \geq p\) and that \(C\) is \(m\)-pseudo-coherent by Cohomology, Lemma 08CD. By induction we see that \(i_*C\) is \(m\)-pseudo-coherent on \(X\). Since \(i_*\mathcal{O}_Z\) is \(m\)-pseudo-coherent on \(X\) as well, we conclude from the distinguished triangle \[i_*\mathcal{O}_Z^{\oplus n}[-p] \to i_*E \to i_*C \to i_*\mathcal{O}_Z^{\oplus n}[-p + 1]\] and Cohomology, Lemma 08CD that \(i_*E\) is \(m\)-pseudo-coherent.

Assume that \(i_*E\) is \(m\)-pseudo-coherent. Let \(z \in Z\). We will find a neighbourhood of \(z\) such that \(E\) is \(m\)-pseudo-coherent on it. We will use that \(U \cap Z\) for \(z \in U \subset X\) open form a fundamental system of neighbourhoods of \(z\) in \(Z\). After shrinking \(X\) we may assume \(i_*E\) and hence \(E\) is bounded above. We will argue by induction on the largest integer \(p\) such that \(H^p(E)\) is nonzero. If \(p < m\), then there is nothing to prove. If \(p \geq m\), then \(H^p(i_*E) = i_*H^p(E)\) is an \(\mathcal{O}_X\)-module of finite type, see Cohomology, Lemma 08DN. Choose a complex \(\mathcal{E}^\bullet\) of \(\mathcal{O}_Z\)-modules representing \(E\). We may choose, after shrinking \(X\), a map \(\alpha : \mathcal{O}_X^{\oplus n}[-p] \to i_*\mathcal{E}^\bullet\) which induces a surjection \(\mathcal{O}_X^{\oplus n} \to i_*H^p(\mathcal{E}^\bullet)\). By adjunction we find a map \(\alpha : \mathcal{O}_Z^{\oplus n}[-p] \to \mathcal{E}^\bullet\) which induces a surjection \(\mathcal{O}_Z^{\oplus n} \to H^p(\mathcal{E}^\bullet)\). Choose a distinguished triangle \[\mathcal{O}_Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_Z^{\oplus n}[-p + 1]\] We see that \(H^j(C) = 0\) for \(j \geq p\). From the distinguished triangle \[i_*\mathcal{O}_Z^{\oplus n}[-p] \to i_*E \to i_*C \to i_*\mathcal{O}_Z^{\oplus n}[-p + 1]\] the fact that \(i_*\mathcal{O}_Z\) is pseudo-coherent and Cohomology, Lemma 08CD we conclude that \(i_*C\) is \(m\)-pseudo-coherent. By induction we conclude that \(C\) is \(m\)-pseudo-coherent. By Cohomology, Lemma 08CD again we conclude that \(E\) is \(m\)-pseudo-coherent.

Lemma

Let \(f : X \to Y\) be a finite morphism of schemes such that \(f_*\mathcal{O}_X\) is pseudo-coherent as an \(\mathcal{O}_Y\)-module². Let \(E \in D_\QCoh(\mathcal{O}_X)\). Then \(E\) is \(m\)-pseudo-coherent if and only if \(Rf_*E\) is \(m\)-pseudo-coherent.

Proof

This is a translation of More on Algebra, Lemma 064Z into the language of schemes. To do the translation, use Lemmas 06Z0 and 08E7.

Lifting complexes

Let \(U \subset X\) be an open subspace of a ringed space and denote \(j : U \to X\) the inclusion morphism. The functor \(D(\mathcal{O}_X) \to D(\mathcal{O}_U)\) is essentially surjective as \(Rj_*\) is a right inverse to restriction. In this section we extend this to complexes with quasi-coherent cohomology sheaves, etc.

Lemma

Let \(X\) be a scheme and let \(j : U \to X\) be a quasi-compact open immersion. The functors \[D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_U) \quad\text{and}\quad D^+_\QCoh(\mathcal{O}_X) \to D^+_\QCoh(\mathcal{O}_U)\] are essentially surjective. If \(X\) is quasi-compact, then the functors \[D^-_\QCoh(\mathcal{O}_X) \to D^-_\QCoh(\mathcal{O}_U) \quad\text{and}\quad D^b_\QCoh(\mathcal{O}_X) \to D^b_\QCoh(\mathcal{O}_U)\] are essentially surjective.

Proof

The argument preceding the lemma applies for the first case because \(Rj_*\) maps \(D_\QCoh(\mathcal{O}_U)\) into \(D_\QCoh(\mathcal{O}_X)\) by Lemma 08D5. It is clear that \(Rj_*\) maps \(D^+_\QCoh(\mathcal{O}_U)\) into \(D^+_\QCoh(\mathcal{O}_X)\) which implies the statement on bounded below complexes. Finally, Lemma 08D5 guarantees that \(Rj_*\) maps \(D^-_\QCoh(\mathcal{O}_U)\) into \(D^-_\QCoh(\mathcal{O}_X)\) if \(X\) is quasi-compact. Combining these two we obtain the last statement.

Lemma

Let \(X\) be a Noetherian scheme and let \(j : U \to X\) be an open immersion. The functor \(D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_U)\) is essentially surjective.

Proof

Let \(K\) be an object of \(D^b_{\textit{Coh}}(\mathcal{O}_U)\). By Proposition 0FDB we can represent \(K\) by a bounded complex \(\mathcal{F}^\bullet\) of coherent \(\mathcal{O}_U\)-modules. Say \(\mathcal{F}^i = 0\) for \(i \not \in [a, b]\) for some \(a \leq b\). Since \(j\) is quasi-compact and separated, the terms of the bounded complex \(j_*\mathcal{F}^\bullet\) are quasi-coherent modules on \(X\), see Schemes, Lemma 01LC. We inductively pick a coherent submodule \(\mathcal{G}^i \subset j_*\mathcal{F}^i\) as follows. For \(i = a\) we pick any coherent submodule \(\mathcal{G}^a \subset j_*\mathcal{F}^a\) whose restriction to \(U\) is \(\mathcal{F}^a\). This is possible by Properties, Lemma 01PF. For \(i > a\) we first pick any coherent submodule \(\mathcal{H}^i \subset j_*\mathcal{F}^i\) whose restriction to \(U\) is \(\mathcal{F}^i\) and then we set \(\mathcal{G}^i = \Im(\mathcal{H}^i \oplus \mathcal{G}^{i - 1} \to j_*\mathcal{F}^i)\). It is clear that \(\mathcal{G}^\bullet \subset j_*\mathcal{F}^\bullet\) is a bounded complex of coherent \(\mathcal{O}_X\)-modules whose restriction to \(U\) is \(\mathcal{F}^\bullet\) as desired.

Lemma

Let \(X\) be an affine scheme and let \(U \subset X\) be a quasi-compact open subscheme. For any pseudo-coherent object \(E\) of \(D(\mathcal{O}_U)\) there exists a bounded above complex of finite free \(\mathcal{O}_X\)-modules whose restriction to \(U\) is isomorphic to \(E\).

Proof

By Lemma 08E5 we see that \(E\) is an object of \(D_\QCoh(\mathcal{O}_U)\). By Lemma 08ED we may assume \(E = E'|U\) for some object \(E'\) of \(D_\QCoh(\mathcal{O}_X)\). Write \(X = \Spec(A)\). By Lemma 06Z0 we can find a complex \(M^\bullet\) of \(A\)-modules whose associated complex of \(\mathcal{O}_X\)-modules is a representative of \(E'\).

Choose \(f_1, \ldots, f_r \in A\) such that \(U = D(f_1) \cup \ldots \cup D(f_r)\). By Lemma 08E7 the complexes \(M^\bullet_{f_j}\) are pseudo-coherent complexes of \(A_{f_j}\)-modules. Let \(n\) be an integer. Assume we have a map of complexes \(\alpha : F^\bullet \to M^\bullet\) where \(F^\bullet\) is bounded above, \(F^i = 0\) for \(i < n\), each \(F^i\) is a finite free \(R\)-module, such that \[H^i(\alpha_{f_j}) : H^i(F^\bullet_{f_j}) \to H^i(M^\bullet_{f_j})\] is an isomorphism for \(i > n\) and surjective for \(i = n\). Picture \[\xymatrix{ & F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ M^{n-1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots }\] Since each \(M^\bullet_{f_j}\) has vanishing cohomology in large degrees we can find such a map for \(n \gg 0\). By induction on \(n\) we are going to extend this to a map of complexes \(F^\bullet \to M^\bullet\) such that \(H^i(\alpha_{f_j})\) is an isomorphism for all \(i\). The lemma will follow by taking \(\widetilde{F^\bullet}\).

The induction step will be to extend the diagram above by adding \(F^{n - 1}\). Let \(C^\bullet\) be the cone on \(\alpha\) (Derived Categories, Definition 014E). The long exact sequence of cohomology shows that \(H^i(C^\bullet_{f_j}) = 0\) for \(i \geq n\). By More on Algebra, Lemma 064R we see that \(C^\bullet_{f_j}\) is \((n - 1)\)-pseudo-coherent. By More on Algebra, Lemma 064S we see that \(H^{n - 1}(C^\bullet_{f_j})\) is a finite \(A_{f_j}\)-module. Choose a finite free \(A\)-module \(F^{n - 1}\) and an \(A\)-module \(\beta : F^{n - 1} \to C^{n - 1}\) such that the composition \(F^{n - 1} \to C^{n - 1} \to C^n\) is zero and such that \(F^{n - 1}_{f_j}\) surjects onto \(H^{n - 1}(C^\bullet_{f_j})\). (Some details omitted; hint: clear denominators.) Since \(C^{n - 1} = M^{n - 1} \oplus F^n\) we can write \(\beta = (\alpha^{n - 1}, -d^{n - 1})\). The vanishing of the composition \(F^{n - 1} \to C^{n - 1} \to C^n\) implies these maps fit into a morphism of complexes \[\xymatrix{ & F^{n - 1} \ar[d]^{\alpha^{n - 1}} \ar[r]_{d^{n - 1}} & F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ \ldots \ar[r] & M^{n - 1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots }\] Moreover, these maps define a morphism of distinguished triangles \[\xymatrix{ (F^n \to \ldots) \ar[r] \ar[d] & (F^{n-1} \to \ldots) \ar[r] \ar[d] & F^{n-1} \ar[r] \ar[d]_\beta & (F^n \to \ldots)[1] \ar[d] \\ (F^n \to \ldots) \ar[r] & M^\bullet \ar[r] & C^\bullet \ar[r] & (F^n \to \ldots)[1] }\] Hence our choice of \(\beta\) implies that the map of complexes \((F^{-1} \to \ldots) \to M^\bullet\) induces an isomorphism on cohomology localized at \(f_j\) in degrees \(\geq n\) and a surjection in degree \(n - 1\). This finishes the proof of the lemma.

The following two lemmas should probably go somewhere else.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(E \in D^b_\QCoh(\mathcal{O}_X)\). There exists an integer \(n_0 > 0\) such that \(\Ext^n_{D(\mathcal{O}_X)}(\mathcal{E}, E) = 0\) for every finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) and every \(n \geq n_0\).

Proof

Recall that \(\Ext^n_{D(\mathcal{O}_X)}(\mathcal{E}, E) = \Hom_{D(\mathcal{O}_X)}(\mathcal{E}, E[n])\). We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma 08BW. Thus if \(X = U \cup V\) and the result of the lemma holds for \(E|_U\), \(E|_V\), and \(E|_{U \cap V}\) for some bound \(n_0\), then the result holds for \(E\) with bound \(n_0 + 1\). Thus it suffices to prove the lemma when \(X\) is affine, see Cohomology of Schemes, Lemma 08DR.

Assume \(X = \Spec(A)\) is affine. Choose a complex of \(A\)-modules \(M^\bullet\) whose associated complex of quasi-coherent modules represents \(E\), see Lemma 06Z0. Write \(\mathcal{E} = \widetilde{P}\) for some \(A\)-module \(P\). Since \(\mathcal{E}\) is finite locally free, we see that \(P\) is a finite projective \(A\)-module. We have \[\begin{align*} \Hom_{D(\mathcal{O}_X)}(\mathcal{E}, E[n]) & = \Hom_{D(A)}(P, M^\bullet[n]) \\ & = \Hom_{K(A)}(P, M^\bullet[n]) \\ & = \Hom_A(P, H^n(M^\bullet)) \end{align*}\] The first equality by Lemma 06Z0, the second equality by Derived Categories, Lemma 064B, and the final equality because \(\Hom_A(P, -)\) is an exact functor. As \(E\) and hence \(M^\bullet\) is bounded we get zero for all sufficiently large \(n\).

The following lemma can be strengthened (there is a uniformity in the vanishing over all \(L\) with nonzero cohomology sheaves only in a fixed range).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(K\) be a perfect object of \(D(\mathcal{O}_X)\). Then

1. there exist integers \(a \leq b\) such that for any \(L \in D_\QCoh(\mathcal{O}_X)\) with \(H^i(L) = 0\) for \(i \in [a, b]\) we have \(\Hom_{D(\mathcal{O}_X)}(K, L) = 0\), and

2. if \(L\) is bounded, then \(\Ext^n_{D(\mathcal{O}_X)}(K, L)\) is zero for all but finitely many \(n\).

Proof

Part (2) follows from (1) as \(\Ext^n_{D(\mathcal{O}_X)}(K, L) = \Hom_{D(\mathcal{O}_X)}(K, L[n])\). We prove (1). Since \(K\) is perfect we have \[\Hom_{D(\mathcal{O}_X)}(K, L) = H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] where \(K^\vee\) is the “dual” perfect complex to \(K\), see Cohomology, Lemma 08DQ. Note that \(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L\) is in \(D_\QCoh(X)\) by Lemmas 08DX and 08E5 (to see that a perfect complex has quasi-coherent cohomology sheaves). Say \(K^\vee\) has tor amplitude in \([a, b]\). Then the spectral sequence \[E_1^{p, q} = H^p(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} H^q(L)) \Rightarrow H^{p + q}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] shows that \(H^j(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\) is zero if \(H^q(L) = 0\) for \(q \in [j - b, j - a]\). Let \(N\) be the integer \(d\) of Cohomology of Schemes, Lemma 071L. Then \(H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\) vanishes if the cohomology sheaves \[H^{-N}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L), \ H^{-N + 1}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L), \ \ldots, \ H^0(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L)\] are zero. Namely, by the lemma cited and Lemma 08D4, we have \[H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L) = H^0(X, \tau_{\geq -N}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L))\] and by the vanishing of cohomology sheaves, this is equal to \(H^0(X, \tau_{\geq 1}(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L))\) which is zero by Derived Categories, Lemma 05TC. It follows that \(\Hom_{D(\mathcal{O}_X)}(K, L)\) is zero if \(H^i(L) = 0\) for \(i \in [-b - N, -a]\).

Lemma

Let \(X\) be an affine scheme. Let \(U \subset X\) be a quasi-compact open. For every perfect object \(E\) of \(D(\mathcal{O}_U)\) there exists an integer \(r\) and a finite locally free sheaf \(\mathcal{F}\) on \(U\) such that \(\mathcal{F}[-r] \oplus E\) is the restriction of a perfect object of \(D(\mathcal{O}_X)\).

Proof

Say \(X = \Spec(A)\). Recall that a perfect complex is pseudo-coherent, see Cohomology, Lemma 08CQ. By Lemma 08EE we can find a bounded above complex \(\mathcal{F}^\bullet\) of finite free \(A\)-modules such that \(E\) is isomorphic to \(\mathcal{F}^\bullet|_U\) in \(D(\mathcal{O}_U)\). By Cohomology, Lemma 08CQ and since \(U\) is quasi-compact, we see that \(E\) has finite tor dimension, say \(E\) has tor amplitude in \([a, b]\). Pick \(r < a\) and set \[\mathcal{K} = \Ker(\mathcal{F}^{r} \to \mathcal{F}^{r + 1}) = \Im(\mathcal{F}^{r - 1} \to \mathcal{F}^r).\] Since \(E\) has tor amplitude in \([a, b]\) we see that \(\mathcal{F} = \mathcal{K}|_U\) is flat (Cohomology, Lemma 08CH). Hence \(\mathcal{F}\) is flat and of finite presentation, thus finite locally free (Properties, Lemma 05P2). It follows that \[\mathcal{F} \to \mathcal{F}^r|_U \to \mathcal{F}^{r + 1}|_U \to \ldots\] is a strictly perfect complex on \(U\) representing \(E\). On the other hand, the complex \(P = (\mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )\) is a perfect complex on \(X\). Using stupid truncations we obtain a distinguished triangle \[P|_U \to E \to \mathcal{F}[-r - 1] \to (P|_U)[1]\] If the map \(E \to \mathcal{F}[-r - 1]\) is zero in \(D(\mathcal{O}_U)\), then \(P|_U = \mathcal{F}[-r - 2] \oplus E\), see Derived Categories, Lemma 05QT. This will be true for \(r \ll 0\) for example by Lemma 09M4.

Lemma

Let \(X\) be an affine scheme. Let \(U \subset X\) be a quasi-compact open. Let \(E, E'\) be objects of \(D_\QCoh(\mathcal{O}_X)\) with \(E\) perfect. For every map \(\alpha : E|_U \to E'|_U\) there exist maps \[E \xleftarrow{\beta} E_1 \xrightarrow{\gamma} E'\] of complexes on \(X\) with \(E_1\) perfect such that \(\beta : E_1 \to E\) restricts to an isomorphism on \(U\) and such that \(\alpha = \gamma|_U \circ \beta|_U^{-1}\). Moreover we can assume \(E_1 = E \otimes_{\mathcal{O}_X}^\mathbf{L} I\) for some perfect complex \(I\) on \(X\).

Proof

Write \(X = \Spec(A)\). Write \(U = D(f_1) \cup \ldots \cup D(f_r)\). Choose finite complex of finite projective \(A\)-modules \(M^\bullet\) representing \(E\) (Lemma 08EB). Choose a complex of \(A\)-modules \((M')^\bullet\) representing \(E'\) (Lemma 06Z0). In this case the complex \(H^\bullet = \Hom_A(M^\bullet, (M')^\bullet)\) is a complex of \(A\)-modules whose associated complex of quasi-coherent \(\mathcal{O}_X\)-modules represents \(R\SheafHom(E, E')\), see Cohomology, Lemma 08DM. Then \(\alpha\) determines an element \(s\) of \(H^0(U, R\SheafHom(E, E'))\), see Cohomology, Lemma 08DK. There exists an \(e\) and a map \[\xi : I^\bullet(f_1^e, \ldots, f_r^e) \to \Hom_A(M^\bullet, (M')^\bullet)\] corresponding to \(s\), see Proposition 08DD. Letting \(E_1\) be the object corresponding to complex of quasi-coherent \(\mathcal{O}_X\)-modules associated to \[\text{Tot}(I^\bullet(f_1^e, \ldots, f_r^e) \otimes_A M^\bullet)\] we obtain \(E_1 \to E\) using the canonical map \(I^\bullet(f_1^e, \ldots, f_r^e) \to A\) and \(E_1 \to E'\) using \(\xi\) and Cohomology, Lemma 08DK.

Lemma

Let \(X\) be an affine scheme. Let \(U \subset X\) be a quasi-compact open. For every perfect object \(F\) of \(D(\mathcal{O}_U)\) the object \(F \oplus F[1]\) is the restriction of a perfect object of \(D(\mathcal{O}_X)\).

Proof

By Lemma 08EG we can find a perfect object \(E\) of \(D(\mathcal{O}_X)\) such that \(E|_U = \mathcal{F}[r] \oplus F\) for some finite locally free \(\mathcal{O}_U\)-module \(\mathcal{F}\). By Lemma 08EH we can find a morphism of perfect complexes \(\alpha : E_1 \to E\) such that \((E_1)|_U \cong E|_U\) and such that \(\alpha|_U\) is the map \[\left( \begin{matrix} \text{id}_{\mathcal{F}[r]} & 0 \\ 0 & 0 \end{matrix} \right) : \mathcal{F}[r] \oplus F \to \mathcal{F}[r] \oplus F\] Then the cone on \(\alpha\) is a solution.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(f \in \Gamma(X, \mathcal{O}_X)\). For any morphism \(\alpha : E \to E'\) in \(D_\QCoh(\mathcal{O}_X)\) such that

1. \(E\) is perfect, and

2. \(E'\) is supported on \(T = V(f)\)

there exists an \(n \geq 0\) such that \(f^n \alpha = 0\).

Proof

We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma 08BW. Thus if \(X = U \cup V\) and the result of the lemma holds for \(f|_U\), \(f|_V\), and \(f|_{U \cap V}\), then the result holds for \(f\). Thus it suffices to prove the lemma when \(X\) is affine, see Cohomology of Schemes, Lemma 08DR.

Let \(X = \Spec(A)\). Then \(f \in A\). We will use the equivalence \(D(A) = D_\QCoh(X)\) of Lemma 06Z0 without further mention. Represent \(E\) by a finite complex of finite projective \(A\)-modules \(P^\bullet\). This is possible by Lemma 08EB. Let \(t\) be the largest integer such that \(P^t\) is nonzero. The distinguished triangle \[P^t[-t] \to P^\bullet \to \sigma_{\leq t - 1}P^\bullet \to P^t[-t + 1]\] shows that by induction on the length of the complex \(P^\bullet\) we can reduce to the case where \(P^\bullet\) has a single nonzero term. This and the shift functor reduces us to the case where \(P^\bullet\) consists of a single finite projective \(A\)-module \(P\) in degree \(0\). Represent \(E'\) by a complex \(M^\bullet\) of \(A\)-modules. Then \(\alpha\) corresponds to a map \(P \to H^0(M^\bullet)\). Since the module \(H^0(M^\bullet)\) is supported on \(V(f)\) by assumption (2) we see that every element of \(H^0(M^\bullet)\) is annihilated by a power of \(f\). Since \(P\) is a finite \(A\)-module the map \(f^n\alpha : P \to H^0(M^\bullet)\) is zero for some \(n\) as desired.

Lemma

Let \(X\) be an affine scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. Let \(U \subset X\) be a quasi-compact open. For every perfect object \(F\) of \(D(\mathcal{O}_U)\) supported on \(T \cap U\) the object \(F \oplus F[1]\) is the restriction of a perfect object \(E\) of \(D(\mathcal{O}_X)\) supported in \(T\).

Proof

Say \(T = V(g_1, \ldots, g_s)\). After replacing \(g_j\) by a power we may assume multiplication by \(g_j\) is zero on \(F\), see Lemma 08EJ. Choose \(E\) as in Lemma 08EI. Note that \(g_j : E \to E\) restricts to zero on \(U\). Choose a distinguished triangle \[E \xrightarrow{g_1} E \to C_1 \to E[1]\] By Derived Categories, Lemma 05QT the object \(C_1\) restricts to \(F \oplus F[1] \oplus F[1] \oplus F[2]\) on \(U\). Moreover, \(g_1 : C_1 \to C_1\) has square zero by Derived Categories, Lemma 05QP. Namely, the diagram \[\xymatrix{ E \ar[r] \ar[d]_0 & C_1 \ar[d]_{g_1} \ar[r] & E[1] \ar[d]_0 \\ E \ar[r] & C_1 \ar[r] & E[1] }\] is commutative since the compositions \(E \xrightarrow{g_1} E \to C_1\) and \(C_1 \to E[1] \xrightarrow{g_1} E[1]\) are zero. Continuing, setting \(C_{i + 1}\) equal to the cone of the map \(g_i : C_i \to C_i\) we obtain a perfect complex \(C_s\) on \(X\) supported on \(T\) whose restriction to \(U\) gives \[F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s]\] Choose morphisms of perfect complexes \(\beta : C' \to C_s\) and \(\gamma : C' \to C_s\) as in Lemma 08EH such that \(\beta|_U\) is an isomorphism and such that \(\gamma|_U \circ \beta|_U^{-1}\) is the morphism \[F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s] \to F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s]\] which is the identity on all summands except for \(F\) where it is zero. By Lemma 08EH we also have \(C' = C_s \otimes^\mathbf{L} I\) for some perfect complex \(I\) on \(X\). Hence the nullity of \(g_j^2\text{id}_{C_s}\) implies the same thing for \(C'\). Thus \(C'\) is supported on \(T\) as well. Then \(\text{Cone}(\gamma)\) is a solution.

A special case of the following lemma can be found in [Neeman-Grothendieck].

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(U \subset X\) be a quasi-compact open. Let \(T \subset X\) be a closed subset with \(X \setminus T\) retro-compact in \(X\). Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). Let \(\alpha : P \to E|_U\) be a map where \(P\) is a perfect object of \(D(\mathcal{O}_U)\) supported on \(T \cap U\). Then there exists a map \(\beta : R \to E\) where \(R\) is a perfect object of \(D(\mathcal{O}_X)\) supported on \(T\) such that \(P\) is a direct summand of \(R|_U\) in \(D(\mathcal{O}_U)\) compatible \(\alpha\) and \(\beta|_U\).

Proof

Since \(X\) is quasi-compact there exists an integer \(m\) such that \(X = U \cup V_1 \cup \ldots \cup V_m\) for some affine opens \(V_j\) of \(X\). Arguing by induction on \(m\) we see that we may assume \(m = 1\). In other words, we may assume that \(X = U \cup V\) with \(V\) affine. By Lemma 08EK we can choose a perfect object \(Q\) in \(D(\mathcal{O}_V)\) supported on \(T \cap V\) and an isomorphism \(Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}\). By Lemma 08EH we can replace \(Q\) by \(Q \otimes^\mathbf{L} I\) (still supported on \(T \cap V\)) and assume that the map \[Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V}\] lifts to \(Q \to E|_V\). By Cohomology, Lemma 08DG we find an morphism \(a : R \to E\) of \(D(\mathcal{O}_X)\) such that \(a|_U\) is isomorphic to \(P \oplus P[1] \to E|_U\) and \(a|_V\) isomorphic to \(Q \to E|_V\). Thus \(R\) is perfect and supported on \(T\) as desired.

Remark

The proof of Lemma 09IM shows that \[R|_U = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m]\] for some \(m \geq 0\) and \(n_j \geq 0\). Thus the highest degree cohomology sheaf of \(R|_U\) equals that of \(P\). By repeating the construction for the map \(P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] \to R|_U\), taking cones, and using induction we can achieve equality of cohomology sheaves of \(R|_U\) and \(P\) above any given degree.

Approximation by perfect complexes

In this section we discuss the observation, due to Neeman and Lipman, that a pseudo-coherent complex can be “approximated” by perfect complexes.

Definition

Let \(X\) be a scheme. Consider triples \((T, E, m)\) where

1. \(T \subset X\) is a closed subset,

2. \(E\) is an object of \(D_\QCoh(\mathcal{O}_X)\), and

3. \(m \in \mathbf{Z}\).

We say approximation holds for the triple \((T, E, m)\) if there exists a perfect object \(P\) of \(D(\mathcal{O}_X)\) supported on \(T\) and a map \(\alpha : P \to E\) which induces isomorphisms \(H^i(P) \to H^i(E)\) for \(i > m\) and a surjection \(H^m(P) \to H^m(E)\).

Approximation cannot hold for every triple. Namely, it is clear that if approximation holds for the triple \((T, E, m)\), then

1. \(E\) is \(m\)-pseudo-coherent, see Cohomology, Definition 08CB, and

2. the cohomology sheaves \(H^i(E)\) are supported on \(T\) for \(i \geq m\).

Moreover, the “support” of a perfect complex is a closed subscheme whose complement is retrocompact in \(X\) (details omitted). Hence we cannot expect approximation to hold without this assumption on \(T\). This partly explains the conditions in the following definition.

Definition

Let \(X\) be a scheme. We say approximation by perfect complexes holds on \(X\) if for any closed subset \(T \subset X\) with \(X \setminus T\) retro-compact in \(X\) there exists an integer \(r\) such that for every triple \((T, E, m)\) as in Definition 08EM with

1. \(E\) is \((m - r)\)-pseudo-coherent, and

2. \(H^i(E)\) is supported on \(T\) for \(i \geq m - r\)

approximation holds.

We will prove that approximation by perfect complexes holds for quasi-compact and quasi-separated schemes. It seems that the second condition is necessary for our method of proof. It is possible that the first condition may be weakened to “\(E\) is \(m\)-pseudo-coherent” by carefully analyzing the arguments below.

Lemma

Let \(X\) be a scheme. Let \(U \subset X\) be an open subscheme. Let \((T, E, m)\) be a triple as in Definition 08EM. If

1. \(T \subset U\),

2. approximation holds for \((T, E|_U, m)\), and

3. the sheaves \(H^i(E)\) for \(i \geq m\) are supported on \(T\),

then approximation holds for \((T, E, m)\).

Proof

Let \(j : U \to X\) be the inclusion morphism. If \(P \to E|_U\) is an approximation of the triple \((T, E|_U, m)\) over \(U\), then \(j_!P = Rj_*P \to j_!(E|_U) \to E\) is an approximation of \((T, E, m)\) over \(X\). See Cohomology, Lemmas 08DF and 08DP.

Lemma

Let \(X\) be an affine scheme. Then approximation holds for every triple \((T, E, m)\) as in Definition 08EM such that there exists an integer \(r \geq 0\) with

1. \(E\) is \(m\)-pseudo-coherent,

2. \(H^i(E)\) is supported on \(T\) for \(i \geq m - r + 1\),

3. \(X \setminus T\) is the union of \(r\) affine opens.

In particular, approximation by perfect complexes holds for affine schemes.

Proof

Say \(X = \Spec(A)\). Write \(T = V(f_1, \ldots, f_r)\). (The case \(r = 0\), i.e., \(T = X\) follows immediately from Lemma 08E7 and the definitions.) Let \((T, E, m)\) be a triple as in the lemma. Let \(t\) be the largest integer such that \(H^t(E)\) is nonzero. We will proceed by induction on \(t\). The base case is \(t < m\); in this case the result is trivial. Now suppose that \(t \geq m\). By Cohomology, Lemma 08DN the sheaf \(H^t(E)\) is of finite type. Since it is quasi-coherent it is generated by finitely many sections (Properties, Lemma 01PB). For every \(s \in \Gamma(X, H^t(E)) = H^t(X, E)\) (see proof of Lemma 06Z0) we can find an \(e > 0\) and a morphism \(K_e[-t] \to E\) such that \(s\) is in the image of \(H^0(K_e) = H^t(K_e[-t]) \to H^t(E)\), see Lemma 08E3. Taking a finite direct sum of these maps we obtain a map \(P \to E\) where \(P\) is a perfect complex supported on \(T\), where \(H^i(P) = 0\) for \(i > t\), and where \(H^t(P) \to E\) is surjective. Choose a distinguished triangle \[P \to E \to E' \to P[1]\] Then \(E'\) is \(m\)-pseudo-coherent (Cohomology, Lemma 08CD), \(H^i(E') = 0\) for \(i \geq t\), and \(H^i(E')\) is supported on \(T\) for \(i \geq m - r + 1\). By induction we find an approximation \(P' \to E'\) of \((T, E', m)\). Fit the composition \(P' \to E' \to P[1]\) into a distinguished triangle \(P \to P'' \to P' \to P[1]\) and extend the morphisms \(P' \to E'\) and \(P[1] \to P[1]\) into a morphism of distinguished triangles \[\xymatrix{ P \ar[r] \ar[d] & P'' \ar[d] \ar[r] & P' \ar[d] \ar[r] & P[1] \ar[d] \\ P \ar[r] & E \ar[r] & E' \ar[r] & P[1] }\] using TR3. Then \(P''\) is a perfect complex (Cohomology, Lemma 08CR) supported on \(T\). An easy diagram chase shows that \(P'' \to E\) is the desired approximation.

Lemma

Let \(X\) be a scheme. Let \(X = U \cup V\) be an open covering with \(U\) quasi-compact, \(V\) affine, and \(U \cap V\) quasi-compact. If approximation by perfect complexes holds on \(U\), then approximation holds on \(X\).

Proof

Let \(T \subset X\) be a closed subset with \(X \setminus T\) retro-compact in \(X\). Let \(r_U\) be the integer of Definition 08EN adapted to the pair \((U, T \cap U)\). Set \(T' = T \setminus U\). Note that \(T' \subset V\) and that \(V \setminus T' = (X \setminus T) \cap U \cap V\) is quasi-compact by our assumption on \(T\). Let \(r'\) be the number of affines needed to cover \(V \setminus T'\). We claim that \(r = \max(r_U, r')\) works for the pair \((X, T)\).

To see this choose a triple \((T, E, m)\) such that \(E\) is \((m - r)\)-pseudo-coherent and \(H^i(E)\) is supported on \(T\) for \(i \geq m - r\). Let \(t\) be the largest integer such that \(H^t(E)|_U\) is nonzero. (Such an integer exists as \(U\) is quasi-compact and \(E|_U\) is \((m - r)\)-pseudo-coherent.) We will prove that \(E\) can be approximated by induction on \(t\).

Base case: \(t \leq m - r'\). This means that \(H^i(E)\) is supported on \(T'\) for \(i \geq m - r'\). Hence Lemma 08EQ guarantees the existence of an approximation \(P \to E|_V\) of \((T', E|_V, m)\) on \(V\). Applying Lemma 08EP we see that \((T', E, m)\) can be approximated. Such an approximation is also an approximation of \((T, E, m)\).

Induction step. Choose an approximation \(P \to E|_U\) of \((T \cap U, E|_U, m)\). This in particular gives a surjection \(H^t(P) \to H^t(E|_U)\). By Lemma 08EK we can choose a perfect object \(Q\) in \(D(\mathcal{O}_V)\) supported on \(T \cap V\) and an isomorphism \(Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}\). By Lemma 08EH we can replace \(Q\) by \(Q \otimes^\mathbf{L} I\) and assume that the map \[Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V}\] lifts to \(Q \to E|_V\). By Cohomology, Lemma 08DG we find an morphism \(a : R \to E\) of \(D(\mathcal{O}_X)\) such that \(a|_U\) is isomorphic to \(P \oplus P[1] \to E|_U\) and \(a|_V\) isomorphic to \(Q \to E|_V\). Thus \(R\) is perfect and supported on \(T\) and the map \(H^t(R) \to H^t(E)\) is surjective on restriction to \(U\). Choose a distinguished triangle \[R \to E \to E' \to R[1]\] Then \(E'\) is \((m - r)\)-pseudo-coherent (Cohomology, Lemma 08CD), \(H^i(E')|_U = 0\) for \(i \geq t\), and \(H^i(E')\) is supported on \(T\) for \(i \geq m - r\). By induction we find an approximation \(R' \to E'\) of \((T, E', m)\). Fit the composition \(R' \to E' \to R[1]\) into a distinguished triangle \(R \to R'' \to R' \to R[1]\) and extend the morphisms \(R' \to E'\) and \(R[1] \to R[1]\) into a morphism of distinguished triangles \[\xymatrix{ R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\ R \ar[r] & E \ar[r] & E' \ar[r] & R[1] }\] using TR3. Then \(R''\) is a perfect complex (Cohomology, Lemma 08CR) supported on \(T\). An easy diagram chase shows that \(R'' \to E\) is the desired approximation.

Theorem

Let \(X\) be a quasi-compact and quasi-separated scheme. Then approximation by perfect complexes holds on \(X\).

Proof

This follows from the induction principle of Cohomology of Schemes, Lemma 08DR and Lemmas 08ER and 08EQ.

Generating derived categories

In this section we prove that the derived category \(D_\QCoh(\mathcal{O}_X)\) of a quasi-compact and quasi-separated scheme can be generated by a single perfect object. We urge the reader to read the proof of this result in the wonderful paper by Bondal and van den Bergh, see [BvdB].

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(U\) be a quasi-compact open subscheme. Let \(P\) be a perfect object of \(D(\mathcal{O}_U)\). Then \(P\) is a direct summand of the restriction of a perfect object of \(D(\mathcal{O}_X)\).

Proof

Special case of Lemma 09IM.

Lemma

In Situation 08CZ denote \(j : U \to X\) the open immersion and let \(K\) be the perfect object of \(D(\mathcal{O}_X)\) corresponding to the Koszul complex on \(f_1, \ldots, f_r\) over \(A\). For \(E \in D_\QCoh(\mathcal{O}_X)\) the following are equivalent

1. \(E = Rj_*(E|_U)\), and

2. \(\Hom_{D(\mathcal{O}_X)}(K[n], E) = 0\) for all \(n \in \mathbf{Z}\).

Proof

Choose a distinguished triangle \(E \to Rj_*(E|_U) \to N \to E[1]\). Observe that \[\Hom_{D(\mathcal{O}_X)}(K[n], Rj_*(E|_U)) = \Hom_{D(\mathcal{O}_U)}(K|_U[n], E) = 0\] for all \(n\) as \(K|_U = 0\). Thus it suffices to prove the result for \(N\). In other words, we may assume that \(E\) restricts to zero on \(U\). Observe that there are distinguished triangles \[K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i}, \ldots, f_r^{e_r}) \to K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i + e''_i}, \ldots, f_r^{e_r}) \to K^\bullet(f_1^{e_1}, \ldots, f_i^{e''_i}, \ldots, f_r^{e_r}) \to \ldots\] of Koszul complexes, see More on Algebra, Lemma 062C. Hence if \(\Hom_{D(\mathcal{O}_X)}(K[n], E) = 0\) for all \(n \in \mathbf{Z}\) then the same thing is true for the \(K\) replaced by \(K_e\) as in Lemma 08E3. Thus our lemma follows immediately from that one and the fact that \(E\) is determined by the complex of \(A\)-modules \(R\Gamma(X, E)\), see Lemma 06Z0.

Theorem

Let \(X\) be a quasi-compact and quasi-separated scheme. The category \(D_\QCoh(\mathcal{O}_X)\) can be generated by a single perfect object. More precisely, there exists a perfect object \(P\) of \(D(\mathcal{O}_X)\) such that for \(E \in D_\QCoh(\mathcal{O}_X)\) the following are equivalent

1. \(E = 0\), and

2. \(\Hom_{D(\mathcal{O}_X)}(P[n], E) = 0\) for all \(n \in \mathbf{Z}\).

Proof

We will prove this using the induction principle of Cohomology of Schemes, Lemma 08DR.

If \(X\) is affine, then \(\mathcal{O}_X\) is a perfect generator. This follows from Lemma 06Z0.

Assume that \(X = U \cup V\) is an open covering with \(U\) quasi-compact such that the theorem holds for \(U\) and \(V\) is an affine open. Let \(P\) be a perfect object of \(D(\mathcal{O}_U)\) which is a generator for \(D_\QCoh(\mathcal{O}_U)\). Using Lemma 09IQ we may choose a perfect object \(Q\) of \(D(\mathcal{O}_X)\) whose restriction to \(U\) is a direct sum one of whose summands is \(P\). Say \(V = \Spec(A)\). Let \(Z = X \setminus U\). This is a closed subset of \(V\) with \(V \setminus Z\) quasi-compact. Choose \(f_1, \ldots, f_r \in A\) such that \(Z = V(f_1, \ldots, f_r)\). Let \(K \in D(\mathcal{O}_V)\) be the perfect object corresponding to the Koszul complex on \(f_1, \ldots, f_r\) over \(A\). Note that since \(K\) is supported on \(Z \subset V\) closed, the pushforward \(K' = R(V \to X)_*K\) is a perfect object of \(D(\mathcal{O}_X)\) whose restriction to \(V\) is \(K\) (see Cohomology, Lemma 08DP). We claim that \(Q \oplus K'\) is a generator for \(D_\QCoh(\mathcal{O}_X)\).

Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\) such that there are no nontrivial maps from any shift of \(Q \oplus K'\) into \(E\). By Cohomology, Lemma 08DF we have \(K' = R(V \to X)_! K\) and hence \[\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)\] Thus by Lemma 09IR the vanishing of these groups implies that \(E|_V\) is isomorphic to \(R(U \cap V \to V)_*E|_{U \cap V}\). This implies that \(E = R(U \to X)_*E|_U\) (small detail omitted). If this is the case then \[\Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U)\] which contains \(\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)\) as a direct summand. Thus by our choice of \(P\) the vanishing of these groups implies that \(E|_U\) is zero. Whence \(E\) is zero.

The following result is an strengthening of Theorem 09IS proved using exactly the same methods. Recall that for a closed subset \(T\) of a scheme \(X\) we denote \(D_T(\mathcal{O}_X)\) the strictly full, saturated, triangulated subcategory of \(D(\mathcal{O}_X)\) consisting of objects supported on \(T\) (Definition 08DA). We similarly denote \(D_{\QCoh, T}(\mathcal{O}_X)\) the strictly full, saturated, triangulated subcategory of \(D(\mathcal{O}_X)\) consisting of those complexes whose cohomology sheaves are quasi-coherent and are supported on \(T\).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. With notation as above, the category \(D_{\QCoh, T}(\mathcal{O}_X)\) is generated by a single perfect object.

Proof

We will prove this using the induction principle of Cohomology of Schemes, Lemma 08DR.

Assume \(X = \Spec(A)\) is affine. In this case there exist \(f_1, \ldots, f_r \in A\) such that \(T = V(f_1, \ldots, f_r)\). Let \(K\) be the Koszul complex on \(f_1, \ldots, f_r\) as in Lemma 09IR. Then \(K\) is a perfect object with cohomology supported on \(T\) and hence a perfect object of \(D_{\QCoh, T}(\mathcal{O}_X)\). On the other hand, if \(E \in D_{\QCoh, T}(\mathcal{O}_X)\) and \(\Hom(K, E[n]) = 0\) for all \(n\), then Lemma 09IR tells us that \(E = Rj_*(E|_{X \setminus T}) = 0\). Hence \(K\) generates \(D_{\QCoh, T}(\mathcal{O}_X)\), (by our definition of generators of triangulated categories in Derived Categories, Definition 09SJ).

Assume that \(X = U \cup V\) is an open covering with \(V\) affine and \(U\) quasi-compact such that the lemma holds for \(U\). Let \(P\) be a perfect object of \(D(\mathcal{O}_U)\) supported on \(T \cap U\) which is a generator for \(D_{\QCoh, T \cap U}(\mathcal{O}_U)\). Using Lemma 09IM we may choose a perfect object \(Q\) of \(D(\mathcal{O}_X)\) supported on \(T\) whose restriction to \(U\) is a direct sum one of whose summands is \(P\). Write \(V = \Spec(B)\). Let \(Z = X \setminus U\). Then \(Z\) is a closed subset of \(V\) such that \(V \setminus Z\) is quasi-compact. As \(X\) is quasi-separated, it follows that \(Z \cap T\) is a closed subset of \(V\) such that \(W = V \setminus (Z \cap T)\) is quasi-compact. Thus we can choose \(g_1, \ldots, g_s \in B\) such that \(Z \cap T = V(g_1, \ldots, g_r)\). Let \(K \in D(\mathcal{O}_V)\) be the perfect object corresponding to the Koszul complex on \(g_1, \ldots, g_s\) over \(B\). Note that since \(K\) is supported on \((Z \cap T) \subset V\) closed, the pushforward \(K' = R(V \to X)_*K\) is a perfect object of \(D(\mathcal{O}_X)\) whose restriction to \(V\) is \(K\) (see Cohomology, Lemma 08DP). We claim that \(Q \oplus K'\) is a generator for \(D_{\QCoh, T}(\mathcal{O}_X)\).

Let \(E\) be an object of \(D_{\QCoh, T}(\mathcal{O}_X)\) such that there are no nontrivial maps from any shift of \(Q \oplus K'\) into \(E\). By Cohomology, Lemma 08DF we have \(K' = R(V \to X)_! K\) and hence \[\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)\] Thus by Lemma 09IR we have \(E|_V = Rj_*E|_W\) where \(j : W \to V\) is the inclusion. Picture \[\xymatrix{ W \ar[r]_j & V & Z \cap T \ar[l] \ar[d] \\ U \cap V \ar[u]^{j'} \ar[ru]_{j''} & & Z \ar[lu] }\] Since \(E\) is supported on \(T\) we see that \(E|_W\) is supported on \(T \cap W = T \cap U \cap V\) which is closed in \(W\). We conclude that \[E|_V = Rj_*(E|_W) = Rj_*(Rj'_*(E|_{U \cap V})) = Rj''_*(E|_{U \cap V})\] where the second equality is part (1) of Cohomology, Lemma 08DF. This implies that \(E = R(U \to X)_*E|_U\) (small detail omitted). If this is the case then \[\Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U)\] which contains \(\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)\) as a direct summand. Thus by our choice of \(P\) the vanishing of these groups implies that \(E|_U\) is zero. Whence \(E\) is zero.

An example generator

In this section we prove that the derived category of projective space over a ring is generated by a vector bundle, in fact a direct sum of shifts of the structure sheaf.

The following lemma says that \(\bigoplus_{n \geq 0} \mathcal{L}^{\otimes -n}\) is a generator if \(\mathcal{L}\) is ample.

Lemma

Let \(X\) be a scheme and \(\mathcal{L}\) an ample invertible \(\mathcal{O}_X\)-module. If \(K\) is a nonzero object of \(D_\QCoh(\mathcal{O}_X)\), then for some \(n \geq 0\) and \(p \in \mathbf{Z}\) the cohomology group \(H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes n})\) is nonzero.

Proof

Recall that as \(X\) has an ample invertible sheaf, it is quasi-compact and separated (Properties, Definition 01PS and Lemma 01PY). Thus we may apply Proposition 08DB and represent \(K\) by a complex \(\mathcal{F}^\bullet\) of quasi-coherent modules. Pick any \(p\) such that \(\mathcal{H}^p = \Ker(\mathcal{F}^p \to \mathcal{F}^{p + 1})/ \Im(\mathcal{F}^{p - 1} \to \mathcal{F}^p)\) is nonzero. Choose a point \(x \in X\) such that the stalk \(\mathcal{H}^p_x\) is nonzero. Choose an \(n \geq 0\) and \(s \in \Gamma(X, \mathcal{L}^{\otimes n})\) such that \(X_s\) is an affine open neighbourhood of \(x\). Choose \(\tau \in \mathcal{H}^p(X_s)\) which maps to a nonzero element of the stalk \(\mathcal{H}^p_x\); this is possible as \(\mathcal{H}^p\) is quasi-coherent and \(X_s\) is affine. Since taking sections over \(X_s\) is an exact functor on quasi-coherent modules, we can find a section \(\tau' \in \mathcal{F}^p(X_s)\) mapping to zero in \(\mathcal{F}^{p + 1}(X_s)\) and mapping to \(\tau\) in \(\mathcal{H}^p(X_s)\). By Properties, Lemma 01PW there exists an \(m\) such that \(\tau' \otimes s^{\otimes m}\) is the image of a section \(\tau'' \in \Gamma(X, \mathcal{F}^p \otimes \mathcal{L}^{\otimes mn})\). Applying the same lemma once more, we find \(l \geq 0\) such that \(\tau'' \otimes s^{\otimes l}\) maps to zero in \(\mathcal{F}^{p + 1} \otimes \mathcal{L}^{\otimes (m + l)n}\). Then \(\tau''\) gives a nonzero class in \(H^p(X, K \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{L}^{(m + l)n})\) as desired.

Lemma

Let \(A\) be a ring. Let \(X = \mathbf{P}^n_A\). For every \(a \in \mathbf{Z}\) there exists an exact complex \[0 \to \mathcal{O}_X(a) \to \ldots \to \mathcal{O}_X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_X(a + n + 1) \to 0\] of vector bundles on \(X\).

Proof

Recall that \(\mathbf{P}^n_A\) is \(\text{Proj}(A[X_0, \ldots, X_n])\), see Constructions, Definition 01NF. Consider the Koszul complex \[K_\bullet = K_\bullet(A[X_0, \ldots, X_n], X_0, \ldots, X_n)\] over \(S = A[X_0, \ldots, X_n]\) on \(X_0, \ldots, X_n\). Since \(X_0, \ldots, X_n\) is clearly a regular sequence in the polynomial ring \(S\), we see that (More on Algebra, Lemma 062F) that the Koszul complex \(K_\bullet\) is exact, except in degree \(0\) where the cohomology is \(S/(X_0, \ldots, X_n)\). Note that \(K_\bullet\) becomes a complex of graded modules if we put the generators of \(K_i\) in degree \(+i\). In other words an exact complex \[0 \to S(-n - 1) \to \ldots \to S(-n - 1 + i)^{\oplus {n \choose i}} \to \ldots \to S \to S/(X_0, \ldots, X_n) \to 0\] Applying the exact functor \(\tilde{\ }\) functor of Constructions, Lemma 01M7 and using that the last term is in the kernel of this functor, we obtain the exact complex \[0 \to \mathcal{O}_X(-n - 1) \to \ldots \to \mathcal{O}_X(-n - 1 + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_X \to 0\] Twisting by the invertible sheaves \(\mathcal{O}_X(n + a + 1)\) we get the exact complexes of the lemma.

Lemma

Let \(A\) be a ring. Let \(X = \mathbf{P}^n_A\). Then \[E = \mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)\] is a generator (Derived Categories, Definition 09SJ) of \(D_\QCoh(X)\).

Proof

Let \(K \in D_\QCoh(\mathcal{O}_X)\). Assume \(\Hom(E, K[p]) = 0\) for all \(p \in \mathbf{Z}\). We have to show that \(K = 0\). By Derived Categories, Lemma 09SK we see that \(\Hom(E', K[p])\) is zero for all \(E' \in \langle E \rangle\) and \(p \in \mathbf{Z}\). By Lemma 0BQS applied with \(a = -n - 1\) we see that \(\mathcal{O}_X(-n - 1) \in \langle E \rangle\) because it is quasi-isomorphic to a finite complex whose terms are finite direct sums of summands of \(E\). Repeating the argument with \(a = -n - 2\) we see that \(\mathcal{O}_X(-n - 2) \in \langle E \rangle\). Arguing by induction we find that \(\mathcal{O}_X(-m) \in \langle E \rangle\) for all \(m \geq 0\). Since \[\Hom(\mathcal{O}_X(-m), K[p]) = H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(m)) = H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(1)^{\otimes m})\] we conclude that \(K = 0\) by Lemma 0BQR. (This also uses that \(\mathcal{O}_X(1)\) is an ample invertible sheaf on \(X\) which follows from Properties, Lemma 01Q2.)

Remark

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated schemes. Let \(E \in D_\QCoh(\mathcal{O}_Y)\) be a generator (see Theorem 09IS). Then the following are equivalent

1. for \(K \in D_\QCoh(\mathcal{O}_X)\) we have \(Rf_*K = 0\) if and only if \(K = 0\),

2. \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) reflects isomorphisms, and

3. \(Lf^*E\) is a generator for \(D_\QCoh(\mathcal{O}_X)\).

The equivalence between (1) and (2) is a formal consequence of the fact that \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)\) is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that \(Lf^*\) is the left adjoint to \(Rf_*\). These conditions hold if \(f\) is affine (Lemma 08I8) or if \(f\) is an open immersion, or if \(f\) is a composition of such. We conclude that

1. if \(X\) is a quasi-affine scheme then \(\mathcal{O}_X\) is a generator for \(D_\QCoh(\mathcal{O}_X)\),

2. if \(X \subset \mathbf{P}^n_A\) is a quasi-compact locally closed subscheme, then \(\mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)\) is a generator for \(D_\QCoh(\mathcal{O}_X)\) by Lemma 0A9V.

Compact and perfect objects

Let \(X\) be a Noetherian scheme of finite dimension. By Cohomology, Proposition 02UZ and Cohomology on Sites, Lemma 094D the sheaves of modules \(j_!\mathcal{O}_U\) are compact objects of \(D(\mathcal{O}_X)\) for all opens \(U \subset X\). These sheaves are typically not quasi-coherent, hence these do not give perfect objects of the derived category \(D(\mathcal{O}_X)\). However, if we restrict ourselves to complexes with quasi-coherent cohomology sheaves, then this does not happen. Here is the precise statement.

Proposition

Let \(X\) be a quasi-compact and quasi-separated scheme. An object of \(D_\QCoh(\mathcal{O}_X)\) is compact if and only if it is perfect.

Proof

If \(K\) is a perfect object of \(D(\mathcal{O}_X)\) with dual \(K^\vee\) (Cohomology, Lemma 08DQ) we have \[\Hom_{D(\mathcal{O}_X)}(K, M) = H^0(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} M)\] functorially in \(M\). Since \(K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} -\) commutes with direct sums and since \(H^0(X, -)\) commutes with direct sums on \(D_\QCoh(\mathcal{O}_X)\) by Lemma 08DZ we conclude that \(K\) is compact in \(D_\QCoh(\mathcal{O}_X)\).

Conversely, let \(K\) be a compact object of \(D_\QCoh(\mathcal{O}_X)\). To show that \(K\) is perfect, it suffices to show that \(K|_U\) is perfect for every affine open \(U \subset X\), see Cohomology, Lemma 08CN. Observe that \(j : U \to X\) is a quasi-compact and separated morphism. Hence \(Rj_* : D_\QCoh(\mathcal{O}_U) \to D_\QCoh(\mathcal{O}_X)\) commutes with direct sums, see Lemma 08DZ. Thus the adjointness of restriction to \(U\) and \(Rj_*\) implies that \(K|_U\) is a compact object of \(D_\QCoh(\mathcal{O}_U)\). Hence we reduce to the case that \(X\) is affine.

Assume \(X = \Spec(A)\) is affine. By Lemma 06Z0 the problem is translated into the same problem for \(D(A)\). For \(D(A)\) the result is More on Algebra, Proposition 07LT.

Remark

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(G\) be a perfect object of \(D(\mathcal{O}_X)\) which is a generator for \(D_\QCoh(\mathcal{O}_X)\). By Theorem 09IS there is at least one of these. Combining Lemma 08DT with Proposition 09M1 and with Derived Categories, Proposition 09SR we see that \(G\) is a classical generator for \(D_{perf}(\mathcal{O}_X)\).

The following result is a strengthening of Proposition 09M1. Let \(T \subset X\) be a closed subset of a scheme \(X\). As before \(D_T(\mathcal{O}_X)\) denotes the strictly full, saturated, triangulated subcategory of \(D(\mathcal{O}_X)\) consisting of objects supported on \(T\) (Definition 08DA). Since taking direct sums commutes with taking cohomology sheaves, it follows that \(D_T(\mathcal{O}_X)\) has direct sums and that they are equal to direct sums in \(D(\mathcal{O}_X)\).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. An object of \(D_{\QCoh, T}(\mathcal{O}_X)\) is compact if and only if it is perfect as an object of \(D(\mathcal{O}_X)\).

Proof

We observe that \(D_{\QCoh, T}(\mathcal{O}_X)\) is a triangulated category with direct sums by the remark preceding the lemma. By Proposition 09M1 the perfect objects define compact objects of \(D(\mathcal{O}_X)\) hence a fortiori of any subcategory preserved under taking direct sums. For the converse we will use there exists a generator \(E \in D_{\QCoh, T}(\mathcal{O}_X)\) which is a perfect complex of \(\mathcal{O}_X\)-modules, see Lemma 0A9A. Hence by the above, \(E\) is compact. Then it follows from Derived Categories, Proposition 09SR that \(E\) is a classical generator of the full subcategory of compact objects of \(D_{\QCoh, T}(\mathcal{O}_X)\). Thus any compact object can be constructed out of \(E\) by a finite sequence of operations consisting of (a) taking shifts, (b) taking finite direct sums, (c) taking cones, and (d) taking direct summands. Each of these operations preserves the property of being perfect and the result follows.

Remark

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. Let \(G\) be a perfect object of \(D_{\QCoh, T}(\mathcal{O}_X)\) which is a generator for \(D_{\QCoh, T}(\mathcal{O}_X)\). By Lemma 0A9A there is at least one of these. Combining the fact that \(D_{\QCoh, T}(\mathcal{O}_X)\) has direct sums with Lemma 0A9B and with Derived Categories, Proposition 09SR we see that \(G\) is a classical generator for \(D_{perf, T}(\mathcal{O}_X)\).

The following lemma is an application of the ideas that go into the proof of the preceding lemma.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(U = X \setminus T\) is quasi-compact. Let \(\alpha : P \to E\) be a morphism of \(D_\QCoh(\mathcal{O}_X)\) with either

1. \(P\) is perfect and \(E\) supported on \(T\), or

2. \(P\) pseudo-coherent, \(E\) supported on \(T\), and \(E\) bounded below.

Then there exists a perfect complex of \(\mathcal{O}_X\)-modules \(I\) and a map \(I \to \mathcal{O}_X[0]\) such that \(I \otimes^\mathbf{L} P \to E\) is zero and such that \(I|_U \to \mathcal{O}_U[0]\) is an isomorphism.

Proof

Set \(\mathcal{D} = D_{\QCoh, T}(\mathcal{O}_X)\). In both cases the complex \(K = R\SheafHom(P, E)\) is an object of \(\mathcal{D}\). See Lemma 0A6H for quasi-coherence. It is clear that \(K\) is supported on \(T\) as formation of \(R\SheafHom\) commutes with restriction to opens. The map \(\alpha\) defines an element of \(H^0(K) = \Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X[0], K)\). Then it suffices to prove the result for the map \(\alpha : \mathcal{O}_X[0] \to K\).

Let \(E \in \mathcal{D}\) be a perfect generator, see Lemma 0A9A. Write \[K = \text{hocolim} K_n\] as in Derived Categories, Lemma 09SN using the generator \(E\). Since the functor \(\mathcal{D} \to D(\mathcal{O}_X)\) commutes with direct sums, we see that \(K = \text{hocolim} K_n\) holds in \(D(\mathcal{O}_X)\). Since \(\mathcal{O}_X\) is a compact object of \(D(\mathcal{O}_X)\) we find an \(n\) and a morphism \(\alpha_n : \mathcal{O}_X \to K_n\) which gives rise to \(\alpha\), see Derived Categories, Lemma 094A. By Derived Categories, Lemma 09SP applied to the morphism \(\mathcal{O}_X[0] \to K_n\) in the ambient category \(D(\mathcal{O}_X)\) we see that \(\alpha_n\) factors as \(\mathcal{O}_X[0] \to Q \to K_n\) where \(Q\) is an object of \(\langle E \rangle\). We conclude that \(Q\) is a perfect complex supported on \(T\).

Choose a distinguished triangle \[I \to \mathcal{O}_X[0] \to Q \to I[1]\] By construction \(I\) is perfect, the map \(I \to \mathcal{O}_X[0]\) restricts to an isomorphism over \(U\), and the composition \(I \to K\) is zero as \(\alpha\) factors through \(Q\). This proves the lemma.

Derived categories as module categories

In this section we draw some conclusions of what has gone before. Before we do so we need a couple more lemmas.

Lemma

Let \(X\) be a scheme. Let \(K^\bullet\) be a complex of \(\mathcal{O}_X\)-modules whose cohomology sheaves are quasi-coherent. Let \((E, d) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)\) be the endomorphism differential graded algebra. Then the functor \[- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X)\] of Differential Graded Algebra, Lemma 09LX has image contained in \(D_\QCoh(\mathcal{O}_X)\).

Proof

Let \(P\) be a differential graded \(E\)-module with property (P) and let \(F_\bullet\) be a filtration on \(P\) as in Differential Graded Algebra, Section 09KK. Then we have \[P \otimes_E K^\bullet = \text{hocolim}\ F_iP \otimes_E K^\bullet\] Each of the \(F_iP\) has a finite filtration whose graded pieces are direct sums of \(E[k]\). The result follows easily.

The following result is taken from [BvdB].

Theorem

Let \(X\) be a quasi-compact and quasi-separated scheme. Then there exist a differential graded algebra \((E, \text{d})\) with only a finite number of nonzero cohomology groups \(H^i(E)\) such that \(D_\QCoh(\mathcal{O}_X)\) is equivalent to \(D(E, \text{d})\).

Proof

Let \(K^\bullet\) be a K-injective complex of \(\mathcal{O}\)-modules which is perfect and generates \(D_\QCoh(\mathcal{O}_X)\). Such a thing exists by Theorem 09IS and the existence of K-injective resolutions. We will show the theorem holds with \[(E, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)\] where \(\text{Comp}^{dg}(\mathcal{O}_X)\) is the differential graded category of complexes of \(\mathcal{O}\)-modules. Please see Differential Graded Algebra, Section 09LU. Since \(K^\bullet\) is K-injective we have [09M6]\[\begin{equation} H^n(E) = \Ext^n_{D(\mathcal{O}_X)}(K^\bullet, K^\bullet) \end{equation}\] for all \(n \in \mathbf{Z}\). Only a finite number of these Exts are nonzero by Lemma 09M4. Consider the functor \[- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X)\] of Differential Graded Algebra, Lemma 09LX. Since \(K^\bullet\) is perfect, it defines a compact object of \(D(\mathcal{O}_X)\), see Proposition 09M1. Combined with (09M6) the functor above is fully faithful as follows from Differential Graded Algebra, Lemmas 09LZ. It has a right adjoint \[R\Hom(K^\bullet, - ) : D(\mathcal{O}_X) \longrightarrow D(E, \text{d})\] by Differential Graded Algebra, Lemmas 09LY which is a left quasi-inverse functor by generalities on adjoint functors. On the other hand, it follows from Lemma 09M3 that we obtain \[- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D_\QCoh(\mathcal{O}_X)\] and by our choice of \(K^\bullet\) as a generator of \(D_\QCoh(\mathcal{O}_X)\) the kernel of the adjoint restricted to \(D_\QCoh(\mathcal{O}_X)\) is zero. A formal argument shows that we obtain the desired equivalence, see Derived Categories, Lemma 09J1.

Remark

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. The analogue of Theorem 09M5 holds for \(D_{\QCoh, T}(\mathcal{O}_X)\). This follows from the exact same argument as in the proof of the theorem, using Lemmas 0A9A and 0A9B and a variant of Lemma 09M3 with supports. If we ever need this, we will precisely state the result here and give a detailed proof.

Remark

Let \(X\) be a quasi-compact and quasi-separated scheme over a ring \(R\). By the construction of the proof of Theorem 09M5 there exists a differential graded algebra \((A, \text{d})\) over \(R\) such that \(D_\QCoh(X)\) is \(R\)-linearly equivalent to \(D(A, \text{d})\) as a triangulated category. One may ask: how unique is \((A, \text{d})\)? The answer is (only) slightly better than just saying that \((A, \text{d})\) is well defined up to derived equivalence. Namely, suppose that \((B, \text{d})\) is a second such pair. Then we have \[(A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)\] and \[(B, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, L^\bullet)\] for some K-injective complexes \(K^\bullet\) and \(L^\bullet\) of \(\mathcal{O}_X\)-modules corresponding to perfect generators of \(D_\QCoh(\mathcal{O}_X)\). Set \[\Omega = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, L^\bullet) \quad \Omega' = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(L^\bullet, K^\bullet)\] Then \(\Omega\) is a differential graded \(B^{opp} \otimes_R A\)-module and \(\Omega'\) is a differential graded \(A^{opp} \otimes_R B\)-module. Moreover, the equivalence \[D(A, \text{d}) \to D_\QCoh(\mathcal{O}_X) \to D(B, \text{d})\] is given by the functor \(- \otimes_A^\mathbf{L} \Omega'\) and similarly for the quasi-inverse. Thus we are in the situation of Differential Graded Algebra, Remark 09ST. If we ever need this remark we will provide a precise statement with a detailed proof here.

Characterizing pseudo-coherent complexes, I

We can use the methods above to characterize pseudo-coherent objects as derived homotopy limits of approximations by perfect objects.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(K \in D(\mathcal{O}_X)\). The following are equivalent

1. \(K\) is pseudo-coherent, and

2. \(K = \text{hocolim} K_n\) where \(K_n\) is perfect and \(\tau_{\geq -n}K_n \to \tau_{\geq -n}K\) is an isomorphism for all \(n\).

Proof

The implication (2) \(\Rightarrow\) (1) is true on any ringed space. Namely, assume (2) holds. Recall that a perfect object of the derived category is pseudo-coherent, see Cohomology, Lemma 08CQ. Then it follows from the definitions that \(\tau_{\geq -n}K_n\) is \((-n + 1)\)-pseudo-coherent and hence \(\tau_{\geq -n}K\) is \((-n + 1)\)-pseudo-coherent, hence \(K\) is \((-n + 1)\)-pseudo-coherent. This is true for all \(n\), hence \(K\) is pseudo-coherent, see Cohomology, Definition 08CB.

Assume (1). We start by choosing an approximation \(K_1 \to K\) of \((X, K, -2)\) by a perfect complex \(K_1\), see Definitions 08EM and 08EN and Theorem 08ES. Suppose by induction we have \[K_1 \to K_2 \to \ldots \to K_n \to K\] with \(K_i\) perfect such that such that \(\tau_{\geq -i}K_i \to \tau_{\geq -i}K\) is an isomorphism for all \(1 \leq i \leq n\). Then we pick \(a \leq b\) as in Lemma 09M4 for the perfect object \(K_n\). Choose an approximation \(K_{n + 1} \to K\) of \((X, K, \min(a - 1, -n - 1))\). Choose a distinguished triangle \[K_{n + 1} \to K \to C \to K_{n + 1}[1]\] Then we see that \(C \in D_\QCoh(\mathcal{O}_X)\) has \(H^i(C) = 0\) for \(i \geq a\). Thus by our choice of \(a, b\) we see that \(\Hom_{D(\mathcal{O}_X)}(K_n, C) = 0\). Hence the composition \(K_n \to K \to C\) is zero. Hence by Derived Categories, Lemma 0149 we can factor \(K_n \to K\) through \(K_{n + 1}\) proving the induction step.

We still have to prove that \(K = \text{hocolim} K_n\). This follows by an application of Derived Categories, Lemma 0CRK to the functors \(H^i( - ) : D(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)\) and our choice of \(K_n\).

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(T \subset X\) be a closed subset such that \(X \setminus T\) is quasi-compact. Let \(K \in D(\mathcal{O}_X)\) supported on \(T\). The following are equivalent

1. \(K\) is pseudo-coherent, and

2. \(K = \text{hocolim} K_n\) where \(K_n\) is perfect, supported on \(T\), and \(\tau_{\geq -n}K_n \to \tau_{\geq -n}K\) is an isomorphism for all \(n\).

Proof

The proof of this lemma is exactly the same as the proof of Lemma 0DJN except that in the choice of the approximations we use the triples \((T, K, m)\).

An example equivalence

In Section 0BQQ we proved that the derived category of projective space \(\mathbf{P}^n_A\) over a ring \(A\) is generated by a vector bundle, in fact a direct sum of shifts of the structure sheaf. In this section we prove this determines an equivalence of \(D_\QCoh(\mathcal{O}_{\mathbf{P}^n_A})\) with the derived category of an \(A\)-algebra.

Before we can state the result we need some notation. Let \(A\) be a ring. Let \(X = \mathbf{P}^n_A = \text{Proj}(S)\) where \(S = A[X_0, \ldots, X_n]\). By Lemma 0A9V we know that [0CS8]\[\begin{equation} P = \mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n) \end{equation}\] is a perfect generator of \(D_\QCoh(\mathcal{O}_X)\). Consider the (noncommutative) \(A\)-algebra [0CS9]\[\begin{equation} R = \Hom_{\mathcal{O}_X}(P, P) = \left( \begin{matrix} S_0 & S_1 & S_2 & \ldots & \ldots \\ 0 & S_0 & S_1 & \ldots & \ldots\\ 0 & 0 & S_0 & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & S_0 \end{matrix} \right) \end{equation}\] with obvious multiplication and addition. If we view \(P\) as a complex of \(\mathcal{O}_X\)-modules in the usual way (i.e., with \(P\) in degree \(0\) and zero in every other degree), then we have \[R = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(P, P)\] where on the right hand side we view \(R\) as a differential graded algebra over \(A\) with zero differential (i.e., with \(R\) in degree \(0\) and zero in every other degree). According to the discussion in Differential Graded Algebra, Section 09LU we obtain a derived functor \[- \otimes_R^\mathbf{L} P : D(R) \longrightarrow D(\mathcal{O}_X),\] see especially Differential Graded Algebra, Lemma 09LX. By Lemma 09M3 we see that the essential image of this functor is contained in \(D_\QCoh(\mathcal{O}_X)\).

Lemma

Let \(A\) be a ring. Let \(X = \mathbf{P}^n_A = \text{Proj}(S)\) where \(S = A[X_0, \ldots, X_n]\). With \(P\) as in (0CS8) and \(R\) as in (0CS9) the functor \[- \otimes_R^\mathbf{L} P : D(R) \longrightarrow D_\QCoh(\mathcal{O}_X)\] is an \(A\)-linear equivalence of triangulated categories sending \(R\) to \(P\).

In words: the derived category of quasi-coherent modules on projective space is equivalent to the derived category of modules over a (noncommutative) algebra. This property of projective space appears to be quite unusual among all projective schemes over \(A\).

Proof

To prove that our functor is fully faithful it suffices to prove that \(\Ext^i_X(P, P)\) is zero for \(i \not = 0\) and equal to \(R\) for \(i = 0\), see Differential Graded Algebra, Lemma 09LZ. As in the proof of Lemma 09M4 we see that \[\Ext^i_X(P, P) = H^i(X, P^\wedge \otimes P) = \bigoplus\nolimits_{0 \leq a, b \leq n} H^i(X, \mathcal{O}_X(a - b))\] By the computation of cohomology of projective space (Cohomology of Schemes, Lemma 01XT) we find that these \(\Ext\)-groups are zero unless \(i = 0\). For \(i = 0\) we recover \(R\) because this is how we defined \(R\) in (0CS9). By Differential Graded Algebra, Lemma 09LY our functor has a right adjoint, namely \(R\Hom(P, -) : D_\QCoh(\mathcal{O}_X) \to D(R)\). Since \(P\) is a generator for \(D_\QCoh(\mathcal{O}_X)\) by Lemma 0A9V we see that the kernel of \(R\Hom(P, -)\) is zero. Hence our functor is an equivalence of triangulated categories by Derived Categories, Lemma 09J1.

The coherator revisited

In Section 08D6 we constructed and studied the right adjoint \(RQ_X\) to the canonical functor \(D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)\). It was constructed as the right derived extension of the coherator \(Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)\). In this section, we study when the inclusion functor \[D_\QCoh(\mathcal{O}_X) \longrightarrow D(\mathcal{O}_X)\] has a right adjoint. If this right adjoint exists, we will denote³ it \[DQ_X : D(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_X)\] It turns out that quasi-compact and quasi-separated schemes have such a right adjoint.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. The inclusion functor \(D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)\) has a right adjoint \(DQ_X\).

Proof

We will use the induction principle as in Cohomology of Schemes, Lemma 08DR to prove this. If \(D(\QCoh(\mathcal{O}_X)) \to D_\QCoh(\mathcal{O}_X)\) is an equivalence, then the lemma is true because the functor \(RQ_X\) of Section 08D6 is a right adjoint to the functor \(D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)\). In particular, our lemma is true for affine schemes, see Lemma 08D9. Thus we see that it suffices to show: if \(X = U \cup V\) is a union of two quasi-compact opens and the lemma holds for \(U\), \(V\), and \(U \cap V\), then the lemma holds for \(X\).

The adjoint exists if and only if for every object \(K\) of \(D(\mathcal{O}_X)\) we can find a distinguished triangle \[E' \to E \to K \to E'[1]\] in \(D(\mathcal{O}_X)\) such that \(E'\) is in \(D_\QCoh(\mathcal{O}_X)\) and such that \(\Hom(M, K) = 0\) for all \(M\) in \(D_\QCoh(\mathcal{O}_X)\). See Derived Categories, Lemma 0CQS. Consider the distinguished triangle \[E \to Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to Rj_{U \cap V, *}E|_{U \cap V} \to E[1]\] in \(D(\mathcal{O}_X)\) of Cohomology, Lemma 08BV. By Derived Categories, Lemma 0CQR it suffices to construct the desired distinguished triangles for \(Rj_{U, *}E|_U\), \(Rj_{V, *}E|_V\), and \(Rj_{U \cap V, *}E|_{U \cap V}\). This reduces us to the statement discussed in the next paragraph.

Let \(j : U \to X\) be an open immersion corresponding with \(U\) a quasi-compact open for which the lemma is true. Let \(L\) be an object of \(D(\mathcal{O}_U)\). Then there exists a distinguished triangle \[E' \to Rj_*L \to K \to E'[1]\] in \(D(\mathcal{O}_X)\) such that \(E'\) is in \(D_\QCoh(\mathcal{O}_X)\) and such that \(\Hom(M, K) = 0\) for all \(M\) in \(D_\QCoh(\mathcal{O}_X)\). To see this we choose a distinguished triangle \[L' \to L \to Q \to L'[1]\] in \(D(\mathcal{O}_U)\) such that \(L'\) is in \(D_\QCoh(\mathcal{O}_U)\) and such that \(\Hom(N, Q) = 0\) for all \(N\) in \(D_\QCoh(\mathcal{O}_U)\). This is possible because the statement in Derived Categories, Lemma 0CQS is an if and only if. We obtain a distinguished triangle \[Rj_*L' \to Rj_*L \to Rj_*Q \to Rj_*L'[1]\] in \(D(\mathcal{O}_X)\). Observe that \(Rj_*L'\) is in \(D_\QCoh(\mathcal{O}_X)\) by Lemma 08D5. On the other hand, if \(M\) in \(D_\QCoh(\mathcal{O}_X)\), then \[\Hom(M, Rj_*Q) = \Hom(Lj^*M, Q) = 0\] because \(Lj^*M\) is in \(D_\QCoh(\mathcal{O}_U)\) by Lemma 08DW. This finishes the proof.

Proof

The adjoint exists by Derived Categories, Proposition 0A8G. The hypotheses are satisfied: First, note that \(D_\QCoh(\mathcal{O}_X)\) has direct sums and direct sums commute with the inclusion functor (Lemma 08DT). On the other hand, \(D_\QCoh(\mathcal{O}_X)\) is compactly generated because it has a perfect generator Theorem 09IS and because perfect objects are compact by Proposition 09M1.

Lemma

Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of schemes. If the right adjoints \(DQ_X\) and \(DQ_Y\) of the inclusion functors \(D_\QCoh \to D\) exist for \(X\) and \(Y\), then \[Rf_* \circ DQ_X = DQ_Y \circ Rf_*\]

Proof

The statement makes sense because \(Rf_*\) sends \(D_\QCoh(\mathcal{O}_X)\) into \(D_\QCoh(\mathcal{O}_Y)\) by Lemma 08D5. The statement is true because \(Lf^*\) similarly maps \(D_\QCoh(\mathcal{O}_Y)\) into \(D_\QCoh(\mathcal{O}_X)\) (Lemma 08DW) and hence both \(Rf_* \circ DQ_X\) and \(DQ_Y \circ Rf_*\) are right adjoint to \(Lf^* : D_\QCoh(\mathcal{O}_Y) \to D(\mathcal{O}_X)\).

Remark

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(X = U \cup V\) with \(U\) and \(V\) quasi-compact open. By Lemma 0CR0 the functors \(DQ_X\), \(DQ_U\), \(DQ_V\), \(DQ_{U \cap V}\) exist. Moreover, there is a canonical distinguished triangle \[DQ_X(K) \to Rj_{U, *}DQ_U(K|_U) \oplus Rj_{V, *}DQ_V(K|_V) \to Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V}) \to\] for any \(K \in D(\mathcal{O}_X)\). This follows by applying the exact functor \(DQ_X\) to the distinguished triangle of Cohomology, Lemma 08BV and using Lemma 0CR1 three times.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. The functor \(DQ_X\) of Lemma 0CR0 has the following boundedness property: there exists an integer \(N = N(X)\) such that, if \(K\) in \(D(\mathcal{O}_X)\) with \(H^i(U, K) = 0\) for \(U\) affine open in \(X\) and \(i \not \in [a, b]\), then the cohomology sheaves \(H^i(DQ_X(K))\) are zero for \(i \not \in [a, b + N]\).

Proof

We will prove this using the induction principle of Cohomology of Schemes, Lemma 08DR.

If \(X\) is affine, then the lemma is true with \(N = 0\) because then \(RQ_X = DQ_X\) is given by taking the complex of quasi-coherent sheaves associated to \(R\Gamma(X, K)\). See Lemmas 06Z0 and 08D9.

Assume \(U, V\) are quasi-compact open in \(X\) and the lemma holds for \(U\), \(V\), and \(U \cap V\). Say with integers \(N(U)\), \(N(V)\), and \(N(U \cap V)\). Now suppose \(K\) is in \(D(\mathcal{O}_X)\) with \(H^i(W, K) = 0\) for all affine open \(W \subset X\) and all \(i \not \in [a, b]\). Then \(K|_U\), \(K|_V\), \(K|_{U \cap V}\) have the same property. Hence we see that \(RQ_U(K|_U)\) and \(RQ_V(K|_V)\) and \(RQ_{U \cap V}(K|_{U \cap V})\) have vanishing cohomology sheaves outside the interval \([a, b + \max(N(U), N(V), N(U \cap V))\). Since the functors \(Rj_{U, *}\), \(Rj_{V, *}\), \(Rj_{U \cap V, *}\) have finite cohomological dimension on \(D_\QCoh\) by Lemma 08D5 we see that there exists an \(N\) such that \(Rj_{U, *}DQ_U(K|_U)\), \(Rj_{V, *}DQ_V(K|_V)\), and \(Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V})\) have vanishing cohomology sheaves outside the interval \([a, b + N]\). Then finally we conclude by the distinguished triangle of Remark 0CR2.

Example

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \((\mathcal{F}_n)\) be an inverse system of quasi-coherent sheaves. Since \(DQ_X\) is a right adjoint it commutes with products and therefore with derived limits. Hence we see that \[DQ_X(R\lim \mathcal{F}_n) = (R\lim\text{ in }D_\QCoh(\mathcal{O}_X))(\mathcal{F}_n)\] where the first \(R\lim\) is taken in \(D(\mathcal{O}_X)\). In fact, let’s write \(K = R\lim \mathcal{F}_n\) for this. For any affine open \(U \subset X\) we have \[H^i(U, K) = H^i(R\Gamma(U, R\lim \mathcal{F}_n)) = H^i(R\lim R\Gamma(U, \mathcal{F}_n)) = H^i(R\lim \Gamma(U, \mathcal{F}_n))\] since cohomology commutes with derived limits and since the quasi-coherent sheaves \(\mathcal{F}_n\) have no higher cohomology on affines. By the computation of \(R\lim\) in the category of abelian groups, we see that \(H^i(U, K) = 0\) unless \(i \in [0, 1]\). Then finally we conclude that the \(R\lim\) in \(D_\QCoh(\mathcal{O}_X)\), which is \(DQ_X(K)\) by the above, is in \(D^b_\QCoh(\mathcal{O}_X)\) by Lemma 0CSA.

Cohomology and base change, IV

This section continues the discussion of Cohomology of Schemes, Section 07VJ. First, we have a very general version of the projection formula for quasi-compact and quasi-separated morphisms of schemes and complexes with quasi-coherent cohomology sheaves.

Lemma

Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of schemes. For \(E\) in \(D_\QCoh(\mathcal{O}_X)\) and \(K\) in \(D_\QCoh(\mathcal{O}_Y)\) the map \[Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K \longrightarrow Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)\] defined in Cohomology, Equation (0B53) is an isomorphism.

Proof

To check the map is an isomorphism we may work locally on \(Y\). Hence we reduce to the case that \(Y\) is affine.

Suppose that \(K = \bigoplus K_i\) is a direct sum of some complexes \(K_i \in D_\QCoh(\mathcal{O}_Y)\). If the statement holds for each \(K_i\), then it holds for \(K\). Namely, the functors \(Lf^*\) and \(\otimes^\mathbf{L}\) preserve direct sums by construction and \(Rf_*\) commutes with direct sums (for complexes with quasi-coherent cohomology sheaves) by Lemma 08DZ. Moreover, suppose that \(K \to L \to M \to K[1]\) is a distinguished triangle in \(D_\QCoh(Y)\). Then if the statement of the lemma holds for two of \(K, L, M\), then it holds for the third (as the functors involved are exact functors of triangulated categories).

Assume \(Y\) affine, say \(Y = \Spec(A)\). The functor \(\widetilde{\ } : D(A) \to D_\QCoh(\mathcal{O}_Y)\) is an equivalence (Lemma 06Z0). Let \(T\) be the property for \(K \in D(A)\) that the statement of the lemma holds for \(\widetilde{K}\). The discussion above and More on Algebra, Remark 09PB shows that it suffices to prove \(T\) holds for \(A[k]\). This finishes the proof, as the statement of the lemma is clear for shifts of the structure sheaf.

Definition

Let \(S\) be a scheme. Let \(X\), \(Y\) be schemes over \(S\). We say \(X\) and \(Y\) are Tor independent over \(S\) if for every \(x \in X\) and \(y \in Y\) mapping to the same point \(s \in S\) the rings \(\mathcal{O}_{X, x}\) and \(\mathcal{O}_{Y, y}\) are Tor independent over \(\mathcal{O}_{S, s}\) (see More on Algebra, Definition 0660).

Lemma

Let \(f : X \to S\) and \(g : Y \to S\) be morphisms of schemes. The following are equivalent

1. \(X\) and \(Y\) are tor independent over \(S\), and

2. for every affine opens \(U \subset X\), \(V \subset Y\), \(W \subset S\) with \(f(U) \subset W\) and \(g(V) \subset W\) the rings \(\mathcal{O}_X(U)\) and \(\mathcal{O}_Y(V)\) are tor independent over \(\mathcal{O}_S(W)\).

3. there exists an affine open overing \(S = \bigcup W_i\) and for each \(i\) affine open coverings \(f^{-1}(W_i) = \bigcup U_{ij}\) and \(g^{-1}(W_i) = \bigcup V_{ik}\) such that the rings \(\mathcal{O}_X(U_{ij})\) and \(\mathcal{O}_Y(V_{ik})\) are tor independent over \(\mathcal{O}_S(W_i)\) for all \(i, j, k\).

Proof

Omitted. Hint: use More on Algebra, Lemma 08HX.

Lemma

Let \(X \to S\) and \(Y \to S\) be morphisms of schemes. Let \(S' \to S\) be a morphism of schemes and denote \(X' = X \times_S S'\) and \(Y' = Y \times_S S'\). If \(X\) and \(Y\) are tor independent over \(S\) and \(S' \to S\) is flat, then \(X'\) and \(Y'\) are tor independent over \(S'\).

Proof

Omitted. Hint: use Lemma 0FXV and on affine opens use More on Algebra, Lemma 0FXF.

Lemma

Let \(g : S' \to S\) be a morphism of schemes. Let \(f : X \to S\) be quasi-compact and quasi-separated. Consider the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] If \(X\) and \(S'\) are Tor independent over \(S\), then for all \(E \in D_\QCoh(\mathcal{O}_X)\) the canonical arrow \(Lg^*Rf_*E \to Rf'_*L(g')^*E\) is an isomorphism.

Proof

For any object \(E\) of \(D(\mathcal{O}_X)\) we can use Cohomology, Remark 08HY to get a canonical base change map \(Lg^*Rf_*E \to Rf'_*L(g')^*E\). To check this is an isomorphism we may work locally on \(S'\). Hence we may assume \(g : S' \to S\) is a morphism of affine schemes. In particular, \(g\) is affine and it suffices to show that \[Rg_*Lg^*Rf_*E \to Rg_*Rf'_*L(g')^*E = Rf_*(Rg'_* L(g')^* E)\] is an isomorphism, see Lemma 08I8 (and use Lemmas 08DW, 08DX, and 08D5 to see that the objects \(Rf'_*L(g')^*E\) and \(Lg^*Rf_*E\) have quasi-coherent cohomology sheaves). Note that \(g'\) is affine as well (Morphisms, Lemma 01SD). By Lemma 08I9 the map becomes a map \[Rf_*E \otimes_{\mathcal{O}_S}^\mathbf{L} g_*\mathcal{O}_{S'} \longrightarrow Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'})\] Observe that \(g'_*\mathcal{O}_{X'} = f^*g_*\mathcal{O}_{S'}\) (by affine base change, see Cohomology of Schemes, Lemma 02KG). Thus by Lemma 08EU it suffices to prove that \(Lf^*g_*\mathcal{O}_{S'} = f^*g_*\mathcal{O}_{S'}\). This follows from our assumption that \(X\) and \(S'\) are Tor independent over \(S\). Namely, to check it we may work locally on \(X\), hence we may also assume \(X\) is affine. Say \(X = \Spec(A)\), \(S = \Spec(R)\) and \(S' = \Spec(R')\). Our assumption implies that \(A\) and \(R'\) are Tor independent over \(R\) (More on Algebra, Lemma 08HX), i.e., \(\text{Tor}_i^R(A, R') = 0\) for \(i > 0\). In other words \(A \otimes_R^\mathbf{L} R' = A \otimes_R R'\) which exactly means that \(Lf^*g_*\mathcal{O}_{S'} = f^*g_*\mathcal{O}_{S'}\) (use Lemma 08DW).

The following lemma will be used in the chapter on dualizing complexes.

Lemma

Consider a cartesian square \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] of quasi-compact and quasi-separated schemes. Assume \(g\) and \(f\) Tor independent and \(S = \Spec(R)\), \(S' = \Spec(R')\) affine. For \(M, K \in D(\mathcal{O}_X)\) the canonical map \[R\Hom_X(M, K) \otimes^\mathbf{L}_R R' \longrightarrow R\Hom_{X'}(L(g')^*M, L(g')^*K)\] in \(D(R')\) is an isomorphism in the following two cases

1. \(M \in D(\mathcal{O}_X)\) is perfect and \(K \in D_\QCoh(X)\), or

2. \(M \in D(\mathcal{O}_X)\) is pseudo-coherent, \(K \in D_\QCoh^+(X)\), and \(R'\) has finite tor dimension over \(R\).

Proof

There is a canonical map \(R\Hom_X(M, K) \to R\Hom_{X'}(L(g')^*M, L(g')^*K)\) in \(D(\Gamma(X, \mathcal{O}_X))\) of global hom complexes, see Cohomology, Section 0B6A. Restricting scalars we can view this as a map in \(D(R)\). Then we can use the adjointness of restriction and \(- \otimes_R^\mathbf{L} R'\) to get the displayed map of the lemma. Having defined the map it suffices to prove it is an isomorphism in the derived category of abelian groups.

The right hand side is equal to \[R\Hom_X(M, R(g')_*L(g')^*K) = R\Hom_X(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'})\] by Lemma 08I9. In both cases the complex \(R\SheafHom(M, K)\) is an object of \(D_\QCoh(\mathcal{O}_X)\) by Lemma 0A6H. There is a natural map \[R\SheafHom(M, K) \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'} \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} g'_*\mathcal{O}_{X'})\] which is an isomorphism in both cases by Lemma 0ATN. To see that this lemma applies in case (2) we note that \(g'_*\mathcal{O}_{X'} = Rg'_*\mathcal{O}_{X'} = Lf^*g_*\mathcal{O}_{S'}\) the second equality by Lemma 08IB. Using Lemma 08E9 and Cohomology, Lemma 09U8 we conclude that \(g'_*\mathcal{O}_{X'}\) has finite Tor dimension. Hence, in both cases by replacing \(K\) by \(R\SheafHom(M, K)\) we reduce to proving \[R\Gamma(X, K) \otimes^\mathbf{L}_A A' \longrightarrow R\Gamma(X, K \otimes^\mathbf{L}_{\mathcal{O}_X} g'_*\mathcal{O}_{X'})\] is an isomorphism. Note that the left hand side is equal to \(R\Gamma(X', L(g')^*K)\) by Lemma 08I9. Hence the result follows from Lemma 08IB.

Remark

With notation as in Lemma 0AA7. The diagram \[\xymatrix{ R\Hom_X(M, Rg'_*L) \otimes_R^\mathbf{L} R' \ar[r] \ar[d]_\mu & R\Hom_{X'}(L(g')^*M, L(g')^*Rg'_*L) \ar[d]^a \\ R\Hom_X(M, R(g')_*L) \ar@{=}[r] & R\Hom_{X'}(L(g')^*M, L) }\] is commutative where the top horizontal arrow is the map from the lemma, \(\mu\) is the multiplication map, and \(a\) comes from the adjunction map \(L(g')^*Rg'_*L \to L\). The multiplication map is the adjunction map \(K' \otimes_R^\mathbf{L} R' \to K'\) for any \(K' \in D(R')\).

Lemma

Consider a cartesian square of schemes \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] Assume \(g\) and \(f\) Tor independent.

1. If \(E \in D(\mathcal{O}_X)\) has tor amplitude in \([a, b]\) as a complex of \(f^{-1}\mathcal{O}_S\)-modules, then \(L(g')^*E\) has tor amplitude in \([a, b]\) as a complex of \(f^{-1}\mathcal{O}_{S'}\)-modules.

2. If \(\mathcal{G}\) is an \(\mathcal{O}_X\)-module flat over \(S\), then \(L(g')^*\mathcal{G} = (g')^*\mathcal{G}\).

Proof

We can compute tor dimension at stalks, see Cohomology, Lemma 09U9. If \(x' \in X'\) with image \(x \in X\), then \[(L(g')^*E)_{x'} = E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X', x'}\] Let \(s' \in S'\) and \(s \in S\) be the image of \(x'\) and \(x\). Since \(X\) and \(S'\) are tor independent over \(S\), we can apply More on Algebra, Lemma 0661 to see that the right hand side of the displayed formula is equal to \(E_x \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'}\) in \(D(\mathcal{O}_{S', s'})\). Thus (1) follows from More on Algebra, Lemma 066L. To see (2) observe that flatness of \(\mathcal{G}\) is equivalent to the condition that \(\mathcal{G}[0]\) has tor amplitude in \([0, 0]\). Applying (1) we conclude.

Lemma

Consider a cartesian diagram of schemes \[\xymatrix{ Z' \ar[r]_{i'} \ar[d]_g & X' \ar[d]^f \\ Z \ar[r]^i & X }\] where \(i\) is a closed immersion. If \(Z\) and \(X'\) are tor independent over \(X\), then \(Ri'_* \circ Lg^* = Lf^* \circ Ri_*\) as functors \(D(\mathcal{O}_Z) \to D(\mathcal{O}_{X'})\).

Proof

Note that the lemma is supposed to hold for all \(K \in D(\mathcal{O}_Z)\). Observe that \(i_*\) and \(i'_*\) are exact functors and hence \(Ri_*\) and \(Ri'_*\) are computed by applying \(i_*\) and \(i'_*\) to any representatives. Thus the base change map \[Lf^*(Ri_*(K)) \longrightarrow Ri'_*(Lg^*(K))\] on stalks at a point \(z' \in Z'\) with image \(z \in Z\) is given by \[K_z \otimes_{\mathcal{O}_{X, z}}^\mathbf{L} \mathcal{O}_{X', z'} \longrightarrow K_z \otimes_{\mathcal{O}_{Z, z}}^\mathbf{L} \mathcal{O}_{Z', z'}\] This map is an isomorphism by More on Algebra, Lemma 0661 and the assumed tor independence.

Künneth formula, II

For the case where the base is a field, please see Varieties, Section 0BEC. Consider a cartesian diagram of schemes \[\xymatrix{ & X \times_S Y \ar[ld]^p \ar[rd]_q \ar[dd]^f \\ X \ar[rd]_a & & Y \ar[ld]^b \\ & S }\] Let \(K \in D(\mathcal{O}_X)\) and \(M \in D(\mathcal{O}_Y)\). There is a canonical map [0FLP]\[\begin{equation} Ra_*K \otimes_{\mathcal{O}_S}^\mathbf{L} Rb_*M \longrightarrow Rf_*(Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M) \end{equation}\] Namely, we can use the maps \(Ra_*K \to Ra_*Rp_* Lp^*K = Rf_*Lp^*K\) and \(Rb_*M \to Rb_*Rq_* Lq^*M = Rf_*Lq^*M\) and then we can use the relative cup product (Cohomology, Remark 0B68).

Set \(A = \Gamma(S, \mathcal{O}_S)\). There is a global Künneth map [0G7V]\[\begin{equation} R\Gamma(X, K) \otimes_A^\mathbf{L} R\Gamma(Y, M) \longrightarrow R\Gamma(X \times_S Y, Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M) \end{equation}\] in \(D(A)\). This map is constructed using the pullback maps \(R\Gamma(X, K) \to R\Gamma(X \times_S Y, Lp^*K)\) and \(R\Gamma(Y, M) \to R\Gamma(X \times_S Y, Lq^*M)\) and the cup product constructed in Cohomology, Section 0FKU.

Lemma

In the situation above, if \(a\) and \(b\) are quasi-compact and quasi-separated and \(X\) and \(Y\) are tor-independent over \(S\), then (0FLP) is an isomorphism for \(K \in D_\QCoh(\mathcal{O}_X)\) and \(M \in D_\QCoh(\mathcal{O}_Y)\). If in addition \(S = \Spec(A)\) is affine, then the map (0G7V) is an isomorphism.

Proof

This follows from the following sequence of isomorphisms \[\begin{align*} Rf_*(Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M) & = Ra_*Rp_*(Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M) \\ & = Ra_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} Rp_*Lq^*M) \\ & = Ra_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} La^*Rb_*M) \\ & = Ra_*K \otimes_{\mathcal{O}_S}^\mathbf{L} Rb_*M \end{align*}\] The first equality holds because \(f = a \circ p\). The second equality by Lemma 08EU. The third equality by Lemma 08IB. The fourth equality by Lemma 08EU. We omit the verification that the composition of these isomorphisms is the same as the map (0FLP). If \(S\) is affine, then the source and target of the arrow (0G7V) are the result of applying \(R\Gamma(S, -)\) to the source and target of (0FLP) and we obtain the final statement; details omitted.

Proof

The construction of the arrow (0FLP) is compatible with restricting to open subschemes of \(S\) as is immediate from the construction of the relative cup product. Thus it suffices to prove that (0FLP) is an isomorphism when \(S\) is affine.

Assume \(S = \Spec(A)\) is affine. By Leray we have \(R\Gamma(S, Rf_*K) = R\Gamma(X, K)\) and similarly for the other cases. By Cohomology, Lemma 0FP6 the map (0FLP) induces the map (0G7V) on taking \(R\Gamma(S, -)\). On the other hand, by Lemmas 08D5 and 08DX the source and target of the map (0FLP) are in \(D_\QCoh(\mathcal{O}_S)\). Thus, by Lemma 06Z0, it suffices to prove that (0G7V) is an isomorphism.

Assume \(S = \Spec(A)\) and \(X = \Spec(B)\) and \(Y = \Spec(C)\) are all affine. We will use Lemma 06Z0 without further mention. In this case we can choose a K-flat complex \(K^\bullet\) of \(B\)-modules whose terms are flat such that \(K\) is represented by \(\widetilde{K}^\bullet\). Similarly, we can choose a K-flat complex \(M^\bullet\) of \(C\)-modules whose terms are flat such that \(M\) is represented by \(\widetilde{M}^\bullet\). See More on Algebra, Lemma 06Y4. Then \(\widetilde{K}^\bullet\) is a K-flat complex of \(\mathcal{O}_X\)-modules and similarly for \(\widetilde{M}^\bullet\), see Lemma 08DV. Thus \(La^*K\) is represented by \[a^*\widetilde{K}^\bullet = \widetilde{K^\bullet \otimes_A C}\] and similarly for \(Lb^*M\). This in turn is a K-flat complex of \(\mathcal{O}_{X \times_S Y}\)-modules by the lemma cited above and More on Algebra, Lemma 06Y1. Thus we finally see that the complex of \(\mathcal{O}_{X \times_S Y}\)-modules associated to \[\text{Tot}((K^\bullet \otimes_A C) \otimes_{B \otimes_A C} (B \otimes_A M^\bullet)) = \text{Tot}(K^\bullet \otimes_A M^\bullet)\] represents \(La^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lb^*M\) in the derived category of \(X \times_S Y\). Taking global sections we obtain \(\text{Tot}(K^\bullet \otimes_A M^\bullet)\) which of course is also the complex representing \(R\Gamma(X, K) \otimes_A^\mathbf{L} R\Gamma(Y, M)\). The fact that the isomorphism is given by cup product follows from the relationship between the genuine cup product and the naive one in Cohomology, Section 0FKU (and in particular Cohomology, Lemma 0FP3 and the discussion following it).

Assume \(S = \Spec(A)\) and \(Y\) are affine. We will use the induction principle of Cohomology of Schemes, Lemma 08DR to prove the statement. To do this we only have to show: if \(X = U \cup V\) is an open covering with \(U\) and \(V\) quasi-compact and if the map (0G7V) \[R\Gamma(U, K) \otimes_A^\mathbf{L} R\Gamma(Y, M) \longrightarrow R\Gamma(U \times_S Y, Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M)\] for \(U\) and \(Y\) over \(S\), the map (0G7V) \[R\Gamma(V, K) \otimes_A^\mathbf{L} R\Gamma(Y, M) \longrightarrow R\Gamma(V \times_S Y, Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M)\] for \(V\) and \(Y\) over \(S\), and the map (0G7V) \[R\Gamma(U \cap V, K) \otimes_A^\mathbf{L} R\Gamma(Y, M) \longrightarrow R\Gamma((U \cap V) \times_S Y, Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M)\] for \(U \cap V\) and \(Y\) over \(S\) are isomorphisms, then so is the map (0G7V) for \(X\) and \(Y\) over \(S\). However, by Cohomology, Lemma 0G6X these maps fit into a map of distinguished triangles with (0G7V) the final leg and hence we conclude by Derived Categories, Lemma 014A.

Assume \(S = \Spec(A)\) is affine. To finish the proof we can use the induction principle of Cohomology of Schemes, Lemma 08DR on \(Y\). Namely, by the above we already know that our map is an isomorphism when \(Y\) is affine. The rest of the argument is exactly the same as in the previous paragraph but with the roles of \(X\) and \(Y\) switched.

Lemma

Let \(a : X \to S\) be a quasi-compact and quasi-separated morphism of schemes. Let \(\mathcal{F}^\bullet\) be a locally bounded complex of \(a^{-1}\mathcal{O}_S\)-modules. Assume for all \(n \in \mathbf{Z}\) the sheaf \(\mathcal{F}^n\) is a flat \(a^{-1}\mathcal{O}_S\)-module and \(\mathcal{F}^n\) has the structure of a quasi-coherent \(\mathcal{O}_X\)-module compatible with the given \(a^{-1}\mathcal{O}_S\)-module structure (but the differentials in the complex \(\mathcal{F}^\bullet\) need not be \(\mathcal{O}_X\)-linear). Then the following hold

1. \(Ra_*\mathcal{F}^\bullet\) is locally bounded,

2. \(Ra_*\mathcal{F}^\bullet\) is in \(D_\QCoh(\mathcal{O}_S)\),

3. \(Ra_*\mathcal{F}^\bullet\) locally has finite tor dimension,

4. \(\mathcal{G} \otimes_{\mathcal{O}_S}^\mathbf{L} Ra_*\mathcal{F}^\bullet = Ra_*(a^{-1}\mathcal{G} \otimes_{a^{-1}\mathcal{O}_S} \mathcal{F}^\bullet)\) for \(\mathcal{G} \in \QCoh(\mathcal{O}_S)\), and

5. \(K \otimes_{\mathcal{O}_S}^\mathbf{L} Ra_*\mathcal{F}^\bullet = Ra_*(a^{-1}K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet)\) for \(K \in D_\QCoh(\mathcal{O}_S)\).

Proof

Parts (1), (2), (3) are local on \(S\) hence we may and do assume \(S\) is affine. Since \(a\) is quasi-compact we conclude that \(X\) is quasi-compact. Since \(\mathcal{F}^\bullet\) is locally bounded, we conclude that \(\mathcal{F}^\bullet\) is bounded.

For (1) and (2) we can use the first spectral sequence \(R^pa_*\mathcal{F}^q \Rightarrow R^{p + q}a_*\mathcal{F}^\bullet\) of Derived Categories, Lemma 015J. Combining Cohomology of Schemes, Lemma 01XJ and Homology, Lemma 012W we conclude.

Let us prove (3) by the induction principle of Cohomology of Schemes, Lemma 08DR. Namely, for a quasi-compact open of \(U\) of \(X\) consider the condition that \(R(a|_U)_*(\mathcal{F}^\bullet|_U)\) has finite tor dimension. If \(U, V\) are quasi-compact open in \(X\), then we have a relative Mayer-Vietoris distinguished triangle \[R(a|_{U \cup V})_*\mathcal{F}^\bullet|_{U \cup V} \to R(a|_U)_*\mathcal{F}^\bullet|_U \oplus R(a|_V)_*\mathcal{F}^\bullet|_V \to R(a|_{U \cap V})_*\mathcal{F}^\bullet|_{U \cap V} \to\] by Cohomology, Lemma 08HZ. By the behaviour of tor amplitude in distinguished triangles (see Cohomology, Lemma 08CJ) we see that if we know the result for \(U\), \(V\), \(U \cap V\), then the result holds for \(U \cup V\). This reduces us to the case where \(X\) is affine. In this case we have \[Ra_*\mathcal{F}^\bullet = a_*\mathcal{F}^\bullet\] by Leray’s acyclicity lemma (Derived Categories, Lemma 015E) and the vanishing of higher direct images of quasi-coherent modules under an affine morphism (Cohomology of Schemes, Lemma 01XC). Since \(\mathcal{F}^n\) is \(S\)-flat by assumption and \(X\) affine, the modules \(a_*\mathcal{F}^n\) are flat for all \(n\). Hence \(a_*\mathcal{F}^\bullet\) is a bounded complex of flat \(\mathcal{O}_S\)-modules and hence has finite tor dimension.

Proof of part (5). Denote \(a' : (X, a^{-1}\mathcal{O}_S) \to (S, \mathcal{O}_S)\) the obvious flat morphism of ringed spaces. Part (5) says that \[K \otimes_{\mathcal{O}_S}^\mathbf{L} Ra'_*\mathcal{F}^\bullet = Ra'_*(L(a')^*K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet)\] Thus Cohomology, Equation (0B53) gives a functorial map from the left to the right and we want to show this map is an isomorphism. This question is local on \(S\) hence we may and do assume \(S\) is affine. The rest of the proof is exactly the same as the proof of Lemma 08EU except that we have to show that the functor \(K \mapsto Ra'_*(L(a')^*K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet)\) commutes with direct sums. This is where we will use \(\mathcal{F}^n\) has the structure of a quasi-coherent \(\mathcal{O}_X\)-module. Namely, observe that \(K \mapsto L(a')^*K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet\) commutes with arbitrary direct sums. Next, if \(\mathcal{F}^\bullet\) consists of a single quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}^\bullet = \mathcal{F}^n[-n]\) then we have \(L(a')^*G \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet = La^*K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}^n[-n]\), see Cohomology, Lemma 08DE. Hence in this case the commutation with direct sums follows from Lemma 08DZ. Now, in general, since \(S\) is affine (hence \(X\) quasi-compact) and \(\mathcal{F}^\bullet\) is locally bounded, we see that \[\mathcal{F}^\bullet = (\mathcal{F}^a \to \ldots \to \mathcal{F}^b)\] is bounded. Arguing by induction on \(b - a\) and considering the distinguished triangle \[\mathcal{F}^b[-b] \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^b) \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^b[-b + 1]\] the proof of this part is finished. Some details omitted.

Proof of part (4). Let \(a' : (X, a^{-1}\mathcal{O}_S) \to (S, \mathcal{O}_S)\) be as above. Since \(\mathcal{F}^\bullet\) is a locally bounded complex of flat \(a^{-1}\mathcal{O}_S\)-modules we see the complex \(a^{-1}\mathcal{G} \otimes_{a^{-1}\mathcal{O}_S} \mathcal{F}^\bullet\) represents \(L(a')^*\mathcal{G} \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet\) in \(D(a^{-1}\mathcal{O}_S)\). Hence (4) follows from (5).

Lemma

Let \(f : X \to Y\) be a morphism of schemes with \(Y = \Spec(A)\) affine. Let \(\mathcal{U} : X = \bigcup_{i \in I} U_i\) be a finite affine open covering such that all the finite intersections \(U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}\) are affine. Let \(\mathcal{F}^\bullet\) be a bounded complex of \(f^{-1}\mathcal{O}_Y\)-modules. Assume for all \(n \in \mathbf{Z}\) the sheaf \(\mathcal{F}^n\) is a flat \(f^{-1}\mathcal{O}_Y\)-module and \(\mathcal{F}^n\) has the structure of a quasi-coherent \(\mathcal{O}_X\)-module compatible with the given \(p^{-1}\mathcal{O}_Y\)-module structure (but the differentials in the complex \(\mathcal{F}^\bullet\) need not be \(\mathcal{O}_X\)-linear). Then the complex \(\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))\) is K-flat as a complex of \(A\)-modules.

Proof

We may write \[\mathcal{F}^\bullet = (\mathcal{F}^a \to \ldots \to \mathcal{F}^b)\] Arguing by induction on \(b - a\) and considering the distinguished triangle \[\mathcal{F}^b[-b] \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^b) \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^b[-b + 1]\] and using More on Algebra, Lemma 06Y2 we reduce to the case where \(\mathcal{F}^\bullet\) consists of a single quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) placed in degree \(0\). In this case the Čech complex for \(\mathcal{F}\) and \(\mathcal{U}\) is homotopy equivalent to the alternating Čech complex, see Cohomology, Lemma 01FM. Since \(U_{i_0 \ldots i_p}\) is always affine, we see that \(\mathcal{F}(U_{i_0 \ldots i_p})\) is \(A\)-flat. Hence \(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})\) is a bounded complex of flat \(A\)-modules and hence K-flat by More on Algebra, Lemma 064K.

Let \(X, Y, S, a, b, p, q, f\) be as in the introduction to this section. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Let \(\mathcal{G}\) be an \(\mathcal{O}_Y\)-module. Set \(A = \Gamma(S, \mathcal{O}_S)\). Consider the map [0G49]\[\begin{equation} R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}) \longrightarrow R\Gamma(X \times_S Y, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}) \end{equation}\] in \(D(A)\). This map is constructed using the pullback maps \(R\Gamma(X, \mathcal{F}) \to R\Gamma(X \times_S Y, p^*\mathcal{F})\) and \(R\Gamma(Y, \mathcal{G}) \to R\Gamma(X \times_S Y, q^*\mathcal{G})\), the cup product constructed in Cohomology, Section 0FKU, and the canonical map \(p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} q^*\mathcal{G} \to p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}\).

Lemma

In the situation above the map (0G49) is an isomorphism if \(S\) is affine, \(\mathcal{F}\) and \(\mathcal{G}\) are \(S\)-flat and quasi-coherent and \(X\) and \(Y\) are quasi-compact with affine diagonal.

Proof

We strongly urge the reader to read the proof of Varieties, Lemma 0BED first. Choose finite affine open coverings \(\mathcal{U} : X = \bigcup_{i \in I} U_i\) and \(\mathcal{V} : Y = \bigcup_{j \in J} V_j\). This determines an affine open covering \(\mathcal{W} : X \times_S Y = \bigcup_{(i, j) \in I \times J} U_i \times_S V_j\). Note that \(\mathcal{W}\) is a refinement of \(\text{pr}_1^{-1}\mathcal{U}\) and of \(\text{pr}_2^{-1}\mathcal{V}\). Thus by the discussion in Cohomology, Section 01FP we obtain maps \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F}) \quad\text{and}\quad \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, q^*\mathcal{G})\] well defined up to homotopy and compatible with pullback maps on cohomology. In Cohomology, Equation (07MB) we have constructed a map of complexes \[\text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F}) \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{W}, q^*\mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G})\] which is compatible with the cup product on cohomology by Cohomology, Lemma 0FKV. Combining the above we obtain a map of complexes [0FLU]\[\begin{equation} \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}) \end{equation}\] We claim this is the map in the statement of the lemma, i.e., the source and target of this arrow are the same as the source and target of (0G49). Namely, by Cohomology of Schemes, Lemma 01XB and Cohomology, Lemma 0FLH the canonical maps \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to R\Gamma(X, \mathcal{F}), \quad \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \to R\Gamma(Y, \mathcal{G})\] and \[\check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}) \to R\Gamma(X \times_S Y, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G})\] are isomorphisms. On the other hand, the complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) is K-flat by Lemma 0FMQ and we conclude that \(\text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}))\) represents the derived tensor product \(R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G})\) as claimed.

We still have to show that (0FLU) is a quasi-isomorphism. We will do this using dimension shifting. Set \(d(\mathcal{F}) = \max \{d \mid H^d(X, \mathcal{F}) \not = 0\}\). Assume \(d(\mathcal{F}) > 0\). Set \(U = \coprod\nolimits_{i \in I} U_i\). This is an affine scheme as \(I\) is finite. Denote \(j : U \to X\) the morphism which is the inclusion \(U_i \to X\) on each \(U_i\). Since the diagonal of \(X\) is affine, the morphism \(j\) is affine, see Morphisms, Lemma 01SG. It follows that \(\mathcal{F}' = j_*j^*\mathcal{F}\) is \(S\)-flat, see Morphisms, Lemma 0FLM. It also follows that \(d(\mathcal{F}') = 0\) by combining Cohomology of Schemes, Lemmas 089W and 01XB. For all \(x \in X\) we have \(\mathcal{F}_x \to \mathcal{F}'_x\) is the inclusion of a direct summand: if \(x \in U_i\), then \(\mathcal{F}' \to (U_i \to X)_*\mathcal{F}|_{U_i}\) gives a splitting. We conclude that \(\mathcal{F} \to \mathcal{F}'\) is injective and \(\mathcal{F}'' = \mathcal{F}'/\mathcal{F}\) is \(S\)-flat as well. The short exact sequence \(0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0\) of flat quasi-coherent \(\mathcal{O}_X\)-modules produces a short exact sequence of complexes \[0 \to \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \to \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}') \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \to \text{Tot}( \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}'') \otimes_A \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})) \to 0\] and a short exact sequence of complexes \[0 \to \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F}' \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}) \to \check{\mathcal{C}}^\bullet(\mathcal{W}, p^*\mathcal{F}'' \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}) \to 0\] Moreover, the maps (0FLU) between these are compatible with these short exact sequences. Hence it suffices to prove (0FLU) is an isomorphism for \(\mathcal{F}'\) and \(\mathcal{F}''\). Finally, we have \(d(\mathcal{F}'') < d(\mathcal{F})\). In this way we reduce to the case \(d(\mathcal{F}) = 0\).

Arguing in the same fashion for \(\mathcal{G}\) we find that we may assume that both \(\mathcal{F}\) and \(\mathcal{G}\) have nonzero cohomology only in degree \(0\). Observe that this means that \(\Gamma(X, \mathcal{F})\) is quasi-isomorphic to the \(K\)-flat complex \(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})\) of \(A\)-modules sitting in degrees \(\geq 0\). It follows that \(\Gamma(X, \mathcal{F})\) is a flat \(A\)-module (because we can compute higher Tor’s against this module by tensoring with the Cech complex). Let \(V \subset Y\) be an affine open. Consider the affine open covering \(\mathcal{U}_V : X \times_S V = \bigcup_{i \in I} U_i \times_S V\). It is immediate that \[\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A \mathcal{G}(V) = \check{\mathcal{C}}^\bullet(\mathcal{U}_V, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G})\] (equality of complexes). By the flatness of \(\mathcal{G}(V)\) over \(A\) we see that \(\Gamma(X, \mathcal{F}) \otimes_A \mathcal{G}(V) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \otimes_A \mathcal{G}(V)\) is a quasi-isomorphism. Since the sheafification of \(V \mapsto \check{\mathcal{C}}^\bullet(\mathcal{U}_V, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G})\) represents \(Rq_*(p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G})\) by Cohomology of Schemes, Lemma 01XL we conclude that \[Rq_*(p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) \cong \Gamma(X, \mathcal{F}) \otimes_A \mathcal{G}\] on \(Y\) where the notation on the right hand side indicates the module \[b^*\widetilde{\Gamma(X, \mathcal{F})} \otimes_{\mathcal{O}_Y} \mathcal{G}\] Using the Leray spectral sequence for \(q\) we find \[H^n(X \times_S Y, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) = H^n(Y, b^*\widetilde{\Gamma(X, \mathcal{F})} \otimes_{\mathcal{O}_Y} \mathcal{G})\] Using Lemma 08EU for the morphism \(b : Y \to S = \Spec(A)\) and using that \(\Gamma(X, \mathcal{F})\) is \(A\)-flat we conclude that \(H^n(X \times_S Y, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G})\) is zero for \(n > 0\) and isomorphic to \(H^0(X, \mathcal{F}) \otimes_A H^0(Y, \mathcal{G})\) for \(n = 0\). Of course, here we also use that \(\mathcal{G}\) only has cohomology in degree \(0\). This finishes the proof (except that we should check that the isomorphism is indeed given by cup product in degree \(0\); we omit the verification).

Remark

Let \(S = \Spec(A)\) be an affine scheme. Let \(a : X \to S\) and \(b : Y \to S\) be morphisms of schemes. Let \(\mathcal{F}\), \(\mathcal{G}\) be quasi-coherent \(\mathcal{O}_X\)-modules and let \(\mathcal{E}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. Let \(\xi \in H^i(X, \mathcal{G})\) with pullback \(p^*\xi \in H^i(X \times_S Y, p^*\mathcal{G})\). Then the following diagram is commutative \[\xymatrix{ R\Gamma(X, \mathcal{F})[-i] \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{E}) \ar[d] \ar[rr]_-{\xi \otimes \text{id}} & & R\Gamma(X, \mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{E}) \ar[d] \\ R\Gamma(X \times_S Y, p^*\mathcal{F} \otimes q^*\mathcal{E})[-i] \ar[rr]^-{p^*\xi} & & R\Gamma(X \times_S Y, p^*(\mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{F}) \otimes q^*\mathcal{E}) }\] where the unadorned tensor products are over \(\mathcal{O}_{X \times_S Y}\). The horizontal arrows are from Cohomology, Remark 0G6W and the vertical arrows are (0G7V) hence given by pulling back followed by cup product on \(X \times_S Y\). The diagram commutes because the global cup product (on \(X \times_S Y\) with the sheaves \(p^*\mathcal{G}\), \(p^*\mathcal{F}\), and \(q^*\mathcal{E}\)) is associative, see Cohomology, Lemma 0FP4.

Künneth formula, III

Let \(X, Y, S, a, b, p, q, f\) be as in the introduction to Section 0FLN. In this section, given an \(\mathcal{O}_X\)-module \(\mathcal{F}\) and a \(\mathcal{O}_Y\)-module \(\mathcal{G}\) let us set \[\mathcal{F} \boxtimes \mathcal{G} = p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}\] Note that, contrary to what happens in a future section, we take the nonderived tensor product here.

On \(X\) let \(\mathcal{F}^\bullet\) be a complex of sheaves of abelian groups whose terms are quasi-coherent \(\mathcal{O}_X\)-modules such that the differentials \(d^i_\mathcal{F} : \mathcal{F}^i \to \mathcal{F}^{i + 1}\) are differential operators on \(X/S\) of finite order, see Morphisms, Section 0G43. Similarly, on \(Y\) let \(\mathcal{G}^\bullet\) be a complex of sheaves of abelian groups whose terms are quasi-coherent \(\mathcal{O}_Y\)-modules such that the differentials \(d^j_\mathcal{G} : \mathcal{G}^j \to \mathcal{G}^{j + 1}\) are differential operators on \(Y/S\) of finite order. Applying the construction of Morphisms, Lemma 0G45 we obtain a double complex \[\xymatrix{ \ldots & \ldots & \ldots & \ldots \\ \ldots \ar[r] & \mathcal{F}^i \boxtimes \mathcal{G}^{j + 1} \ar[r]^{d_1^{i, j + 1}} \ar[u] & \mathcal{F}^{i + 1} \boxtimes \mathcal{G}^{j + 1} \ar[r] \ar[u] & \ldots \\ \ldots \ar[r] & \mathcal{F}^i \boxtimes \mathcal{G}^j \ar[r]^{d_1^{i, j}} \ar[u]^{d_2^{i, j}} & \mathcal{F}^{i + 1} \boxtimes \mathcal{G}^j \ar[r] \ar[u]_{d_2^{i + 1, j}} & \ldots \\ \ldots & \ldots \ar[u] & \ldots \ar[u] & \ldots }\] of quasi-coherent modules whose maps are differential operators of finite order on \(X \times_S Y / S\). Please see the discussion in Morphisms, Remark 0G46 and Homology, Example 0A5J. To be explicit, we set \[d_1^{i, j} = d^i_\mathcal{F} \boxtimes 1 \quad\text{and}\quad d_2^{i, j} = 1 \boxtimes d^j_\mathcal{G}\] In the discussion below the notation \[\text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet)\] refers to the total complex associated to this double complex. This complex has terms which are quasi-coherent \(\mathcal{O}_{X \times_S Y}\)-modules and whose differentials are differential operators of finite order on \(X \times_S Y / S\).

In the situation above there exists a “relative cup product” map [0G4B]\[\begin{equation} Ra_*(\mathcal{F}^\bullet) \otimes_{\mathcal{O}_S}^\mathbf{L} Rb_*(\mathcal{G}^\bullet) \longrightarrow Rf_*\left(\text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet)\right) \end{equation}\] Namely, we can construct this map by combining

1. \(Ra_*(\mathcal{F}^\bullet) \to Rf_*(p^{-1}\mathcal{F}^\bullet)\),

2. \(Rb_*(\mathcal{G}^\bullet) \to Rf_*(q^{-1}\mathcal{G}^\bullet)\),

3. \(Rf_*(p^{-1}\mathcal{F}^\bullet) \otimes_{\mathcal{O}_S}^\mathbf{L} Rf_*(q^{-1}\mathcal{G}^\bullet) \to Rf_*(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} q^{-1}\mathcal{G}^\bullet)\),

4. \(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} q^{-1}\mathcal{G}^\bullet \to \text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^\bullet)\)

5. \(\text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^\bullet) \to \text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet)\).

Maps (1) and (2) are pullback maps, map (3) is the relative cup product, see Cohomology, Remark 0B68, map (4) compares the derived and nonderived tensor products, and map (5) is given by the obvious maps \(p^{-1}\mathcal{F}^i \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^j \to \mathcal{F}^i \boxtimes \mathcal{G}^j\) on the underlying double complexes.

Set \(A = \Gamma(S, \mathcal{O}_S)\). There exists a “global cup product” map [0FLR]\[\begin{equation} R\Gamma(X, \mathcal{F}^\bullet) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}^\bullet) \longrightarrow R\Gamma(X \times_S Y, \text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet)) \end{equation}\] in \(D(A)\). This is constructed similarly to the relative cup product above using

1. \(R\Gamma(X, \mathcal{F}^\bullet) \to R\Gamma(X \times_S Y, p^{-1}\mathcal{F}^\bullet)\)

2. \(R\Gamma(Y, \mathcal{G}^\bullet) \to R\Gamma(X \times_S Y, q^{-1}\mathcal{G}^\bullet)\),

3. \(R\Gamma(X \times_S Y, p^{-1}\mathcal{F}^\bullet) \otimes_A^\mathbf{L} R\Gamma(X \times_S Y, q^{-1}\mathcal{G}^\bullet) \to R\Gamma(X \times_S Y, p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} q^{-1}\mathcal{G}^\bullet)\),

4. \(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} q^{-1}\mathcal{G}^\bullet \to \text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^\bullet)\)

5. \(\text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^\bullet) \to \text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet)\).

Here maps (1) and (2) are the pullback maps, map (3) is the cup product constructed in Cohomology, Section 0FKU. Maps (4) and (5) are as indicated in the previous paragraph.

Lemma

In the situation above the cup product (0FLR) is an isomorphism in \(D(A)\) if the following assumptions hold

1. \(S = \Spec(A)\) is affine,

2. \(X\) and \(Y\) are quasi-compact with affine diagonal,

3. \(\mathcal{F}^\bullet\) is bounded,

4. \(\mathcal{G}^\bullet\) is bounded below,

5. \(\mathcal{F}^n\) is \(S\)-flat, and

6. \(\mathcal{G}^m\) is \(S\)-flat.

Proof

We will use the notation \(\mathcal{A}_{X/S}\) and \(\mathcal{A}_{Y/S}\) introduced in Morphisms, Remark 0G46. Suppose that we have maps of complexes \[\mathcal{F}_1^\bullet \to \mathcal{F}_2^\bullet \to \mathcal{F}_3^\bullet \to \mathcal{F}_1^\bullet[1]\] in the category \(\mathcal{A}_{X/S}\). Then by the functoriality of the cup product we obtain a commutative diagram \[\xymatrix{ R\Gamma(X, \mathcal{F}_1^\bullet) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}^\bullet) \ar[r] \ar[d] & R\Gamma(X \times_S Y, \text{Tot}(\mathcal{F}_1^\bullet \boxtimes \mathcal{G}^\bullet)) \ar[d] \\ R\Gamma(X, \mathcal{F}_2^\bullet) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}^\bullet) \ar[r] \ar[d] & R\Gamma(X \times_S Y, \text{Tot}(\mathcal{F}_2^\bullet \boxtimes \mathcal{G}^\bullet)) \ar[d] \\ R\Gamma(X, \mathcal{F}_3^\bullet) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}^\bullet) \ar[r] \ar[d] & R\Gamma(X \times_S Y, \text{Tot}(\mathcal{F}_3^\bullet \boxtimes \mathcal{G}^\bullet)) \ar[d] \\ R\Gamma(X, \mathcal{F}_1^\bullet[1]) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G}^\bullet) \ar[r] & R\Gamma(X \times_S Y, \text{Tot}(\mathcal{F}_1^\bullet[1] \boxtimes \mathcal{G}^\bullet)) }\] If the original maps form a distinguished triangle in the homotopy category of \(\mathcal{A}_{X/S}\), then the columns of this diagram form distinguished triangles in \(D(A)\).

In the situation of the lemma, suppose that \(\mathcal{F}^n = 0\) for \(n < i\). Then we may consider the termwise split short exact sequence of complexes \[0 \to \sigma_{\geq i + 1}\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to \mathcal{F}^i[-i] \to 0\] where the truncation is as in Homology, Section 0118. This produces the distinguished triangle \[\sigma_{\geq i + 1}\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to \mathcal{F}^i[-i] \to (\sigma_{\geq i + 1}\mathcal{F}^\bullet)[1]\] in the homotopy category of \(\mathcal{A}_{X/S}\) where the final arrow is given by the boundary map \(\mathcal{F}^i \to \mathcal{F}^{i + 1}\). It follows from the discussion above that it suffices to prove the lemma for \(\mathcal{F}^i[-i]\) and \(\sigma_{\geq i + 1}\mathcal{F}^\bullet\). Since \(\sigma_{\geq i + 1}\mathcal{F}^\bullet\) has fewer nonzero terms, by induction, if we can prove the lemma if \(\mathcal{F}^\bullet\) is nonzero only in single degree, then the lemma follows. Thus we may assume \(\mathcal{F}^\bullet\) is nonzero only in one degree.

Assume \(\mathcal{F}^\bullet\) is the complex which has an \(S\)-flat quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) sitting in degree \(0\) and is zero in other degrees. Observe that \(R\Gamma(X, \mathcal{F})\) has finite tor dimension by Lemma 0FML for example. Say it has tor amplitude in \([i, j]\). Pick \(N \gg 0\) and consider the distinguished triangle \[\sigma_{\geq N + 1}\mathcal{G}^\bullet \to \mathcal{G}^\bullet \to \sigma_{\leq N}\mathcal{G}^\bullet \to (\sigma_{\geq N + 1}\mathcal{G}^\bullet)[1]\] in the homotopy category of \(\mathcal{A}_{Y/S}\). Now observe that both \[R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \sigma_{\geq N + 1}\mathcal{G}^\bullet) \quad\text{and}\quad R\Gamma(X \times_S Y, \text{Tot}(\mathcal{F} \boxtimes \sigma_{\geq N + 1}\mathcal{G}^\bullet))\] have vanishing cohomology in degrees \(\leq N + i\). Thus, using the arguments given above, if we want to prove our statement in a given degree, then we may assume \(\mathcal{G}^\bullet\) is bounded. Repeating the arguments above one more time we may also assume \(\mathcal{G}^\bullet\) is nonzero only in one degree. This case is handled by Lemma 0FU4.

Künneth formula for Ext

Consider a cartesian diagram of schemes \[\xymatrix{ & X \times_S Y \ar[ld]^p \ar[rd]_q \ar[dd]^f \\ X \ar[rd]_a & & Y \ar[ld]^b \\ & S }\] For \(K \in D(\mathcal{O}_X)\) and \(M \in D(\mathcal{O}_Y)\) in this section let us define \[K \boxtimes M = Lp^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^*M\] We claim there is a canonical map [0FXY]\[\begin{equation} Ra_*R\SheafHom(K, K') \otimes_{\mathcal{O}_S}^\mathbf{L} Rb_*R\SheafHom(M, M') \longrightarrow Rf_*(R\SheafHom(K \boxtimes M, K' \boxtimes M')) \end{equation}\] for \(K, K' \in D(\mathcal{O}_X)\) and \(M, M' \in D(\mathcal{O}_Y)\). Namely, we can take the map adjoint to the map \[\begin{matrix} Lf^*\left(Ra_*R\SheafHom(K, K') \otimes_{\mathcal{O}_S}^\mathbf{L} Rb_* R\SheafHom(M, M')\right) = \\ Lf^* Ra_* R\SheafHom(K, K') \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lf^* Rb_* R\SheafHom(M, M') = \\ Lp^* La^* Ra_* R\SheafHom(K, K') \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^* Lb^* Rb_* R\SheafHom(M, M') \to \\ Lp^* R\SheafHom(K, K') \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} Lq^* R\SheafHom(M, M') \to \\ R\SheafHom(Lp^*K, Lp^*K') \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} R\SheafHom(Lq^*M, Lq^*M') \to \\ R\SheafHom(K \boxtimes M, K' \boxtimes M') \end{matrix}\] Here the first equality is compatibility of pullbacks with tensor products, Cohomology, Lemma 079U. The second equality is \(f = a \circ p = b \circ q\) and composition of pullbacks, Cohomology, Lemma 0D5S. The first arrow is given by the adjunction maps \(La^* Ra_* \to \text{id}\) and \(Lb^* Rb_* \to \text{id}\) because pushforward and pullback are adjoint, Cohomology, Lemma 079W. The second arrow is given by Cohomology, Remark 08I3. The third and final arrow is Cohomology, Remark 0FXP. A simple special case of this is the following result.

Lemma

In the situation above, assume \(a\) and \(b\) are quasi-compact and quasi-separated and \(X\) and \(Y\) are tor independent over \(S\). If \(K\) is perfect, \(K' \in D_\QCoh(\mathcal{O}_X)\), \(M\) is perfect, and \(M' \in D_\QCoh(\mathcal{O}_Y)\), then (0FXY) is an isomorphism.

Proof

In this case we have \(R\SheafHom(K, K') = K' \otimes^\mathbf{L} K^\vee\), \(R\SheafHom(M, M') = M' \otimes^\mathbf{L} M^\vee\), and \[R\SheafHom(K \boxtimes M, K' \boxtimes M') = (K' \otimes^\mathbf{L} K^\vee) \boxtimes (M' \otimes^\mathbf{L} M^\vee)\] See Cohomology, Lemma 08DQ and we also use that being perfect is preserved by pullback and by tensor products. Hence this case follows from Lemma 0FLQ. (We omit the verification that with these identifications we obtain the same map.)

Cohomology and base change, V

In Section 08ET we saw a base change theorem holds when the morphisms are tor independent. Even in the affine case there cannot be a base change theorem without such a condition, see More on Algebra, Section 065Y. In this section we analyze when one can get a base change result “one complex at a time”.

To make this work, suppose we have a commutative diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] of schemes (usually we will assume it is cartesian). Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). For a point \(x' \in X'\) set \(x = g'(x') \in X\), \(s' = f'(x') \in S'\) and \(s = f(x) = g(s')\). Then we can consider the maps \[K_x \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'} \to K_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X', x'} \to K'_{x'}\] where the first arrow is More on Algebra, Equation (065Z) and the second comes from \((L(g')^*K)_{x'} = K_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X', x'}\) and the given map \(L(g')^*K \to K'\). For each \(i \in \mathbf{Z}\) we obtain a \(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S', s'}\)-module structure on \(H^i(K_x \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'})\). Putting everything together we obtain canonical maps [0DJ7]\[\begin{equation} H^i(K_x \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'}) \otimes_{(\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S', s'})} \mathcal{O}_{X', x'} \longrightarrow H^i(K'_{x'}) \end{equation}\] of \(\mathcal{O}_{X', x'}\)-modules.

Lemma

Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] be a cartesian diagram of schemes. Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). The following are equivalent

1. for any \(x' \in X'\) and \(i \in \mathbf{Z}\) the map (0DJ7) is an isomorphism,

2. for \(U \subset X\), \(V' \subset S'\) affine open both mapping into the affine open \(V \subset S\) with \(U' = V' \times_V U\) the composition \[R\Gamma(U, K) \otimes_{\mathcal{O}_S(U)}^\mathbf{L} \mathcal{O}_{S'}(V') \to R\Gamma(U, K) \otimes_{\mathcal{O}_X(U)}^\mathbf{L} \mathcal{O}_{X'}(U') \to R\Gamma(U', K')\] is an isomorphism in \(D(\mathcal{O}_{S'}(V'))\), and

3. there is a set \(I\) of quadruples \(U_i, V_i', V_i, U_i'\), \(i \in I\) as in (2) with \(X' = \bigcup U'_i\).

Proof

The second arrow in (2) comes from the equality \[R\Gamma(U, K) \otimes_{\mathcal{O}_X(U)}^\mathbf{L} \mathcal{O}_{X'}(U') = R\Gamma(U', L(g')^*K)\] of Lemma 08DW and the given arrow \(L(g')^*K \to K'\). The first arrow of (2) is More on Algebra, Equation (065Z). It is clear that (2) implies (3). Observe that (1) is local on \(X'\). Therefore it suffices to show that if \(X\), \(S\), \(S'\), \(X'\) are affine, then (1) is equivalent to the condition that \[R\Gamma(X, K) \otimes_{\mathcal{O}_S(S)}^\mathbf{L} \mathcal{O}_{S'}(S') \to R\Gamma(X, K) \otimes_{\mathcal{O}_X(X)}^\mathbf{L} \mathcal{O}_{X'}(X') \to R\Gamma(X', K')\] is an isomorphism in \(D(\mathcal{O}_{S'}(S'))\). Say \(S = \Spec(R)\), \(X = \Spec(A)\), \(S' = \Spec(R')\), \(X' = \Spec(A')\), \(K\) corresponds to the complex \(M^\bullet\) of \(A\)-modules, and \(K'\) corresponds to the complex \(N^\bullet\) of \(A'\)-modules. Note that \(A' = A \otimes_R R'\). The condition above is that the composition \[M^\bullet \otimes_R^\mathbf{L} R' \to M^\bullet \otimes_A^\mathbf{L} A' \to N^\bullet\] is an isomorphism in \(D(R')\). Equivalently, it is that for all \(i \in \mathbf{Z}\) the map \[H^i(M^\bullet \otimes_R^\mathbf{L} R') \to H^i(M^\bullet \otimes_A^\mathbf{L} A') \to H^i(N^\bullet)\] is an isomorphism. Observe that this is a map of \(A \otimes_R R'\)-modules, i.e., of \(A'\)-modules. On the other hand, (1) is the requirement that for compatible primes \(\mathfrak q' \subset A'\), \(\mathfrak q \subset A\), \(\mathfrak p' \subset R'\), \(\mathfrak p \subset R\) the composition \[H^i(M^\bullet_\mathfrak q \otimes_{R_\mathfrak p}^\mathbf{L} R'_{\mathfrak p'}) \otimes_{(A_\mathfrak q \otimes_{R_\mathfrak p} R'_{\mathfrak p'})} A'_{\mathfrak q'} \to H^i(M^\bullet_{\mathfrak q} \otimes_{A_\mathfrak q}^\mathbf{L} A'_{\mathfrak q'}) \to H^i(N^\bullet_{\mathfrak q'})\] is an isomorphism. Since \[H^i(M^\bullet_\mathfrak q \otimes_{R_\mathfrak p}^\mathbf{L} R'_{\mathfrak p'}) \otimes_{(A_\mathfrak q \otimes_{R_\mathfrak p} R'_{\mathfrak p'})} A'_{\mathfrak q'} = H^i(M^\bullet \otimes_R^\mathbf{L} R') \otimes_{A'} A'_{\mathfrak q'}\] is the localization at \(\mathfrak q'\), we see that these two conditions are equivalent by Algebra, Lemma 00HN.

Lemma

Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] be a cartesian diagram of schemes. Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). If

1. the equivalent conditions of Lemma 0DJ8 hold, and

2. \(f\) is quasi-compact and quasi-separated,

then the composition \(Lg^*Rf_*K \to Rf'_*L(g')^*K \to Rf'_*K'\) is an isomorphism.

Proof

We could prove this using the same method as in the proof of Lemma 08IB but instead we will prove it using the induction principle and relative Mayer-Vietoris.

To check the map is an isomorphism we may work locally on \(S'\). Hence we may assume \(g : S' \to S\) is a morphism of affine schemes. In particular \(X\) is a quasi-compact and quasi-separated scheme. We will use the induction principle of Cohomology of Schemes, Lemma 08DR to prove that for any quasi-compact open \(U \subset X\) the similarly constructed map \(Lg^*R(U \to S)_*K|_U \to R(U' \to S')_*K'|_{U'}\) is an isomorphism. Here \(U' = (g')^{-1}(U)\).

If \(U \subset X\) is an affine open, then we find that the result is true by assumption, see Lemma 0DJ8 part (2) and the translation into algebra afforded to us by Lemmas 06Z0 and 08DW.

The induction step. Suppose that \(X = U \cup V\) is an open covering with \(U\), \(V\), \(U \cap V\) quasi-compact such that the result holds for \(U\), \(V\), and \(U \cap V\). Denote \(a = f|_U\), \(b = f|_V\) and \(c = f|_{U \cap V}\). Let \(a' : U' \to S'\), \(b' : V' \to S'\) and \(c' : U' \cap V' \to S'\) be the base changes of \(a\), \(b\), and \(c\). Using the distinguished triangles from relative Mayer-Vietoris (Cohomology, Lemma 08HZ) we obtain a commutative diagram \[\xymatrix{ Lg^*Rf_*K \ar[r] \ar[d] & Rf'_* K' \ar[d] \\ Lg^*Ra_* K|_U \oplus Lg^*Rb_* K|_V \ar[r] \ar[d] & Ra'_* K'|_{U'} \oplus Rb'_* K'|_{V'} \ar[d] \\ Lg^*Rc_* K|_{U \cap V} \ar[r] \ar[d] & Rc'_* K'|_{U' \cap V'} \ar[d] \\ Lg^*Rf_* K[1] \ar[r] & Rf'_* K'[1] }\] Since the 2nd and 3rd horizontal arrows are isomorphisms so is the first (Derived Categories, Lemma 014A) and the proof of the lemma is finished.

Lemma

Let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] be a cartesian diagram of schemes. Let \(K \in D_\QCoh(\mathcal{O}_X)\) and let \(L(g')^*K \to K'\) be a map in \(D_\QCoh(\mathcal{O}_{X'})\). If the equivalent conditions of Lemma 0DJ8 hold, then

1. for \(E \in D_\QCoh(\mathcal{O}_X)\) the equivalent conditions of Lemma 0DJ8 hold for \(L(g')^*(E \otimes^\mathbf{L} K) \to L(g')^*E \otimes^\mathbf{L} K'\),

2. if \(E\) in \(D(\mathcal{O}_X)\) is perfect the equivalent conditions of Lemma 0DJ8 hold for \(L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')\), and

3. if \(K\) is bounded below and \(E\) in \(D(\mathcal{O}_X)\) pseudo-coherent the equivalent conditions of Lemma 0DJ8 hold for \(L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')\).

Proof

The statement makes sense as the complexes involved have quasi-coherent cohomology sheaves by Lemmas 08DW, 08DX, and 0A6H and Cohomology, Lemmas 09U7 and 09UA. Having said this, we can check the maps (0DJ7) are isomorphisms in case (1) by computing the source and target of (0DJ7) using the transitive property of tensor product, see More on Algebra, Lemma 0BYJ. The map in (2) and (3) is the composition \[L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, L(g')^*K) \to R\SheafHom(L(g')^*E, K')\] where the first arrow is Cohomology, Remark 08I3 and the second arrow comes from the given map \(L(g')^*K \to K'\). To prove the maps (0DJ7) are isomorphisms one represents \(E_x\) by a bounded complex of finite projective \(\mathcal{O}_{X. x}\)-modules in case (2) or by a bounded above complex of finite free modules in case (3) and computes the source and target of the arrow. Some details omitted.

Lemma

Let \(f : X \to S\) be a quasi-compact and quasi-separated morphism of schemes. Let \(E \in D_\QCoh(\mathcal{O}_X)\). Let \(\mathcal{G}^\bullet\) be a bounded above complex of quasi-coherent \(\mathcal{O}_X\)-modules flat over \(S\). Then formation of \[Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet)\] commutes with arbitrary base change (see proof for precise statement).

Proof

The statement means the following. Let \(g : S' \to S\) be a morphism of schemes and consider the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] in other words \(X' = S' \times_S X\). The lemma asserts that \[Lg^*Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) \longrightarrow Rf'_*\left( L(g')^*E \otimes^\mathbf{L}_{\mathcal{O}_{X'}} (g')^*\mathcal{G}^\bullet \right)\] is an isomorphism. Observe that on the right hand side we do not use the derived pullback on \(\mathcal{G}^\bullet\). To prove this, we apply Lemmas 0DJ9 and 0DJA to see that it suffices to prove the canonical map \[L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet\] satisfies the equivalent conditions of Lemma 0DJ8. This follows by checking the condition on stalks, where it immediately follows from the fact that \(\mathcal{G}^\bullet_x \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{S', s'}\) computes the derived tensor product by our assumptions on the complex \(\mathcal{G}^\bullet\).

Lemma

Let \(f : X \to S\) be a quasi-compact and quasi-separated morphism of schemes. Let \(E\) be an object of \(D(\mathcal{O}_X)\). Let \(\mathcal{G}^\bullet\) be a complex of quasi-coherent \(\mathcal{O}_X\)-modules. If

1. \(E\) is perfect, \(\mathcal{G}^\bullet\) is a bounded above, and \(\mathcal{G}^n\) is flat over \(S\), or

2. \(E\) is pseudo-coherent, \(\mathcal{G}^\bullet\) is bounded, and \(\mathcal{G}^n\) is flat over \(S\),

then formation of \[Rf_*R\SheafHom(E, \mathcal{G}^\bullet)\] commutes with arbitrary base change (see proof for precise statement).

Proof

The statement means the following. Let \(g : S' \to S\) be a morphism of schemes and consider the base change diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] in other words \(X' = S' \times_S X\). The lemma asserts that \[Lg^*Rf_*R\SheafHom(E, \mathcal{G}^\bullet) \longrightarrow R(f')_*R\SheafHom(L(g')^*E, (g')^*\mathcal{G}^\bullet)\] is an isomorphism. Observe that on the right hand side we do not use the derived pullback on \(\mathcal{G}^\bullet\). To prove this, we apply Lemmas 0DJ9 and 0DJA to see that it suffices to prove the canonical map \[L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet\] satisfies the equivalent conditions of Lemma 0DJ8. This was shown in the proof of Lemma 0A1D.

Producing perfect complexes

The following lemma is our main technical tool for producing perfect complexes. Later versions of this result will reduce to this by Noetherian approximation, see Section 0A1G.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E \in D(\mathcal{O}_X)\) such that

1. \(E \in D^b_{\textit{Coh}}(\mathcal{O}_X)\),

2. the support of \(H^i(E)\) is proper over \(S\) for all \(i\), and

3. \(E\) has finite tor dimension as an object of \(D(f^{-1}\mathcal{O}_S)\).

Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_S)\).

Proof

By Lemma 08E2 we see that \(Rf_*E\) is an object of \(D^b_{\textit{Coh}}(\mathcal{O}_S)\). Hence \(Rf_*E\) is pseudo-coherent (Lemma 08E8). Hence it suffices to show that \(Rf_*E\) has finite tor dimension, see Cohomology, Lemma 08CQ. By Lemma 08EA it suffices to check that \(Rf_*(E) \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{F}\) has universally bounded cohomology for all quasi-coherent sheaves \(\mathcal{F}\) on \(S\). Bounded from above is clear as \(Rf_*(E)\) is bounded from above. Let \(T \subset X\) be the union of the supports of \(H^i(E)\) for all \(i\). Then \(T\) is proper over \(S\) by assumptions (1) and (2), see Cohomology of Schemes, Lemma 0CYR. In particular there exists a quasi-compact open \(X' \subset X\) containing \(T\). Setting \(f' = f|_{X'}\) we have \(Rf_*(E) = Rf'_*(E|_{X'})\) because \(E\) restricts to zero on \(X \setminus T\). Thus we may replace \(X\) by \(X'\) and assume \(f\) is quasi-compact. Moreover, \(f\) is quasi-separated by Morphisms, Lemma 01T7. Now \[Rf_*(E) \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{F} = Rf_*\left(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}\right) = Rf_*\left(E \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} f^{-1}\mathcal{F}\right)\] by Lemma 08EU and Cohomology, Lemma 08DE. By assumption (3) the complex \(E \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} f^{-1}\mathcal{F}\) has cohomology sheaves in a given finite range, say \([a, b]\). Then \(Rf_*\) of it has cohomology in the range \([a, \infty)\) and we win.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E \in D(\mathcal{O}_X)\) be perfect. Let \(\mathcal{G}^\bullet\) be a bounded complex of coherent \(\mathcal{O}_X\)-modules flat over \(S\) with support proper over \(S\). Then \(K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\) is a perfect object of \(D(\mathcal{O}_S)\).

Proof

The object \(K\) is perfect by Lemma 08EV. We check the lemma applies: Locally \(E\) is isomorphic to a finite complex of finite free \(\mathcal{O}_X\)-modules. Hence locally \(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet\) is isomorphic to a finite complex whose terms are of the form \[\bigoplus\nolimits_{i = a, \ldots, b} (\mathcal{G}^i)^{\oplus r_i}\] for some integers \(a, b, r_a, \ldots, r_b\). This immediately implies the cohomology sheaves \(H^i(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G})\) are coherent. The hypothesis on the tor dimension also follows as \(\mathcal{G}^i\) is flat over \(f^{-1}\mathcal{O}_S\).

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a morphism of schemes which is locally of finite type. Let \(E \in D(\mathcal{O}_X)\) be perfect. Let \(\mathcal{G}^\bullet\) be a bounded complex of coherent \(\mathcal{O}_X\)-modules flat over \(S\) with support proper over \(S\). Then \(K = Rf_*R\SheafHom(E, \mathcal{G}^\bullet)\) is a perfect object of \(D(\mathcal{O}_S)\).

Proof

Since \(E\) is a perfect complex there exists a dual perfect complex \(E^\vee\), see Cohomology, Lemma 08DQ. Observe that \(R\SheafHom(E, \mathcal{G}^\bullet) = E^\vee \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet\). Thus the perfectness of \(K\) follows from Lemma 0DJQ.

We will generalize the following lemma to flat and proper morphisms over general bases in Lemma 0B91 and to perfect proper morphisms in More on Morphisms, Lemma 0B6G.

Lemma

Let \(S\) be a Noetherian scheme. Let \(f : X \to S\) be a flat proper morphism of schemes. Let \(E \in D(\mathcal{O}_X)\) be perfect. Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_S)\).

Proof

We claim that Lemma 08EV applies. Conditions (1) and (2) are immediate. Condition (3) is local on \(X\). Thus we may assume \(X\) and \(S\) affine and \(E\) represented by a strictly perfect complex of \(\mathcal{O}_X\)-modules. Since \(\mathcal{O}_X\) is flat as a sheaf of \(f^{-1}\mathcal{O}_S\)-modules we find that condition (3) is satisfied.

A projection formula for Ext

Lemma 08IF (or similar results in the literature) is sometimes used to verify one of Artin’s criteria for Quot functors, Hilbert schemes, and other moduli problems. Suppose that \(f : X \to S\) is a proper, flat, finitely presented morphism of schemes and \(E \in D(\mathcal{O}_X)\) is perfect. Here the lemma says \[\Ext^i_X(E, f^*\mathcal{F}) = \Ext^i_S((Rf_*E^\vee)^\vee, \mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(S\). Writing it this way makes it look like a projection formula for Ext and indeed the result follows rather easily from Lemma 08EU.

Lemma

Assumptions and notation as in Lemma 0DJQ. Then there are functorial isomorphisms \[H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) \longrightarrow H^i(X, E \otimes_{\mathcal{O}_X}^\mathbf{L} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}))\] for \(\mathcal{F}\) quasi-coherent on \(S\) compatible with boundary maps (see proof).

Proof

We have \[\mathcal{G}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F} = \mathcal{G}^\bullet \otimes_{f^{-1}\mathcal{O}_S}^\mathbf{L} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes_{f^{-1}\mathcal{O}_S} f^{-1}\mathcal{F} = \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}\] the first equality by Cohomology, Lemma 08DE, the second as \(\mathcal{G}^n\) is a flat \(f^{-1}\mathcal{O}_S\)-module, and the third by definition of pullbacks. Hence we obtain \[\begin{align*} H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})) & = H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}) \\ & = H^i(S, Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet \otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*\mathcal{F})) \\ & = H^i(S, Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) \\ & = H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) \end{align*}\] The first equality by the above, the second by Leray (Cohomology, Lemma 01EZ), and the third equality by Lemma 08EU. The statement on boundary maps means the following: Given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) of quasi-coherent \(\mathcal{O}_S\)-modules, the isomorphisms fit into commutative diagrams \[\xymatrix{ H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_3) \ar[r] \ar[d]_\delta & H^i(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3)) \ar[d]^\delta \\ H^{i + 1}(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_1) \ar[r] & H^{i + 1}(X, E \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1)) }\] where the boundary maps come from the distinguished triangle \[K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_1 \to K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_2 \to K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_3 \to K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_1[1]\] and the distinguished triangle in \(D(\mathcal{O}_X)\) associated to the short exact sequence \[0 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3 \to 0\] of complexes of \(\mathcal{O}_X\)-modules. This sequence is exact because \(\mathcal{G}^n\) is flat over \(S\). We omit the verification of the commutativity of the displayed diagram.

Lemma

Assumptions and notation as in Lemma 0DJR. Then there are functorial isomorphisms \[H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(S\) compatible with boundary maps (see proof).

Proof

As in the proof of Lemma 0DJR let \(E^\vee\) be the dual perfect complex and recall that \(K = Rf_*(E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\). Since we also have \[\Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) = H^i(X, E^\vee \otimes^\mathbf{L}_{\mathcal{O}_X} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}))\] by construction of \(E^\vee\), the existence of the isomorphisms follows from Lemma 0A1F applied to \(E^\vee\) and \(\mathcal{G}^\bullet\). The statement on boundary maps means the following: Given a short exact sequence \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) then the isomorphisms fit into commutative diagrams \[\xymatrix{ H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_3) \ar[r] \ar[d]_\delta & \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3) \ar[d]^\delta \\ H^{i + 1}(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_1) \ar[r] & \Ext^{i + 1}_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1) }\] where the boundary maps come from the distinguished triangle \[K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_1 \to K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_2 \to K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_3 \to K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}_1[1]\] and the distinguished triangle in \(D(\mathcal{O}_X)\) associated to the short exact sequence \[0 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_2 \to \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3 \to 0\] of complexes. This sequence is exact because \(\mathcal{G}\) is flat over \(S\). We omit the verification of the commutativity of the displayed diagram.

Lemma

Let \(f : X \to S\) be a morphism of schemes, \(E \in D(\mathcal{O}_X)\) and \(\mathcal{G}^\bullet\) a complex of \(\mathcal{O}_X\)-modules. Assume

1. \(S\) is Noetherian,

2. \(f\) is locally of finite type,

3. \(E \in D^-_{\textit{Coh}}(\mathcal{O}_X)\),

4. \(\mathcal{G}^\bullet\) is a bounded complex of coherent \(\mathcal{O}_X\)-modules flat over \(S\) with support proper over \(S\).

Then the following two statements are true

1. for every \(m \in \mathbf{Z}\) there exists a perfect object \(K\) of \(D(\mathcal{O}_S)\) and functorial maps \[\alpha^i_\mathcal{F} : \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(S\) compatible with boundary maps (see proof) such that \(\alpha^i_\mathcal{F}\) is an isomorphism for \(i \leq m\)

2. there exists a pseudo-coherent \(L \in D(\mathcal{O}_S)\) and functorial isomorphisms \[\Ext^i_{\mathcal{O}_S}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] for \(\mathcal{F}\) quasi-coherent on \(S\) compatible with boundary maps.

Proof

Proof of (A). Suppose \(\mathcal{G}^i\) is nonzero only for \(i \in [a, b]\). We may replace \(X\) by a quasi-compact open neighbourhood of the union of the supports of \(\mathcal{G}^i\). Hence we may assume \(X\) is Noetherian. In this case \(X\) and \(f\) are quasi-compact and quasi-separated. Choose an approximation \(P \to E\) by a perfect complex \(P\) of \((X, E, -m - 1 + a)\) (possible by Theorem 08ES). Then the induced map \[\Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(P, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] is an isomorphism for \(i \leq m\). Namely, the kernel, resp. cokernel of this map is a quotient, resp. submodule of \[\Ext^i_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \quad\text{resp.}\quad \Ext^{i + 1}_{\mathcal{O}_X}(C, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})\] where \(C\) is the cone of \(P \to E\). Since \(C\) has vanishing cohomology sheaves in degrees \(\geq -m - 1 + a\) these \(\Ext\)-groups are zero for \(i \leq m + 1\) by Derived Categories, Lemma 06XS. This reduces us to the case that \(E\) is a perfect complex which is Lemma 08ID. The statement on boundaries is explained in the proof of Lemma 08ID.

Proof of (B). As in the proof of (A) we may assume \(X\) is Noetherian. Observe that \(E\) is pseudo-coherent by Lemma 08E8. By Lemma 0DJN we can write \(E = \text{hocolim} E_n\) with \(E_n\) perfect and \(E_n \to E\) inducing an isomorphism on truncations \(\tau_{\geq -n}\). Let \(E_n^\vee\) be the dual perfect complex (Cohomology, Lemma 08DQ). We obtain an inverse system \(\ldots \to E_3^\vee \to E_2^\vee \to E_1^\vee\) of perfect objects. This in turn gives rise to an inverse system \[\ldots \to K_3 \to K_2 \to K_1\quad\text{with}\quad K_n = Rf_*(E_n^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\] perfect on \(S\), see Lemma 0DJQ. By Lemma 08ID and its proof and by the arguments in the previous paragraph (with \(P = E_n\)) for any quasi-coherent \(\mathcal{F}\) on \(S\) we have functorial canonical maps \[\xymatrix{ & \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \ar[ld] \ar[rd] \\ H^i(S, K_{n + 1} \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{F}) \ar[rr] & & H^i(S, K_n \otimes_{\mathcal{O}_S}^\mathbf{L} \mathcal{F}) }\] which are isomorphisms for \(i \leq n + a\). Let \(L_n = K_n^\vee\) be the dual perfect complex. Then we see that \(L_1 \to L_2 \to L_3 \to \ldots\) is a system of perfect objects in \(D(\mathcal{O}_S)\) such that for any quasi-coherent \(\mathcal{F}\) on \(S\) the maps \[\Ext^i_{\mathcal{O}_S}(L_{n + 1}, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_S}(L_n, \mathcal{F})\] are isomorphisms for \(i \leq n + a - 1\). This implies that \(L_n \to L_{n + 1}\) induces an isomorphism on truncations \(\tau_{\geq -n - a + 2}\) (hint: take cone of \(L_n \to L_{n + 1}\) and look at its last nonvanishing cohomology sheaf). Thus \(L = \text{hocolim} L_n\) is pseudo-coherent, see Lemma 0DJN. The mapping property of homotopy colimits gives that \(\Ext^i_{\mathcal{O}_S}(L, \mathcal{F}) = \Ext^i_{\mathcal{O}_S}(L_n, \mathcal{F})\) for \(i \leq n + a - 3\) which finishes the proof.

Remark

The pseudo-coherent complex \(L\) of part (B) of Lemma 08IF is canonically associated to the situation. For example, formation of \(L\) as in (B) is compatible with base change. In other words, given a cartesian diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] of schemes we have canonical functorial isomorphisms \[\Ext^i_{\mathcal{O}_{S'}}(Lg^*L, \mathcal{F}') \longrightarrow \Ext^i_{\mathcal{O}_X}(L(g')^*E, (g')^*\mathcal{G}^\bullet \otimes_{\mathcal{O}_{X'}} (f')^*\mathcal{F}')\] for \(\mathcal{F}'\) quasi-coherent on \(S'\). Obsere that we do not use derived pullback on \(\mathcal{G}^\bullet\) on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof.

Limits and derived categories

In this section we collect some results about the derived category of a scheme which is the limit of an inverse system of schemes. More precisely, we will work in the following setting.

Situation

Let \(S = \lim_{i \in I} S_i\) be a limit of a directed system of schemes with affine transition morphisms \(f_{i'i} : S_{i'} \to S_i\). We assume that \(S_i\) is quasi-compact and quasi-separated for all \(i \in I\). We denote \(f_i : S \to S_i\) the projection. We also fix an element \(0 \in I\).

Lemma

In Situation 09RD. Let \(E_0\) and \(K_0\) be objects of \(D(\mathcal{O}_{S_0})\). Set \(E_i = Lf_{i0}^*E_0\) and \(K_i = Lf_{i0}^*K_0\) for \(i \geq 0\) and set \(E = Lf_0^*E_0\) and \(K = Lf_0^*K_0\). Then the map \[\colim_{i \geq 0} \Hom_{D(\mathcal{O}_{S_i})}(E_i, K_i) \longrightarrow \Hom_{D(\mathcal{O}_S)}(E, K)\] is an isomorphism if either

1. \(E_0\) is perfect and \(K_0 \in D_\QCoh(\mathcal{O}_{S_0})\), or

2. \(E_0\) is pseudo-coherent and \(K_0 \in D_\QCoh(\mathcal{O}_{S_0})\) has finite tor dimension.

Proof

For every open \(U_0 \subset S_0\) consider the condition \(P\) that the canonical map \[\colim_{i \geq 0} \Hom_{D(\mathcal{O}_{U_i})}(E_i|_{U_i}, K_i|_{U_i}) \longrightarrow \Hom_{D(\mathcal{O}_U)}(E|_U, K|_U)\] is an isomorphism, where \(U = f_0^{-1}(U_0)\) and \(U_i = f_{i0}^{-1}(U_0)\). We will prove \(P\) holds for all quasi-compact opens \(U_0\) by the induction principle of Cohomology of Schemes, Lemma 08DR. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Cohomology, Lemma 08BW. Thus it suffices to prove the lemma when \(S_0\) is affine.

Assume \(S_0\) is affine. Say \(S_0 = \Spec(A_0)\), \(S_i = \Spec(A_i)\), and \(S = \Spec(A)\). We will use Lemma 06Z0 without further mention.

In case (1) the object \(E_0^\bullet\) corresponds to a finite complex of finite projective \(A_0\)-modules, see Lemma 08EB. We may represent the object \(K_0\) by a K-flat complex \(K_0^\bullet\) of \(A_0\)-modules. In this situation we are trying to prove \[\colim_{i \geq 0} \Hom_{D(A_i)}(E_0^\bullet \otimes_{A_0} A_i, K_0^\bullet \otimes_{A_0} A_i) \longrightarrow \Hom_{D(A)}(E_0^\bullet \otimes_{A_0} A, K_0^\bullet \otimes_{A_0} A)\] Because \(E_0^\bullet\) is a bounded above complex of projective modules we can rewrite this as \[\colim_{i \geq 0} \Hom_{K(A_0)}(E_0^\bullet, K_0^\bullet \otimes_{A_0} A_i) \longrightarrow \Hom_{K(A_0)}(E_0^\bullet, K_0^\bullet \otimes_{A_0} A)\] Since there are only a finite number of nonzero modules \(E_0^n\) and since these are all finitely presented modules, this map is an isomorphism.

In case (2) the object \(E_0\) corresponds to a bounded above complex \(E_0^\bullet\) of finite free \(A_0\)-modules, see Lemma 08E7. We may represent \(K_0\) by a finite complex \(K_0^\bullet\) of flat \(A_0\)-modules, see Lemma 08E9 and More on Algebra, Lemma 0654. In particular \(K_0^\bullet\) is K-flat and we can argue as before to arrive at the map \[\colim_{i \geq 0} \Hom_{K(A_0)}(E_0^\bullet, K_0^\bullet \otimes_{A_0} A_i) \longrightarrow \Hom_{K(A_0)}(E_0^\bullet, K_0^\bullet \otimes_{A_0} A)\] It is clear that this map is an isomorphism (only a finite number of terms are involved since \(K_0^\bullet\) is bounded).

Lemma

In Situation 09RD the category of perfect objects of \(D(\mathcal{O}_S)\) is the colimit of the categories of perfect objects of \(D(\mathcal{O}_{S_i})\).

Proof

For every open \(U_0 \subset S_0\) consider the condition \(P\) that the functor \[\colim_{i \geq 0} D_{perf}(\mathcal{O}_{U_i}) \longrightarrow D_{perf}(\mathcal{O}_U)\] is an equivalence where \({}_{perf}\) indicates the full subcategory of perfect objects and where \(U = f_0^{-1}(U_0)\) and \(U_i = f_{i0}^{-1}(U_0)\). We will prove \(P\) holds for all quasi-compact opens \(U_0\) by the induction principle of Cohomology of Schemes, Lemma 08DR. First, we observe that we already know the functor is fully faithful by Lemma 09RE. Thus it suffices to prove essential surjectivity.

We first check condition (2) of the induction principle. Thus suppose that we have \(S_0 = U_0 \cup V_0\) and that \(P\) holds for \(U_0\), \(V_0\), and \(U_0 \cap V_0\). Let \(E\) be a perfect object of \(D(\mathcal{O}_S)\). We can find \(i \geq 0\) and \(E_{U, i}\) perfect on \(U_i\) and \(E_{V, i}\) perfect on \(V_i\) whose pullback to \(U\) and \(V\) are isomorphic to \(E|_U\) and \(E|_V\). Denote \[a : E_{U, i} \to (Rf_{i, *}E)|_{U_i} \quad\text{and}\quad b : E_{V, i} \to (Rf_{i, *}E)|_{V_i}\] the maps adjoint to the isomorphisms \(Lf_i^*E_{U, i} \to E|_U\) and \(Lf_i^*E_{V, i} \to E|_V\). By fully faithfulness, after increasing \(i\), we can find an isomorphism \(c : E_{U, i}|_{U_i \cap V_i} \to E_{V, i}|_{U_i \cap V_i}\) which pulls back to the identifications \[Lf_i^*E_{U, i}|_{U \cap V} \to E|_{U \cap V} \to Lf_i^*E_{V, i}|_{U \cap V}.\] Apply Cohomology, Lemma 08DG to get an object \(E_i\) on \(S_i\) and a map \(d : E_i \to Rf_{i, *}E\) which restricts to the maps \(a\) and \(b\) over \(U_i\) and \(V_i\). Then it is clear that \(E_i\) is perfect and that \(d\) is adjoint to an isomorphism \(Lf_i^*E_i \to E\).

Finally, we check condition (1) of the induction principle, in other words, we check the lemma holds when \(S_0\) is affine. Say \(S_0 = \Spec(A_0)\), \(S_i = \Spec(A_i)\), and \(S = \Spec(A)\). Using Lemmas 06Z0 and 08EB we see that we have to show that \[D_{perf}(A) = \colim D_{perf}(A_i)\] This is clear from the fact that perfect complexes over rings are given by finite complexes of finite projective (hence finitely presented) modules. See More on Algebra, Lemma 0BC7 for details.

Cohomology and base change, VI

A final section on cohomology and base change continuing the discussion of Sections 08ET, 0DJ6, and 0A1E. An easy to grok special case is given in Remark 0A1I.

Lemma

Let \(f : X \to S\) be a morphism of finite presentation. Let \(E \in D(\mathcal{O}_X)\) be a perfect object. Let \(\mathcal{G}^\bullet\) be a bounded complex of finitely presented \(\mathcal{O}_X\)-modules, flat over \(S\), with support proper over \(S\). Then \[K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\] is a perfect object of \(D(\mathcal{O}_S)\) and its formation commutes with arbitrary base change.

Proof

The statement on base change is Lemma 0A1D. Thus it suffices to show that \(K\) is a perfect object. If \(S\) is Noetherian, then this follows from Lemma 0DJQ. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on \(S\), hence we may assume \(S\) is affine. Say \(S = \Spec(R)\). We write \(R = \colim R_i\) as a filtered colimit of Noetherian rings \(R_i\). By Limits, Lemma 01ZM there exists an \(i\) and a scheme \(X_i\) of finite presentation over \(R_i\) whose base change to \(R\) is \(X\). By Limits, Lemma 01ZR we may assume after increasing \(i\), that there exists a bounded complex of finitely presented \(\mathcal{O}_{X_i}\)-modules \(\mathcal{G}_i^\bullet\) whose pullback to \(X\) is \(\mathcal{G}^\bullet\). After increasing \(i\) we may assume \(\mathcal{G}_i^n\) is flat over \(R_i\), see Limits, Lemma 05LY. After increasing \(i\) we may assume the support of \(\mathcal{G}_i^n\) is proper over \(R_i\), see Limits, Lemma 081G and Cohomology of Schemes, Lemma 0CYS. Finally, by Lemma 09RF we may, after increasing \(i\), assume there exists a perfect object \(E_i\) of \(D(\mathcal{O}_{X_i})\) whose pullback to \(X\) is \(E\). Applying Lemma 0DJQ to \(X_i \to \Spec(R_i)\), \(E_i\), \(\mathcal{G}_i^\bullet\) and using the base change property already shown we obtain the result.

Remark

Let \(R\) be a ring. Let \(X\) be a scheme of finite presentation over \(R\). Let \(\mathcal{G}\) be a finitely presented \(\mathcal{O}_X\)-module flat over \(R\) with support proper over \(R\). By Lemma 0A1H there exists a finite complex of finite projective \(R\)-modules \(M^\bullet\) such that we have \[R\Gamma(X_{R'}, \mathcal{G}_{R'}) = M^\bullet \otimes_R R'\] functorially in the \(R\)-algebra \(R'\).

Lemma

Let \(f : X \to S\) be a morphism of finite presentation. Let \(E \in D(\mathcal{O}_X)\) be a pseudo-coherent object. Let \(\mathcal{G}^\bullet\) be a bounded above complex of finitely presented \(\mathcal{O}_X\)-modules, flat over \(S\), with support proper over \(S\). Then \[K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)\] is a pseudo-coherent object of \(D(\mathcal{O}_S)\) and its formation commutes with arbitrary base change.

Proof

The statement on base change is Lemma 0A1D. Thus it suffices to show that \(K\) is a pseudo-coherent object. This will follow from Lemma 0A1H by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.

The question is local on \(S\), hence we may assume \(S\) is affine. Then \(X\) is quasi-compact and quasi-separated. Moreover, there exists an integer \(N\) such that total direct image \(Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)\) has cohomological dimension \(N\) as explained in Lemma 08D5. Choose an integer \(b\) such that \(\mathcal{G}^i = 0\) for \(i > b\). It suffices to show that \(K\) is \(m\)-pseudo-coherent for every \(m\). Choose an approximation \(P \to E\) by a perfect complex \(P\) of \((X, E, m - N - 1 - b)\). This is possible by Theorem 08ES. Choose a distinguished triangle \[P \to E \to C \to P[1]\] in \(D_\QCoh(\mathcal{O}_X)\). The cohomology sheaves of \(C\) are zero in degrees \(\geq m - N - 1 - b\). Hence the cohomology sheaves of \(C \otimes^\mathbf{L} \mathcal{G}^\bullet\) are zero in degrees \(\geq m - N - 1\). Thus the cohomology sheaves of \(Rf_*(C \otimes^\mathbf{L} \mathcal{G}^\bullet)\) are zero in degrees \(\geq m - 1\). Hence \[Rf_*(P \otimes^\mathbf{L} \mathcal{G}^\bullet) \to Rf_*(E \otimes^\mathbf{L} \mathcal{G}^\bullet)\] is an isomorphism on cohomology sheaves in degrees \(\geq m\). Next, suppose that \(H^i(P) = 0\) for \(i > a\). Then \(P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet \longrightarrow P \otimes^\mathbf{L} \mathcal{G}^\bullet\) is an isomorphism on cohomology sheaves in degrees \(\geq m - N - 1\). Thus again we find that \[Rf_*(P \otimes^\mathbf{L} \sigma_{\geq m - N - 1 - a}\mathcal{G}^\bullet) \to Rf_*(P \otimes^\mathbf{L} \mathcal{G}^\bullet)\] is an isomorphism on cohomology sheaves in degrees \(\geq m\). By Lemma 0A1H the source is a perfect complex. We conclude that \(K\) is \(m\)-pseudo-coherent as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to S\) be a proper morphism of finite presentation.

1. Let \(E \in D(\mathcal{O}_X)\) be perfect and \(f\) flat. Then \(Rf_*E\) is a perfect object of \(D(\mathcal{O}_S)\) and its formation commutes with arbitrary base change.

2. Let \(\mathcal{G}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). Then \(Rf_*\mathcal{G}\) is a perfect object of \(D(\mathcal{O}_S)\) and its formation commutes with arbitrary base change.

Proof

Special cases of Lemma 0A1H applied with (1) \(\mathcal{G}^\bullet\) equal to \(\mathcal{O}_X\) in degree \(0\) and (2) \(E = \mathcal{O}_X\) and \(\mathcal{G}^\bullet\) consisting of \(\mathcal{G}\) sitting in degree \(0\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to S\) be a flat proper morphism of finite presentation. Let \(E \in D(\mathcal{O}_X)\) be pseudo-coherent. Then \(Rf_*E\) is a pseudo-coherent object of \(D(\mathcal{O}_S)\) and its formation commutes with arbitrary base change.

More generally, if \(f : X \to S\) is proper and \(E\) on \(X\) is pseudo-coherent relative to \(S\) (More on Morphisms, Definition 09UI), then \(Rf_*E\) is pseudo-coherent (but formation does not commute with base change in this generality). See [Kiehl].

Proof

Special case of Lemma 0CSC applied with \(\mathcal{G}^\bullet\) equal to \(\mathcal{O}_X\) in degree \(0\).

Lemma

Let \(R\) be a ring. Let \(X\) be a scheme and let \(f : X \to \Spec(R)\) be proper, flat, and of finite presentation. Let \((M_n)\) be an inverse system of \(R\)-modules with surjective transition maps. Then the canonical map \[\mathcal{O}_X \otimes_R (\lim M_n) \longrightarrow \lim \mathcal{O}_X \otimes_R M_n\] induces an isomorphism from the source to \(DQ_X\) applied to the target.

Proof

The statement means that for any object \(E\) of \(D_\QCoh(\mathcal{O}_X)\) the induced map \[\Hom(E, \mathcal{O}_X \otimes_R (\lim M_n)) \longrightarrow \Hom(E, \lim \mathcal{O}_X \otimes_R M_n)\] is an isomorphism. Since \(D_\QCoh(\mathcal{O}_X)\) has a perfect generator (Theorem 09IS) it suffices to check this for perfect \(E\). By Lemma 0A0J we have \(\lim \mathcal{O}_X \otimes_R M_n = R\lim \mathcal{O}_X \otimes_R M_n\). The exact functor \(R\Hom_X(E, -) : D_\QCoh(\mathcal{O}_X) \to D(R)\) of Cohomology, Section 0B6A commutes with products and hence with derived limits, whence \[R\Hom_X(E, \lim \mathcal{O}_X \otimes_R M_n) = R\lim R\Hom_X(E, \mathcal{O}_X \otimes_R M_n)\] Let \(E^\vee\) be the dual perfect complex, see Cohomology, Lemma 08DQ. We have \[R\Hom_X(E, \mathcal{O}_X \otimes_R M_n) = R\Gamma(X, E^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*M_n) = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n\] by Lemma 08EU. From Lemma 0B91 we see \(R\Gamma(X, E^\vee)\) is a perfect complex of \(R\)-modules. In particular it is a pseudo-coherent complex and by More on Algebra, Lemma 0D2L we obtain \[R\lim R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} M_n = R\Gamma(X, E^\vee) \otimes_R^\mathbf{L} \lim M_n\] as desired.

Lemma

Let \(f : X \to S\) be a morphism of finite presentation. Let \(E \in D(\mathcal{O}_X)\) be a perfect object. Let \(\mathcal{G}^\bullet\) be a bounded complex of finitely presented \(\mathcal{O}_X\)-modules, flat over \(S\), with support proper over \(S\). Then \[K = Rf_*R\SheafHom(E, \mathcal{G}^\bullet)\] is a perfect object of \(D(\mathcal{O}_S)\) and its formation commutes with arbitrary base change.

Proof

The statement on base change is Lemma 08IE. Thus it suffices to show that \(K\) is a perfect object. If \(S\) is Noetherian, then this follows from Lemma 0DJR. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on \(S\), hence we may assume \(S\) is affine. Say \(S = \Spec(R)\). We write \(R = \colim R_i\) as a filtered colimit of Noetherian rings \(R_i\). By Limits, Lemma 01ZM there exists an \(i\) and a scheme \(X_i\) of finite presentation over \(R_i\) whose base change to \(R\) is \(X\). By Limits, Lemma 01ZR we may assume after increasing \(i\), that there exists a bounded complex of finitely presented \(\mathcal{O}_{X_i}\)-modules \(\mathcal{G}_i^\bullet\) whose pullback to \(X\) is \(\mathcal{G}^\bullet\). After increasing \(i\) we may assume \(\mathcal{G}_i^n\) is flat over \(R_i\), see Limits, Lemma 05LY. After increasing \(i\) we may assume the support of \(\mathcal{G}_i^n\) is proper over \(R_i\), see Limits, Lemma 081G and Cohomology of Schemes, Lemma 0CYS. Finally, by Lemma 09RF we may, after increasing \(i\), assume there exists a perfect object \(E_i\) of \(D(\mathcal{O}_{X_i})\) whose pullback to \(X\) is \(E\). Applying Lemma 0DJR to \(X_i \to \Spec(R_i)\), \(E_i\), \(\mathcal{G}_i^\bullet\) and using the base change property already shown we obtain the result.

Perfect complexes

We first talk about jumping loci for betti numbers of perfect complexes. Given a complex \(E\) on a scheme \(X\) and a point \(x\) of \(X\) we often write \(E \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x)\) instead of the more correct \(Li_x^*E\), where \(i_x : x \to X\) is the canonical morphism.

Lemma

Let \(X\) be a scheme. Let \(E \in D(\mathcal{O}_X)\) be pseudo-coherent (for example perfect). For any \(i \in \mathbf{Z}\) consider the function \[\beta_i : X \longrightarrow \{0, 1, 2, \ldots\},\quad x \longmapsto \dim_{\kappa(x)} H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x))\] Then we have

1. formation of \(\beta_i\) commutes with arbitrary base change,

2. the functions \(\beta_i\) are upper semi-continuous, and

3. the level sets of \(\beta_i\) are locally constructible in \(X\).

Proof

Consider a morphism of schemes \(f : Y \to X\) and a point \(y \in Y\). Let \(x\) be the image of \(y\) and consider the commutative diagram \[\xymatrix{ y \ar[r]_j \ar[d]_g & Y \ar[d]^f \\ x \ar[r]^i & X }\] Then we see that \(Lg^* \circ Li^* = Lj^* \circ Lf^*\). This implies that the function \(\beta'_i\) associated to the pseudo-coherent complex \(Lf^*E\) is the pullback of the function \(\beta_i\), in a formula: \(\beta'_i = \beta_i \circ f\). This is the meaning of (1).

Fix \(i\) and let \(x \in X\). It is enough to prove (2) and (3) holds in an open neighbourhood of \(x\), hence we may assume \(X\) affine. Then we can represent \(E\) by a bounded above complex \(\mathcal{F}^\bullet\) of finite free modules (Lemma 08EE). Then \(P = \sigma_{\geq i - 1}\mathcal{F}^\bullet\) is a perfect object and \(P \to E\) induces an isomorphism \[H^i(P \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x')) \to H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x'))\] for all \(x' \in X\). Thus we may assume \(E\) is perfect. In this case by More on Algebra, Lemma 0BCD there exists an affine open neighbourhood \(U\) of \(x\) and \(a \leq b\) such that \(E|_U\) is represented by a complex \[\ldots \to 0 \to \mathcal{O}_U^{\oplus \beta_a(x)} \to \mathcal{O}_U^{\oplus \beta_{a + 1}(x)} \to \ldots \to \mathcal{O}_U^{\oplus \beta_{b - 1}(x)} \to \mathcal{O}_U^{\oplus \beta_b(x)} \to 0 \to \ldots\] (This also uses earlier results to turn the problem into algebra, for example Lemmas 06Z0 and 08EB.) It follows immediately that \(\beta_i(x') \leq \beta_i(x)\) for all \(x' \in U\). This proves that \(\beta_i\) is upper semi-continuous.

To prove (3) we may assume that \(X\) is affine and \(E\) is given by a complex of finite free \(\mathcal{O}_X\)-modules (for example by arguing as in the previous paragraph, or by using Cohomology, Lemma 0BCJ). Thus we have to show that given a complex \[\mathcal{O}_X^{\oplus a} \to \mathcal{O}_X^{\oplus b} \to \mathcal{O}_X^{\oplus c}\] the function associated to a point \(x \in X\) the dimension of the cohomology of \(\kappa_x^{\oplus a} \to \kappa_x^{\oplus b} \to \kappa_x^{\oplus c}\) in the middle has constructible level sets. Let \(A \in \text{Mat}(a \times b, \Gamma(X, \mathcal{O}_X))\) be the matrix of the first arrow. The rank of the image of \(A\) in \(\text{Mat}(a \times b, \kappa(x))\) is equal to \(r\) if all \((r + 1) \times (r + 1)\)-minors of \(A\) vanish at \(x\) and there is some \(r \times r\)-minor of \(A\) which does not vanish at \(x\). Thus the set of points where the rank is \(r\) is a constructible locally closed set. Arguing similarly for the second arrow and putting everything together we obtain the desired result.

Lemma

Let \(X\) be a scheme. Let \(E \in D(\mathcal{O}_X)\) be perfect. The function \[\chi_E : X \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^i \dim_{\kappa(x)} H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x))\] is locally constant on \(X\).

Proof

By Cohomology, Lemma 0BCJ we see that we can, locally on \(X\), represent \(E\) by a finite complex \(\mathcal{E}^\bullet\) of finite free \(\mathcal{O}_X\)-modules. On such an open the function \(\chi_E\) is constant with value \(\sum (-1)^i \text{rank}(\mathcal{E}^i)\).

Lemma

Let \(X\) be a scheme. Let \(E \in D(\mathcal{O}_X)\) be perfect. Given \(i, r \in \mathbf{Z}\), there exists an open subscheme \(U \subset X\) characterized by the following

1. \(E|_U \cong H^i(E|_U)[-i]\) and \(H^i(E|_U)\) is a locally free \(\mathcal{O}_U\)-module of rank \(r\),

2. a morphism \(f : Y \to X\) factors through \(U\) if and only if \(Lf^*E\) is isomorphic to a locally free module of rank \(r\) placed in degree \(i\).

Proof

Let \(\beta_j : X \to \{0, 1, 2, \ldots\}\) for \(j \in \mathbf{Z}\) be the functions of Lemma 0BDI. Then the set \[W = \{x \in X \mid \beta_j(x) \leq 0\text{ for all }j \not = i\}\] is open in \(X\) and its formation commutes with pullback to any \(Y\) over \(X\). This follows from the lemma using that apriori in a neighbourhood of any point only a finite number of the \(\beta_j\) are nonzero. Thus we may replace \(X\) by \(W\) and assume that \(\beta_j(x) = 0\) for all \(x \in X\) and all \(j \not = i\). In this case \(H^i(E)\) is a finite locally free module and \(E \cong H^i(E)[-i]\), see for example More on Algebra, Lemma 0BCD. Thus \(X\) is the disjoint union of the open subschemes where the rank of \(H^i(E)\) is fixed and we win.

Lemma

Let \(X\) be a scheme. Let \(E \in D(\mathcal{O}_X)\) be perfect of tor-amplitude in \([a, b]\) for some \(a, b \in \mathbf{Z}\). Let \(r \geq 0\). Then there exists a locally closed subscheme \(j : Z \to X\) characterized by the following

1. \(H^a(Lj^*E)\) is a locally free \(\mathcal{O}_Z\)-module of rank \(r\), and

2. a morphism \(f : Y \to X\) factors through \(Z\) if and only if for all morphisms \(g : Y' \to Y\) the \(\mathcal{O}_{Y'}\)-module \(H^a(L(f \circ g)^*E)\) is locally free of rank \(r\).

Moreover, \(j : Z \to X\) is of finite presentation and we have

1. if \(f : Y \to X\) factors as \(Y \xrightarrow{g} Z \to X\), then \(H^a(Lf^*E) = g^*H^a(Lj^*E)\),

2. if \(\beta_a(x) \leq r\) for all \(x \in X\), then \(j\) is a closed immersion and given \(f : Y \to X\) the following are equivalent

   1. \(f : Y \to X\) factors through \(Z\),

   2. \(H^a(Lf^*E)\) is a locally free \(\mathcal{O}_Y\)-module of rank \(r\),

   and if \(r = 1\) these are also equivalent to

   1. \(\mathcal{O}_Y \to \SheafHom_{\mathcal{O}_Y}(H^a(Lf^*E), H^a(Lf^*E))\) is injective.

Proof

First, let \(U \subset X\) be the locally constructible open subscheme where the function \(\beta_a\) of Lemma 0BDI has values \(\leq r\). Let \(f : Y \to X\) be as in (2). Then for any \(y \in Y\) we have \(\beta_a(Lf^*E) = r\) hence \(y\) maps into \(U\) by Lemma 0BDI. Hence \(f\) as in (2) factors through \(U\). Thus we may replace \(X\) by \(U\) and assume that \(\beta_a(x) \in \{0, 1, \ldots, r\}\) for all \(x \in X\). We will show that in this case there is a closed subscheme \(Z \subset X\) cut out by a finite type quasi-coherent ideal characterized by the equivalence of (4) (a), (b) and (4)(c) if \(r = 1\) and that (3) holds. This will finish the proof because it will a fortiori show that morphisms as in (2) factor through \(Z\).

If \(x \in X\) and \(\beta_a(x) < r\), then there is an open neighbourhood of \(x\) where \(\beta_a < r\) (Lemma 0BDI). In this way we see that set theoretically at least \(Z\) is a closed subset.

To get a scheme theoretic structure, consider a point \(x \in X\) with \(\beta_a(x) = r\). Set \(\beta = \beta_{a + 1}(x)\). By More on Algebra, Lemma 0BCD there exists an affine open neighbourhood \(U\) of \(x\) such that \(K|_U\) is represented by a complex \[\ldots \to 0 \to \mathcal{O}_U^{\oplus r} \xrightarrow{(f_{ij})} \mathcal{O}_U^{\oplus \beta} \to \ldots \to \mathcal{O}_U^{\oplus \beta_{b - 1}(x)} \to \mathcal{O}_U^{\oplus \beta_b(x)} \to 0 \to \ldots\] (This also uses earlier results to turn the problem into algebra, for example Lemmas 06Z0 and 08EB.) Now, if \(g : Y \to U\) is any morphism of schemes such that \(g^\sharp(f_{ij})\) is nonzero for some pair \(i, j\), then \(H^a(Lg^*E)\) is not a locally free \(\mathcal{O}_Y\)-module of rank \(r\). See More on Algebra, Lemma 0EWY. Trivially \(H^a(Lg^*E)\) is a locally free \(\mathcal{O}_Y\)-module if \(g^\sharp(f_{ij}) = 0\) for all \(i, j\). Thus we see that over \(U\) the closed subscheme cut out by all \(f_{ij}\) satisfies (3) and we have the equivalence of (4)(a) and (b). The characterization of \(Z\) shows that the locally constructed patches glue (details omitted). Finally, if \(r = 1\) then (4)(c) is equivalent to (4)(b) because in this case locally \(H^a(Lg^*E) \subset \mathcal{O}_Y\) is the annihilator of the ideal generated by the elements \(g^\sharp(f_{ij})\).

Applications

Mostly applications of cohomology and base change. In the future we may generalize these results to the situation discussed in Lemma 0A1H.

Lemma

Let \(f : X \to S\) be a proper morphism of finite presentation. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). For fixed \(i \in \mathbf{Z}\) consider the function \[\beta_i : S \to \{0, 1, 2, \ldots\},\quad s \longmapsto \dim_{\kappa(s)} H^i(X_s, \mathcal{F}_s)\] Then we have

1. formation of \(\beta_i\) commutes with arbitrary base change,

2. the functions \(\beta_i\) are upper semi-continuous, and

3. the level sets of \(\beta_i\) are locally constructible in \(S\).

Proof

By cohomology and base change (more precisely by Lemma 0B91) the object \(K = Rf_*\mathcal{F}\) is a perfect object of the derived category of \(S\) whose formation commutes with arbitrary base change. In particular we have \[H^i(X_s, \mathcal{F}_s) = H^i(K \otimes_{\mathcal{O}_S}^\mathbf{L} \kappa(s))\] Thus the lemma follows from Lemma 0BDI.

Lemma

Let \(f : X \to S\) be a proper morphism of finite presentation. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). The function \[s \longmapsto \chi(X_s, \mathcal{F}_s)\] is locally constant on \(S\). Formation of this function commutes with base change.

Proof

By cohomology and base change (more precisely by Lemma 0B91) the object \(K = Rf_*\mathcal{F}\) is a perfect object of the derived category of \(S\) whose formation commutes with arbitrary base change. Thus we have to show the map \[s \longmapsto \sum (-1)^i \dim_{\kappa(s)} H^i(K \otimes^\mathbf{L}_{\mathcal{O}_S} \kappa(s))\] is locally constant on \(S\). This is Lemma 0BDJ.

Lemma

Let \(f : X \to S\) be a proper morphism of finite presentation. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\). Fix \(i, r \in \mathbf{Z}\). Then there exists an open subscheme \(U \subset S\) with the following property: A morphism \(T \to S\) factors through \(U\) if and only if \(Rf_{T, *}\mathcal{F}_T\) is isomorphic to a finite locally free module of rank \(r\) placed in degree \(i\).

Proof

By cohomology and base change (more precisely by Lemma 0B91) the object \(K = Rf_*\mathcal{F}\) is a perfect object of the derived category of \(S\) whose formation commutes with arbitrary base change. Thus this lemma follows immediately from Lemma 0BDK.

Lemma

Let \(f : X \to S\) be a morphism of finite presentation. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(S\) with support proper over \(S\). If \(R^if_*\mathcal{F} = 0\) for \(i > 0\), then \(f_*\mathcal{F}\) is locally free and its formation commutes with arbitrary base change (see proof for explanation).

Proof

By Lemma 0A1H the object \(E = Rf_*\mathcal{F}\) of \(D(\mathcal{O}_S)\) is perfect and its formation commutes with arbitrary base change, in the sense that \(Rf'_*(g')^*\mathcal{F} = Lg^*E\) for any cartesian diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }\] of schemes. Since there is never any cohomology in degrees \(< 0\), we see that \(E\) (locally) has tor-amplitude in \([0, b]\) for some \(b\). If \(H^i(E) = R^if_*\mathcal{F} = 0\) for \(i > 0\), then \(E\) has tor amplitude in \([0, 0]\). Whence \(E = H^0(E)[0]\). We conclude \(H^0(E) = f_*\mathcal{F}\) is finite locally free by More on Algebra, Lemma 0658 (and the characterization of finite projective modules in Algebra, Lemma 00NX). Commutation with base change means that \(g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}\) for a diagram as above and it follows from the already established commutation of base change for \(E\).

Lemma

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

1. \(f\) is proper, flat, and of finite presentation, and

2. for all \(s \in S\) we have \(\kappa(s) = H^0(X_s, \mathcal{O}_{X_s})\).

Then we have

1. \(f_*\mathcal{O}_X = \mathcal{O}_S\) and this holds after any base change,

2. locally on \(S\) we have \[Rf_*\mathcal{O}_X = \mathcal{O}_S \oplus P\] in \(D(\mathcal{O}_S)\) where \(P\) is perfect of tor amplitude in \([1, \infty)\).

Proof

By cohomology and base change (Lemma 0B91) the complex \(E = Rf_*\mathcal{O}_X\) is perfect and its formation commutes with arbitrary base change. This first implies that \(E\) has tor amplitude in \([0, \infty)\). Second, it implies that for \(s \in S\) we have \(H^0(E \otimes^\mathbf{L} \kappa(s)) = H^0(X_s, \mathcal{O}_{X_s}) = \kappa(s)\). It follows that the map \(\mathcal{O}_S \to Rf_*\mathcal{O}_X = E\) induces an isomorphism \(\mathcal{O}_S \otimes \kappa(s) \to H^0(E \otimes^\mathbf{L} \kappa(s))\). Hence \(H^0(E) \otimes \kappa(s) \to H^0(E \otimes^\mathbf{L} \kappa(s))\) is surjective and we may apply More on Algebra, Lemma 0A1U to see that, after replacing \(S\) by an affine open neighbourhood of \(s\), we have a decomposition \(E = H^0(E) \oplus \tau_{\geq 1}E\) with \(\tau_{\geq 1}E\) perfect of tor amplitude in \([1, \infty)\). Since \(E\) has tor amplitude in \([0, \infty)\) we find that \(H^0(E)\) is a flat \(\mathcal{O}_S\)-module. It follows that \(H^0(E)\) is a flat, perfect \(\mathcal{O}_S\)-module, hence finite locally free, see More on Algebra, Lemma 0658 (and the fact that finite projective modules are finite locally free by Algebra, Lemma 00NX). It follows that the map \(\mathcal{O}_S \to H^0(E)\) is an isomorphism as we can check this after tensoring with residue fields (Algebra, Lemma 00O0).

Lemma

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

1. \(f\) is proper, flat, and of finite presentation, and

2. the geometric fibres of \(f\) are reduced and connected.

Then \(f_*\mathcal{O}_X = \mathcal{O}_S\) and this holds after any base change.

Proof

By Lemma 0E62 it suffices to show that \(\kappa(s) = H^0(X_s, \mathcal{O}_{X_s})\) for all \(s \in S\). This follows from Varieties, Lemma 0BUG and the fact that \(X_s\) is geometrically connected and geometrically reduced.

Lemma

Let \(f : X \to S\) be a proper morphism of schemes. Let \(s \in S\) and let \(e \in H^0(X_s, \mathcal{O}_{X_s})\) be an idempotent. Then \(e\) is in the image of the map \((f_*\mathcal{O}_X)_s \to H^0(X_s, \mathcal{O}_{X_s})\).

Proof

Let \(X_s = T_1 \amalg T_2\) be the disjoint union decomposition with \(T_1\) and \(T_2\) nonempty and open and closed in \(X_s\) corresponding to \(e\), i.e., such that \(e\) is identitically \(1\) on \(T_1\) and identically \(0\) on \(T_2\).

Assume \(S\) is Noetherian. We will use the theorem on formal functions in the form of Cohomology of Schemes, Lemma 02OD. It tells us that \[(f_*\mathcal{O}_X)_s^\wedge = \lim_n H^0(X_n, \mathcal{O}_{X_n})\] where \(X_n\) is the \(n\)th infinitesimal neighbourhood of \(X_s\). Since the underlying topological space of \(X_n\) is equal to that of \(X_s\) we obtain for all \(n\) a disjoint union decomposition of schemes \(X_n = T_{1, n} \amalg T_{2, n}\) where the underlying topological space of \(T_{i, n}\) is \(T_i\) for \(i = 1, 2\). This means \(H^0(X_n, \mathcal{O}_{X_n})\) contains a nontrivial idempotent \(e_n\), namely the function which is identically \(1\) on \(T_{1, n}\) and identically \(0\) on \(T_{2, n}\). It is clear that \(e_{n + 1}\) restricts to \(e_n\) on \(X_n\). Hence \(e_\infty = \lim e_n\) is a nontrivial idempotent of the limit. Thus \(e_\infty\) is an element of the completion of \((f_*\mathcal{O}_X)_s\) mapping to \(e\) in \(H^0(X_s, \mathcal{O}_{X_s})\). Since the map \((f_*\mathcal{O}_X)_s^\wedge \to H^0(X_s, \mathcal{O}_{X_s})\) factors through \((f_*\mathcal{O}_X)^\wedge_s / \mathfrak m_s (f_*\mathcal{O}_X)_s^\wedge = (f_*\mathcal{O}_X)_s / \mathfrak m_s (f_*\mathcal{O}_X)_s\) (Algebra, Lemma 05GG) we conclude that \(e\) is in the image of the map \((f_*\mathcal{O}_X)_s \to H^0(X_s, \mathcal{O}_{X_s})\) as desired.

General case: we reduce the general case to the Noetherian case by limit arguments. We urge the reader to skip the proof. We may replace \(S\) by an affine open neighbourhood of \(s\). Thus we may and do assume that \(S\) is affine. By Limits, Lemma 0A0P we can write \((f : X \to S) = \lim (f_i : X_i \to S_i)\) with \(f_i\) proper and \(S_i\) Noetherian. Denote \(s_i \in S_i\) the image of \(s\). Then \(s = \lim s_i\), see Limits, Lemma 0CUG. Then \(X_s = X \times_S s = \lim X_i \times_{S_i} s_i = \lim X_{i, s_i}\) because limits commute with limits (Categories, Lemma 002M). Hence \(e\) is the image of some idempotent \(e_i \in H^0(X_{i, s_i}, \mathcal{O}_{X_{i, s_i}})\) by Limits, Lemma 01Z0. By the Noetherian case there is an element \(\tilde e_i\) in the stalk \((f_{i, *}\mathcal{O}_{X_i})_{s_i}\) mapping to \(e_i\). Taking the pullback of \(\tilde e_i\) we get an element \(\tilde e\) of \((f_*\mathcal{O}_X)_s\) mapping to \(e\) and the proof is complete.

Lemma

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

1. \(f\) is proper, flat, and of finite presentation, and

2. the fibre \(X_s\) is geometrically reduced.

Then, after replacing \(S\) by an open neighbourhood of \(s\), there exists a direct sum decomposition \(Rf_*\mathcal{O}_X = f_*\mathcal{O}_X \oplus P\) in \(D(\mathcal{O}_S)\) where \(f_*\mathcal{O}_X\) is a finite étale \(\mathcal{O}_S\)-algebra and \(P\) is a perfect of tor amplitude in \([1, \infty)\).

Proof

The proof of this lemma is similar to the proof of Lemma 0E62 which we suggest the reader read first. By cohomology and base change (Lemma 0B91) the complex \(E = Rf_*\mathcal{O}_X\) is perfect and its formation commutes with arbitrary base change. This first implies that \(E\) has tor amplitude in \([0, \infty)\).

We claim that after replacing \(S\) by an open neighbourhood of \(s\) we can find a direct sum decomposition \(E = H^0(E) \oplus \tau_{\geq 1}E\) in \(D(\mathcal{O}_S)\) with \(\tau_{\geq 1}E\) of tor amplitude in \([1, \infty)\). Assume the claim is true for now and assume we’ve made the replacement so we have the direct sum decomposition. Since \(E\) has tor amplitude in \([0, \infty)\) we find that \(H^0(E)\) is a flat \(\mathcal{O}_S\)-module. Hence \(H^0(E)\) is a flat, perfect \(\mathcal{O}_S\)-module, hence finite locally free, see More on Algebra, Lemma 0658 (and the fact that finite projective modules are finite locally free by Algebra, Lemma 00NX). Of course \(H^0(E) = f_*\mathcal{O}_X\) is an \(\mathcal{O}_S\)-algebra. By cohomology and base change we obtain \(H^0(E) \otimes \kappa(s) = H^0(X_s, \mathcal{O}_{X_s})\). By Varieties, Lemma 0BUG and the assumption that \(X_s\) is geometrically reduced, we see that \(\kappa(s) \to H^0(E) \otimes \kappa(s)\) is finite étale. By Morphisms, Lemma 0476 applied to the finite locally free morphism \(\underline{\Spec}_S(H^0(E)) \to S\), we conclude that after shrinking \(S\) the \(\mathcal{O}_S\)-algebra \(H^0(E)\) is finite étale.

It remains to prove the claim. For this it suffices to prove that the map \[(f_*\mathcal{O}_X)_s \longrightarrow H^0(X_s, \mathcal{O}_{X_s}) = H^0(E \otimes^\mathbf{L} \kappa(s))\] is surjective, see More on Algebra, Lemma 0A1U. Choose a flat local ring homomorphism \(\mathcal{O}_{S, s} \to A\) such that the residue field \(k\) of \(A\) is algebraically closed, see Algebra, Lemma 03C3. By flat base change (Cohomology of Schemes, Lemma 02KH) we get \(H^0(X_A, \mathcal{O}_{X_A}) = (f_*\mathcal{O}_X)_s \otimes_{\mathcal{O}_{S, s}} A\) and \(H^0(X_k, \mathcal{O}_{X_k}) = H^0(X_s, \mathcal{O}_{X_s}) \otimes_{\kappa(s)} k\). Hence it suffices to prove that \(H^0(X_A, \mathcal{O}_{X_A}) \to H^0(X_k, \mathcal{O}_{X_k})\) is surjective. Since \(X_k\) is a reduced proper scheme over \(k\) and since \(k\) is algebraically closed, we see that \(H^0(X_k, \mathcal{O}_{X_k})\) is a finite product of copies of \(k\) by the already used Varieties, Lemma 0BUG. Since by Lemma 0G7X the idempotents of this \(k\)-algebra are in the image of \(H^0(X_A, \mathcal{O}_{X_A}) \to H^0(X_k, \mathcal{O}_{X_k})\) we conclude.

Other applications

In this section we state and prove some results that can be deduced from the theory worked out above.

Lemma

Let \(R\) be a coherent ring. Let \(X\) be a scheme of finite presentation over \(R\). Let \(\mathcal{G}\) be an \(\mathcal{O}_X\)-module of finite presentation, flat over \(R\), with support proper over \(R\). Then \(H^i(X, \mathcal{G})\) is a coherent \(R\)-module.

Proof

Combine Lemma 0A1H with More on Algebra, Lemmas 0EWZ and 0658.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme. Let \(K\) be an object of \(D_\QCoh(\mathcal{O}_X)\) such that the cohomology sheaves \(H^i(K)\) have countable sets of sections over affine opens. Then for any quasi-compact open \(U \subset X\) and any perfect object \(E\) in \(D(\mathcal{O}_X)\) the sets \[H^i(U, K \otimes^\mathbf{L} E),\quad \Ext^i(E|_U, K|_U)\] are countable.

Proof

Using Cohomology, Lemma 08DQ we see that it suffices to prove the result for the groups \(H^i(U, K \otimes^\mathbf{L} E)\). We will use the induction principle to prove the lemma, see Cohomology of Schemes, Lemma 08DR.

First we show that it holds when \(U = \Spec(A)\) is affine. Namely, we can represent \(K\) by a complex of \(A\)-modules \(K^\bullet\) and \(E\) by a finite complex of finite projective \(A\)-modules \(P^\bullet\). See Lemmas 06Z0 and 08EB and our definition of perfect complexes of \(A\)-modules (More on Algebra, Definition 0657). Then \((E \otimes^\mathbf{L} K)|_U\) is represented by the total complex associated to the double complex \(P^\bullet \otimes_A K^\bullet\) (Lemma 08DX). Using induction on the length of the complex \(P^\bullet\) (or using a suitable spectral sequence) we see that it suffices to show that \(H^i(P^a \otimes_A K^\bullet)\) is countable for each \(a\). Since \(P^a\) is a direct summand of \(A^{\oplus n}\) for some \(n\) this follows from the assumption that the cohomology group \(H^i(K^\bullet)\) is countable.

To finish the proof it suffices to show: if \(U = V \cup W\) and the result holds for \(V\), \(W\), and \(V \cap W\), then the result holds for \(U\). This is an immediate consequence of the Mayer-Vietoris sequence, see Cohomology, Lemma 08BX.

Lemma

Let \(X\) be a quasi-compact and quasi-separated scheme such that the sets of sections of \(\mathcal{O}_X\) over affine opens are countable. Let \(K\) be an object of \(D_\QCoh(\mathcal{O}_X)\). The following are equivalent

1. \(K = \text{hocolim} E_n\) with \(E_n\) a perfect object of \(D(\mathcal{O}_X)\), and

2. the cohomology sheaves \(H^i(K)\) have countable sets of sections over affine opens.

Proof

If (1) is true, then (2) is true because homotopy colimits commutes with taking cohomology sheaves (by Derived Categories, Lemma 0CRK) and because a perfect complex is locally isomorphic to a finite complex of finite free \(\mathcal{O}_X\)-modules and therefore satisfies (2) by assumption on \(X\).

Assume (2). Choose a K-injective complex \(\mathcal{K}^\bullet\) representing \(K\). Choose a perfect generator \(E\) of \(D_\QCoh(\mathcal{O}_X)\) and represent it by a K-injective complex \(\mathcal{I}^\bullet\). According to Theorem 09M5 and its proof there is an equivalence of triangulated categories \(F : D_\QCoh(\mathcal{O}_X) \to D(A, \text{d})\) where \((A, \text{d})\) is the differential graded algebra \[(A, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{I}^\bullet)\] which maps \(K\) to the differential graded module \[M = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)} (\mathcal{I}^\bullet, \mathcal{K}^\bullet)\] Note that \(H^i(A) = \Ext^i(E, E)\) and \(H^i(M) = \Ext^i(E, K)\). Moreover, since \(F\) is an equivalence it and its quasi-inverse commute with homotopy colimits. Therefore, it suffices to write \(M\) as a homotopy colimit of compact objects of \(D(A, \text{d})\). By Differential Graded Algebra, Lemma 0CRM it suffices show that \(\Ext^i(E, E)\) and \(\Ext^i(E, K)\) are countable for each \(i\). This follows from Lemma 0CRP.

Lemma

Let \(A\) be a ring. Let \(X\) be a scheme of finite presentation over \(A\). Let \(f : U \to X\) be a flat morphism of finite presentation. Then

1. there exists an inverse system of perfect objects \(L_n\) of \(D(\mathcal{O}_X)\) such that \[R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Hom_X(L_n, K)\] in \(D(A)\) functorially in \(K\) in \(D_\QCoh(\mathcal{O}_X)\), and

2. there exists a system of perfect objects \(E_n\) of \(D(\mathcal{O}_X)\) such that \[R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Gamma(X, E_n \otimes^\mathbf{L} K)\] in \(D(A)\) functorially in \(K\) in \(D_\QCoh(\mathcal{O}_X)\).

Proof

By Lemma 08EU we have \[R\Gamma(U, Lf^*K) = R\Gamma(X, Rf_*\mathcal{O}_U \otimes^\mathbf{L} K)\] functorially in \(K\). Observe that \(R\Gamma(X, -)\) commutes with homotopy colimits because it commutes with direct sums by Lemma 08DZ. Similarly, \(- \otimes^\mathbf{L} K\) commutes with derived colimits because \(- \otimes^\mathbf{L} K\) commutes with direct sums (because direct sums in \(D(\mathcal{O}_X)\) are given by direct sums of representing complexes). Hence to prove (2) it suffices to write \(Rf_*\mathcal{O}_U = \text{hocolim} E_n\) for a system of perfect objects \(E_n\) of \(D(\mathcal{O}_X)\). Once this is done we obtain (1) by setting \(L_n = E_n^\vee\), see Cohomology, Lemma 08DQ.

Write \(A = \colim A_i\) with \(A_i\) of finite type over \(\mathbf{Z}\). By Limits, Lemma 01ZM we can find an \(i\) and morphisms \(U_i \to X_i \to \Spec(A_i)\) of finite presentation whose base change to \(\Spec(A)\) recovers \(U \to X \to \Spec(A)\). After increasing \(i\) we may assume that \(f_i : U_i \to X_i\) is flat, see Limits, Lemma 04AI. By Lemma 08IB the derived pullback of \(Rf_{i, *}\mathcal{O}_{U_i}\) by \(g : X \to X_i\) is equal to \(Rf_*\mathcal{O}_U\). Since \(Lg^*\) commutes with derived colimits, it suffices to prove what we want for \(f_i\). Hence we may assume that \(U\) and \(X\) are of finite type over \(\mathbf{Z}\).

Assume \(f : U \to X\) is a morphism of schemes of finite type over \(\mathbf{Z}\). To finish the proof we will show that \(Rf_*\mathcal{O}_U\) is a homotopy colimit of perfect complexes. To see this we apply Lemma 0CRQ. Thus it suffices to show that \(R^if_*\mathcal{O}_U\) has countable sets of sections over affine opens. This follows from Lemma 0CRP applied to the structure sheaf.

Characterizing pseudo-coherent complexes, II

This section is a continuation of Section 0DJM. In this section we discuss characterizations of pseudo-coherent complexes in terms of cohomology. More results of this nature can be found in More on Morphisms, Section 0CSI.

Lemma

Let \(A\) be a ring. Let \(R\) be a (possibly noncommutative) \(A\)-algebra which is finite free as an \(A\)-module. Then any object \(M\) of \(D(R)\) which is pseudo-coherent in \(D(A)\) can be represented by a bounded above complex of finite free (right) \(R\)-modules.

Proof

Choose a complex \(M^\bullet\) of right \(R\)-modules representing \(M\). Since \(M\) is pseudo-coherent we have \(H^i(M) = 0\) for large enough \(i\). Let \(m\) be the smallest index such that \(H^m(M)\) is nonzero. Then \(H^m(M)\) is a finite \(A\)-module by More on Algebra, Lemma 064S. Thus we can choose a finite free \(R\)-module \(F^m\) and a map \(F^m \to M^m\) such that \(F^m \to M^m \to M^{m + 1}\) is zero and such that \(F^m \to H^m(M)\) is surjective. Picture: \[\xymatrix{ & F^m \ar[d]^\alpha \ar[r] & 0 \ar[d] \ar[r] & \ldots \\ M^{m - 1} \ar[r] & M^m \ar[r] & M^{m + 1} \ar[r] & \ldots }\] By descending induction on \(n \leq m\) we are going to construct finite free \(R\)-modules \(F^i\) for \(i \geq n\), differentials \(d^i : F^i \to F^{i + 1}\) for \(i \geq n\), maps \(\alpha : F^i \to K^i\) compatible with differentials, such that (1) \(H^i(\alpha)\) is an isomorphism for \(i > n\) and surjective for \(i = n\), and (2) \(F^i = 0\) for \(i > m\). Picture \[\xymatrix{ & F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \ar[r] & F^i \ar[d]^\alpha \ar[r] & 0 \ar[d] \ar[r] & \ldots \\ M^{n - 1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots \ar[r] & M^i \ar[r] & M^{i + 1} \ar[r] & \ldots }\] The base case is \(n = m\) which we’ve done above. Induction step. Let \(C^\bullet\) be the cone on \(\alpha\) (Derived Categories, Definition 014E). The long exact sequence of cohomology shows that \(H^i(C^\bullet) = 0\) for \(i \geq n\). Observe that \(F^\bullet\) is pseudo-coherent as a complex of \(A\)-modules because \(R\) is finite free as an \(A\)-module. Hence by More on Algebra, Lemma 064R we see that \(C^\bullet\) is \((n - 1)\)-pseudo-coherent as a complex of \(A\)-modules. By More on Algebra, Lemma 064S we see that \(H^{n - 1}(C^\bullet)\) is a finite \(A\)-module. Choose a finite free \(R\)-module \(F^{n - 1}\) and a map \(\beta : F^{n - 1} \to C^{n - 1}\) such that the composition \(F^{n - 1} \to C^{n - 1} \to C^n\) is zero and such that \(F^{n - 1}\) surjects onto \(H^{n - 1}(C^\bullet)\). Since \(C^{n - 1} = M^{n - 1} \oplus F^n\) we can write \(\beta = (\alpha^{n - 1}, -d^{n - 1})\). The vanishing of the composition \(F^{n - 1} \to C^{n - 1} \to C^n\) implies these maps fit into a morphism of complexes \[\xymatrix{ & F^{n - 1} \ar[d]^{\alpha^{n - 1}} \ar[r]_{d^{n - 1}} & F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ \ldots \ar[r] & M^{n - 1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots }\] Moreover, these maps define a morphism of distinguished triangles \[\xymatrix{ (F^n \to \ldots) \ar[r] \ar[d] & (F^{n - 1} \to \ldots) \ar[r] \ar[d] & F^{n - 1} \ar[r] \ar[d]_\beta & (F^n \to \ldots)[1] \ar[d] \\ (F^n \to \ldots) \ar[r] & M^\bullet \ar[r] & C^\bullet \ar[r] & (F^n \to \ldots)[1] }\] Hence our choice of \(\beta\) implies that the map of complexes \((F^{n - 1} \to \ldots) \to M^\bullet\) induces an isomorphism on cohomology in degrees \(\geq n\) and a surjection in degree \(n - 1\). This finishes the proof of the lemma.

Lemma

Let \(A\) be a ring. Let \(n \geq 0\). Let \(K \in D_\QCoh(\mathcal{O}_{\mathbf{P}^n_A})\). The following are equivalent

1. \(K\) is pseudo-coherent,

2. \(R\Gamma(\mathbf{P}^n_A, E \otimes^\mathbf{L} K)\) is a pseudo-coherent object of \(D(A)\) for each pseudo-coherent object \(E\) of \(D(\mathcal{O}_{\mathbf{P}^n_A})\),

3. \(R\Gamma(\mathbf{P}^n_A, E \otimes^\mathbf{L} K)\) is a pseudo-coherent object of \(D(A)\) for each perfect object \(E\) of \(D(\mathcal{O}_{\mathbf{P}^n_A})\),

4. \(R\Hom_{\mathbf{P}^n_A}(E, K)\) is a pseudo-coherent object of \(D(A)\) for each perfect object \(E\) of \(D(\mathcal{O}_{\mathbf{P}^n_A})\),

5. \(R\Gamma(\mathbf{P}^n_A, K \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_A}(d))\) is pseudo-coherent object of \(D(A)\) for \(d = 0, 1, \ldots, n\).

Proof

Recall that \[R\Hom_{\mathbf{P}^n_A}(E, K) = R\Gamma(\mathbf{P}^n_A, R\SheafHom_{\mathcal{O}_{\mathbf{P}^n_A}}(E, K))\] by definition, see Cohomology, Section 0B6A. Thus parts (4) and (3) are equivalent by Cohomology, Lemma 08DQ.

Since every perfect complex is pseudo-coherent, it is clear that (2) implies (3).

Assume (1) holds. Then \(E \otimes^\mathbf{L} K\) is pseudo-coherent for every pseudo-coherent \(E\), see Cohomology, Lemma 09J3. By Lemma 0CSD the direct image of such a pseudo-coherent complex is pseudo-coherent and we see that (2) is true.

Part (3) implies (5) because we can take \(E = \mathcal{O}_{\mathbf{P}^n_A}(d)\) for \(d = 0, 1, \ldots, n\).

To finish the proof we have to show that (5) implies (1). Let \(P\) be as in (0CS8) and \(R\) as in (0CS9). By Lemma 0BQU we have an equivalence \[- \otimes^\mathbf{L}_R P : D(R) \longrightarrow D_\QCoh(\mathcal{O}_{\mathbf{P}^n_A})\] Let \(M \in D(R)\) be an object such that \(M \otimes^\mathbf{L} P = K\). By Differential Graded Algebra, Lemma 0CS6 there is an isomorphism \[R\Hom(R, M) = R\Hom_{\mathbf{P}^n_A}(P, K)\] in \(D(A)\). Arguing as above we obtain \[R\Hom_{\mathbf{P}^n_A}(P, K) = R\Gamma(\mathbf{P}^n_A, R\SheafHom_{\mathcal{O}_{\mathbf{P}^n_A}}(E, K)) = R\Gamma(\mathbf{P}^n_A, P^\vee \otimes^\mathbf{L}_{\mathcal{O}_{\mathbf{P}^n_A}} K).\] Using that \(P^\vee\) is the direct sum of \(\mathcal{O}_{\mathbf{P}^n_A}(d)\) for \(d = 0, 1, \ldots, n\) and (5) we conclude \(R\Hom(R, M)\) is pseudo-coherent as a complex of \(A\)-modules. Of course \(M = R\Hom(R, M)\) in \(D(A)\). Thus \(M\) is pseudo-coherent as a complex of \(A\)-modules. By Lemma 0CSF we may represent \(M\) by a bounded above complex \(F^\bullet\) of finite free \(R\)-modules. Then \(F^\bullet = \bigcup_{p \geq 0} \sigma_{\geq p}F^\bullet\) is a filtration which shows that \(F^\bullet\) is a differential graded \(R\)-module with property (P), see Differential Graded Algebra, Section 09KK. Hence \(K = M \otimes^\mathbf{L}_R P\) is represented by \(F^\bullet \otimes_R P\) (follows from the construction of the derived tensor functor, see for example the proof of Differential Graded Algebra, Lemma 09LX). Since \(F^\bullet \otimes_R P\) is a bounded above complex whose terms are direct sums of copies of \(P\) we conclude that the lemma is true.

Lemma

Let \(A\) be a ring. Let \(X\) be a scheme over \(A\) which is quasi-compact and quasi-separated. Let \(K \in D^-_\QCoh(\mathcal{O}_X)\). If \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every perfect \(E\) in \(D(\mathcal{O}_X)\), then \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is pseudo-coherent in \(D(A)\) for every pseudo-coherent \(E\) in \(D(\mathcal{O}_X)\).

Proof

There exists an integer \(N\) such that \(R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \to D(A)\) has cohomological dimension \(N\) as explained in Lemma 08D5. Let \(b \in \mathbf{Z}\) be such that \(H^i(K) = 0\) for \(i > b\). Let \(E\) be pseudo-coherent on \(X\). It suffices to show that \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is \(m\)-pseudo-coherent for every \(m\). Choose an approximation \(P \to E\) by a perfect complex \(P\) of \((X, E, m - N - 1 - b)\). This is possible by Theorem 08ES. Choose a distinguished triangle \[P \to E \to C \to P[1]\] in \(D_\QCoh(\mathcal{O}_X)\). The cohomology sheaves of \(C\) are zero in degrees \(\geq m - N - 1 - b\). Hence the cohomology sheaves of \(C \otimes^\mathbf{L} K\) are zero in degrees \(\geq m - N - 1\). Thus the cohomology of \(R\Gamma(X, C \otimes^\mathbf{L} K)\) are zero in degrees \(\geq m - 1\). Hence \[R\Gamma(X, P \otimes^\mathbf{L} K) \to R\Gamma(X, E \otimes^\mathbf{L} K)\] is an isomorphism on cohomology in degrees \(\geq m\). By assumption the source is pseudo-coherent. We conclude that \(R\Gamma(X, E \otimes^\mathbf{L} K)\) is \(m\)-pseudo-coherent as desired.

Relatively perfect objects

In this section we introduce a notion from [lieblich-complexes].

Definition

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. An object \(E\) of \(D(\mathcal{O}_X)\) is perfect relative to \(S\) or \(S\)-perfect if \(E\) is pseudo-coherent (Cohomology, Definition 08CB) and \(E\) locally has finite tor dimension as an object of \(D(f^{-1}\mathcal{O}_S)\) (Cohomology, Definition 08CG).

Please see Remark 0DI9 for a discussion.

Example

Let \(k\) be a field. Let \(X\) be a scheme of finite presentation over \(k\) (in particular \(X\) is quasi-compact). Then an object \(E\) of \(D(\mathcal{O}_X)\) is \(k\)-perfect if and only if it is bounded and pseudo-coherent (by definition), i.e., if and only if it is in \(D^b_{\textit{Coh}}(X)\) (by Lemma 08E8). Thus being relatively perfect does not mean “perfect on the fibres”.

The corresponding algebra concept is studied in More on Algebra, Section 0DHR. We can link the notion for schemes with the algebraic notion as follows.

Lemma

Let \(f : X \to S\) be a morphism of schemes which is flat and locally of finite presentation. Let \(E\) be an object of \(D_\QCoh(\mathcal{O}_X)\). The following are equivalent

1. \(E\) is \(S\)-perfect,

2. for any affine open \(U \subset X\) mapping into an affine open \(V \subset S\) the complex \(R\Gamma(U, E)\) is \(\mathcal{O}_S(V)\)-perfect.

3. there exists an affine open covering \(S = \bigcup V_i\) and for each \(i\) an affine open covering \(f^{-1}(V_i) = \bigcup U_{ij}\) such that the complex \(R\Gamma(U_{ij}, E)\) is \(\mathcal{O}_S(V_i)\)-perfect.

Proof

Being pseudo-coherent is a local property and “locally having finite tor dimension” is a local property. Hence this lemma immediately reduces to the statement: if \(X\) and \(S\) are affine, then \(E\) is \(S\)-perfect if and only if \(K = R\Gamma(X, E)\) is \(\mathcal{O}_S(S)\)-perfect. Say \(X = \Spec(A)\), \(S = \Spec(R)\) and \(E\) corresponds to \(K \in D(A)\), i.e., \(K = R\Gamma(X, E)\), see Lemma 06Z0.

Observe that \(K\) is \(R\)-perfect if and only if \(K\) is pseudo-coherent and has finite tor dimension as a complex of \(R\)-modules (More on Algebra, Definition 0DHS). By Lemma 08E7 we see that \(E\) is pseudo-coherent if and only if \(K\) is pseudo-coherent. By Lemma 0DHY we see that \(E\) has finite tor dimension over \(f^{-1}\mathcal{O}_S\) if and only if \(K\) has finite tor dimension as a complex of \(R\)-modules.