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36.12 Descent finiteness properties of complexes

This section is the analogue of Descent, Section 35.7 for objects of the derived category of a scheme. The easiest such result is probably the following.

Lemma 36.12.1. 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 20.48.4. We may check tor-amplitude in [a, b] on stalks, see Cohomology, Lemma 20.48.5. A flat local ring homomorphism is faithfully flat by Algebra, Lemma 10.39.17. Thus the result follows from More on Algebra, Lemma 15.66.17. \square

Lemma 36.12.2. Let \{ f_ i : X_ i \to X\} be an fpqc covering of schemes. Let E \in D_\mathit{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 20.47.3. 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 15.64.15 via Lemmas 36.3.5 and 36.10.2. \square

Lemma 36.12.3. 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 20.47.3. 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 20.47.9. Then by Descent, Lemma 35.7.2 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 20.47.4 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 20.47.4 again we conclude. \square

Lemma 36.12.4. 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 20.49.6. 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 36.10.1 and Cohomology, Lemma 20.49.5. 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 35.5.2. Having said this the lemma follows formally from Cohomology, Lemma 20.49.5 and Lemmas 36.12.1 and 36.12.2. \square

Lemma 36.12.5. 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 17.6.1.

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 20.47.9. 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 20.47.4. 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 20.47.4 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 20.47.9. 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 20.47.4 we conclude that i_*C is m-pseudo-coherent. By induction we conclude that C is m-pseudo-coherent. By Cohomology, Lemma 20.47.4 again we conclude that E is m-pseudo-coherent. \square

Lemma 36.12.6. 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-module1. Let E \in D_\mathit{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 15.64.11 into the language of schemes. To do the translation, use Lemmas 36.3.5 and 36.10.2. \square

[1] This means that f is pseudo-coherent, see More on Morphisms, Lemma 37.60.8.

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