The Stacks project

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 20.45.4. We may check tor-amplitude in $[a, b]$ on stalks, see Cohomology, Lemma 20.45.5. A flat local ring homomorphism is faithfully flat by Algebra, Lemma 10.39.17. Thus the result follows from More on Algebra, Lemma 15.65.17. $\square$

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 20.44.3. 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 15.63.15 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 20.44.3. 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 20.44.9. 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 20.44.4 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 20.44.4 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 20.46.6. 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 20.46.5. 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 20.46.5 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 20.44.9. 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 20.44.4. 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 20.44.4 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 20.44.9. 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 20.44.4 we conclude that $i_*C$ is $m$-pseudo-coherent. By induction we conclude that $C$ is $m$-pseudo-coherent. By Cohomology, Lemma 20.44.4 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 15.63.11 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 37.54.8.

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