The Stacks project

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 20.45.3. 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 20.45.9. 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 20.45.4 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 20.45.4 again we conclude. $\square$

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