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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09UF. Beware of the difference between the letter 'O' and the digit '0'.