The Stacks project

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 20.45.3. 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 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. $\square$

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Your email address will not be published.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 09UE. Beware of the difference between the letter 'O' and the digit '0'.