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. $\square$


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