The Stacks project

Lemma 37.67.3. Let $f : X \to S$ be locally of finite type. Let $\{ S_ i \to S\} $ be an fppf covering of schemes. Denote $f_ i : X_ i \to S_ i$ the base change of $f$ and $g_ i : X_ i \to X$ the projection. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if each $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$.

Proof. This follows formally from Lemmas 37.67.1 and 37.67.2. Namely, if $E$ is $m$-pseudo-coherent relative to $S$, then $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma), hence $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$ (by the second). Conversely, if $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$, then $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S$ (by the second lemma), hence $E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma). $\square$


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