The Stacks project

Lemma 36.35.6. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $g : S' \to S$ be a morphism of schemes. Set $X' = S' \times _ S X$ and denote $g' : X' \to X$ the projection. If $K \in D(\mathcal{O}_ X)$ is $S$-perfect, then $L(g')^*K$ is $S'$-perfect.

Proof. First proof: reduce to the affine case using Lemma 36.35.3 and then apply More on Algebra, Lemma 15.82.5.

Second proof: $L(g')^*K$ is pseudo-coherent by Cohomology, Lemma 20.44.3 and the bounded tor dimension property follows from Lemma 36.22.8. $\square$


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