The Stacks project

Lemma 37.56.15. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ X)$ which is $m$-pseudo-coherent relative to $S$. Let $S' \to S$ be a morphism of schemes. Set $X' = X \times _ S S'$ and denote $E'$ the derived pullback of $E$ to $X'$. If $S'$ and $X$ are Tor independent over $S$, then $E'$ is $m$-pseudo-coherent relative to $S'$.

Proof. The problem is local on $X$ and $X'$ hence we may assume $X$, $S$, $S'$, and $X'$ are affine. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$ and denote $i' : X' \to \mathbf{A}^ n_{S'}$ the base change to $S'$. Denote $g : X' \to X$ and $g' : \mathbf{A}^ n_{S'} \to \mathbf{A}^ n_ S$ the projections, so $E' = Lg^*E$. Since $X$ and $S'$ are tor-independent over $S$, the base change map (Cohomology, Remark 20.28.3) induces an isomorphism

\[ Ri'_*(Lg^*E) = L(g')^*Ri_*E \]

Namely, for a point $x' \in X'$ lying over $x \in X$ the base change map on stalks at $x'$ is the map

\[ E_ x \otimes _{\mathcal{O}_{\mathbf{A}^ n_ S, x}}^\mathbf {L} \mathcal{O}_{\mathbf{A}^ n_{S'}, x'} \longrightarrow E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{O}_{X', x'} \]

coming from the closed immersions $i$ and $i'$. Note that the source is quasi-isomorphic to a localization of $E_ x \otimes _{\mathcal{O}_{S, s}}^\mathbf {L} \mathcal{O}_{S', s'}$ which is isomorphic to the target as $\mathcal{O}_{X', x'}$ is isomorphic to (the same) localization of $\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{S, s}}^\mathbf {L} \mathcal{O}_{S', s'}$ by assumption. We conclude the lemma holds by an application of Cohomology, Lemma 20.44.3. $\square$


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