The Stacks project

Lemma 36.30.1. Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Let $\mathcal{G}^\bullet $ be a bounded complex of finitely presented $\mathcal{O}_ X$-modules, flat over $S$, with support proper over $S$. Then

\[ K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]

is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 36.26.4. Thus it suffices to show that $K$ is a perfect object. If $S$ is Noetherian, then this follows from Lemma 36.27.2. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on $S$, hence we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits, Lemma 32.10.1 there exists an $i$ and a scheme $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits, Lemma 32.10.2 we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_ i}$-modules $\mathcal{G}_ i^\bullet $ whose pullback to $X$ is $\mathcal{G}^\bullet $. After increasing $i$ we may assume $\mathcal{G}_ i^ n$ is flat over $R_ i$, see Limits, Lemma 32.10.4. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits, Lemma 32.13.5 and Cohomology of Schemes, Lemma 30.26.7. Finally, by Lemma 36.29.3 we may, after increasing $i$, assume there exists a perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Lemma 36.27.2 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$, $E_ i$, $\mathcal{G}_ i^\bullet $ and using the base change property already shown we obtain the result. $\square$

Comments (3)

Comment #515 by Dylan on

Suppose E is concentrated in degree zero and locally free. Is there any way to recover from this lemma a condition which guarantees that, say, (not derived) is locally free? Of course we have such a thing when there's local Noetherian hypotheses, but my question is if we can remove them. For example, is it enough to check that R^1f_*E has constant rank?

Comment #517 by on

Not sure what criterion for the Noetherian case you have in mind, but if I understand you correctly, then the result of the lemma reduces your question to a question cohomology modules of a perfect complex over a ring . In this situation quite a bit of what Hartshorne does in Chapter III, Section 12 carries over to the non-Noetherian case (you do have to modify the statements somewhat).

Comment #523 by on

@#515: Take a look at Lemma 15.76.3 for an example of something from hartshorne that carries over.

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