The Stacks project

Lemma 73.25.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. 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 $Y$, with support proper over $Y$. Then

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

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

Proof. The statement on base change is Lemma 73.21.4. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma 73.22.2. 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 $Y$, hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits of Spaces, Lemma 68.7.1 there exists an $i$ and an algebraic space $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemma 68.7.2 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 of Spaces, Lemma 68.6.12. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits of Spaces, Lemma 68.12.3. Finally, by Lemma 73.24.3 we may, after increasing $i$, assume there exists a perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. By Lemma 73.22.2 we have that $K_ i = Rf_{i, *}(E_ i \otimes _{\mathcal{O}_{X_ i}}^\mathbf {L} \mathcal{G}_ i^\bullet )$ is perfect on $\mathop{\mathrm{Spec}}(R_ i)$ where $f_ i : X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ is the structure morphism. By the base change result (Lemma 73.21.4) the pullback of $K_ i$ to $Y = \mathop{\mathrm{Spec}}(R)$ is $K$ and we conclude. $\square$

Comments (3)

Comment #560 by BB on

The word 'quasi-coherent' should be added as a hypothesis on (as the lemma on base change has that hypothesis).

Comment #561 by on

Nope... because a finitely presented module is always a quasi-coherent module. This is clear from the definitions, see Lemma 17.11.2. The corresponding statement for ringed topoi seems to be missing. Maybe we should fix this by adding a reference to the module in the proof?

Comment #563 by BB on

Haha, oops, yes, you're right, sorry. I somehow assumed modules of finite presentation are the same thing as compact objects in all -modules...

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