The Stacks project

Lemma 96.21.3. Let $S$ be a scheme. Let $\tau \in \{ Zariski,\linebreak[0] {\acute{e}tale},\linebreak[0] smooth,\linebreak[0] syntomic,\linebreak[0] fppf\} $. Let

\[ \xymatrix{ \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \ar[r]_{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Y}' \ar[r]^ g & \mathcal{Y} } \]

be a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change map is an isomorphism

\[ g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F} \]

functorial for $\mathcal{F}$ in $\textit{Ab}(\mathcal{X}_\tau )$ or $\mathcal{F}$ in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.

Proof. The isomorphism $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ is Lemma 96.5.1 (and it holds for arbitrary presheaves). For the total direct images, there is a base change map because the morphisms $g$ and $g'$ are flat, see Cohomology on Sites, Section 21.15. To see that this map is a quasi-isomorphism we can use that for an object $y'$ of $\mathcal{Y}'$ over a scheme $V$ there is an equivalence

\[ (\mathit{Sch}/V)_{fppf} \times _{g(y'), \mathcal{Y}} \mathcal{X} = (\mathit{Sch}/V)_{fppf} \times _{y', \mathcal{Y}'} (\mathcal{Y}' \times _\mathcal {Y} \mathcal{X}) \]

We conclude that the induced map $g^{-1}R^ if_*\mathcal{F} \to R^ if'_*(g')^{-1}\mathcal{F}$ is an isomorphism by Lemma 96.21.2. $\square$

Comments (2)

Comment #5830 by Georg Oberdieck on

In the statement, is it asumed that and hence is flat?

Comment #5831 by on

@#5830: No, not in the sense you mean. But here we are discussing sheaves of modules on the big sites (because this is the default in the Stacks project) and the morphism of ringed topoi associated to any -morphism of algebraic stacks is flat because the inverse image of the structure sheaf is the structure sheaf, see Lemma 96.6.2.

We only start writing about quasi-coherent modules (in case you were looking for this) and their higher direct images in the Chapter 103. Unfortunately, we haven't yet written much about topics such as flat base change of higher direct images of quasi-coherent modules.

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