The Stacks project

Lemma 100.7.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ be an object of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ which is locally quasi-coherent and has the flat base change property. Then each $R^ if_*\mathcal{F}$ (computed in the étale topology) has the flat base change property.

Proof. We will use Lemma 100.5.1 to prove this. For every algebraic stack $\mathcal{X}$ let $\mathcal{M}_\mathcal {X}$ denote the full subcategory of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ consisting of locally quasi-coherent sheaves with the flat base change property. Once we verify conditions (1) – (4) of Lemma 100.5.1 the lemma will follow. Properties (1), (2), and (3) follow from Sheaves on Stacks, Lemmas 93.11.7 and 93.11.8 and Lemmas 100.6.1 and 100.7.2. Thus it suffices to show part (4).

Suppose $f : \mathcal{X} \to \mathcal{Y}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes $X$ and $Y$. In this case, suppose that $\psi : y \to y'$ is a morphism of $\mathcal{Y}$ lying over a flat morphism $b : V \to V'$ of schemes. For clarity denote $\mathcal{V} = (\mathit{Sch}/V)_{fppf}$ and $\mathcal{V}' = (\mathit{Sch}/V')_{fppf}$ the corresponding algebraic stacks. Consider the diagram of algebraic stacks

\[ \xymatrix{ \mathcal{Z} \ar[d]_{f''} \ar[r]_ a & \mathcal{Z}' \ar[r]_{x'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{V} \ar[r]^ b & \mathcal{V}' \ar[r]^{y'} & \mathcal{Y} } \]

with both squares cartesian. As $f$ is representable by schemes (and quasi-compact and separated – even affine) we see that $\mathcal{Z}$ and $\mathcal{Z}'$ are representable by schemes $Z$ and $Z'$ and in fact $Z = V \times _{V'} Z'$. Since $\mathcal{F}$ has the flat base change property we see that

\[ a_{small}^*\big (\mathcal{F}|_{Z'_{\acute{e}tale}}\big ) \longrightarrow \mathcal{F}|_{Z_{\acute{e}tale}} \]

is an isomorphism. Moreover,

\[ R^ if_*\mathcal{F}|_{V'_{\acute{e}tale}} = R^ i(f')_{small, *}\big (\mathcal{F}|_{Z'_{\acute{e}tale}}\big ) \]

and

\[ R^ if_*\mathcal{F}|_{V_{\acute{e}tale}} = R^ i(f'')_{small, *}\big (\mathcal{F}|_{Z_{\acute{e}tale}}\big ) \]

by Sheaves on Stacks, Lemma 93.21.3. Hence we see that the comparison map

\[ c_\psi : b_{small}^*(R^ if_*\mathcal{F}|_{V'_{\acute{e}tale}}) \longrightarrow R^ if_*\mathcal{F}|_{V_{\acute{e}tale}} \]

is an isomorphism by Cohomology of Spaces, Lemma 66.11.2. Thus $R^ if_*\mathcal{F}$ has the flat base change property. Since $R^ if_*\mathcal{F}$ is locally quasi-coherent by Lemma 100.6.2 we win. $\square$


Comments (2)

Comment #3185 by anonymous on

typo in formulation of lemma: should be


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