The Stacks project

Lemma 101.7.3. 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 101.5.1 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 101.5.1 the lemma will follow. Properties (1), (2), and (3) follow from Sheaves on Stacks, Lemmas 94.11.7 and 94.11.8 and Lemmas 101.6.1 and 101.7.2. 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,

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


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

by Sheaves on Stacks, Lemma 94.21.3. 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 67.11.2. Thus $R^ if_*\mathcal{F}$ has the flat base change property. Since $R^ if_*\mathcal{F}$ is locally quasi-coherent by Lemma 101.6.2 we win. $\square$

Comments (2)

Comment #3185 by anonymous on

typo in formulation of lemma: should be

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