Lemma 103.15.3 (0GR3)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 103: Cohomology of Algebraic Stacks
Section 103.15: Functoriality of the lisse-étale and flat-fppf sites
Lemma 103.15.3 (cite)

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Lemma 103.15.3. With assumptions and notation as in Lemma 103.15.1 the canonical (base change) map

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

is an isomorphism for any abelian sheaf $\mathcal{F}$ on $\mathcal{X}_{\acute{e}tale}$ (resp. $\mathcal{X}_{fppf}$ ).

Proof. Comparing the formula for $g^{-1}R^ pf_*\mathcal{F}$ and $R^ pf'_*(g')^{-1}\mathcal{F}$ given in Sheaves on Stacks, Lemma 96.21.2 and Lemma 103.15.2 we see that it suffices to show

$H^ p((V \times _{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times _{y, \mathcal{Y}} \mathcal{X})'}) = H^ p_\tau (V \times _{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})$

where $\tau = {\acute{e}tale}$ (resp. $\tau = fppf$ ). Here $y$ is an object of $\mathcal{Y}$ lying over a scheme $V$ such that the morphism $y : V \to \mathcal{Y}$ is smooth (resp. flat). This equality follows from Sheaves on Stacks, Lemma 96.23.3. Although we omit the verification of the assumptions of the lemma, we note that the fact that $V \to \mathcal{Y}$ is smooth (resp. flat) is used to verify the second condition. $\square$

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