The Stacks project

Lemma 102.15.3. With assumptions and notation as in Lemma 102.15.1 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 95.21.2 and Lemma 102.15.2 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 95.23.3. 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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