Lemma 103.15.2. With assumptions and notation as in Lemma 103.15.1. Let \mathcal{H} be an abelian sheaf on \mathcal{X}_{lisse,{\acute{e}tale}} (resp. \mathcal{X}_{flat,fppf}). Then
103.15.2.1
\begin{equation} \label{stacks-cohomology-equation-higher-direct-image-lisse-etale} R^ pf'_*\mathcal{H} = \text{sheaf associated to }y \longmapsto H^ p((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{H}) \end{equation}
Here y is an object of \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{flat,fppf}) lying over the scheme V and the notation (V \times _{y, \mathcal{Y}} \mathcal{X})' and \text{pr}' are explained in the proof.
Proof.
As in the proof of Lemma 103.15.1 let (V \times _{y, \mathcal{Y}} \mathcal{X})' \subset V \times _{y, \mathcal{Y}} \mathcal{X} be the full subcategory consisting of objects (x, \varphi ) where x is an object of \mathcal{X}_{lisse,{\acute{e}tale}} (resp. \mathcal{X}_{flat,fppf}) and \varphi : f(x) \to y is a morphism in \mathcal{Y}. By Equation (103.15.1.1) we have
f'_*\mathcal{H}(y) = \Gamma ((V \times _{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H})
where \text{pr}' is the projection. For an object (x, \varphi ) of (V \times _{y, \mathcal{Y}} \mathcal{X})' we can think of \varphi as a section of (f')^{-1}h_ y over x. Thus (V \times _\mathcal {Y} \mathcal{X})' is the localization of the site \mathcal{X}_{lisse,{\acute{e}tale}} (resp. \mathcal{X}_{flat,fppf}) at the sheaf of sets (f')^{-1}h_ y, see Sites, Lemma 7.30.3. The morphism
\text{pr}' : (V \times _{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{lisse,{\acute{e}tale}} \ (\text{resp. } \text{pr}' : (V \times _{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{flat,fppf})
is the localization morphism. In particular, the pullback (\text{pr}')^{-1} preserves injective abelian sheaves, see Cohomology on Sites, Lemma 21.13.3.
Choose an injective resolution \mathcal{H} \to \mathcal{I}^\bullet on \mathcal{X}_{lisse,{\acute{e}tale}} (resp. \mathcal{X}_{flat,fppf}). By the formula for pushforward we see that R^ if'_*\mathcal{H} is the sheaf associated to the presheaf which associates to y the cohomology of the complex
\begin{matrix} \Gamma \Big((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^{i - 1}\Big)
\\ \downarrow
\\ \Gamma \Big((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^ i\Big)
\\ \downarrow
\\ \Gamma \Big((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{I}^{i + 1}\Big)
\end{matrix}
Since (\text{pr}')^{-1} is exact and preserves injectives the complex (\text{pr}')^{-1}\mathcal{I}^\bullet is an injective resolution of (\text{pr}')^{-1}\mathcal{H} and the proof is complete.
\square
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