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
$$\label{stacks-cohomology-equation-higher-direct-image-lisse-etale} R^ pf'_*\mathcal{H} = \text{sheaf associated to }y \longmapsto H^ p((V \times _{y, \mathcal{Y}} \mathcal{X})', (\text{pr}')^{-1}\mathcal{H})$$

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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