The Stacks project

Proof. We will use the equivalence of Lemma 57.87.6 without further mention. Let $\ell $ be a prime number. Let $\mathcal{I}$ be an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $X_{\acute{e}tale}$. Then $f_*\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $Y_{\acute{e}tale}$ (Cohomology on Sites, Lemma 21.14.2). The result follows formally from this, but we will also spell it out.

Let $Z' \to Z$ be a morphism of schemes and set $Y' = Z' \times _ Z Y$ and $X' = Z' \times _ Z X = Y' \times _ Y X$. Denote $a : X' \to X$, $b : Y' \to Y$, and $c : Z' \to Z$ the projections. Similarly for $f' : X' \to Y'$ and $g' : Y' \to Z'$. By Lemma 57.87.5 we have $b^{-1}f_*\mathcal{I} = f'_*a^{-1}\mathcal{I}$. On the other hand, we know that $R^ qf'_*a^{-1}\mathcal{I}$ and $R^ q(g')_*b^{-1}f_*\mathcal{I}$ are zero for $q > 0$. Using the spectral sequence (Cohomology on Sites, Lemma 21.14.7)

we conclude that $R^ p(g' \circ f')_*a^{-1}\mathcal{I} = 0$ for $p > 0$ as desired. $\square$