The Stacks project

Lemma 57.87.7. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume

  1. cohomology commutes with base change for $f$,

  2. cohomology commutes with base change for $g \circ f$, and

  3. $f$ is surjective.

Then cohomology commutes with base change for $g$.

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 $Y_{\acute{e}tale}$. Choose an injective map of sheaves $f^{-1}\mathcal{I} \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $Z_{\acute{e}tale}$. Since $f$ is surjective the map $\mathcal{I} \to f_*\mathcal{J}$ is injective (look at stalks in geometric points). Since $\mathcal{I}$ is injective we see that $\mathcal{I}$ is a direct summand of $f_*\mathcal{J}$. Thus it suffices to prove the desired vanishing for $f_*\mathcal{J}$.

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 $b^{-1}f_*\mathcal{J} = f'_*a^{-1}\mathcal{J}$. On the other hand, we know that $R^ qf'_*a^{-1}\mathcal{J}$ and $R^ q(g' \circ f')_*a^{-1}\mathcal{J}$ are zero for $q > 0$. Using the spectral sequence (Cohomology on Sites, Lemma 21.14.7)

\[ R^ pg'_* R^ qf'_* a^{-1}\mathcal{J} \Rightarrow R^{p + q}(g' \circ f')_* a^{-1}\mathcal{J} \]

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

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