Lemma 59.91.8. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume
cohomology commutes with base change for $f$, and
cohomology commutes with base change for $g$.
Then cohomology commutes with base change for $g \circ f$.
Lemma 59.91.8. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume
cohomology commutes with base change for $f$, and
cohomology commutes with base change for $g$.
Then cohomology commutes with base change for $g \circ f$.
Proof. We will use the equivalence of Lemma 59.91.6 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 59.91.5 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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #3213 by Alexander Schmidt on
Comment #3314 by Johan on
There are also: