The Stacks project

Lemma 82.4.4. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$ be a morphisms of algebraic spaces over $S$. Assume $f$ is proper. Set $X' = Y' \times _ Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_{\acute{e}tale}$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.

Proof. The question is étale local on $Y'$. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $V'$ and a surjective étale morphism $V' \to V \times _ Y Y'$. Then we may replace $Y'$ by $V'$ and $Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes. Then we may work Zariski locally on $Y$ and $Y'$ and hence we may assume $Y$ and $Y'$ are affine schemes.

Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism $h_1 : X_1 \to X$ where $X_1$ is a scheme, see Cohomology of Spaces, Lemma 67.18.1. Set $X_2 = X_1 \times _ X X_1$ and denote $h_2 : X_2 \to X$ the structure morphism. Observe this is a scheme. By the case of schemes (Étale Cohomology, Lemma 58.87.5) we know the lemma is true for the cartesian diagrams

\[ \vcenter { \xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\ Y' \ar[r] & Y } } \quad \text{and}\quad \vcenter { \xymatrix{ X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\ Y' \ar[r] & Y } } \]

and the sheaves $\mathcal{F}_ i = (X_ i \to X)^{-1}\mathcal{F}$. By Lemma 82.4.1 we have an exact sequence $0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2$ and similarly for $(g')^{-1}\mathcal{F}$ because $X'_2 = X'_1 \times _{X'} X'_1$. Hence we conlude that the lemma is true (some details omitted). $\square$


Comments (0)


Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DG0. Beware of the difference between the letter 'O' and the digit '0'.