The Stacks project

Lemma 83.4.2. Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$ such that the structure morphism $f : X \to \mathop{\mathrm{Spec}}(A)$ is proper. Let $i : X_0 \to X$ be the inclusion of $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (X_0, i^{-1}\mathcal{F})$.

Proof. Choose a surjective proper morphism $Y \to X$ where $Y$ is a scheme, see Cohomology of Spaces, Lemma 68.18.1. Consider the diagram

\[ \xymatrix{ \Gamma (X_0, \mathcal{F}_0) \ar[r] & \Gamma (Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] & \Gamma ((Y \times _ X Y)_0, \mathcal{H}_0) \\ \Gamma (X, \mathcal{F}) \ar[r] \ar[u] & \Gamma (Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[u] & \Gamma (Y \times _ X Y, \mathcal{H}) \ar[u] } \]

Here $\mathcal{G}$, resp. $\mathcal{H}$ is the pullbackf or $\mathcal{F}$ to $Y$, resp. $Y \times _ X Y$ and the index $0$ indicates base change to $\mathop{\mathrm{Spec}}(A/I)$. By the case of schemes (Étale Cohomology, Lemma 59.91.2) we see that the middle and right vertical arrows are bijective. By Lemma 83.4.1 it follows that the left one is too. $\square$

Comments (2)

Comment #5919 by Harry Gindi on

I'm pretty sure that this diagram is upside-down. The map on global sections goes Γ(X,F)→Γ(X_0,F_0) (and similarly with the higher terms). Otherwise, the proof looks good!

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Your email address will not be published.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 0DFY. Beware of the difference between the letter 'O' and the digit '0'.