Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 59.91.2. Let $(A, I)$ be a henselian pair. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Let $i : Z \to X$ be the closed immersion of $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$ into $X$. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (Z, i_{small}^{-1}\mathcal{F})$.

Proof. This follows from Lemma 59.82.2 and 59.91.1 and the fact that any scheme finite over $X$ is proper over $\mathop{\mathrm{Spec}}(A)$. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 59.91: The proper base change theorem

Your email address will not be published.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.