The Stacks project

Theorem 59.82.7 (Gabber). Let $(A, I)$ be a henselian pair. Set $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$. For any torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(Z, \mathcal{F}|_ Z)$.

Proof. The result holds for $q = 0$ by Lemma 59.82.6. Let $q \geq 1$. Suppose the result has been shown in all degrees $< q$. Let $\mathcal{F}$ be a torsion abelian sheaf. Let $\mathcal{F} \to \mathcal{F}'$ be an injective map of torsion abelian sheaves (to be chosen later) with cokernel $\mathcal{Q}$ so that we have the short exact sequence

\[ 0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{Q} \to 0 \]

of torsion abelian sheaves on $X_{\acute{e}tale}$. This gives a map of long exact cohomology sequences over $X$ and $Z$ part of which looks like

\[ \xymatrix{ H^{q - 1}_{\acute{e}tale}(X, \mathcal{F}') \ar[d] \ar[r] & H^{q - 1}_{\acute{e}tale}(X, \mathcal{Q}) \ar[d] \ar[r] & H^ q_{\acute{e}tale}(X, \mathcal{F}) \ar[d] \ar[r] & H^ q_{\acute{e}tale}(X, \mathcal{F}') \ar[d] \\ H^{q - 1}_{\acute{e}tale}(Z, \mathcal{F}'|_ Z) \ar[r] & H^{q - 1}_{\acute{e}tale}(Z, \mathcal{Q}|_ Z) \ar[r] & H^ q_{\acute{e}tale}(Z, \mathcal{F}|_ Z) \ar[r] & H^ q_{\acute{e}tale}(Z, \mathcal{F}'|_ Z) } \]

Using this commutative diagram of abelian groups with exact rows we will finish the proof.

Injectivity for $\mathcal{F}$. Let $\xi $ be a nonzero element of $H^ q_{\acute{e}tale}(X, \mathcal{F})$. By Lemma 59.82.1 applied with $Z = X$ (!) we can find $\mathcal{F} \subset \mathcal{F}'$ such that $\xi $ maps to zero to the right. Then $\xi $ is the image of an element of $H^{q - 1}_{\acute{e}tale}(X, \mathcal{Q})$ and bijectivity for $q - 1$ implies $\xi $ does not map to zero in $H^ q_{\acute{e}tale}(Z, \mathcal{F}|_ Z)$.

Surjectivity for $\mathcal{F}$. Let $\xi $ be an element of $H^ q_{\acute{e}tale}(Z, \mathcal{F}|_ Z)$. By Lemma 59.82.1 applied with $Z = Z$ we can find $\mathcal{F} \subset \mathcal{F}'$ such that $\xi $ maps to zero to the right. Then $\xi $ is the image of an element of $H^{q - 1}_{\acute{e}tale}(Z, \mathcal{Q}|_ Z)$ and bijectivity for $q - 1$ implies $\xi $ is in the image of the vertical map. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 59.82: Affine analog of proper base change

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Your email address will not be published.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 09ZI. Beware of the difference between the letter 'O' and the digit '0'.