The Stacks project

Theorem 57.79.7 (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 57.79.6. 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 57.79.1 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 57.79.1 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. $\square$


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