The Stacks project

Lemma 59.85.1. Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$ with closed fibre $X_0$. Let $M$ be a finite abelian group. Then $H^1_{\acute{e}tale}(X, \underline{M}) = H^1_{\acute{e}tale}(X_0, \underline{M})$.

Proof. By Cohomology on Sites, Lemma 21.4.3 an element of $H^1_{\acute{e}tale}(X, \underline{M})$ corresponds to a $\underline{M}$-torsor $\mathcal{F}$ on $X_{\acute{e}tale}$. Such a torsor is clearly a finite locally constant sheaf. Hence $\mathcal{F}$ is representable by a scheme $V$ finite étale over $X$, Lemma 59.64.4. Conversely, a scheme $V$ finite étale over $X$ with an $M$-action which turns it into an $M$-torsor over $X$ gives rise to a cohomology class. The same translation between cohomology classes over $X_0$ and torsors finite étale over $X_0$ holds. Thus the lemma is a consequence of the equivalence of categories of Fundamental Groups, Lemma 58.9.1. $\square$

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