Lemma 57.29.1. Let $(A, \mathfrak m)$ be a regular local ring which contains a field. Let $f : V \to \mathop{\mathrm{Spec}}(A)$ be étale and quasi-compact. Assume that $\mathfrak m \not\in f(V)$ and assume that $g : V \to \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $ is affine. Then $H^ i(V, \mathcal{O}_ V)$, $i > 0$ is isomorphic to a direct sum of copies of the injective hull of the residue field of $A$.

**Proof.**
Denote $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $ the punctured spectrum. Thus $g : V \to U$ is affine. We have $H^ i(V, \mathcal{O}_ V) = H^ i(U, g_*\mathcal{O}_ V)$ by Cohomology of Schemes, Lemma 30.2.4. The $\mathcal{O}_ U$-module $g_*\mathcal{O}_ V$ is quasi-coherent by Schemes, Lemma 26.24.1. For any quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}$ the cohomology $H^ i(U, \mathcal{F})$, $i > 0$ is $\mathfrak m$-power torsion, see for example Local Cohomology, Lemma 51.2.2. In particular, the $A$-modules $H^ i(V, \mathcal{O}_ V)$, $i > 0$ are $\mathfrak m$-power torsion. For any flat ring map $A \to A'$ we have $H^ i(V, \mathcal{O}_ V) \otimes _ A A' = H^ i(V', \mathcal{O}_{V'})$ where $V' = V \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A')$ by flat base change Cohomology of Schemes, Lemma 30.5.2. If we take $A'$ to be the completion of $A$ (flat by More on Algebra, Section 15.43), then we see that

The first equality by the torsion property we just proved and More on Algebra, Lemma 15.88.3. Moreover, the injective hull of the residue field $k$ is the same for $A$ and $A'$, see Dualizing Complexes, Lemma 47.7.4. In this way we reduce to the case $A = k[[x_1, \ldots , x_ d]]$, see Algebra, Section 10.159.

Assume the characteristic of $k$ is $p > 0$. Since $F : A \to A$, $a \mapsto a^ p$ is flat (Local Cohomology, Lemma 51.17.6) and since $V \times _{\mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(F)} \mathop{\mathrm{Spec}}(A) \cong V$ as schemes over $\mathop{\mathrm{Spec}}(A)$ by Étale Morphisms, Lemma 41.14.3 the above gives $H^ i(V, \mathcal{O}_ V) \otimes _{A, F} A \cong H^ i(V, \mathcal{O}_ V)$. Thus we get the result by Local Cohomology, Lemma 51.18.2.

Assume the characteristic of $k$ is $0$. By Local Cohomology, Lemma 51.19.3 there are additive operators $D_ j$, $j = 1, \ldots , d$ on $H^ i(V, \mathcal{O}_ V)$ satisfying the Leibniz rule with respect to $\partial _ j = \partial /\partial x_ j$. Thus we get the result by Local Cohomology, Lemma 51.18.1. $\square$

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