Lemma 58.29.1. 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
\[ H^ i(V, \mathcal{O}_ V) = H^ i(V, \mathcal{O}_ V) \otimes _ A A' = H^ i(V', \mathcal{O}_{V'}),\quad \text{for } i > 0 \]
The first equality by the torsion property we just proved and More on Algebra, Lemma 15.89.3. 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 10.160.
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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