The Stacks project

Lemma 59.59.4. Let $R$ be a local ring of dimension $0$. Let $S = \mathop{\mathrm{Spec}}(R)$. Then every $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$ is quasi-coherent.

Proof. Let $\mathcal{F}$ be an $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$. We have to show that $\mathcal{F}$ is determined by the $R$-module $M = \Gamma (S, \mathcal{F})$. More precisely, if $\pi : X \to S$ is étale we have to show that $\Gamma (X, \mathcal{F}) = \Gamma (X, \pi ^*\widetilde{M})$.

Let $\mathfrak m \subset R$ be the maximal ideal and let $\kappa $ be the residue field. By Algebra, Lemma 10.153.10 the local ring $R$ is henselian. If $X \to S$ is étale, then the underlying topological space of $X$ is discrete by Morphisms, Lemma 29.36.7 and hence $X$ is a disjoint union of affine schemes each having one point. Moreover, if $X = \mathop{\mathrm{Spec}}(A)$ is affine and has one point, then $R \to A$ is finite étale by Algebra, Lemma 10.153.5. We have to show that $\Gamma (X, \mathcal{F}) = M \otimes _ R A$ in this case.

The functor $A \mapsto A/\mathfrak m A$ defines an equivalence of the category of finite étale $R$-algebras with the category of finite separable $\kappa $-algebras by Algebra, Lemma 10.153.7. Let us first consider the case where $A/\mathfrak m A$ is a Galois extension of $\kappa $ with Galois group $G$. For each $\sigma \in G$ let $\sigma : A \to A$ denote the corresponding automorphism of $A$ over $R$. Let $N = \Gamma (X, \mathcal{F})$. Then $\mathop{\mathrm{Spec}}(\sigma ) : X \to X$ is an automorphism over $S$ and hence pullback by this defines a map $\sigma : N \to N$ which is a $\sigma $-linear map: $\sigma (an) = \sigma (a) \sigma (n)$ for $a \in A$ and $n \in N$. We will apply Galois descent to the quasi-coherent module $\widetilde{N}$ on $X$ endowed with the isomorphisms coming from the action on $\sigma $ on $N$. See Descent, Lemma 35.6.2. This lemma tells us there is an isomorphism $N = N^ G \otimes _ R A$. On the other hand, it is clear that $N^ G = M$ by the sheaf property for $\mathcal{F}$. Thus the required isomorphism holds.

The general case (with $A$ local and finite étale over $R$) is deduced from the Galois case as follows. Choose $A \to B$ finite étale such that $B$ is local with residue field Galois over $\kappa $. Let $G = \text{Aut}(B/R) = \text{Gal}(\kappa _ B/\kappa )$. Let $H \subset G$ be the Galois group corresponding to the Galois extension $\kappa _ B/\kappa _ A$. Then as above one shows that $\Gamma (X, \mathcal{F}) = \Gamma (\mathop{\mathrm{Spec}}(B), \mathcal{F})^ H$. By the result for Galois extensions (used twice) we get

\[ \Gamma (X, \mathcal{F}) = (M \otimes _ R B)^ H = M \otimes _ R A \]

as desired. $\square$

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