Number Rings and (Galois) Descent

Question

In algebraic number theory, one chooses for each finite étale $\mathbb{Q}$-algebra $K$ a finite $\mathbb{Z}$-algebra $\mathcal{O}_K$. Usually one simply speaks of the finite $\mathbb{Q}$-algebras which are fields, but the étale $\mathbb{Q}$-algebras are just products of these.

Now $- \otimes \mathbb{Q}$ takes the ring of integers $\mathcal{O}_K$ to $K$. Moreover, any integral basis for $\mathcal{O}_K$ is a $\mathbb{Q}$-basis for $K$. In terms of descent theory, we might think of $\mathcal{O}_K$ as a $\mathbb{Z}$-form of $K$. Indeed, this seems similar to the situation for Galois descent, or descent in general. However, as far as I know, descent theory applies to Galois extensions $L/K$ of fields. In that case, a descent datum for an $L$-algebra $A$ is the structure of an $G$-module, where $G$ is the Galois group of the extension, but here, any map of $K$ fixing $\mathcal{O}_K$ seems to also fix $K$.

I guess I'll still ask: is it possible to do descent theory in a way that applies to this context? More precisely, what kind of descent data would we need to recover $\mathcal{O}_K$ from $K$?

Answer 1

The categorical Galois theory of Borceux and Janelidze given in chapter 4 for commutative rings applies to your situation.

In particular, it applies to any 'effective Galois descent morphism' defined as follows:

Definition 1. Let $\mathcal{C}$ be a category. An arrow $f:X\to Y\in \mathsf{Hom}_\mathcal{C}$ is an effective descent morphism iff the pullback along $f$ functor $-_f:\mathcal{C}/Y\to\mathcal{C}/X$ is monadic.

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Definition 2. Let $\sigma:R\to S$ be a morphism of rings. Write $\eta$ for the unit of the adjunction $$\mathsf{Sp}_S:(\mathsf{S}\text{-Alg})^{op}\rightleftarrows\mathsf{Prof}/\mathsf{Sp}(S):\mathcal{C}_S,$$ where $\mathsf{Sp}_S$ and $\mathcal{C}_S$ are defined by $$\mathsf{Sp}_S(A)=\big(\mathsf{Sp}(A)\longrightarrow\mathsf{Sp}(S)\big),$$ $$\mathcal{C}_S(X,f)=\mathsf{Hom}\big((X,f),(\coprod_MS/M,p)\big),$$ with $p:\coprod_MS/M\to\mathsf{Sp}(S)$ the projection of the Pierce structural space of the ring $S$. An $R$-algebra $A$ is split by $\sigma$ when the morphism $$\eta_{S\otimes_RA}:\mathcal{C}_S\mathsf{Sp}_S(S\otimes_R A)\longrightarrow S\otimes_RA$$ is an isomorphism.

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Definition 3. A morphism of rings $\sigma:R\to S$ is of effective Galois descent iff

1. $\sigma$ is an effective descent morphism in $\mathsf{Ring}^{op}$, and

2. For every object $(X,\psi)\in\mathsf{Prof}/\mathsf{Sp}(S)$, the $R$-algebra $\mathcal{C}_S(X,\psi)$ is split by $\sigma$.

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They then give the following Galois theorem for commutative rings.

Theorem 1. Let $\sigma:R\to S$ be an effective Galois descent morphism, with $Gal[\sigma]$ the corresponding Galois groupoid in the category of profinite spaces. Then there exists an equivalence of categories $$\big(\mathsf{Split}_R(\sigma)\big)^{op}\simeq\mathsf{Prof}^{Gal[\sigma]}$$ between the dual of the category of $R$-algebras split by $\sigma$ and the category of internal covariant presheaves on $Gal[\sigma]$ in the category of profinite topological spaces.

The reference to étale stuff was for étale morphisms of topological spaces, but your situation should fall under the scope of above theorem.