# Number Rings and (Galois) Descent

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 $$K$$?

• "the number rings O_K are precisely the finite Z-algebras such that O_K⊗Z Q_sep splits into the product of rank_Z(O_K) many Q_sep's. " I don't think this is right: the property you've stated also holds for any order in a number field (or more generally any order in an etale Q-algebra) since it only depends on O_K ⊗Z Q. Jan 31, 2019 at 23:59
• Sorry, you're right. I forgot those have the right rank. Feb 1, 2019 at 0:04
• I think all these is in Waterhouse "Affine Group Schemes" (Springer) and it seems to me that the descent data that generalizes Galois descent in you case is faithfully flat descent. Feb 1, 2019 at 1:09
• I thought about this for a while, but I think $\mathbb{Q} / \mathbb{Z}$ is not faithfully flat, which can be seen from the fact that $\mathbb{Z} / n \mathbb{Z}$ collapses. Something close may be the case though. I will look into the reference you provided, thanks very much. Feb 1, 2019 at 1:44
• You are right. I think I can see that that the étale condition gives covering data, but I don't see how to define the cocycle condition to get descent data in a non faithfully flat situation. Feb 1, 2019 at 1:48

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.

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.

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 $$\sigma$$.

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.

• Thanks, I'll check that out. Maybe you can say if they give conditions for when that morphism $f : X \rightarrow Y$ defines some sort of categorical equivalence, and if they characterize the one's that split. You see, $\mathcal{O}_K$ splits when we tensor up with $\mathbb{Q}^{sep}$, but so does any order of $K$. Edit: just saw your modification- wow, thanks so much! Feb 1, 2019 at 1:03
• @DeanYoung No problem, let me know if I can add anything else to help. Feb 1, 2019 at 1:05
• Hmm... $\mathbb{Q} / \mathbb{Z}$ is not faithfully flat, which (maybe?) is the right condition for monadicity of the pushout functor. E.g. $\mathbb{Z} / n \mathbb{Z}$ collapses. However, we can probably modify this by restricting $\mathbb{Z}$-algebras. The key thing to notice is that any two distinct orders with the same number field would also go to the same structure in $\text{Prof}^{\text{Gal} [\sigma]}$, so there must be some further restriction besides characteristic zero to make this work. I hope (!) that we can just restrict to the integrally closed orders here to make this work. Feb 1, 2019 at 1:30
• I also notice that the next chapter of this book, chapter 6, provides generalizations which may be able to accomidate to this. Feb 1, 2019 at 1:32
• @DeanYoung Yes, they ultimately culminate with a Non-Galoisian galois theorem for arbitrary extensions of fields in chapter $7$ which is what I was after, so I'm admittedly a bit fuzzy on the details of chapter $4$. I'll see if I can address your comments after some thought tonight, but the book is excellent and I recommend it highly. Feb 1, 2019 at 1:35