## June 6, 2023

### Commutative Separable Algebras III

#### Posted by John Baez I wrote two blog articles on this theme back in 2010:

Now for rather different reasons I’m returning to it.

A separable algebra $A$ over a commutative ring $R$ is an algebra that’s projective as $A \otimes_R A^{\text{op}}$-module. That may sound dry, but you can see three other equivalent definitions here. For example, a separable algebra is an algebra that can be given a comultiplication obeying the Frobenius laws: and the special law: Lately I’ve been trying to understand Azumaya algebras, which are separable algebras over $R$ whose center is just $R$. The interesting Azumaya algebras are of course the noncommutative ones, since the only commutative one is $R$ itself.

But commutative separable algebras are also interesting. They are important in Grothendieck’s approach to Galois theory. So, I want to understand them better, to better understand how Azumaya algebras are connected to Galois theory.

So, back to commutative separable algebras! And this time we’ll see how they show up in the Fundamental Theorem of Grothendieck Galois Theory!

First let me talk about commutative separable algebras over a field $k$. These are just the finite direct sums of finite-dimensional separable extensions of $k$. If you don’t know what a separable extension is, well for starters every finite-dimensional extension of a finite field, or a field of characteristic zero, is separable. So to a zeroth approximation, you can think of a separable extension as one that’s not really weird.

But in algebraic geometry, we think of commutative algebras as being spaces, with the arrows between them turned around. Then each field corresponds to a ‘kind of point’ $B$, and an extension of that field is a fatter kind of point $E$ that maps down to $B$:

$p : E \to B$

Then the basic idea is that $E$ is separable if and only if this map $p: E \to B$ doesn’t map any nonzero tangent vectors on $E$ to zero tangent vectors down on $B$. (More precisely, this is true when $E$ is finite-dimensional over $B$.)

Of course it’s weird to think that one kind of point could map in an interesting way down to another kind of point — or that a point could have nonzero tangent vectors! But this is part of the fun of algebraic geometry.

Anyway, if a field $k$ has no finite-dimensional separable extensions except itself, we say it’s separably closed. This happens, for example, if it’s algebraically closed. When this happens, the study of commutative separable algebras over $k$ simplifies immensely. In this case the only such algebras are the finite products $k \times \cdots \times k$.

Geometrically speaking, in this case there’s only one kind of point that can sit over our point $k$, namely the same kind of point! Thus, the most general sort of zero-dimensional space that can sit over this point is just a finite collection of points of this kind. The algebra of functions on this finite set is then $k \times \cdots \times k$.

So we get a result:

Theorem. If a field $k$ is separably closed, the opposite of the category of commutative separable algebras over $k$ is equivalent to $FinSet$.

In the case $k = \mathbb{C}$ some related results later became important in relating quantum mechanics to set theory:

but these were formulated in terms of commutative Frobenius algebras rather than commutative separable algebras.

Right now I’m eager to look at fields $k$ that aren’t separably closed! Then we get different kinds of point that can sit over the kind of point corresponding to $k$: one for each finite-dimensional separable extension of $k$. And using some ideas from Galois theory, Grothendieck proved this:

The Fundamental Theorem of Grothendieck Galois Theory. The opposite of the category of commutative separable algebras over a field $k$ is equivalent to the topos of continuous actions on finite sets of the absolute Galois group of $k$.

The absolute Galois group of $k$ is the Galois group of the separable closure of $k$ over $k$. We should think of the separable closure as a kind of ‘universal cover’ of the point corresponding to $k$, and the absolute Galois group as a group of ‘deck transformations’ of this cover. (How a point could have an interesting fundamental group is another one of the mysteries of algebraic geometry!) This absolute Galois group naturally gets a structure of a profinite group, so it gets a topology.

Now, while this is fascinating, it’s tempting to generalize. First we could generalize from fields to commutative rings, and look at commutative separable algebras over such rings. Grothendieck did this, I’m pretty sure.

But we could also go further and look at commutative monoids in some sufficiently nice symmetric monoidal category $V$. (When $V = AbGp$ these are just commutative rings.)

One nice thing is that the opposite of the category of commutative monoids in a symmetric monoidal category $V$ is always a cartesian category. This exhibits the duality between ‘commutative algebras’ and ‘spaces’ in a very general, simple way.

But we can also define commutative separable monoids in $V$. Many of the different equivalent definitions of commutative separable algebra generalize straightforwardly. And it turns out that Aurelio Carboni generalized the Fundamental Theorem of Grothendieck Galois Theory to commutative separable monoids in any sufficiently nice symmetric monoidal category!

He did it here:

(A deceptively elementary-sounding title!)

Here is what he proved:

Theorem. Let $V$ be a compact closed additive category with coequalizers. Then the opposite of the category of separable commutative monoids in $V$ is an essentially small Boolean pretopos.

Even though I’m not advanced enough to find a Boolean pretopos heartwarming, I know it’s a generalization of the category of finite sets. So this is nice. But do any of you have anything more to say about this, like examples of Boolean pretopoi that show up this way, which aren’t topoi?

Someone who likes string diagrams should rewrite large sections of Carboni’s paper using those. There are a lot of intimidating calculations that should become very pretty.

By the way, I heard about Carboni’s theorem here:

I found this paper quite fun to read, mainly because it clarified some connections I’ve been trying to make lately.

Posted at June 6, 2023 10:15 PM UTC

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### Re: Commutative Separable Algebras III

Posted by: Metatron on June 9, 2023 8:21 PM | Permalink | Reply to this

### Re: Commutative Separable Algebras III

I fixed your links for you—you were using overly curly quotes around the URLs, like “this”.

Yes, this stuff is pretty intriguing, even though I’m about a hundred times more interested in Azumaya algebras than D-branes. As far as I can tell, the basic idea is something like this.

We can define Azumaya monoids very generally in any symmetric monoidal category $C$: they’re invertible objects in the bicategory of

• monoid objects in $C$,
• bimodules, and
• bimodule homomorphisms.

Taking $C$ to be something like the symmetric monoidal category of coherent sheaves on a scheme, this lets us define Azumaya algebras over a scheme. But I like smooth manifolds better so let’s look at the symmetric monoidal category of complex vector bundles on a manifold $X$. Then our Azumaya algebras amount to bundles of algebras over $X$ that are locally isomorphic to $n \times n$ complex matrix algebras. These are classified by $H^1(X,PGL(n,\mathbb{C}))$, since the automorphism group of an $n \times n$ complex matrix algebra is $PGL(n,\mathbb{C})$. If we take the limit $n \to \infty$ we get

$H^1(X,PGL(\infty,\mathbb{C})) \cong H^2(X,\mathbb{C}^\ast)$

which classifies gerbes over $X$! So, what’s really going on is that we can take the trivial bundle of complex matrix algebras and ‘twist it by a gerbe’ to get an Azumaya algebra over our manifold — at least in some $n \to \infty$ limit, I guess.

I’m leaving out everything about algebraic geometry here, or holomorphic geometry. But the overall picture is already quite pretty.

By the way, here’s a book you can get online, which gets into the algebraic geometry.

For example Section 3.2 defines the ‘Brauer–Grothendieck group’ of a scheme to be $H^2_{et}(X,\mathbb{G}_{m,X})$, which is the algebraic geometer’s version of my $H^2(X,\mathbb{C}^\ast)$, and Theorem 3.2.2, due to Gabber, says that the ‘Brauer–Azumaya’ group is the torsion part of the Brauer–Grothendieck group. Apparently Toen got the whole Brauer–Grothendieck group using the derived category of coherent sheaves.

Posted by: John Baez on June 9, 2023 10:18 PM | Permalink | Reply to this

### Re: Commutative Separable Algebras III

“Links” to zulipchat might as well not exist. They require registration and then login to even view. Facebook is heinous and utterly hateful, but even FB links aren’t that bad.

And what are the odds any of these Zulip chat “links” work one year from now, let alone two or, at the outside, three?

Posted by: Richard on June 10, 2023 8:48 AM | Permalink | Reply to this

### Re: Commutative Separable Algebras III

Sorry: many posts on the Category Theory Community Server are publicly visible and publicly archived, including on the Internet Archive, and I thought my posts on Azumaya algebras were among those. I see now that they were not, because I put them in the section “Learning: questions”: this section is protected to spare newbie questioners potential embarrassment. I’ve moved them to the section “General: mathematics”, so now they’re visible to the world.

Posted by: John Baez on June 10, 2023 4:48 PM | Permalink | Reply to this

### Re: Commutative Separable Algebras III

Very interesting. Trying to understand the Fundamental Theorem of Grothendieck Galois theory better. What does it mean for a topological group to act “continuously” on a mere finite set? I think I am missing something. Probably has to do with the topos structure.

Posted by: Bruce Bartlett on June 15, 2023 11:05 PM | Permalink | Reply to this

### Re: Commutative Separable Algebras III

Yeah, it’s weird, but I believe the definition is exactly the usual definition: a topological group $G$ acts continuously on a topological space $X$ if the action $\alpha: G \times X \to X$ is a continuous map. If $X$ is discrete it’s automatically true that $\alpha(g,x)$ depends continuously on $x$, but it might not depend continuously on $g$. So, we are imposing a nontrivial constraint on the action to demand that it’s continuous! I believe that if $G$ is a profinite group with its usual topology, saying that the action $\alpha: G \times X \to X$ is continuous is equivalent to saying the action factors through some finite subgroup of $G$.

Someone will correct me if this is wrong! But the basic idea seems to be this: we are really interested in Galois groups $Gal(K|k)$ of Galois extensions $K$ of a field $k$. To handle them all at once, we use the fact that these groups are precisely the finite quotients of the ‘absolute’ Galois group of $k$. So, we treat that absolute group as a profinite group and give it the profinite topology, so its continuous actions are the same as actions of $Gal(K|k)$ for arbitrary Galois extensions $K$ of $k$. This lets us study all the Galois extensions of $k$, and their Galois groups, and their actions, all at once.

Posted by: John Baez on June 16, 2023 12:48 AM | Permalink | Reply to this

### Re: Commutative Separable Algebras III

To address Bruce Bartlett’s comment, yes, requiring $\alpha$ to be a continuous map is the “correct” definition topos-theoretically, justified by the fact that the category $Cont(G)$ of such continuous $G$-sets (morphisms are $G$-equivariant maps) is a Grothendieck topos “classifying principal $G$-bundles”. This means that for every topological space $X$, there is an equivalence between the category of geometric morphisms $Sh(X)\to Cont(G)$ and the category $Bund_G(X)$ of principal $G$-bundles on $X$.

What John Baez intuitively describes as the procedure for constructing an “absolute Galois group” out of the finite groups arising from finite Galois extensions can be thought more generally as setting up a “pro-system of finite groups”. The theorem here is that there is an equivalence between the category $Pro\;Grp_f$ of such pro-systems and the category $StoneGrp$ of Stone (= compact Hausdorff totally disconnected) topological groups. These topological groups $\pi$ are determined by their discrete quotients, and the connection with Grothendieck’s parallel world of Galois categories is that these quotients can be characterized categorically as the normal objects in the (Galois) category $Cont_f(\pi)$ of finite continuous $\pi$-sets.

Posted by: Georgios Chara-Lambous on June 18, 2023 9:12 AM | Permalink | Reply to this

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