Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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 AA over a commutative ring RR is an algebra that’s projective as A RA opA \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 RR whose center is just RR. The interesting Azumaya algebras are of course the noncommutative ones, since the only commutative one is RR 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 kk. These are just the finite direct sums of finite-dimensional separable extensions of kk. 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’ BB, and an extension of that field is a fatter kind of point EE that maps down to BB:

p:EBp : E \to B

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

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 kk 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 kk simplifies immensely. In this case the only such algebras are the finite products k××kk \times \cdots \times k.

Geometrically speaking, in this case there’s only one kind of point that can sit over our point kk, 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××kk \times \cdots \times k.

So we get a result:

Theorem. If a field kk is separably closed, the opposite of the category of commutative separable algebras over kk is equivalent to FinSetFinSet.

In the case k=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 kk that aren’t separably closed! Then we get different kinds of point that can sit over the kind of point corresponding to kk: one for each finite-dimensional separable extension of kk. 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 kk is equivalent to the topos of continuous actions on finite sets of the absolute Galois group of kk.

The absolute Galois group of kk is the Galois group of the separable closure of kk over kk. We should think of the separable closure as a kind of ‘universal cover’ of the point corresponding to kk, 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 VV. (When V=AbGpV = AbGp these are just commutative rings.)

One nice thing is that the opposite of the category of commutative monoids in a symmetric monoidal category VV 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 VV. 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 VV be a compact closed additive category with coequalizers. Then the opposite of the category of separable commutative monoids in VV 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

TrackBack URL for this Entry:

7 Comments & 0 Trackbacks

Re: Commutative Separable Algebras III

For applications of Azumaya algebras to physics, see:

D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck


Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes

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 CC: they’re invertible objects in the bicategory of

  • monoid objects in CC,
  • bimodules, and
  • bimodule homomorphisms.

Taking CC 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 XX. Then our Azumaya algebras amount to bundles of algebras over XX that are locally isomorphic to n×nn \times n complex matrix algebras. These are classified by H 1(X,PGL(n,))H^1(X,PGL(n,\mathbb{C})), since the automorphism group of an n×nn \times n complex matrix algebra is PGL(n,)PGL(n,\mathbb{C}). If we take the limit nn \to \infty we get

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

which classifies gerbes over XX! 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 nn \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 et 2(X,𝔾 m,X)H^2_{et}(X,\mathbb{G}_{m,X}), which is the algebraic geometer’s version of my H 2(X, *)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 GG acts continuously on a topological space XX if the action α:G×XX\alpha: G \times X \to X is a continuous map. If XX is discrete it’s automatically true that α(g,x)\alpha(g,x) depends continuously on xx, but it might not depend continuously on gg. So, we are imposing a nontrivial constraint on the action to demand that it’s continuous! I believe that if GG is a profinite group with its usual topology, saying that the action α:G×XX\alpha: G \times X \to X is continuous is equivalent to saying the action factors through some finite subgroup of GG.

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)Gal(K|k) of Galois extensions KK of a field kk. To handle them all at once, we use the fact that these groups are precisely the finite quotients of the ‘absolute’ Galois group of kk. 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)Gal(K|k) for arbitrary Galois extensions KK of kk. This lets us study all the Galois extensions of kk, 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 α\alpha to be a continuous map is the “correct” definition topos-theoretically, justified by the fact that the category Cont(G)Cont(G) of such continuous GG-sets (morphisms are GG-equivariant maps) is a Grothendieck topos “classifying principal GG-bundles”. This means that for every topological space XX, there is an equivalence between the category of geometric morphisms Sh(X)Cont(G)Sh(X)\to Cont(G) and the category Bund G(X)Bund_G(X) of principal GG-bundles on XX.

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 ProGrp fPro\;Grp_f of such pro-systems and the category StoneGrpStoneGrp of Stone (= compact Hausdorff totally disconnected) topological groups. These topological groups π\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(π)Cont_f(\pi) of finite continuous π\pi-sets.

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

Post a New Comment