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Stream: deprecated: mathematics

Topic: Separable algebras and Galois theory

John Baez (Jun 06 2023 at 20:32):

I've been trying to understand Azumaya algebras over a commutative ring R. These are separable algebras over R whose center is 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 important! If R is a field these are just the finite direct sums of finite-dimensional separable extensions of R. 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, each field corresponds to a 'kind of point' $B$, and an extension of that field is a different kind of point $E$ that maps down to $B$:

$p : E \to B$

I think the basic idea is that when the extension is separable, this map $p: E \to B$ has the property that there aren't any nonzero tangent vectors on $E$ that map to zero tangent vectors down on $B$.

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.

John Baez (Jun 06 2023 at 20:33):

Anyway, if a field k is separably closed - for example if it's algebraically closed! - then the study of commutative separable algebras over it simplifies immensely.

John Baez (Jun 06 2023 at 20:35):

Then the only commutative separable algebras over k are the finite products k $\times \cdots \times$ k.

John Baez (Jun 06 2023 at 20:37):

Morally 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 finite 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.

John Baez (Jun 06 2023 at 20:37):

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.

John Baez (Jun 06 2023 at 20:39):

But it's much more exciting 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:

John Baez (Jun 06 2023 at 20:40):

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.

John Baez (Jun 06 2023 at 20:41):

The absolute Galois group of k is the Galois group of the separable closure of k over k. It naturally gets a structure of a profinite group, so it gets a topology.

John Baez (Jun 06 2023 at 20:43):

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.

John Baez (Jun 06 2023 at 20:44):

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.)

John Baez (Jun 06 2023 at 20:45):

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.

John Baez (Jun 06 2023 at 20:47):

But we can also define commutative separable monoids in any symmetric monoidal category. I've discussed a bunch of equivalent ways to define them in the thread on Azumaya algebras, so I won't repeat that stuff here.

John Baez (Jun 06 2023 at 20:50):

Anyway, it turns out that Aurelio Carboni has generalized the Fundamental Theorem of Grothendieck Galois Theory to commutative separable monoids in any sufficiently nice symmetric monoidal category!

John Baez (Jun 06 2023 at 20:51):

He did it here:

• Aurelio Carboni, Matrices, relations, and group representations.

John Baez (Jun 06 2023 at 20:52):

(A deceptively elementary-sounding title!)

John Baez (Jun 06 2023 at 20:53):

Here is what he apparently proved:

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

John Baez (Jun 06 2023 at 20:55):

I say "apparently" because I got this version of the theorem from here:

• Georgios Chara-Lambous, On the relationship between Galois and Tannakian categories: an open letter.

John Baez (Jun 06 2023 at 21:03):

Looking at Carboni's paper, I see his Theorem 1 looks like the above theorem with the "essentially small" part removed from the hypothesis and the conclusion.

Morgan Rogers (he/him) (Jun 07 2023 at 05:50):

This is sounding increasingly like stuff I am interested in, which is exciting.

John Baez (Jun 07 2023 at 19:16):

Great! Do you have anything you can say about the overlap between this stuff and your interests? As you can probably tell, I'm shying away from a deep dive either into hardcore algebraic geometry (schemes, etale cohomology groups etc.) or topos theory, and trying to pluck out the ideas that live at the level of either symmetric monoidal categories, or symmetric monoidal bicategories of enriched profunctors. I'm doing this in service of some "quest" that I'm on, trying to more deeply understand generalizations of the Brauer group, but the circle of ideas seems to keep expanding.

Morgan Rogers (he/him) (Jun 08 2023 at 07:27):

For one thing, generalizations of Galois theory are up my street, and producing a pretopos is a nice bonus (although not all that surprising, I'm betting the result comes from checking each of the required categorical properties). I've also interacted with the formalism of Tannakian categories a little bit.

One thing I would say is that I expect "separable" to be the property determining both "group" and "Boolean" - you can likely get a less constrained theorem with "monoid" in place of "group" by dropping the property of being separable. This isn't necessarily helpful - we understand groups a lot better than general monoids - but on the other hand it isn't necessarily easy to check separability so it might be worth thinking about.