Group Cohomology and Homotopy Fixed Points
Posted by John Baez
I now have a better understanding of crossed homomorphisms and why they show up so prominently in Gille and Szamuely’s Central Simple Algebras and Galois Cohomology. I told the tale of my enlightenment on Twitter. Basically I just read Qiaochu Yuan’s blog posts on this subject, and discovered that I’d been struggling to understand exactly the things he had figured out and written about:
But I didn’t say enough about what I learned, since Twitter is not so good for that. So let me do that now.
First, I’ll completely eliminate the context that led me to get interested in crossed homomorphisms in the first place, since — as I suspected — something very general is at work here. So I won’t say anything out loud about ‘Galois descent’ or ‘Galois cohomology’ — at least, not at first.
Second, I will take a beautiful question in higher category theory and brutally truncate it and strictify it and skeletalize it, just to reach a conceptual explanation of ‘crossed homomorphisms’ as rapidly as possible. For a more civilized approach, see Qiaochu Yuan’s posts.
Crossed homomorphisms
The idea is that crossed homomorphisms arise naturally when we categorify the concept of ‘fixed point’.
As a warmup, suppose we have a group acting on a set . Thus, we have a homomorphism
We might be interested in the set of fixed points of : that is, the set of such that
for all . This set is sometimes called , where you’re implicitly supposed to know that here is a set equipped with an action of .
Why such a weird name for the set of fixed points? Well, this is a degenerate special case of group cohomology. And one way to understand the group cohomology is to understand it as a categorification of this degenerate special case. One can go further and understand for by repeated categorification, but I won’t do that here.
How can we categorify the concept of fixed point?
There are several things we could do. The most important is to replace the set by a category. But this opens the door for other tricks. For example we could replace by a 2-group, which is the categorical analogue of a group. There’s also another, independent generalization we could try: we could ‘weaken’ the concept of group action, replacing the usual equations and by isomorphisms satisfying some equations.
All these directions of generalization are interesting. We really should do all of them. But I want to go in a very classical direction. So, I’ll take a very low-budget approach! I’ll replace by a category, but I’ll keep as a group, and keep the action of on strict.
So, assume we have a group , a category , and a group homomorphism
where is just the group of invertible functors from to itself.
This is pretty simple, but now we’ll do something a bit more interesting: we’ll look at homotopy fixed points of the action of on .
What are these?
For starters, a homotopy fixed point is an object equipped with isomorphisms
for each . But that’s not good enough: we want these isomorphisms to obey some ‘coherence laws’. Given we have two isomorphisms from to , the obvious one:
and the composite
so for a homotopy fixed point we demand that these are equal. We also have two isomorphisms from to , namely
and the identity
So, for a homotopy fixed point we also demand that these are equal.
Besides defining homotopy fixed points, we can define morphisms between homotopy fixed points, so we get a category with homotopy fixed points as its objects. Qiaochu Yuan explains this. But following the bargain-basement approach I have in mind here, let me simplify the situation further!
I’ll assume is skeletal, so isomorphic objects are equal. Then our homotopy fixed point actually has equal to for all . This may seem crazy, like we’re throwing out the whole idea of homotopy fixed point and going back to a plain old fixed point. But it’s not! It doesn’t imply that the isomorphism is the identity: it’s just some automorphism of . So a homotopy fixed point is still more interesting than a plain old fixed point.
Given that is skeletal, if we’re looking at a single homotopy fixed point, or even a bunch of them with isomorphic underlying objects, we can take our category and throw out all the objects except one, say , since they play no role in the story. This makes into a one-object category.
And since we’re only working with isomorphisms, we can also throw out all the noninvertible morphisms in . This makes into a one-object groupoid. Such a category is called the ‘delooping’ of a group… but let’s just switch to thinking of as a group.
What are we left with now? We’re left with an action of a group as automorphisms of a group :
A homotopy fixed point of this now consists of a function
obeying two equations, the ‘coherence laws’ I wrote down earlier. Translated into the language we’re using now, the first, more interesting coherence law says that
The second says that
And now we’ve explained crossed homomorphisms! Given a group acting on a group via , a crossed homomorphism is defined to be a function obeying the equation
If it also obeys we say it’s normalized.
So, we’ve proved:
Theorem. Suppose is a group acting strictly on some skeletal category, let be an object of that category, and let be its automorphism group. Then ways of making into a homotopy fixed point of the action of correspond bijectively to normalized crossed homomorphisms
This is great, but we can go further. I mentioned that there’s actually a category of homotopy fixed points. So, we can talk about isomorphisms between homotopy fixed points… and figure out what look like in this simplified context, where we have a group acting strictly on a skeletal category, and we focus on one object , so that homotopy fixed points can be described using crossed homomorphisms.
I think the answer is this — I haven’t checked all the details. Suppose we have two normalized crossed homomorphisms:
Then a morphism from to is an element such that
Composition of morphisms is just multiplication in .
The set of isomorphism classes of normalized crossed homomorphisms is called . But now we have a conceptual interpretation of this:
Theorem. Suppose is a group acting strictly on some skeletal category, let be an object of that category, and let be its automorphism group. Then the set of isomorphism classes of homotopy fixed points of the action of having as their underlying object is in bijection with .
Galois descent
Now I can’t resist sketching an application, just to give a little taste of ‘Galois descent’.
Suppose we have a Galois extension of a field , and let be the Galois group of over . Remember, this consists of automorphisms of the field that fix .
You can usually take an algebraic gadget defined over — like a vector space over , or an algebra over , or a variety over — and get a gadget defined over , by ‘extension of scalars’. Galois descent is all about reversing this. If you have a gadget defined over , what are the different gadgets over that it can come from? There could be more than one.
To answer this question, the trick is to notice that acts on the category of gadgets over . And here’s the magic: we can think of gadgets over as weak fixed points of the action of on gadgets over !
Let’s do an easy example, where our gadgets are just vector spaces.
The Galois group acts strictly on the category of vector spaces over . How?
Here’s how it acts on objects. Given and , we get a new vector space that has the same underlying set and the same underlying addition, but where we multiply by scalars in a new way: instead of multiplying by , we multiply by .
Here’s how it acts on morphisms. If is linear, the very same function is a linear map from to , and we call this .
Now, it’s easy to turn a vector space over into a vector space over : just tensor it with , getting . This is called ‘extension of scalars’. This gives a functor from to .
But here’s the interesting point: in this situation you can go further and make into a homotopy fixed point of the action of . Even better, the category of homotopy fixed points of this action is equivalent to .
As a consequence we get this, which is a simple case of ‘Galois descent’:
Ways of making into a homotopy fixed point of are ‘the same’ as ways of writing as for some .
‘The same’ means we have an equivalence of categories, and thus a bijection between isomorphism classes.
We can classify these isomorphism classes using common sense, or using all the fancy machinery we’ve developed.
Using common sense, there’s clearly just one. That is: all vector spaces that become isomorphic to when we tensor them with were already isomorphic. They had to have the same dimension, after all.
So that was easy. But now let’s use our fancy machinery!
First, I believe (but haven’t carefully checked so I’m still worried) that we can replace with a skeletal category while keeping the action of strict. Given this, we can apply the general theorem from the previous section and see the following:
For any vector space , the set of isomorphism classes of ways of making into a homotopy fixed point for the action of is in bijection with
Using common sense, we already know this set has just one element. So what’s the payoff for bringing all the fancy machinery?
Here’s the payoff: we’ve computed the first cohomology of with coefficients in without doing any real work — just thinking. And we’ve seen it’s a set with one element.
The most famous example is when . Then , the group of invertible linear transformations of , is just , the multiplicative group of . So we’ve shown
And this is called Hilbert’s Theorem 90. It’s a famous result from his Zahlbericht — his report on number theory, which came out in 1897. Needless to say, he didn’t phrase it in terms of cohomology. I read that Andreas Speiser did it in 1919, but from a brief glance I couldn’t recognize cohomology in his paper. Noether wrote about it in 1933, and her paper looks a lot easier to understand.
For other more interesting cases of Galois descent, becomes nontrivial. To compute it, homological algebra takes over where common sense leaves off. Read Gille and Szamuely’s Central Simple Algebras and Galois Cohomology for nice examples, starting in Section 2.3 — the section on Galois descent.
Re: Group Cohomology and Homotopy Fixed Points
It looks like Andreas Speiser did that in 1919. did that in 1933.
There is something missing, presumably the name of Emmy Noether.