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.

April 28, 2020

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 GG acting on a set AA. Thus, we have a homomorphism

α:GAut(A) \alpha \colon G \to \mathrm{Aut}(A)

We might be interested in the set of fixed points of α\alpha: that is, the set of aAa \in A such that

α(g)a=a \alpha(g) a = a

for all gGg \in G. This set is sometimes called H 0(G,A)H^0(G,A), where you’re implicitly supposed to know that AA here is a set equipped with an action of GG.

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 H 1(G,A)H^1(G,A) is to understand it as a categorification of this degenerate special case. One can go further and understand H n(G,A)H^n(G,A) for n>1n > 1 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 AA by a category. But this opens the door for other tricks. For example we could replace GG 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 α(gh)=α(g)α(h)\alpha(g h) = \alpha(g) \alpha(h) and α(1)=1\alpha(1) = 1 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 AA by a category, but I’ll keep GG as a group, and keep the action of GG on AA strict.

So, assume we have a group GG, a category AA, and a group homomorphism

α:GAut(A) \alpha \colon G \to Aut(A)

where Aut(A)Aut(A) is just the group of invertible functors from AA 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 GG on AA.

What are these?

For starters, a homotopy fixed point is an object aAa \in A equipped with isomorphisms

ϕ(g):α(g)aa \phi(g) \colon \alpha(g) a \stackrel{\sim}{\to} a

for each gGg \in G. But that’s not good enough: we want these isomorphisms to obey some ‘coherence laws’. Given g,hGg,h \in G we have two isomorphisms from α(gh)a\alpha(g h)a to aa, the obvious one:

α(gh)aϕ(gh)a \alpha(g h) a \stackrel{\phi(g h)}{\longrightarrow} a

and the composite

α(gh)aα(g)(ϕ(h))α(g)aϕ(g)a \alpha(g h) a \stackrel{\alpha(g)(\phi(h))}{\longrightarrow} \alpha(g) a \stackrel{\phi(g)}{\longrightarrow} a

so for a homotopy fixed point we demand that these are equal. We also have two isomorphisms from α(1)a\alpha(1) a to aa, namely

α(1)aϕ(1)a \alpha(1) a \stackrel{\phi(1)}{\longrightarrow} a

and the identity

α(1)a=a1a \alpha(1) a = a \stackrel{1}{\longrightarrow} a

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 AA is skeletal, so isomorphic objects are equal. Then our homotopy fixed point aa actually has α(g)a\alpha(g) a equal to aa for all gg. 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 ϕ(g):α(g)a\phi(g) \colon \alpha(g) \to a is the identity: it’s just some automorphism of aa. So a homotopy fixed point is still more interesting than a plain old fixed point.

Given that AA 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 AA and throw out all the objects except one, say aa, since they play no role in the story. This makes AA into a one-object category.

And since we’re only working with isomorphisms, we can also throw out all the noninvertible morphisms in AA. This makes AA into a one-object groupoid. Such a category is called the ‘delooping’ of a group… but let’s just switch to thinking of AA as a group.

What are we left with now? We’re left with an action of a group GG as automorphisms of a group AA:

α:GAut(A) \alpha \colon G \to \mathrm{Aut}(A)

A homotopy fixed point of this now consists of a function

ϕ:GA \phi \colon G \to A

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

ϕ(gh)=ϕ(g)(α(g)(ϕ(h))) \phi(g h) = \phi(g) \; (\alpha(g) (\phi(h)))

The second says that

ϕ(1)=1 \phi(1) = 1

And now we’ve explained crossed homomorphisms! Given a group GG acting on a group AA via α\alpha, a crossed homomorphism is defined to be a function ϕ:GA\phi \colon G \to A obeying the equation

ϕ(gh)=ϕ(g)(α(g)(ϕ(h))) \phi(g h) = \phi(g) \; (\alpha(g) (\phi(h)))

If it also obeys ϕ(1)=1 \phi(1) = 1 we say it’s normalized.

So, we’ve proved:

Theorem. Suppose GG is a group acting strictly on some skeletal category, let aa be an object of that category, and let A=Aut(a)A = \mathrm{Aut}(a) be its automorphism group. Then ways of making aa into a homotopy fixed point of the action of GG correspond bijectively to normalized crossed homomorphisms

ϕ:GA \phi \colon G \to A

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 aa, 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:

ϕ,ψ:GA \phi, \psi \colon G \to A

Then a morphism from ϕ\phi to ψ\psi is an element aAa \in A such that

aϕ(g)=ψ(g)(α(g)(a)) a \; \phi(g) = \psi(g) \; (\alpha(g)(a))

Composition of morphisms is just multiplication in AA.

The set of isomorphism classes of normalized crossed homomorphisms is called H 1(G,A)H^1(G,A). But now we have a conceptual interpretation of this:

Theorem. Suppose GG is a group acting strictly on some skeletal category, let aa be an object of that category, and let A=Aut(a)A = \mathrm{Aut}(a) be its automorphism group. Then the set of isomorphism classes of homotopy fixed points of the action of GG having aa as their underlying object is in bijection with H 1(G,A)H^1(G,A).

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 KK of a field kk, and let Gal(K|k)Gal(K|k) be the Galois group of KK over kk. Remember, this consists of automorphisms of the field KK that fix kKk \subseteq K.

You can usually take an algebraic gadget defined over kk — like a vector space over kk, or an algebra over kk, or a variety over kk — and get a gadget defined over KK, by ‘extension of scalars’. Galois descent is all about reversing this. If you have a gadget defined over KK, what are the different gadgets over kk that it can come from? There could be more than one.

To answer this question, the trick is to notice that Gal(K|k)Gal(K|k) acts on the category of gadgets over KK. And here’s the magic: we can think of gadgets over kk as weak fixed points of the action of Gal(K|k)Gal(K|k) on gadgets over KK!

Let’s do an easy example, where our gadgets are just vector spaces.

The Galois group Gal(K|k)Gal(K|k) acts strictly on the category Vect KVect_K of vector spaces over KK. How?

Here’s how it acts on objects. Given VVect KV \in Vect_K and gGal(K|k)g \in Gal(K|k), we get a new vector space α(g)(V)Vect K\alpha(g) (V) \in Vect_K that has the same underlying set and the same underlying addition, but where we multiply by scalars cKc \in K in a new way: instead of multiplying by cc, we multiply by g(c)g(c).

Here’s how it acts on morphisms. If f:VWf \colon V \to W is linear, the very same function is a linear map from α(g)V\alpha(g) V to α(g)W\alpha(g) W, and we call this α(g)(f)\alpha(g) (f).

Now, it’s easy to turn a vector space LL over kk into a vector space over KK: just tensor it with KK, getting K kLK \otimes_k L. This is called ‘extension of scalars’. This gives a functor from Vect kVect_k to Vect KVect_K.

But here’s the interesting point: in this situation you can go further and make K kLK \otimes_k L into a homotopy fixed point of the action of Gal(K|k)Gal(K|k). Even better, the category of homotopy fixed points of this action is equivalent to Vect kVect_k.

As a consequence we get this, which is a simple case of ‘Galois descent’:

Ways of making VVect KV \in Vect_K into a homotopy fixed point of Gal(K|k)Gal(K|k) are ‘the same’ as ways of writing VV as K kLK \otimes_k L for some LVect kL \in \Vect_k.

‘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 LVect kL \in Vect_k that become isomorphic to VVect KV \in Vect_K when we tensor them with KK 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 Vect KVect_K with a skeletal category while keeping the action of Gal(K|k)Gal(K|k) strict. Given this, we can apply the general theorem from the previous section and see the following:

For any vector space VVect KV \in \mathrm{Vect}_K, the set of isomorphism classes of ways of making VV into a homotopy fixed point for the action of Gal(K|k)Gal(K|k) is in bijection with

H 1(Gal(K|k),Aut(V)) H^1(Gal(K|k), \mathrm{Aut}(V))

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 Gal(K|k)Gal(K|k) with coefficients in Aut(V)\mathrm{Aut}(V) without doing any real work — just thinking. And we’ve seen it’s a set with one element.

The most famous example is when V=KV = K. Then Aut(V)\mathrm{Aut}(V), the group of invertible linear transformations of VV, is just K ×K^\times, the multiplicative group of KK. So we’ve shown

H 1(Gal(K|k),K ×)={1} H^1(Gal(K|k), K^\times) = \{1\}

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, H 1H^1 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.

Posted at April 28, 2020 1:34 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3219

2 Comments & 0 Trackbacks

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.

Posted by: Mathlog on April 30, 2020 5:02 PM | Permalink | Reply to this

Re: Group Cohomology and Homotopy Fixed Points

Thanks! Yes, Noether. I’ve decided to improve this paragraph:

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.

Posted by: John Baez on April 30, 2020 5:33 PM | Permalink | Reply to this

Post a New Comment