I finally got around to writing a bit about my paper with Geordie, and realized that this was considerably more than a one post story. I figured I would start by saying a bit about Hochschild homology. This is a pretty standard bit of homological algebra, but has suddenly bursted out of the deformation theory ghetto into link homology.
So, what is Hochschild homology? Unfortunately, in most books, you seem to get a horrifying and useless definition (a bit like what happens with group cohomology). So let me attempt to explain what it really is. It’s a homology theory (for some interpretation of “homology theory”) of -bimodules for a ring .
Hochschild cohomology (a theory in some sense dual to Hochschild homology) appears very naturally in deformation theory of algebras, but since that’s not particularly relevant to what we’re doing here, I’ll leave that to the reader to investigate.
Whenever one has a -bimodule, one can think of this just as well as -module (here I’m using that is commutative, but for non-commutative rings not so much changes). And for any commutative ring , there is an obvious map called “multiplication.”
The extension of scalars by this ring homomorphism (which is just the annoying commutative algebra way of saying the tensor product ) is what people seem to insist on calling “coinvariants” (making that word a very overloaded operator in this field). This no help in picking a name to say for it, but the best written notation for me is the somewhat suggestive .
Another way to think of this functor is as the largest quotient of $M$ on which the left and right actions coincide.
Hochschild homology is what happens when we take this extension of scalars in the only proper way to do anything in homological algebra: in the derived sense.
Of course, derived functors are a big topic, and not one to properly be explained in the course of a blog post. To a certain degree, you just have to take some time to get used to them. But the main point is that you should take the tensor product above and replace one (or both) of its factors with a free (or more generally, flat) resolution. You’ll get a complex in the place of your tensor product, and the right exactness The Hochschild homology is the homology of this complex. As you can easily check, you get back as the 0th homology of this complex.
The definition you see in books is one particular resolution of as an bimodule, which is ugly and hard to use for many purposes, but is completely general. It works for every ring. But polynomial rings are much nicer than just any old ring. In particular, they have a much smaller resolution as bimodules over themselves, called the Koszul resolution. This has the distinct advantage of being finite rank over , and of finite length (in fact, its length is the number of variables of ), neither of which are true of the more general resolution (in fact, for many rings, there is no upper bound on the for which might be non-zero. In polynomial rings, we know that the number of variables gives an upper bound).
So, we can calculate by simply tensoring with this complex, but this is not as nice as we might hope, since taking the homology of a complex whose terms are complicated modules is hard, even if the differentials are very explicit. This may be good enough for a computer, for people, it’s a bit dissatisfying.
But as I mentioned before, you can resolve either side, and is useful for some purposes to do one, and for the purposes the other. For example, if one is lucky and can find a nice resolution of the module one is Hochschild homologizing, then tensoring with , one has a complex whose terms are free modules over and whose differentials are easy to calculate from those of the original complex (just hit the matrices of the map with the multiplication map).
If one is really lucky, then this complex will have trivial differentials after extension of scalars. This sounds like too much ask, I suppose, but in fact this is true for any modules whose Hochschild homology is free (for example, for Soergel bimodules in type A, by results of Jake Rasmussen).
So, why do we care? Well, if we have a category of bimodules over which is closed under tensor product, then functions as a categorical trace: We have an isomorphism (as vector spaces, not necessarily as modules). It turns out that if you apply this to the category of Soergel bimodules (a categorification of the Hecke algebra), then you get back a categorification of the Jones-Ocneanu trace on the Hecke algebra.
All of which suggests (correctly) that you can build a knot invariant using Hochschild homology, just as Khovanov does in “Triply-graded link homology and Hochschild homology of Soergel bimodules.”
Hooray! Triply graded link homology!
As I mention above, this is installment one of at least two posts on my recent paper with Geordie (though funnily enough, it contains essentially no original information from that paper. Go figure). Next time, we’ll have a little geometry.