My first (real) post will be devoted to a pet project of mine that has been on a backburner for several months.
The general idea is that a chain complex can be thought of as nothing more than a graded module, with being a degree 1 or -1 element (depending on whether you want your boundary maps going up or down). This begs the natural question, how much of homological algebra can be stated module-theoretically?
Before I start, however, I should make a case for why such a thing should be interesting. Fundamentally, I think exercises like this are useful for exploring the ‘true’ nature of a rather mysterious concept like homology, but thats a bit hard to sell. On a more practical level, I am curious to see how one can generalize this concept. In particular, the category of chain complexes of sheaves on a scheme X can be seen to be the category of sheaves of (appropriately) graded modules on the scheme , which is kind of like a trivial line bundle over X where you throw away the data beyond first order. What happens if you take a non-trivial line bundle over X, and throw out the data beyond first order? Is there a corresponding notion of twisted homology?
Alright, back to the math.
My first step will be to throw out the grading. It is really a computational convienence that will be helpful later, but isn’t necessary now. This leads to the oddly neglected notion of an ungraded chain complex: a vector space with a distinguished automorphism that squares to zero, which I will think of interchangibly as a module.
Some basic concepts work out straight-forwardly; for instance, chain maps between two complexes are exactly module maps.
Alright, this brings us to the first hard question about these things: how do you say ‘homology’ module-theoretically?
We need some context. First, let be the functor from -mod to -mod which gives a vector space a trivial action. The pursuit of a homology functor from -mod to -mod really will boil down to finding a ‘best’ way of approximating a chain complex by a trivial chain complex, those modules in the image of .
Given a module M, there are two immediate candidates for the module that best approximates it (thought of as sitting inside the category of ungraded chain complexes). I can think about all vector spaces (thought of as modules) and their chain maps into M. It turns out there is a universal vector space with a map into M such that all other vector spaces factor through it; a simple computation shows that it is . Dually, there is a universal vector space factoring all maps from M to vector spaces; it is .
Neither of these vector spaces by themselves truly capture all the ‘vector space’-ish information in M, but together they do. The trick is to compose their maps to and from M to get a map . The image of this map is then , which is the homology!
However, it might not be terribly clear at this juncture that this homology construction is functorial. This can be shown pretty easily by restating the previous construction categorically. The statement that is the universal vector space that maps into M for any M is a plain-spoken way of saying that the functor has a right adjoint , . Dually, has a left adjoint such that .
The map from to M is then exactly the natural transformation given by the adjunction map , and the other map is the adjunction map . Composing them gives a natural transformation from . Now notice that is a fully faithful functor, and so this natural transformation can be pulled back to a natural transformation from . Since functor categories are still abelian, this map factors through an image, a functor I will call H.
I would like to point out that the above construction only required that was a fully faithful functor between two abelian categories with a left and right adjoint. Given any such functor, I can cook up an analogous H, which I like to call the ‘crude homology’ of the functor. It’s still not clear to me how useful this notion is outside the current context, but it certainly seems like a fun thing to explore.
Hopefully that wasn’t more categorical than people would like. I would like to say more, but this is already a pretty long article, so I think I’ll break it up and do another post later. Join me next time as I ask the important question “What about long exact sequences of homology?”, and think about things like bicomplexes.