## Chain Complexes as Graded C[\epsilon] Modules

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 $\mathbb{C}[\epsilon]:=\mathbb{C}[x]/x^2$ module, with $\epsilon$ 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 $X\times_\mathbb{C} Spec(\mathbb{C}[\epsilon])$, 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 $\delta$ that squares to zero, which I will think of interchangibly as a $\mathbb{C}[\epsilon]$ 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 $f_*$ be the functor from $\mathbb{C}$-mod to $\mathbb{C}[\epsilon]$-mod which gives a vector space a trivial $\epsilon$ action. The pursuit of a homology functor from $\mathbb{C}[\epsilon]$-mod to $\mathbb{C}$-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 $f_*$.

Given a $\mathbb{C}[\epsilon]$ module M, there are two immediate candidates for the $\mathbb{C}$ 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 $\mathbb{C}[\epsilon]$ 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 $ker(\delta)$. Dually, there is a universal vector space factoring all maps from M to vector spaces; it is $M/im(\delta)$.

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 $h:ker(\delta)\rightarrow M/im(\delta)$. The image of this map is then $ker(\delta)/im(\delta)$, 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 $im(\delta)$ is the universal vector space that maps into M for any M is a plain-spoken way of saying that the functor $f_*$ has a right adjoint $f^!$, $f^!(M)=ker(\delta)$. Dually, $f_*$ has a left adjoint $f^*$ such that $f^*(M)=M/im(\delta)$.

The map from $ker(\delta)$ to M is then exactly the natural transformation given by the adjunction map $f_*\circ f^!\rightarrow Id$, and the other map is the adjunction map $Id\rightarrow f_*\circ f^*$. Composing them gives a natural transformation from $f_*\circ f^!\rightarrow f_*\circ f^*$. Now notice that $f_*$ is a fully faithful functor, and so this natural transformation can be pulled back to a natural transformation from $f^!\rightarrow f^*$. 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 $f_*$ 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.

> Continue to Part 2

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### 6 Responses to “Chain Complexes as Graded C[\epsilon] Modules”

1. John Armstrong Says:

More categorical? If anything it was less categorical 😀

Any thoughts on how this squares with the picture painted in CWM using the simplicial category?

2. jkamnitz Says:

Here is another interesting aspect to thinking about complexes as graded $\mathbb{C}[\epsilon]$ modules: you can try find other graded rings and study modules over them as if you were doing homological algebra.

Here is an example. Fix a commutative ring $A$ and an element $W \in A$. Let $R$ be the ring $A[x]/(x^2-W)$. This is a $\mathbb{Z}/2$-graded ring where we put $A$ in degree $0$ and $Ax$ in degree $1$.

Now consider $\mathbb{Z}/2$ graded $R$-modules. If you throw in the additional criterion that the modules are free as $A$ modules, then you have exactly the concept of a matrix factorization with potential $W$. You can do a sort of homological algebra with this concept. These matrix factorizations were invented by Eisenbud and have recently been used quite a bit. See recent work by Orlov or Khovanov-Rozansky for example. (If you think for a bit you will see why they are called matrix factorizations, though a factorization of $W$ by matrices would be a more appropriate name.)

I wonder if there are any other good examples out there.

3. Greg Muller Says:

John: I don’t know Categories for the Working Mathematician, actually. What part of it are you refering to? If I had to guess blindly, I would guess that you are talking about the Dold-Kan correspondence, which says that bounded-below chain complexes are the same as simplicial objects.

Cursorily, I don’t know of any connection between the two. I have actively discarded gradings, which means things like bounded-below don’t make sense (even if you adopt a $\mathbb{Z}_2$-grading like I do in part 3). The Dold-Kan construction uses the bounded-below-ness in a very necessary way, by building the simplicial object inductively.

That said, we can always re-introduce gradings, which I avoided since they don’t seem to correspond to anything geometric: $Proj(\mathbb{C}[\epsilon])$ is empty. It’s possible that the Dold-Kan construction can be turned into something in terms of pullbacks and push-forwards, I don’t know it well enough to answer on the spot.

Joel: Can you give me a reference or two here? I am very interested in this idea, except that I can’t for the life of me see how to make a similar kind of thing work. $x$ not squaring to zero pretty much kills every direct analog of the things I’ve said, homology, superalgebras, etc. I checked Orlov on the arxiv, and found lots of interesting looking stuff (particularly when he mentioned fermions and schur functors, which might be in subsequent posts on this stuff), but nothing that talked about matrix factorizations.

4. John Armstrong Says:

It’s in his section on monoidal categories. He constructs the simplicial category, shows it’s the universal monoid, uses this to give a functor to any monad, and uses monads to set up (co)homology theories. Unfortunately, I don’t have time to explain in much depth, but I’m sure it’s in your library.

5. Greg Muller Says:

Ah, then yes, that’s what I was thinking it was. Though, you don’t need a monad to do it; a functor from the simplicial category to an abelian category can be turned into a complex by taking the differentials to be the alternating sum of the (co)face maps.

I suppose you can (kinda) say something about $\mathbb{C}[\epsilon]$ here. You can turn $\mathbb{C}[\epsilon]$ into an abelian category with an object $O_i$ for every non-zero integer, and $Hom(O_i,O_j)$ equal to $\mathbb{C}$ if $i= j$ or $j-1$ and 0 otherwise (composition is unambigious). This category has the property that functors from it into abelian categories are the same as bounded-below chain complexes in that category. This category has a ‘best’ functor to the abelianization of the simplicial category, which is the obvious index-preserving isomorphism on objects, and it takes the generator of $Hom(O_i,O_{i+1})$ to the alternating sum of the face maps from $\Delta_i$ to $\Delta_{i+1}$. Then the chain complex of any simplicial object in $\mathcal{A}$ is just the composition of the functors from $\mathbb{C}[\epsilon]$ to $\mathcal{A}$. This isn’t really saying a ton yet, though I am curious as to see if there is a nice graded-commutative explanation of the need for an alternating sum.

The nice fact is that this construction is (almost) an equivalence of categories. Theres an annoying problem, in that there are two complexes to associate to a simplicial object: the unnormalized one (constructed above), and the normalized one, which is the one that comes up in the Dold-Kan construction. They are almost the same complex, in that they are quasi-isomorphic. However, I am still not at the point where quasi-isomorphism are a natural thing to think about as $\mathbb{C}[\epsilon]$ modules.

6. Joel Kamnitzer Says:

Here are some references for recent uses of matrix factorizations:

math.QA/0401268 Matrix factorizations and link homology. Mikhail Khovanov, Lev Rozansky.
(they have a nice section (section 3) explaining the formal properties of matrix factorizations)

math.AG/0503632 Derived categories of coherent sheaves and triangulated categories of singularities. Dmitri Orlov.
(section 3 is the relevant one, though it is perhaps not so readable)

For a more commutative algebra perspective, you could look at papers by Buchweitz.