## Homological interpretation of the supertrace

Let $\mathcal{V} = V_0 \oplus V_1$ be a $\mathbb{Z}_2$-graded vector space, where both $V_0$ and $V_1$ are finite-dimensional. Suppose that $\phi$ is a degree-preserving endomorphism of $\mathcal{V}$ (so it consists of endomorphisms $\phi_i$ for both $V_i$). Then the supertrace of $\phi$ is defined as the difference of the two traces:

$\mathrm{str}\phi = \mathrm{tr}\phi_0 - \mathrm{tr}\phi_1.$

Obviously, the definition of the supertrace is very simple, but it begs for a justification. For example, why is this better than just adding the two traces? I’ve seen two facts offered in defense of the supertrace (see the Wikipedia article): 1) The supertrace is invariant when applied to endomorphisms of modules over commutative superalgebras, and 2) The supertrace of a supercommutator is 0.

I feel like neither of these justifications is particularly satisfying; the first requires a much more complicated setup than what I described above, and the second passes the burden of proof onto the notion of supercommutator. So I’d like to offer another way of thinking about the supertrace, which should be reasonably convincing to anyone who knows a little bit of homological algebra. I feel like this is one of those ideas that the practitioners in the field all know, but nobody bothers to write down.

## What is a graded manifold?

There are several variations on the definition of “graded manifold”, and it can be a bit confusing for a newcomer. The goal of this post is to describe one definition that rules them all. This definition hopefully sheds some light on the relationships between the various types of graded manifolds.

I thought I had seen the definition in Kostant’s early treatise “Graded manifolds, graded Lie theory, and prequantization,” but, looking back at the paper now, I can’t find the definition there. In any case, I don’t claim any originality here. If anyone can remind me of the correct reference, I would appreciate it.