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.

Suppose that \mathcal{V} is actually a chain complex. This means that it has an odd operator \partial such that \partial^2 = 0. If you prefer, you could imagine a bounded \mathbb{Z}-graded chain complex, where all the even-degree parts are combined to make V_0, and all the odd-degree parts are combined to make V_1.

As usual, you can define the homology H(\mathcal{V}). If \phi is a chain map from \mathcal{V} to itself, then it induces an endomorphism \phi_* of the homology. It’s a fairly easy exercise to show that \mathrm{str} \phi = \mathrm{str} \phi_*. So the point is that the supertrace gives you information about how \phi acts on the homology (or, more precisely, what the homology would be if you had a differential that was compatible with \phi), without requiring you to actually compute the homology. If you’re comfortable with this language, it may be better to say that the supertrace gives you homotopy-invariant information about \phi.

For example, the supertrace of the identity map is just the Euler characteristic \dim V_0 - \dim V_1, which is well-known to be equal to \dim H_0 - \dim H_1, regardless of what \partial is. So you could think of the supertrace as being a generalization of the Euler characteristic in the presence of an endomorphism.

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