Can we categorify the equation (1 - t)(1 + t + t^2 + ...) = 1?

Question

Polynomials in $\mathbb Z[t]$ are categorified by considering Euler characteristics of complexes of finite-dimensional graded vector spaces. Now, given a rational function that has a power series expansion with integer coefficients, it seems natural to consider complexes of (locally finite-dimensional) graded vector spaces.

Are there nice examples of this in nature?

Answer 1

Yes, the particular equation you wrote is categorified by the free resolution of k as module over k[x] by the complex $k[x] \overset{x}\longrightarrow k[x]$ given by multiplication by x. It also appears in the numerical criterion for Koszulity of k[x] (see the paper of Beilinson, Ginzburg and Soergel).

Answer 2

I think the following does what you want:

Let R be a polynomial ring in one variable x over a field with its usual grading; i.e. x has degree $1$. Next consider the graded complex that is R in degree $0$ and $R[1]$ in degree $-1$ --- here by R[1] we mean
$R[1]_ i=R_{i-1}$, and with the non-trivial map given by multiplication by $x$.

I guess that you would understand the Euler characteristic of this complex to be

$$(1-t)(1+t+t^2+t^3+...)$$since you might consider the "graded dimension" of the degree 0 part to be $1+t+t^2+...$ and the "graded dimension" of the degree $-1$ part to be $(t+t^2+...)$.

However when you take homology you get $0$ in degree $-1$ and just $k$ in degree $0$ and this $k$ has "graded dimension" $1$.

You do get things like this in nature when you try to compute Euler characteristics of $p$-torsion modules for Iwasawa algebras: see K0 and the dimension filtration for $p$-torsion Iwasawa modules (by Konstantin Ardakov and Simon Wadsley, DOI: 10.1112/plms/pdm053) for some more calculations in this context. The notion of graded Brauer character there replaces the simpler notion of graded dimension.

Answer 3

I don't know whether or not this satisfies the criterion of "categorification" (what that?), but the equation (1 - t)(1 + t + t² + ...) = 1 and its relation to vector spaces is well-known to differential topologists and geometers. We use it all the time in K-theory and index theory where it becomes the identity Λ_-1V ⊗ S₁V = ℂ. Here, Λ_-1V denotes the alternating sum of the exterior powers of V whilst S₁V is the sum of the symmetric powers.

Of course, the sum of the symmetric powers isn't a class in K-theory as it is an infinite sum. To get round this, we work in K[[t]] and allow parameters, whereupon the equation becomes Λ_-tV ⊗ S_tV = ℂ. Here, the t means formally multiply the kth exterior or symmetric power by t^k.

Where this breaks out of mere formalism and becomes very powerful is in equivariant K-theory. Then the parameter becomes a way of measuring the action of the group, which is usually S¹ or a finite cyclic group for index theory calculations. In particular, in Witten's original adaptation of index theory to loop spaces, we end up with positive energy representations of S¹ which become vector bundles over the original (finite dimensional) manifold with circle actions preserving the fibres. One can decompose these according to the circle action whereupon one has a vector bundle for each k ∈ ℤ. The positive energy criterion means that these are trivial below a certain integer and are always finite dimensional. However, as there are an infinite number of them then the total dimension can be infinite dimensional. Then the identity Λ_-tV ⊗ S_tV = ℂ has real meaning as the power of the t parameter indicates how the circle acts on that component of the vector bundle. That is, if the circle action on V is the standard action then it is multiplication by t^k on Λ^k V and on S^k V.