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

Polynomials in 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?

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Could you clarify what you mean by the Euler characteristic of a complex of finite-dimensional graded vector spaces? I just need to make sure I know where t comes from. –  Qiaochu Yuan Oct 20 '09 at 19:15
@Qiaochu. The graded dimension of a graded vector space V=\oplus_i V_i is dim_t(V) = \sum_i t^i dim(V_i). The graded Euler characteristic of a complex of graded vector spaces is just the alternating sum of the graded dimensions of the terms of the complex. –  Scott Morrison Oct 20 '09 at 21:06

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).

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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 http://www.dpmms.cam.ac.uk/~sjw47/rankskzero.pdf for some more calculations in this context. The notion of graded Brauer character there replaces the simpler notion of graded dimension.

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Aaah. To slow. That's what comes from writing at length... –  Simon Wadsley Oct 20 '09 at 20:19