Nilpotent Elements in Hochschild Cohomology

Abstract

We study the algebra $A=K\langle x, y\rangle /(x^2, y^2, (xy)^k+q(yx)^k)$ over the field K where $k\ge 1$ and where $0\ne q \in K$. We determine a minimal projective bimodule resolution of A. In the case when q is not a root of unity, we compute its Hochschild cohomology. In particular, we show that for $n\ge 3$, the nth part $HH^n(A)$ has dimension $k-1$ if char(K) does not divide k. We also show that every element in $HH^n(A)$ for $n\ge 1$ is nilpotent. This is motivated by the problem of understanding why the finite generation condition (Fg) fails, which is needed to ensure the existence of support varieties.

To Dave