Finite generation of the Gerstenhaber-nilpotent quotient for all finite-dimensional algebras

Determine whether, for every finite-dimensional algebra A over a field, the quotient HH^*(A)/G by the Gerstenhaber ideal generated by all homogeneous nilpotent elements is a finitely generated algebra.

Background

For a finite-dimensional algebra A, the paper studies the Hochschild cohomology ring HH^*(A) together with its cup product and Gerstenhaber bracket. The ideal G is generated by all homogeneous nilpotent elements and is required to be stable under the relevant Gerstenhaber operations.

For the Xu–Snashall algebra, the paper proves that the weak Gerstenhaber ideal generated by homogeneous nilpotents coincides with the nilpotent ideal and that the corresponding quotient is not finitely generated; it also proves that the quotient by the Gerstenhaber ideal is isomorphic to the base field K. The authors then ask whether finite generation of HH^*(A)/G nevertheless holds for every finite-dimensional algebra.