Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra

Let $A$ be a finite dimensional algebra and let $\mathrm{H}{H}^{\ast }(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\text{calN}$ (resp. $G$, $\text{calG}$) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that $\mathrm{H}{H}^{\ast }(A)\mathrm{/}\text{calN}$ is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether $\mathrm{H}{H}^{\ast }(A)\mathrm{/}G$ is a finitely generated algebra and suggested considering first the Xu–Snashall algebra. In this paper, we answer this question for the Xu–Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that $G=\text{calN}$; hence $\mathrm{H}{H}^{\ast }(A)\mathrm{/}G=\mathrm{H}{H}^{\ast }(A)\mathrm{/}\text{calN}$ is not a finitely generated algebra. Furthermore, we show that $\mathrm{H}{H}^{\ast }(A)\mathrm{/}\text{calG}\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\mathrm{H}{H}^{\ast }(A)\mathrm{/}\text{calG}$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.