Hermann’s finite-generation question for the Xu–Snashall algebra

Hermann asked whether the Hochschild-cohomology quotient by the weak Gerstenhaber ideal generated by homogeneous nilpotents is finitely generated. For the Xu–Snashall algebra, the latest preprint claims a complete computation of the Gerstenhaber structure and a negative answer for that quotient.

Known results

Oke showed that the weak Gerstenhaber ideal equals the ideal $\mathcal{N}$ generated by homogeneous nilpotent elements, yielding a negative answer for the Xu–Snashall case.

September 2026 claimed resolution

Qi Long, Ziyang Shi, and Guodong Zhou report that $\mathrm{HH}^*/\mathcal{G}$ is not finitely generated, whereas $\mathrm{HH}^*/\mathcal{G}_{\!\mathrm{full}}\cong k$. This distinguishes the weak and full Gerstenhaber quotients and settles the canonical example if the preprint’s computation is correct; the claim is unverified.

Current status (as of September 2026): The Xu–Snashall case is claimed resolved negatively for the weak quotient and identified with $k$ for the full quotient, but the claim is unverified and the question for arbitrary finite-dimensional algebras remains open.