Xu--Snashall 代数的 Hochschild 上同调环上的 Gerstenhaber 代数结构
Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra
- East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过显式计算 Xu--Snashall 代数的 Hochschild 上同调环的 Gerstenhaber 代数结构，证明其弱 Gerstenhaber 理想等于幂零理想，从而否定有限生成性，并证明其 Gerstenhaber 理想商同构于基域。
AI中文摘要：
设 $A$ 为有限维代数，$\rmHH^*(A)$ 为其 Hochschild 上同调环，它是一个 Gerstenhaber 代数。记 $\calN$（分别地，$G$，$\calG$）为由所有齐次幂零元生成的理想（分别地，弱 Gerstenhaber 理想，Gerstenhaber 理想）。受他们通过 Hochschild 上同调研究支撑簇的工作启发，Snashall 和 Solberg 猜想 $\rmHH^*(A)/\calN$ 是有限生成代数。Xu 在特征为二的基域上构造了 Snashall-Solberg 猜想的一个反例，Snashall 将此例推广到任意特征。Hermann 进一步询问 $\rmHH^*(A)/G$ 是否为有限生成代数，并建议首先考虑 Xu--Snashall 代数。本文回答了关于 Xu--Snashall 代数的这一问题。事实上，通过显式计算 Hochschild 上同调环上的 Gerstenhaber 代数结构，我们证明 $G=\calN$；因此 $\rmHH^*(A)/G=\rmHH^*(A)/\calN$ 不是有限生成代数。进一步，我们证明 $\rmHH^*(A)/\calG\cong K$。因此，人们可以询问：对于有限维代数 $A$，$\rmHH^*(A)/\calG$ 是否总是有限生成代数。我们的主要工具是双边 Anick 分解和弱自同伦。
英文摘要：
Let $A$ be a finite dimensional algebra and let $\rmHH^*(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\calN$ (resp. $G$, $\calG$) 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 $\rmHH^*(A)/\calN$ 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 $\rmHH^*(A)/G$ 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 $G=\calN$; hence $\rmHH^*(A)/G=\rmHH^*(A)/\calN$ is not a finitely generated algebra. Furthermore, we show that $\rmHH^*(A)/\calG\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\rmHH^*(A)/\calG$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.
