Finite gentle repetitions of gentle algebras and their Avella-Alaminos--Geiss invariants
Finite gentle repetitions of gentle algebras and their Avella-Alaminos--Geiss invariants
复制标题
温和代数及其 Avella-Alaminos-Geiss 不变量的有限温和重复
DOI：
10.1080/00927872.2021.2008412
复制
发表时间：
2022
影响因子：
0.7
通讯作者：
Hiroyuki Nakaoka
中科院分区：
文献类型：
--
作者：
Akiyama Shigeki;Kaneko Hajime;Hiranouchi Toshiro;早坂太;Hiroyuki Nakaoka
在域 K 上的有限维代数中，已知温和代数类是由导出等价封闭的。虽然导出等价的分类通常是一个困难的问题，但 Avella-Alaminos 和 Geiss 引入了温和代数 A 的导出不变量，可以从它们的绑定箭袋组合计算，适用于此类分类。 Ladkani 给出了一个公式，根据 Avella-Alaminos-Geiss 不变量的某些值来描述 A 的 Hochschild 上同调的维数。这反过来又暗示了这些值的上同调意义。由于大多数其他值没有出现在该公式中，因此自然会问这些值是否存在类似的上同调含义。在本文中，我们通过称为有限温和重复的过程构造了一个由正整数索引的温和代数序列，以便将 A 的 Avella-Alaminos-Geiss 不变量的这些值与 A 的 Hochschild 上同调的维数联系起来。这样我们就会看到 的 Avella-Alaminos-Geiss 不变量完全由 A 的值决定。因此，人们可能会期望有限的温和重复会在良好的情况下保留派生的等价性。在本文的最后部分，我们将看到广义的 Auslander-Platzeck-Reiten 反射如何与有限温和重复相关。
Among finite dimensional algebras over a fieldK, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebrasA, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies ofAin terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebrasindexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants ofAto the dimensions of Hochschild cohomologies ofOn the way we will see that the Avella-Alaminos–Geiss invariants ofare completely determined by those ofA. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition.