Classifying torsion classes for algebras with radical square zero via sign decomposition
Classifying torsion classes for algebras with radical square zero via sign decomposition
复制标题
通过符号分解对根式平方零代数的挠率类进行分类
DOI：
10.1016/j.jalgebra.2022.06.032
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
Toshitaka Aoki
中科院分区：
文献类型：
--
作者：
Toshitaka Aoki
为了研究域上有限维基本代数的挠率类集合，我们使用称为符号分解的分解，由 {±1} n 的元素参数化，其中 n 是简单模的数量。如果 A 是一个基数平方为零的代数，那么对于每个 ϵε{±1} n 都有一个遗传代数 A ϵ!根平方为零以及与 ϵ 关联的 A 的扭转类集合和 A ϵ! 的忠实扭转类集合之间的双射。此外，这种双射保留了函数有限的性质。从倾斜理论的角度来看，这意味着与 ϵ 相关的 A 的基本两项淤积复合体的同构类集合与基本倾斜 A ϵ!-模的同构类集合之间存在双射。作为一个应用，我们证明了具有 n 个边的布劳尔线代数（布劳尔循环代数）上的两项倾斜复数为 (2 n n)（如果 n 为奇数，则分别为 2 2 n− 1；如果 n 为偶数，则为 ∞）。
To study the set of torsion classes of a finite dimensional basic algebra over a field, we use a decomposition, called sign-decomposition, parameterized by elements of {±1} n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ∈{±1} n there is a hereditary algebra A ϵ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of A ϵ!. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting A ϵ!-modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is (2 n n)(respectively, 2 2 n− 1 if n is odd, and∞ if n is even).