Noncommutative matrix factorizations with an application to skew exterior algebras
Noncommutative matrix factorizations with an application to skew exterior algebras
复制标题
非交换矩阵分解及其在倾斜外代数中的应用
DOI：
10.1016/j.jalgebra.2021.07.012
复制
发表时间：
2021
影响因子：
0.9
通讯作者：
Ueyama Kenta
中科院分区：
文献类型：
--
作者：
Mori Izuru;Ueyama Kenta
矩阵分解理论对于研究交换代数中的超曲面很有用。为了研究非交换超曲面（非交换代数几何中的重要研究对象），我们引入了环的任意非零非单位元素的非交换矩阵分解的概念。首先，我们证明非交换分级矩阵分解的类别在称为扭曲的运算下是不变的（该结果是 Cassidy-Conner-Kirkman-Moore 结果的推广）。然后我们给出涉及非交换矩阵分解和全自反模的两个范畴等价（这个结果类似于艾森巴德关于交换超曲面的著名结果）。作为一个应用，我们描述了斜外代数上的不可分解的非交换分级矩阵分解。
Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.