Representation schemes and rigid maximal Cohen–Macaulay modules

详细信息    查看全文

• 作者：Hailong Dao ; Ian Shipman
• 关键词：Mathematics Subject ClassificationPrimary 13C14 ; Secondary 14D20
• 刊名：Selecta Mathematica
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：23
• 期：1
• 页码：1-14
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-9020
• 卷排序：23

文摘

Let \(\mathbf {k}\) be an algebraically closed field and A be a finitely generated, centrally finite, nonnegatively graded (not necessarily commutative) \(\mathbf {k}\)-algebra. In this note we construct a representation scheme for graded maximal Cohen–Macaulay A modules. Our main application asserts that when A is commutative with an isolated singularity, for a fixed multiplicity, there are only finitely many indecomposable rigid (i.e, with no nontrivial self-extensions) MCM modules up to shifting and isomorphism. We appeal to a result by Keller, Murfet, and Van den Bergh to prove a similar result for rings that are completion of graded rings. Finally, we discuss how finiteness results for rigid MCM modules are related to recent work by Iyama and Wemyss on maximal modifying modules over compound Du Val singularities.