Representation schemes and rigid maximal Cohen-Macaulay modules

TL;DR

This paper constructs a representation scheme for graded maximal Cohen-Macaulay modules over certain algebras and proves finiteness of indecomposable rigid modules in specific cases, linking to singularity theory and recent algebraic research.

Contribution

It introduces a new representation scheme for graded MCM modules and establishes finiteness results for indecomposable rigid modules over commutative isolated singularities.

Findings

Finiteness of indecomposable rigid MCM modules for fixed multiplicity

Construction of a representation scheme for graded MCM modules

Connection to recent work on maximal modifying modules

Abstract

Let k be an algebraically closed field and A be a finitely generated, centrally finite, non- negatively graded (not necessarily commutative) 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.