Representation schemes and rigid maximal Cohen-Macaulay modules

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.

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