Selecta Mathematica

, Volume 23, Issue 1, pp 1–14 | Cite as

Representation schemes and rigid maximal Cohen–Macaulay modules

  • Hailong Dao
  • Ian Shipman


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.

Mathematics Subject Classification

Primary 13C14 Secondary 14D20 



We are delighted to thank Bhargav Bhatt and Igor Burban for interesting conversations and correspondence, and Srikanth Iyengar and Michael Wemyss for many helpful comments on an earlier version of this article. The authors are partially supported by NSF awards DMS-1104017 and DMS-1204733.


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.LawrenceUSA
  2. 2.Salt Lake CityUSA

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