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Representation schemes and rigid maximal Cohen–Macaulay modules

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Abstract

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.

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Acknowledgments

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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  1. Lawrence, KS, USA

    Hailong Dao

  2. Salt Lake City, UT, USA

    Ian Shipman

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  1. Hailong Dao
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  2. Ian Shipman
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Correspondence to Hailong Dao.

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Dao, H., Shipman, I. Representation schemes and rigid maximal Cohen–Macaulay modules. Sel. Math. New Ser. 23, 1–14 (2017). https://doi.org/10.1007/s00029-016-0226-1

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