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Non-Gorenstein Isolated Singularities of Graded Countable Cohen–Macaulay Type

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Connections Between Algebra, Combinatorics, and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 76))

Abstract

In this article we show a partial answer to a question of Huneke and Leuschke (Proc. Am. Math. Soc. 131(10):3003–3007, 2003): Let R be a standard graded Cohen–Macaulay ring of graded countable Cohen–Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen–Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen–Macaulay rings of graded countable Cohen–Macaulay type.

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Acknowledgments

The author would like to thank the anonymous referee for valuable feedback and suggestions regarding the content of this paper. For instance, the referee pointed out the existence of the ring (17) as well as some discrepancies in the initial version of Sect. 4. Further, the author is especially thankful to Craig Huneke for several useful conversations. Additionally, results in this note were inspired by many Macaulay2 [10] computations.

The author was partially funded by the NSF grant, Kansas Partnership for Graduate Fellows in K-12 Education (DGE-0742523).

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Authors and Affiliations

  1. Bard College, 5000, Annandale-on-Hudson, NY, USA

    Branden Stone

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  1. Branden Stone
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Correspondence to Branden Stone .

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Editors and Affiliations

  1. Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan, USA

    Susan M. Cooper

  2. Department of Mathematics, North Dakota State University, Fargo, North Dakota, USA

    Sean Sather-Wagstaff

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Dedicated to Tony Geramita for his role in shaping mathematics

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Stone, B. (2014). Non-Gorenstein Isolated Singularities of Graded Countable Cohen–Macaulay Type. In: Cooper, S., Sather-Wagstaff, S. (eds) Connections Between Algebra, Combinatorics, and Geometry. Springer Proceedings in Mathematics & Statistics, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0626-0_9

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