Abstract
This paper studies the relationship between Serre’s condition (R n ) and Auslander–Buchweitz’s maximal Cohen–Macaulay approximations. It is proved that a Gorenstein local ring satisfies (R n ) if and only if every maximal Cohen–Macaulay module is a direct summand of a maximal Cohen–Macaulay approximation of a (Cohen–Macaulay) module of codimension n+1.
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Acknowledgments
The authors are grateful to Olgur Celikbas for his helpful comments. The authors also thank the referee for his/her careful reading. R. Takahashi was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400038.
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Dedicated to Professor Ngo Viet Trung on the occasion of his sixtieth birthday
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Matsui, H., Takahashi, R. Maximal Cohen–Macaulay Approximations and Serre’s Condition. Acta Math Vietnam 40, 197–203 (2015). https://doi.org/10.1007/s40306-014-0105-9
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DOI: https://doi.org/10.1007/s40306-014-0105-9
Keywords
- Cohen–Macaulay ring
- Gorenstein ring
- Maximal Cohen–Macaulay module
- Maximal Cohen–Macaulay approximation
- Serre’s condition
- Nonfree locus
- Singular locus
- Syzygy
- Transpose