### Abstract

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and Hom _{R}(C,C) ≅R. We prove that a Cohen-Macaulay ring R with dualizing module D admits a semidualizing module C satisfying R ≇ C ≇ D if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that R admits a dualizing module if and only if R is Cohen-Macaulay and a homomorphic image of a local Gorenstein ring.

Language | English (US) |
---|---|

Pages | 165-180 |

Number of pages | 16 |

Journal | Collectanea Mathematica |

Volume | 63 |

Issue number | 2 |

DOIs | |

State | Published - May 2012 |

### Fingerprint

### Keywords

- Gorenstein rings
- Self-orthogonal modules
- Semidualizing modules
- Tate Ext
- Tate Tor
- Tor-independence

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Collectanea Mathematica*,

*63*(2), 165-180. DOI: 10.1007/s13348-010-0024-6

**Presentations of rings with non-trivial semidualizing modules.** / Jorgensen, David A.; Leuschke, Graham J.; Sather-Wagstaff, Sean.

Research output: Contribution to journal › Article

*Collectanea Mathematica*, vol. 63, no. 2, pp. 165-180. DOI: 10.1007/s13348-010-0024-6

}

TY - JOUR

T1 - Presentations of rings with non-trivial semidualizing modules

AU - Jorgensen,David A.

AU - Leuschke,Graham J.

AU - Sather-Wagstaff,Sean

PY - 2012/5

Y1 - 2012/5

N2 - Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and Hom R(C,C) ≅R. We prove that a Cohen-Macaulay ring R with dualizing module D admits a semidualizing module C satisfying R ≇ C ≇ D if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that R admits a dualizing module if and only if R is Cohen-Macaulay and a homomorphic image of a local Gorenstein ring.

AB - Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and Hom R(C,C) ≅R. We prove that a Cohen-Macaulay ring R with dualizing module D admits a semidualizing module C satisfying R ≇ C ≇ D if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that R admits a dualizing module if and only if R is Cohen-Macaulay and a homomorphic image of a local Gorenstein ring.

KW - Gorenstein rings

KW - Self-orthogonal modules

KW - Semidualizing modules

KW - Tate Ext

KW - Tate Tor

KW - Tor-independence

UR - http://www.scopus.com/inward/record.url?scp=84859639286&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859639286&partnerID=8YFLogxK

U2 - 10.1007/s13348-010-0024-6

DO - 10.1007/s13348-010-0024-6

M3 - Article

VL - 63

SP - 165

EP - 180

JO - Collectanea Mathematica

T2 - Collectanea Mathematica

JF - Collectanea Mathematica

SN - 0010-0757

IS - 2

ER -