Finiteness of Gorenstein injective dimension of modules
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- by Leila Khatami, Massoud Tousi and Siamak Yassemi PDF
- Proc. Amer. Math. Soc. 137 (2009), 2201-2207 Request permission
Abstract:
The Chouinard formula for the injective dimension of a module over a noetherian ring is extended to Gorenstein injective dimension. Specifically, if $M$ is a module of finite positive Gorenstein injective dimension over a commutative noetherian ring $R$, then its Gorenstein injective dimension is the supremum of ${depth} R_{\mathfrak {p}}- {width} _{R_\mathfrak {p}}M_{\mathfrak {p}}$, where $\mathfrak {p}$ runs through all prime ideals of $R$. It is also proved that if $M$ is finitely generated and non-zero, then its Gorenstein injective dimension is equal to the depth of the base ring. This generalizes the classical Bass formula for injective dimension.References
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Additional Information
- Leila Khatami
- Affiliation: Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts 02115
- Email: l.khatami@neu.edu
- Massoud Tousi
- Affiliation: Department of Mathematics, Shahid Beheshti University, Tehran, Iran — and — School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
- Email: mtousi@ipm.ir
- Siamak Yassemi
- Affiliation: Department of Mathematics, University of Tehran, Tehran, Iran — and — School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
- MR Author ID: 352988
- Email: yassemi@ipm.ir
- Received by editor(s): February 4, 2008
- Received by editor(s) in revised form: September 9, 2008
- Published electronically: January 26, 2009
- Additional Notes: The second author was supported by a grant from the IPM, No. 870130214
The third author was supported by a grant from the IPM, No. 870130211 - Communicated by: Bernd Ulrich
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 2201-2207
- MSC (2000): Primary 13C11, 13D05, 13H10, 13D45
- DOI: https://doi.org/10.1090/S0002-9939-09-09784-6
- MathSciNet review: 2495252