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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finiteness of Gorenstein injective dimension of modules


Authors: Leila Khatami, Massoud Tousi and Siamak Yassemi
Journal: Proc. Amer. Math. Soc. 137 (2009), 2201-2207
MSC (2000): Primary 13C11, 13D05, 13H10, 13D45
Published electronically: January 26, 2009
MathSciNet review: 2495252
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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.


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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
Email: yassemi@ipm.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09784-6
PII: S 0002-9939(09)09784-6
Keywords: Cohen-Macaulay ring, Gorenstein injective dimension, Bass theorem
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
Article copyright: © Copyright 2009 American Mathematical Society