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On the existence of embeddings into modules of finite homological dimensions
Author(s):
Ryo
Takahashi;
Siamak
Yassemi;
Yuji
Yoshino
Journal:
Proc. Amer. Math. Soc.
138
(2010),
2265-2268.
MSC (2010):
Primary 13D05, 13H10
Posted:
February 23, 2010
MathSciNet review:
2607854
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Abstract:
Let  be a commutative Noetherian local ring. We show that  is Gorenstein if and only if every finitely generated  -module can be embedded in a finitely generated  -module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.
References:
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- 1.
- M. AUSLANDER; M. BRIDGER, Stable module theory. Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685 (42:4580)
- 2.
- W. BRUNS; J. HERZOG, Cohen-Macaulay rings. Revised edition. Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998. MR 1251956 (95h:13020)
- 3.
- R. FOSSUM; H.-B. FOXBY; P. GRIFFITH; I. REITEN, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules. Inst. Hautes Études Sci. Publ. Math. 45 (1975), 193-215. MR 0396529 (53:392)
- 4.
- H.-B. FOXBY, Embedding of modules over Gorenstein rings. Proc. Amer. Math. Soc. 36 (1972), 336-340. MR 0309930 (46:9034)
- 5.
- E. S. GOLOD, G-dimension and generalized perfect ideals (Russian). Algebraic geometry and its applications. Trudy Mat. Inst. Steklov. 165 (1984), 62-66. MR 752933 (85m:13011)
- 6.
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Additional Information:
Ryo
Takahashi
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan
Email:
takahasi@math.shinshu-u.ac.jp
Siamak
Yassemi
Affiliation:
Department of Mathematics, University of Tehran, P. O. Box 13145-448, Tehran, Iran - and - School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
Email:
yassemi@ipm.ir
Yuji
Yoshino
Affiliation:
Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan
Email:
yoshino@math.okayama-u.ac.jp
DOI:
10.1090/S0002-9939-10-10323-2
PII:
S 0002-9939(10)10323-2
Keywords:
Gorenstein ring,
Cohen-Macaulay ring,
projective dimension,
injective dimension,
(semi)dualizing module
Received by editor(s):
November 26, 2008,
Received by editor(s) in revised form:
May 24, 2009
Posted:
February 23, 2010
Additional Notes:
The first and second authors were supported in part by Grant-in-Aid for Young Scientists (B) 19740008 from JSPS and by grant No. 88013211 from IPM, respectively
Communicated by:
Bernd Ulrich
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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