Rings with finite Gorenstein injective dimension

Holm, Henrik

doi:10.1090/S0002-9939-03-07466-5

Rings with finite Gorenstein injective dimension
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Proc. Amer. Math. Soc. 132 (2004), 1279-1283 Request permission

Abstract:

In this paper we prove that for any associative ring $R$, and for any left $R$-module $M$ with finite projective dimension, the Gorenstein injective dimension $\mathrm {Gid}_R M$ equals the usual injective dimension $\mathrm {id}_R M$. In particular, if $\mathrm {Gid}_R R$ is finite, then also $\mathrm {id}_R R$ is finite, and thus $R$ is Gorenstein (provided that $R$ is commutative and Noetherian).

References

Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866, DOI 10.1007/BFb0103980
Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611–633. MR 1363858, DOI 10.1007/BF02572634
Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein balance of Hom and tensor, Tsukuba J. Math. 19 (1995), no. 1, 1–13. MR 1346750, DOI 10.21099/tkbjm/1496162796
Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein injective and flat dimensions, Math. Japon. 44 (1996), no. 2, 261–268. MR 1416263
Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. MR 1753146, DOI 10.1515/9783110803662
H.-B. Foxby, Hyperhomological Algebra & Commutative Rings, notes in preparation.
Hans-Bjørn Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra 15 (1979), no. 2, 149–172. MR 535182, DOI 10.1016/0022-4049(79)90030-6
Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI 10.1007/BF01390094
H. Holm, Gorenstein Homological Dimensions, J. Pure and Appl. Algebra (to appear)
C. U. Jensen, On the vanishing of $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$, J. Algebra 15 (1970), 151–166. MR 260839, DOI 10.1016/0021-8693(70)90071-2
Jinzhong Xu, Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. MR 1438789, DOI 10.1007/BFb0094173