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Rings with finite Gorenstein injective dimension

Author(s): Henrik Holm
Journal: Proc. Amer. Math. Soc. 132 (2004), 1279-1283.
MSC (2000): Primary 13D02, 13D05, 13D07, 13H10; Secondary 16E05, 16E10, 16E30
Posted: November 7, 2003
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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).

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Additional Information:

Henrik Holm
Affiliation: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, 2100 K\obenhavn Ø, Danmark
Email: holm@math.ku.dk

DOI: 10.1090/S0002-9939-03-07466-5
PII: S 0002-9939(03)07466-5
Keywords: Gorenstein dimensions, homological dimensions, Gorenstein rings
Received by editor(s): January 28, 2003
Posted: November 7, 2003
Communicated by: Bernd Ulrich
Copyright of article: Copyright 2003, American Mathematical Society

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