Gorenstein injective dimension, Bass formula and Gorenstein rings

2003/12/31 by Leila Khatami, Siamak Yassemi, Khatami, Leila +1
Mathematics · #13D05 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13D05 #msc:13H10

paper · pdf · doi:10.48550/arxiv.math/0312513

13 pages, Main theorem in section 3 generalized, with new corollaries

arxiv created 2004/06/01 · arxiv updated 2009/12/01

Abstract

Let (R,\frak m, k) be a noetherian local ring. It is well-known that R is regular if and only if the injective dimension of k is finite. In this paper it is shown that R is Gorenstein if and only if the Gorenstein injective dimension of k is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. It also improves Christensen's generalized Bass formula (cf. "Gorenstein dimensions", volume 1747 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000).

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• A Bass equality for Gorenstein injective dimension of modules finite over homomorphisms
• Transfer of Gorenstein dimensions along ring homomorphisms
• A generalization of a theorem of Foxby
• The existence of finitely generated modules of finite Gorenstein injective dimension