Gorenstein Injective Dimension and Cohen-Macaulayness

Throughout this paper‎, (R, m) is a‎ commutative Noetherian local ring with the maximal ideal m. ‎The following conjecture proposed by Bass [1]‎, ‎has been‎ proved by Peskin and Szpiro [2] for almost all rings: ‎
(B) If R admits a finitely generated R-module of‎ finite injective dimension‎, ‎then R is Cohen-Macaulay.
‎The problems treated in this paper are closely related to the‎ following generalization of  Bass conjecture which is still wide‎ open:‎
(GB) If R admits a finitely generated R-module of‎ finite Gorenstein-injective dimension‎, ‎then R is‎ Cohen-Macaulay.
‎  Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a‎ finitely generated R-module of injective dimension 1‎.
Levin and Vasconcelos indicate that if‎ is a non-zerodivisor‎, ‎then for‎ every finitely generated R/xR-module M‎, ‎there is‎ ‎. ‎Using this fact‎, ‎they construct a‎ finitely generated R-module of finite injective dimension in‎ the case where R is Cohen-Macaulay (the converse of Conjecture‎ B)‎.
‎ In this paper we study the Gorenstein injective dimension of local cohomology‎. ‎We also show that if R is Cohen-Macaulay‎ with minimal multiplicity‎, ‎then every finitely generated module‎ of finite Gorenstein injective dimension has finite injective‎ dimension.‎
‎We prove that a Cohen-Macaulay local ring‎ has a finitely generated module of‎ finite Gorenstein injective dimension.‎
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