Let R be a commutative Noetherian local ring with residue field k. We show that if a finite direct sum of syzygy modules of k surjects onto 'a semidualizing module' or 'a non-zero maximal Cohen-Macaulay module of finite injective dimension', then R is regular. We also prove that R is regular if and only if some syzygy module of k has a non-zero direct summand of finite injective dimension.2010 Mathematics Subject Classification. Primary 13D02; Secondary 13D05, 13H05.
Let R be a d-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f ), where f := f 1 , . . . , fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if Ext i S (M, N ) = 0 for some (d + c + 1) consecutive values of i 2, then Ext i S (M, N ) = 0 for all i 1. Moreover, if this holds true, then either projdim R (M ) or injdim R (N ) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.
Abstract. Let A be a Noetherian standard N-graded algebra over an Artinian local ring A 0 . Let I 1 , . . . , It be homogeneous ideals of A and M a finitely generated N-graded A-module. We prove that there exist two integers k, k ′ such that′ for all n 1 , . . . , nt ∈ N.
For a finitely generated module M over a commutative Noetherian ring R, we settle the Auslander-Reiten conjecture when at least one of Hom R (M, R) and Hom R (M, M ) has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.
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