Tate local cohomology and Gorenstein theory, which are important generalizations of the classical cohomology, has been investigated. It found that they have such vanishing properties long exact sequences. However, for homology, what about duality? In this paper we concerned with homology homology. first part generalize as study properties, artinianness some sequence modules. We find an artiania...

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify two special cases of a question of Takahashi and White.

In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.

R ring (always commutative and Noetherian) (R,m,k) local ring with maximal ideal m and k = R/m L,M,N, . . . R-modules (always finitely generated) M HomR(M,R), the dual of M D(M) the Auslander dual of M (Definition 2) σM : M wM∗∗ the natural evaluation map; KM = Ker(σM ), CM = Coker(σM ) G-dimR(M),G-dim(M) Gorenstein dimension of M (Definition 16) G-dim(M) <loc ∞ M has locally finite Gorenstein ...

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f -vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f -vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f -vectors of flag homology spheres and the f -vectors of Gorenstein* order comp...

In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results est...

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f -vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f -vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f -vectors of flag homology spheres and the f -vectors of Gorenstein* order comp...

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...

In terms of the duality property of injective preenvelopes and flat precovers, we get an equivalent characterization of left Noetherian rings. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of RR. Then we get that the injective envelope of RR is (Gor...

It is a well-known result of Auslander and Reiten that contravariant finiteness the class P ∞ fin (of finitely generated modules finite projective dimension) over an Artin algebra sufficient condition for validity finitistic dimension conjectures. Motivated by fact dimensions can alternatively be computed Gorenstein dimension, we examine in this work counterpart Auslander–Reiten condition, name...