Canonical Filtrations of Gorenstein Injective Modules

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the principle. Over commutative, noetherian rings injective modules have direct sum decompositions into indecomposable modules. We will show that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have filtrations analogous to those provided by these decompositions. This result will then provide us with the tools to prove that all tensor products of Gorenstein injective modules over these rings are Gorenstein injective.Comment: 9 pages; It has been accepted for publication in Proceedings of the American Mathematical Societ

Balance with Unbounded Complexes

Given a double complex $X$ there are spectral sequences with the $E_2$ terms being either H$_I$ (H$_{II}(X))$ or H$_{II}($H$_I (X))$. But if $H_I(X)=H_{II}(X)=0$ both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with $X$. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article we give a different way of constructing homology groups of $X$ when H$_I(X)=$H$_{II}(X)=0$. With this result we give a new and elementary proof of balance of Tate homology and cohomology

On Cohen-Macaulay rings

summary:In this paper, we use a characterization of $R$-modules $N$ such that $fd_RN = pd_RN$ to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting $N$ to be the $dth$ local cohomology functor of $R$ with respect to the maximal ideal where $d$ is the Krull dimension of $R$

Cotorsion Pairs in C(R-Mod)

In [8] Salce introduced the notion of a co-torsion pair (A, B) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proved useful in a variety of settings. In this article we will consider complete cotorsion pairs (C, D)in the category C(R-Mod) of complexes of left R-modules over some ring R.If(C, D) is such a pair, and if C is closed un-der taking suspensions, we will show when we regard K(C) and K(D) as subcategories of the homotopy category K(R-Mod), then the embedding functors K(C) → K(R-Mod) and K(D) → K(R-Mod) have left and right adjoints, respectively. In finding examples of such pairs, we will describe a procedure for using Hoveys results in [5] to find a new model structure on C(R-Mod)

Abelian groups which have trivial absolute coGalois group

summary:In this article we characterize those abelian groups for which the coGalois group (associated to a torsion free cover) is equal to the identity