Approximation Presentations of Modules and Homological Conjectures
نویسنده
چکیده
Zaks (1969) proved that the answer is affirmative for a left and right noetherian ring if both dimensions are finite. Such rings are called Gorenstein. For a positive integer k, Auslander and Reiten (1994) initiated the study of k-Gorenstein algebras, which has stimulated several investigations. They showed that the answer to the question above is positive in case is an artin -Gorenstein algebra (i.e., is artin k-Gorenstein for all k). In Auslander and Reiten (1991), they also gave the relationship between the question above and the finitistic dimension conjecture (resp., the contravariant finiteness of the full subcategory of mod consisting of the modules M with Ext M = 0 for any i ≥ 1). It follows from Auslander and Reiten (1996, Theorem 0.1) and Hoshino and Nishida (2005, Theorem 4.1) that the following conditions are equivalent for a left and right noetherian ring :
منابع مشابه
ar X iv : m at h / 04 09 17 5 v 2 [ m at h . R A ] 2 4 M ar 2 00 7 Approximation Presentations of Modules and Homological Conjectures
In this paper we give a sufficient condition of the existence of W-approximation presentations. We also introduce property (W). As an application of the existence of W-approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander-Reiten conjecture and property (W).
متن کاملCopresented Dimension of Modules
In this paper, a new homological dimension of modules, copresented dimension, is defined. We study some basic properties of this homological dimension. Some ring extensions are considered, too. For instance, we prove that if $Sgeq R$ is a finite normalizing extension and $S_R$ is a projective module, then for each right $S$-module $M_S$, the copresented dimension of $M_S$ does not exceed the c...
متن کاملHomological dimensions of complexes of R-modules
Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.
متن کاملOplus-supplemented modules with respect to images of a fully invariant submodule
Lifting modules and their various generalizations as some main concepts in module theory have been studied and investigated extensively in recent decades. Some authors tried to present some homological aspects of lifting modules and -supplemented modules. In this work, we shall present a homological approach to -supplemented modules via fully invariant submodules. Lifting modules and H-suppleme...
متن کاملThe Homological Conjectures
1. Introduction The term \Homological Conjectures" is used here to refer to a certain set of related conjectures about homological properties of commutative rings. While there are numerous conjectures in this area, the ones discussed here are those collected in a monograph of Mel Hochster in 1975 entitled \Topics in the homological theory of modules over commutative rings" 24], as well as sever...
متن کامل