Related papers: Complete Homology over associative rings
We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a…
Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of…
The focus of this paper is on a poorly understood invariant of a commutative noetherian local ring $R$ with residue field $k$: the stable cohomology modules $\hat{Ext}^{n}_R(k,k)$, defined for each $n\in\mathbb{Z}$ by Benson and Carlson,…
In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable…
We investigate Tate cohomology of modules over a commutative noetherian ring with respect to semidualizing modules. We identify classes of modules admitting Tate resolutions and analyze the interaction between the corresponding relative and…
Tate cohomology has been generalised by several authors using different constructions that have applications in group theory, ring theory and homotopical algebra. Therefore, there is a need for a uniform account that explains why their…
We study the interplay between the notions of $n$-coherent rings and finitely $n$-presented modules, and also study the relative homological algebra associated to them. We show that the $n$-coherency of a ring is equivalent to the thickness…
Tate cohomology was originally defined over finite groups. More recently, Avramov and Martsinkovsky showed how to extend the definition so that it now works well over Gorenstein rings. This paper improves the theory further by giving a new…
The study of rings and modules with homological criteria is a cornerstone of commutative algebra. Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. In this paper, a relative…
Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…
We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for…
Let $R$ be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian…
Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local…
We investigate stable homology of modules over a commutative noetherian ring $R$ with respect to a semidualzing module $C$, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of…
We establish the flat cohomology version of the Gabber-Thomason purity for \'{e}tale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak{m})$ and a commutative, finite, flat $R$-group $G$, the flat cohomology…
Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian…
We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with…
We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that,…
We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely…
This paper concerns homological notions of regularity for noncommutative algebras. Properties of an algebra $A$ are reflected in the regularities of certain (complexes of) $A$-modules. We study the classical Tor-regularity and…