Stable cohomology over local rings
type: Item
instance of: scholarly article
title: Stable cohomology over local rings
zbMATH Open document ID: 1127.13012
author: Luchezar L. Avramov
author: Oana Veliche
published in: Advances in Mathematics
publication date: 2007-06-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0508021
review text: Let \(R\) be a noetherian local ring with residue field \(k\). In the present paper, the authors study the absolute cohomology module \(\mathcal E = \text{Ext}_R(k, k)\) and the stable cohomology module \(\mathcal S = \widehat{\text{Ext}}_R(k, k)\). This paper consists of ten sections excluding the introduction. In the first five sections, the authors develop general properties of stable cohomology theory. In Section 6, they give a criterion for \(R\) to be regular, complete intersection, or Gorenstein by using a homogeneous component \(\widehat{\text{Ext}}^n_R(k, k)\) of \(\mathcal S\). In Sections 7--9, they study the structure of \(\mathcal S\). Assume that \(R\) is not regular. It is known that the natural map \(\iota \: \mathcal E \to \mathcal S\) of graded \(k\)-algebras is injective. The authors compute the cokernel of \(\iota\) and show that \(\iota\) splits if \(\text{depth\,} \mathcal E \geq 2\) (in general \(\text{depth\,} \mathcal E \geq 1\)). The condition \(\text{depth\,} \mathcal E \geq 2\) is not strong. They give several classes of rings such that \(\text{depth} \,\mathcal E \geq 2\). On the other hand, \(\text{depth} \,\mathcal E = 1\) if \(R\) is a Golod ring. In Section 10, the authors studied the structure of \(\mathcal S\) when \(R\) is Golod.
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 5166629
zbMATH Keywords: local ring
zbMATH Keywords: stable cohomology
zbMATH Keywords: Tate cohomology
MaRDI profile type: Publication
OpenAlex ID: W2061130102
arXiv ID: math/0508021
cites work: Small homomorphisms of local rings
cites work: Q3717179
cites work: Modules of finite virtual projective dimension
cites work: Q4251930
cites work: Q3715232
cites work: ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION
cites work: The homological theory of contravariantly finite subcategories:auslander-buchweitz contexts, gorenstein categories and (co-)stabilization
cites work: Products in negative cohomology
cites work: Realizability of modules over Tate cohomology
cites work: Q3701517
cites work: Q4143495
cites work: Q3240881
cites work: On complete resolutions
cites work: Homological Algebra on a Complete Intersection, with an Application to Group Representations
cites work: The Radical of the Homotopy Lie Algebra
cites work: Q4505020
cites work: Determination of a class of Poincaré series.
cites work: Equivariant Tate-Vogel homology
cites work: A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity.
cites work: Free summands of conormal modules and central elements in homotopy Lie algebras of local rings
cites work: Q4867086
cites work: Finitely generated Ext algebras.
cites work: Factoring out the socle of a Gorenstein ring
cites work: A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring
cites work: Tate cohomology for arbitrary groups via satellites
cites work: Q3050539
cites work: The Poincaré series of a local ring
cites work: Hopf algebras and derivations
cites work: Q4875159
cites work: Homology of Noetherian rings and local rings
cites work: Gorenstein projective dimension for complexes
cites work: Q4863380
DOI: 10.1016/J.AIM.2006.11.012
description: scientific article