Acyclicity of symmetric and exterior powers of complexes
type: Item
instance of: scholarly article
title: Acyclicity of symmetric and exterior powers of complexes
zbMATH Open document ID: 0876.13006
author: Alexandre B. Tchernev
published in: Journal of Algebra
publication date: 1997-11-18T00:00:00Z
review text: In this paper, for a given finite complex \(F\) of free \(R\)-modules of arbitrary length with \(H_0 (F)=M\), for a given \(R\)-module \(M\), the author constructs complexes \(S^iF\) and \(L^iF\) of free \(R\)-modules whose zero-th homology is \(S^iM(i\)-th symmetric power of \(M)\) and \(\bigwedge^i M(i\)-th exterior power of \(M)\), respectively. In the main theorems (theorems 2.1 and 2.2) the author derives necessary and sufficient conditions for the acyclicity of the complexes \(S^iF\) and \(L^iF\) for a given finite free complex \(F\) with \(H_0 (F)=M\). Next the author constructs complexes \(G^iF\) and \(D^iF\) for a finite free complex \(F\) and derives necessary and sufficient conditions for acyclicity of these complexes in theorems 3.11 and 3.12. In section 4, acyclicity of complexes \(D^kF\) and \(G^kF\) when \(F\) has no gaps and has length at most 2, is considered. Finally, in section 5, using his acyclicity criterion, the author generalizes a result of Avramov on the \(q\)-torsion-freeness of the symmetric powers of a finite module of projective dimension 1 over a Noetherian ring.
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 941283
zbMATH Keywords: acyclicity of complexes
zbMATH Keywords: symmetric powers
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: jabr.1996.0302
OpenAlex ID: W2031531652
DOI: 10.1006/JABR.1996.0302
Recommended article: Acyclic complexes related to noncommutative symmetric functions
Recommended article: Cohomology of composed symmetric powers
Recommended article: Acyclicity criteria for complexes associated with an alternating map
Recommended article: Complexes acyclic up to integral closure
Recommended article: Extensions between exterior powers and between symmetric powers
Recommended article: Acyclicity versus total acyclicity for complexes over noetherian rings
Recommended article: Duality and symmetry of complexity over complete intersections via exterior homology
Recommended article: Homotopy theory of symmetric powers
Recommended article: Symmetric and exterior powers of homogeneous vector bundles
Recommended article: QUELQUES CALCULS DE COHOMOLOGIE DE COMPOSITIONS DE PUISSANCES SYMÉTRIQUES
description: scientific article; zbMATH DE number 941283