On the acyclicity of the Tate complex
type: Item
instance of: scholarly article
title: On the acyclicity of the Tate complex
zbMATH Open document ID: 0953.13006
DOI: 10.1016/S0022-4049(97)00103-5
author: Amalia Blanco
author: Antonio G. Rodicio
author: Javier Majadas
published in: Journal of Pure and Applied Algebra
publication date: 2001-01-23T00:00:00Z
review text: Let \(A\) be a commutative ring with unit, \(I\) an ideal generated by a set \(t\), \(B = A/I\) and \(E\) the Koszul complex associated to \(t\). Let \(s\) be a set of generators for \(H_1(E)\) as a \(B\)-module. This paper gives a proof that the second stage of the Tate complex of \(B\) over \(A\) is acyclic if and only if \(s\) is a basis of the \(B\)-module \(H_1(E)\) and the canonical map \(\wedge^* H_1(E) \to H_*(E)\) is an isomorphism. A similar theorem for a simplicial resolution of the \(A\)-algebra \(B\) is also proved. These theorems occur also in an earlier paper by the same authors [\textit{A. Blanco, J. Majadas} and \textit{A. G. Rodicio}, Invent. Math. 123, No.~1, 123-140 (1996; Zbl 0856.13009); (theorems 2 and 13)], however the proofs given there rely on results about simplicial rings which are not known to hold in complete generality. The present paper gives alternative proofs based on the spectral sequence associated to a certain bicomplex. As an application a new proof is given of a vanishing theorem for André-Quillen cohomology.
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13D03
zbMATH DE Number: 1308229
zbMATH Keywords: Koszul complex
zbMATH Keywords: Tate resolution
zbMATH Keywords: André-Quillen cohomology
zbMATH Keywords: spectral sequence
zbMATH Keywords: vanishing theorem
MaRDI profile type: Publication
cites work: Q4771408
cites work: Produit tensoriel et complexe cotangent. (Tensor product and cotangent complex)
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description: scientific article; zbMATH DE number 1308229