On the vanishing of the homology of the exterior powers of the cotangent complex
type: Item
instance of: scholarly article
title: On the vanishing of the homology of the exterior powers of the cotangent complex
zbMATH Open document ID: 0820.13013
DOI: 10.1007/BF01195130
published in: Archiv der Mathematik
publication date: 1995-09-11T00:00:00Z
review text: Let \(A\) be a commutative noetherian ring and \(B\) a flat commutative \(A\)- algebra essentially of finite type. The following are equivalent: (i) \(B\) is a smooth \(A\)-algebra, (ii) \(H_ p (\bigwedge^ q_ B \mathbb{L}_{B | A}) = 0\) for all \(p \geq 1\), \(q \geq 0\), (iii) There exists some \(p \geq 1\) such that \(H_ p (\bigwedge^ q_ B \mathbb{L}_{B | A}) = 0\) for all \(q \in [p + 1, p + \text{ext.rk.} (\Omega_{B | A})]\). Here \(\mathbb{L}_{B | A}\) is the cotangent complex [see \textit{M. André}, ``Homologie des algèbres commutatives'' (Berlin 1974; Zbl 0284.18009) or \textit{D. Quillen}, in: Appl. categorical algebra, Proc. Symp. Pure Math. 17, 65-87 (1970; Zbl 0234.18010)], \(\bigwedge^ q_ B \mathbb{L}_{B | A}\) is the simplicial \(B\)-module resulting from applying the \(q\)-th exterior power functor \(\bigwedge^ q_ B\) to the simplicial \(B\)-module \(\mathbb{L}_{B | A}\), and \(\text{ext.rk.} (\Omega_{B | A})\) is the maximum \(n\) such that \(\bigwedge^ n_ B \Omega_{B | A} \neq 0\).
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 769535
zbMATH Keywords: André-Quillen homology
zbMATH Keywords: algebra essentially of finite type
zbMATH Keywords: cotangent complex
author: Javier Majadas
reviewed by: Javier Majadas
MaRDI profile type: Publication
cites work: A Hochschild homology criterium for the smoothness of an algebra
cites work: Decomposition of the Hochschild and cyclic homology of commutative differential graded algebras
cites work: Homologie nicht-additiver Funktoren. Anwendungen
cites work: On the vanishing of the second André-Quillen homology of a local homomorphism
cites work: A note on the Hochschild homology and cyclic homology of a topological algebra
cites work: Hochschild homology of a topological algebra and formal smoothness
cites work: Q5642708
cites work: Smooth algebras and vanishing of Hochschild homology
cites work: Commutative augmented algebras with two vanishing homology modules
description: scientific article; zbMATH DE number 769535