André-Quillen homology in degree five
type: Item
instance of: scholarly article
title: André-Quillen homology in degree five
zbMATH Open document ID: 0848.13016
DOI: 10.1016/0022-4049(94)90033-7
author: Yves Pitteloud
published in: Journal of Pure and Applied Algebra
publication date: 1994-08-07T00:00:00Z
review text: The author studies the homology module \(H_5 (A,B,C,B \otimes C)\) for commutative \(A\)-algebras \(B\) and \(C\). As a consequence he obtains some new results in the theory of cotangent complex. In section 1, series of definitions are given and natural homomorphisms \(\varphi (Y)\), \(\psi (Y)\), \(\alpha (Y)\) and \(\beta (Y)\) are defined for a differential \(\Gamma\)-algebra \(Y\). In section 2, the main theorem, theorem 2.1, is stated whose proof is given in sections 3 and 4. The theorem connects the homology modules \(H_4 (A,B,C,D)\), \(H_5 (A,B,C,D)\) and the torsion module \(\text{Tor}^A_3 (B,C)\) where \(D = B \otimes C\). The following results in the theory of cotangent complex are deduced: (i) If \(I\) is an ideal of \(C\) and \(\tau = \{\tau_m\}\) is a system of generators of \(I\), then there is a connection between \(H_5 (C,C/I,C/I)\) and the Koszul homology module \(H_3 (\tau)\). -- In particular if \(I\) is a principal ideal generated by \(\tau\), then \(H_5 (C,C/I,C/I)\) is expressible in terms of the annihilator of \(\tau\). (ii) If \(I\) is an almost complete intersection ideal in a local ring \(C\), then \(H_5 (C,C/I,C/I)\) is \(C/I\)-isomorphic to \(F(H_1)\) where \(H_1\) is the first Koszul homology module of a minimal set of generators of \(I\).
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 18E99
zbMATH DE Number: 612147
zbMATH Keywords: differential algebra
zbMATH Keywords: homology module
zbMATH Keywords: cotangent complex
zbMATH Keywords: torsion module
zbMATH Keywords: Koszul homology
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90033-7
OpenAlex ID: W1978641622
cites work: Méthode simpliciale en algèbre homologique et algèbre commutative
cites work: Q4771408
cites work: Produit tensoriel et complexe cotangent. (Tensor product and cotangent complex)
cites work: Q4047648
cites work: Q5610428
cites work: Autour de la platitude
cites work: Q3876031
cites work: Homology of symmetric algebras
cites work: Lois polynomes et lois formelles en théorie des modules
cites work: A certain exact sequence
description: scientific article; zbMATH DE number 612147