Introduction to \(A\)-infinity algebras and modules
type: Item
instance of: scholarly article
title: Introduction to \(A\)-infinity algebras and modules
zbMATH Open document ID: 0989.18009
author: Bernhard Keller
published in: Homology, Homotopy and Applications
publication date: 2001-05-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9910179
full work available at URL: https://eudml.org/doc/229468
full work available at URL: volume2-1.htm
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18G40
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 1598721
zbMATH Keywords: \(A\) infinity-algebras
zbMATH Keywords: \(A_\infty\)-algebras
zbMATH Keywords: derived category
zbMATH Keywords: differential graded algebra
zbMATH Keywords: differential graded modules
zbMATH Keywords: module category
MaRDI profile type: Publication
arXiv ID: math/9910179
OpenAlex ID: W1941114527
DOI: 10.4310/HHA.2001.V3.N1.A1
review text: In recent years the notion of \(A_\infty\)-algebras and modules became popular in algebra. The theory was already widespread in algebraic topology, but was almost unknown to algebraists until very recently. The paper under review gives an introduction to this theory from the point of view of homological algebra. People who have a background in algebra and who want to know about \(A_\infty\)-algebras and modules should read this paper. The paper will certainly become one of the main references for this theory.NEWLINENEWLINENEWLINEThe paper starts with a motivation from the topological origins, written so that every mathematician can understand the problem. From there the definition of an \(A_\infty\)-algebra is natural and a clear and precise definition is given. Then, \(A_\infty\)-modules over an \(A_\infty\)-algebra are introduced as well as the derived category of an \(A_\infty\)-algebra. The natural functors Hom and \(\otimes\) on \(A_\infty\)-modules are defined and studied, as well as the functor given by pulling back along an \(A_\infty\)-algebra morphism. Along the presentation the links to the underlying differential graded algebra and differential graded modules are clearly marked.NEWLINENEWLINENEWLINEGuiding applications are the problem of reconstructing a complex from its homology and reconstructing a module category from the Ext-algebra of simple modules. The paper explains how these problems can be solved using the theory. At the end indications for the proof of the various main results are given. Most of the full proofs will be given in the thesis of \textit{Kenji Lefevre}.
description: scientific article; zbMATH DE number 1598721