Introduction to \(A\)-infinity algebras and modules
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}.
scientific article; zbMATH DE number 1598721