Homology
type: Item
instance of: scholarly article
title: Homology
zbMATH Open document ID: 0818.18001
published in: Classics in Mathematics
publication date: 1995-05-23T00:00:00Z
review text: Homological algebra has evolved over the last half-century into a fundamental tool in algebra, algebraic topology, algebraic geometry, complex-analytic geometry, and algebraic number theory. The methods of homological algebra, which have their origin in the homology and cohomology theories for topological spaces, provide a powerful machinery for proving nonconstructive existence theorems in these areas of mathematics, on the one hand, and they allow, in many cases, to establish algebraic obstructions to carrying out certain constructions, on the other hand. The first stormy phase of development of homological algebra, touching almost every topic in algebra and related areas, reached its culmination in 1956 with the publication of the systematic monograph of \textit{H. Cartan} and \textit{S. Eilenberg} [Homological algebra. Princeton, New Jersey: Princeton University Press (1956; Zbl 0075.24305)]. The second research monograph and textbook on this more and more popular subject was \textit{S. MacLane}'s work ``Homology'', which appeared in 1963 [Berlin etc.: Springer-Verlag, Zbl 0133.26502]. At that time, homological algebra had reached a higher stage of maturity and ubiquity, also due to the contributions that \textit{S. MacLane} himself had made in the meantime, and his (henceforth classical) book reflected the state of homological algebra at the time of writing in a masterly, brilliant manner. This and the outstanding didactic features have made \textit{S. MacLane}'s ``Homology'' into one of the great classics on homological algebra. The third edition of this book appeared in 1975, and here the author had corrected several errors and inaccuracies. The most important corrections concern the proof of the Homotopy Classification Theorem (Theorem III, 4.3. on p. 78), and a suitable modification of the axioms for allowable short exact sequences in abelian categories (p. 260), so that they can be applied in a direct way to further constructions. The present edition, appe…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 18Gxx
Mathematics Subject Classification ID: 18-01
Mathematics Subject Classification ID: 18Exx
zbMATH DE Number: 758278
zbMATH Keywords: categories
zbMATH Keywords: functors
zbMATH Keywords: rings
zbMATH Keywords: modules
zbMATH Keywords: algebras
zbMATH Keywords: complexes
zbMATH Keywords: homology theories
zbMATH Keywords: Ext and Tor
zbMATH Keywords: cohomology of groups
zbMATH Keywords: homological dimension
zbMATH Keywords: spectral sequences
zbMATH Keywords: derived functors
zbMATH Keywords: homological algebra
MaRDI profile type: Publication
author: Saunders Mac Lane
description: scientific article; zbMATH DE number 758278