History of homological algebra. (English) Zbl 0966.55002

James, I. M. (ed.), History of topology. Amsterdam: Elsevier. 797-836 (1999).
The author gives a survey over the development of homological algebra, organized in five chapters as follows:
1. Betti numbers, torsion coefficients and the rise of homology.
2. Homology and cohomology of algebraic systems: Ext and Tor, homology and cohomology of groups and Lie algebras, cohomology of associative algebras, sheaf cohomology and spectral sequences.
3. The Cartan-Eilenberg revolution: projective and injective resolutions of modules, derived functors, Abelian categories, algebraic \(K\)-theory.
4. After the Cartan-Eilenberg revolution: regular local rings, Grothendieck’s theory of sheaf cohomology, cohomology theories in algebraic geometry.
5. Simplicial methods: homotopical algebra, non-Abelian derived functors, André-Quillen cohomology for commutative rings, higher algebraic \(K\)-theory, Hochschild and cyclic homology.
For the entire collection see [Zbl 0922.54003].


55-03 History of algebraic topology
01A60 History of mathematics in the 20th century
18F99 Categories in geometry and topology
55U99 Applied homological algebra and category theory in algebraic topology
01A55 History of mathematics in the 19th century