Homological algebra

Homological algebra is that branch of mathematics which studies the methods of homology and cohomology in a general setting. These concepts originated in algebraic topology.

Cohomology theories have been described for topological spaces, sheaves, and groupss; also for Lie algebras, C-star algebras. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology.

There are also other homological functors that take their place in the theory, such as Ext and Tor. There have been attempts at 'non-commutative' theories, which extend first cohomology as torsors (which is important in Galois cohomology).

Foundational aspects

The methods of homological algebra start with use of the exact sequence to perform actual calculations. With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts, before the subject settled down. An approximate history can be stated as follows:

• Cartan-Eilenberg: as in their eponymous book, used projective and injective module resolutions.
• 'Tohoku': the approach in a celebrated paper by Alexander Grothendieck using the abelian category concept (to include sheaves of abelian groups).
• The derived category of Grothendieck and Verdier, used in a number of modern theories.

These move from computability to generality.