homological algebra

Homological algebra is the 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 defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C-star algebras. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology. Central to homological algebra is the notion of exact sequence; these can be used to perform actual calculations. A classical tool of homological algebra is that of derived functor; the most basic examples are Ext and Tor.

Foundational aspects

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: In their 1956 book "Homological Algebra", these authors used projective and injective module resolutions.
'Tohoku': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the The Tohoku Mathematical Journal in 1957, using the abelian category concept (to include sheaves of abelian groups).
The derived category of Grothendieck and Verdier. Derived categories date back to Verdier's 1967 thesis. They are examples of triangulated categories used in a number of modern theories.

These move from computability to generality. The computational sledgehammer par excellence is the spectral sequence; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary. There have been attempts at 'non-commutative' theories which extend first cohomology as torsors (important in Galois cohomology).

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