Homological algebra
|
'''Homological algebra''' is the branch of mathematics which studies the methods of homology (mathematics)|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, sheaf (mathematics)|sheaves, group (mathematics)|groups, ring (mathematics)|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 functors|Ext and Tor functors|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:
Henri Cartan|Cartan-Samuel Eilenberg|Eilenberg: In their 1956 book "Homological Algebra", these authors used projective resolution|projective and Injective resolution|injective module resolutions.
'Tohoku': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of The Tohoku Mathematical Journal in 1957, using the abelian category concept (to include sheaf (mathematics)|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 category|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).
|