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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 groups; 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 (important in Galois cohomologyIn mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L ''K acts in a natural way on some ab).

Foundational aspects

The methods of homological algebra start with use of the exact sequenceHomological algebra In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the i 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 GrothendieckAlexander Grothendieck (born March 28, 1928, Berlin) is one of the leading mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966 and coawar using the abelian categoryIn mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abe 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. The computational sledgehammer par excellence is the spectral sequence; in the derived category approach these don't appear at all, in an explicit way.