Editing Homological algebra (section)

== Foundational aspects ==
Cohomology theories have been defined for many different objects such as [[topological space]]s, [[sheaf (mathematics)|sheaves]], [[group (mathematics)|group]]s, [[ring (mathematics)|ring]]s, [[Lie algebra]]s, and [[C*-algebra]]s. 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 functors [[Ext functors|Ext]] and [[Tor functor|Tor]].

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 and injective [[resolution of a module|module resolutions]].
* 'Tohoku': The approach in a [[Grothendieck's Tôhoku paper|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 [[Jean-Louis Verdier|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 ''[[torsor]]s'' (important in [[Galois cohomology]]).