Editing Homological algebra (section)

{{Short description|Branch of mathematics}}
[[File:Snake lemma origin.svg|thumb|350px|A diagram used in the [[snake lemma]], a basic result in homological algebra.]]

'''Homological algebra''' is the branch of [[mathematics]] that studies [[homology (mathematics)|homology]] in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in [[combinatorial topology]] (a precursor to [[algebraic topology]]) and [[abstract algebra]] (theory of [[module (mathematics)|modules]] and [[Syzygy (mathematics)|syzygies]]) at the end of the 19th century, chiefly by [[Henri Poincaré]] and [[David Hilbert]]. 

Homological algebra is the study of homological [[functor]]s and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of [[category theory]]. A central concept is that of [[chain complex]]es, which can be studied through their homology and [[cohomology]].

Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological [[invariant (mathematics)|invariants]] of [[ring (mathematics)|rings]], modules, [[topological space]]s, and other "tangible" mathematical objects. A [[spectral sequence]] is a powerful tool for this.

It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes [[commutative algebra]], [[algebraic geometry]], [[algebraic number theory]], [[representation theory]], [[mathematical physics]], [[operator algebra]]s, [[complex analysis]], and the theory of [[partial differential equation]]s. [[K-theory|''K''-theory]] is an independent discipline which draws upon methods of homological algebra, as does the [[noncommutative geometry]] of [[Alain Connes]].