Homological algebra MOC

Exact sequence

An exact sequence¹ is a sequence of group-like objects² and a sequence of morphisms such that for all .³ #m/def/homology Thus for modules it is a chain complex with only trivial chain homologies, and is a measure of the failure of a chain complex to be exact.

Further terminology

An exact sequence with no more than three non-trivial objects is a Short exact sequence.

Properties

Any guarantees injective , since .
Any guarantees surjective , since .
Any partial exact sequence may be extended to by duplicating the ends and adding trivial tails.
Five lemma

#state/tidy | #lang/en | #SemBr

German exakte Sequenz ↩
The objects have some kind of group structure, hence groups, modules, and thus any objects of an Abelian category will do. ↩
2010, Algebraische Topologie, ¶3.1.6ff, p. 129 ↩