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 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

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Footnotes

1. German exakte Sequenz ↩

2. The objects have some kind of group structure, hence groups, modules, and thus any objects of an Abelian category will do. ↩

3. 2010, Algebraische Topologie, ¶3.1.6ff, p. 129 ↩