Homological algebra MOC

Exact sequence

An exact sequence¹ (𝑀•,𝑓•) is a sequence (𝑀𝑘)𝑘∈ℤ of group-like objects² and a sequence of morphisms (𝑓𝑘 :𝑀𝑘 →𝑀𝑘−1) such that 𝑓𝑘+1(𝑀𝑘+1) =ker⁡𝑓𝑘 for all 𝑘.³ homology Thus for modules it is a chain complex with only trivial chain homologies, and 𝐻𝑘(𝑀) 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 0 →𝐴𝑓→𝐵 guarantees injective 𝑓, since ker⁡𝑓 =0.
• Any 𝐴𝑓→𝐵 →0 guarantees surjective 𝑓, since 𝑓(𝐴) =𝐵.
• Any partial exact sequence may be extended to ℤ 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 ↩