Homological algebra MOC

Exact sequence

An exact sequence¹ $(𝑀 •, 𝑓 •)$ is a sequence $(𝑀 𝑘) 𝑘 \in ℤ$ of group-like objects² and a sequence of morphisms $(𝑓 𝑘 : 𝑀 𝑘 \to 𝑀 𝑘 - 1)$ such that $𝑓 𝑘 + 1 (𝑀 𝑘 + 1) = k e r 𝑓 𝑘$ for all $𝑘$ .³ #m/def/homology Thus for modules it is a chain complex with only trivial chain homologies, and $𝐻 𝑘 (𝑀)$ 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 $0 \to 𝐴 𝑓 \to 𝐵$ guarantees injective $𝑓$ , since $k e r 𝑓 = 0$ .
Any $𝐴 𝑓 \to 𝐵 \to 0$ 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

#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 ↩