Exact Sequences (of R-Modules)  

In General
* Idea: One of the main calculational tools in algebraic topology, used to find homology groups, and so on.
$ Def: A sequence of R-modules and homomorphisms, possibly infinite on both sides,

... → An+1 → fn+1 An →fn An−1 →fn−1 An−2 →fn−2 ...

is exact at An if the image of fn+1 coincides with the kernel of fn; In particular, the composition fn fn+1 = 0.
* Remark: The sequence is just called exact if it is exact everywhere.
> Online resources: see MathWorld page; Wikipedia page.

Applications
* Five lemma: If we have two exact sequences connected by homomorphisms,

\[ \matrix{\ldots & \longrightarrow & A_1 & \longrightarrow & A_2 & \longrightarrow & A_3
& \longrightarrow & A_4 & \longrightarrow & A_5 & \longrightarrow & \ldots\cr
& & \downarrow\ f_1 & & \downarrow\ f_2 & & \downarrow\ f_3 & & \downarrow\ f_4 & & \downarrow\ f_5 \cr
\ldots & \longrightarrow & B_1 & \longrightarrow & B_2 & \longrightarrow & B_3
& \longrightarrow & B_4 & \longrightarrow & \longrightarrow & B_5 & \ldots} \]

which is a commuting diagram, and f1 is surjective, f5 is injective, and f2, f4 are isomorphisms, then f3 is also an isomorphism [> see MathWorld page; Wikipedia page].
> Related topics: see group theory [extension]; homology; Mayer-Vietoris Sequence.

Short Exact Sequence
$ Def: An exact sequence of the form 0 →f1 K →f2 G →f3 H →f4 0.
* Properties: From the properties of exact sequences, it follows that f2 is a monomorphism, f3 is an epimorphism with ker f3 = K, and ker(f4) = H.
* Example: A particularly simple example is 0 → G → H → 0, where G and H have to be isomorphic.
* Remark: Any link in an exact sequence can be replaced by a short exact sequence, e.g., An−1 →fn−1 by

0 → ker(fn−1) → An−1 →fn−1 im(fn−1) → 0 .

* Pure sequence: A short exact sequence of Abelian groups 0 → A →f B →g C → 0 is pure if for all a ∈ A such that f(a) = mb, m ∈ \(\mathbb N\), b ∈ B, ∃ a' ∈ A such that a = ma'; Equivalently, the short sequence is exact if 0 → A/mA → B/mB → C/mC is exact; Or, ∀c ∈ C such that mc = 0, ∃ b ∈ B such that ε(b) = c and mb = 0; or Hom(\(\mathbb Z\)m, · ) preserves exactness for all m.

Splitting Sequence
$ Def: A short exact sequence 0 → A →f B →g C → 0 splits if g has a left inverse, i.e., a map h: C → B such that hg = idB.
* Remark: In this case, B is isomorphic to A ⊕ C (this explains the origin of the name).
* Example: 0 → A →i A ⊕ C →p C → 0, with p the projection map, which splits by the inclusion map of C.

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