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+1fn+1 Anfn An–1fn–1 An–2fn–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 Kf2 Gf3 Hf4 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 → GH → 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–1fn–1 by

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

* Pure sequence: A short exact sequence of Abelian groups 0 → Af Bg C → 0 is pure if for all aA such that f(a) = mb, m ∈ $$\mathbb N$$, bB, ∃ a' ∈ A such that a = ma'; Equivalently, the short sequence is exact if 0 → A/mAB/mBC/mC is exact; Or, ∀cC such that mc = 0, ∃ bB 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 → Af Bg C → 0 splits if g has a left inverse, i.e., a map h: CB such that hg = idB.
* Remark: In this case, B is isomorphic to AC (this explains the origin of the name).
* Example: 0 → Ai ACp C → 0, with p the projection map, which splits by the inclusion map of C.