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,

... → *A*_{n+1}
→_{}_{fn}_{+1}
*A*_{n} →_{fn} * A*_{n–1} →_{fn–1}* A*_{n–2} →_{fn–2} ...

is exact at *A*_{n} if the
image of *f*_{n+1} coincides
with the kernel of *f*_{n};
In particular, the composition *f*_{n} *f*_{n+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 *f*_{1} is
surjective,* f*_{5} is injective,
and* f*_{2},* f*_{4} are
isomorphisms, then *f*_{3} 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 *f*_{2} is
a monomorphism, *f*_{3} is an
epimorphism with ker *f*_{3} = *K*,
and ker(*f*_{4}) = *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., *A*_{n–1}
→_{fn–1} by

0 → ker(*f*_{n–1}) → *A*_{n–1} →_{fn–1} im(*f*_{n–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* = id_{B}.

* __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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send feedback and suggestions to bombelli at olemiss.edu – modified 14
feb 2016