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