Editing Exact sequence (section)

== Properties ==
The [[splitting lemma]] states that, for a  short exact sequence
:<math>0 \to A \;\xrightarrow{\ f\ }\; B \;\xrightarrow{\ g\ }\; C \to 0,</math> the following conditions are equivalent.
*There exists a morphism {{math|''t'' : ''B'' → ''A''}} such that {{math|''t'' ∘ ''f''}} is the identity on {{math|''A''}}.
*There exists a morphism {{math|''u'': ''C'' → ''B''}} such that {{math|''g'' ∘ ''u''}} is the identity on {{math|''C''}}.
*There exists a morphism {{math|''u'': ''C'' → ''B''}} such that {{math|''B''}} is the [[direct sum]] of {{math|''f''(''A'')}} and {{math|''u''(''C'')}}. 

For non-commutative groups, the splitting lemma does not apply, and one has only the equivalence between the two last conditions, with "the direct sum" replaced with "a [[semidirect product]]".

In both cases, one says that such a short exact sequence ''splits''.

The [[snake lemma]] shows how a [[commutative diagram]] with two exact rows gives rise to a longer exact sequence. The [[nine lemma]] is a special case.

The [[five lemma]] gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the [[short five lemma]] is a special case thereof applying to short exact sequences.

===Weaving lemma===

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

:<math>A_1\to A_2\to A_3\to A_4\to A_5\to A_6</math>

which implies that there exist objects ''C<sub>k</sub>'' in the category such that

:<math>C_k \cong \ker (A_k\to A_{k+1}) \cong \operatorname{im} (A_{k-1}\to A_k)</math>.

Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:

:<math>C_k \cong \operatorname{coker} (A_{k-2}\to A_{k-1})</math>

(This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the [[category of groups]], in which coker(''f'') : ''G'' → ''H'' is not ''H''/im(''f'') but <math>H / {\left\langle \operatorname{im} f \right\rangle}^H</math>, the quotient of ''H'' by the [[conjugate closure]] of im(''f'').)  Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

:[[Image:long short exact sequences.png]]
The only portion of this diagram that depends on the cokernel condition is the object <math display="inline">C_7</math> and the final pair of morphisms <math display="inline">A_6 \to C_7\to 0</math>.  If there exists any object <math>A_{k+1}</math> and morphism <math>A_k \to A_{k+1}</math> such that <math>A_{k-1} \to A_k \to A_{k+1}</math> is exact, then the exactness of <math>0 \to  C_k \to A_k \to C_{k+1} \to  0</math> is ensured.  Again taking the example of the category of groups, the fact that im(''f'') is the kernel of some homomorphism on ''H'' implies that it is a [[normal subgroup]], which coincides with its conjugate closure; thus coker(''f'') is isomorphic to the image ''H''/im(''f'') of the next morphism.

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.