Editing Exact sequence (section)

==Definition==
In the context of group theory, a sequence
:<math>G_0\;\xrightarrow{\ f_1\ }\; G_1 \;\xrightarrow{\ f_2\ }\; G_2 \;\xrightarrow{\ f_3\ }\; \cdots \;\xrightarrow{\ f_n\ }\; G_n</math>
of groups and [[group homomorphism]]s is said to be '''exact''' '''at''' <math>G_i</math> if <math>\operatorname{im}(f_i)=\ker(f_{i+1})</math>. The sequence is called '''exact''' if it is exact at each <math>G_i</math> for all <math>1\leq i<n</math>, i.e., if the image of each homomorphism is equal to the kernel of the next.

The sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for other [[algebraic structure]]s.  For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of modules and [[module homomorphism]]s.  More generally, the notion of an exact sequence makes sense in any [[category (mathematics)|category]] with [[kernel (category theory)|kernel]]s and [[cokernel]]s, and more specially in [[abelian categories]], where it is widely used. 

===Simple cases===
To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the [[trivial group]]. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).

* Consider the sequence 0 → ''A'' → ''B''. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from ''A'' to ''B'') has kernel {0}; that is, if and only if that map is a [[monomorphism]] (injective, or one-to-one).
* Consider the dual sequence ''B'' → ''C'' → 0. The kernel of the rightmost map is ''C''. Therefore the sequence is exact if and only if the image of the leftmost map (from ''B'' to ''C'') is all of ''C''; that is, if and only if that map is an [[epimorphism]] (surjective, or onto).
* Therefore, the sequence 0 → ''X'' → ''Y'' → 0 is exact if and only if the map from ''X'' to ''Y'' is both a monomorphism and epimorphism (that is, a [[bimorphism]]), and so usually an [[isomorphism]] from ''X'' to ''Y'' (this always holds in [[exact categories]] like '''Set''').

===Short exact sequence===
Short exact sequences are exact sequences of the form
:<math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0.</math>
As established above, for any such short exact sequence, ''f'' is a monomorphism and ''g'' is an epimorphism. Furthermore, the image of ''f'' is equal to the kernel of ''g''. It is helpful to think of ''A'' as a [[subobject]] of ''B'' with ''f'' embedding ''A'' into ''B'', and of ''C'' as the corresponding factor object (or [[Quotient object|quotient]]), ''B''/''A'', with ''g'' inducing an isomorphism
:<math>C \cong B/\operatorname{im}(f) = B/\operatorname{ker}(g)</math>

The short exact sequence
:<math>0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0\,</math>
is called '''[[split exact sequence|split]]''' if there exists a homomorphism ''h'' : ''C'' → ''B'' such that the composition ''g'' ∘ ''h'' is the identity map on ''C''. It follows that if these are [[abelian group]]s, ''B'' is isomorphic to the [[direct sum]] of ''A'' and ''C'':
:<math>B \cong A \oplus C.</math>

===Long exact sequence===
A general exact sequence is sometimes called a '''long exact sequence''', to distinguish from the special case of a short exact sequence.<ref>{{Cite web|title=exact sequence in nLab, Remark 2.3|url=https://ncatlab.org/nlab/show/exact+sequence#Definition|access-date=2021-09-05|website=ncatlab.org}}</ref>

A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence

{{Equation|1=A_0\;\xrightarrow{\ f_1\ }\; A_1 \;\xrightarrow{\ f_2\ }\; A_2 \;\xrightarrow{\ f_3\ }\; \cdots \;\xrightarrow{\ f_n\ }\; A_n,|2=1}}

with ''n ≥'' 2, we can split it up into the short sequences

{{Equation|1=\begin{align}
      0 \rightarrow K_1 \rightarrow {} & A_1 \rightarrow  K_2 \rightarrow  0 ,\\
      0 \rightarrow K_2 \rightarrow {} & A_2 \rightarrow  K_3 \rightarrow  0 ,\\
                                       & \ \,\vdots \\
  0 \rightarrow K_{n-1} \rightarrow {} & A_{n-1} \rightarrow  K_n \rightarrow  0 ,\\
\end{align}|2=2}}
where <math>K_i = \operatorname{im}(f_i)</math> for every <math>i</math>. By construction, the sequences ''(2)'' are exact at the <math>K_i</math>'s (regardless of the exactness of ''(1)''). Furthermore, ''(1)'' is a long exact sequence if and only if ''(2)'' are all short exact sequences.

See [[#Weaving lemma|weaving lemma]] for details on how to re-form the long exact sequence from the short exact sequences.