Editing Exact sequence (section)

===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.