Editing Chain complex (section)

===Exact sequences===
{{main|Exact sequence}}
An '''exact sequence''' (or '''exact''' complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A '''short exact sequence''' is a bounded exact sequence in which only the groups ''A''<sub>''k''</sub>, ''A''<sub>''k''+1</sub>, ''A''<sub>''k''+2</sub> may be nonzero. For example, the following chain complex is a short exact sequence.
:<math>
\cdots
\xrightarrow{} \; 0 \;
\xrightarrow{} \; \mathbf{Z} \;
\xrightarrow{\times p} \; \mathbf{Z}
\twoheadrightarrow \mathbf{Z}/p\mathbf{Z} \;
\xrightarrow{} \; 0 \;
\xrightarrow{}
\cdots
</math>
In the middle group, the closed elements are the elements p'''Z'''; these are clearly the exact elements in this group.