# Short exact sequence of chain complexes gives long exact sequence of homology

From Topospaces

## Statement

Suppose are chain complexes and we have a short exact sequence of chain complexes, i.e., a *short exact sequence* in the category of chain complexes with chain maps given by:

Here, *exactness* means that for each integer , the induced sequence of maps:

is a short exact sequence of abelian groups (or modules, if this is a chain complex of modules).

Then, we can obtain (canonically) a *long exact sequence* of homology groups (respectively, modules):

The maps from a given homology of to the lower homology of are connecting homomorphisms and arise via an application of the snake lemma.