Short exact sequence of chain complexes

Definition

A short exact sequence of chain complexes is a short exact sequence in the category of chain complexes with chain maps, viewed in an obvious way as an abelian category. More explicitly it is a collection of data of the form:

• chain complexes $A,B,C$.
• chain maps $i:A\to B$ and $p:B\to C$

satisfying the following: For every $n$, the induced sequence of maps:

$0\to A_{n}{\stackrel {i_{n}}{\to }}B_{n}{\stackrel {p_{n}}{\to }}C_{n}\to 0$

is a short exact sequence. (If we are working over the category of abelian groups, then this must be a short exact sequence of abelian groups; if we are working over the category of modules over a commutative unital ring, then this must be a short exact sequence of modules. Note that exactness depends only on the underlying abelian group structure so we can view everything as abelian groups).

Facts

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