Vipul: Created page with '==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 ...'

2011-01-11T21:37:56Z

Created page with '==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 ...'

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==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 complex]]es <math>A,B,C</math>.
* [[chain map]]s <math>i:A \to B</math> and <math>p:B \to C</math>

satisfying the following: For every <math>n</math>, the induced sequence of maps:

<math>0 \to A_n \stackrel{i_n}{\to} B_n \stackrel{p_n}{\to} C_n \to 0</math>

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]]

Short exact sequence of chain complexes - Revision history

Vipul: Created page with '==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 ...'