Editing Chain complex (section)

==Category of chain complexes==
Chain complexes of ''K''-modules with chain maps form a [[category (mathematics)|category]] Ch<sub>''K''</sub>, where ''K'' is a commutative ring.

If ''V'' = ''V''<math>{}_*</math> and ''W'' = ''W''<math>{}_*</math> are chain complexes, their '''tensor product''' <math> V \otimes W </math> is a chain complex with degree ''n'' elements given by 
:<math> (V \otimes W)_n = \bigoplus_{\{i,j|i+j=n\}} V_i \otimes W_j </math>

and differential given by 
: <math> \partial (a \otimes b) = \partial a \otimes b + (-1)^{\left|a\right|} a \otimes \partial b </math> 
where ''a'' and ''b'' are any two homogeneous vectors in ''V'' and ''W'' respectively, and <math> \left|a\right| </math> denotes the degree of ''a''.

This tensor product makes the category Ch<sub>''K''</sub> into a [[symmetric monoidal category]]. The identity object with respect to this monoidal product is the base ring ''K'' viewed as a chain complex in degree 0. The [[braided monoidal category|braiding]] is given on simple tensors of homogeneous elements by 
:<math> a \otimes b \mapsto (-1)^{\left|a\right|\left|b\right|} b \otimes a </math>
The sign is necessary for the braiding to be a chain map.

Moreover, the category of chain complexes of ''K''-modules also has [[closed monoidal category|internal Hom]]: given chain complexes ''V'' and ''W'', the internal Hom of ''V'' and ''W'', denoted Hom(''V'',''W''), is the chain complex with degree ''n'' elements given by <math>\Pi_{i}\text{Hom}_K (V_i,W_{i+n})</math> and differential given by
: <math> (\partial f)(v) = \partial(f(v)) - (-1)^{\left|f\right|} f(\partial(v)) </math>.
We have a [[natural isomorphism]]
:<math>\text{Hom}(A\otimes B, C) \cong \text{Hom}(A,\text{Hom}(B,C))</math>