PlanetMath: tensor product of chain complexes

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tensor product of chain complexes

(Definition)

Let $C'=\left\{ C_n',\partial_{n} ' \right\}$ and $C''=\left\{ C_n'',\partial_{n} '' \right\}$ be two chain complexes of -modules, where is a commutative ring with unity. Their tensor product $C'\otimes_R C''=\left\{ (C'\otimes_R C'')_n,\partial_{n} \right\}$ is the chain complex defined by

$\displaystyle (C'\otimes_R C'')_n = \bigoplus_{i+j=n}(C_i'\otimes_R C_j''),$

$\displaystyle \partial_{n}(t'_i\otimes_R s''_j) = \partial_{i} '(t'_i)\otimes_R... ...R \partial_{j} ''(s''_j),\ \ \ \forall t'_i\in C_i',\ s''_j\in C_j'',\ (i+j=n),$

where $C_i'\otimes_R C_j''$ denotes the tensor product of

-modules

and

Indeed, this defines a chain complex, because for each $t'_i\otimes_R s''_j\in C_i'\otimes_R C_j''\subseteq (C'\otimes_R C'')_{i+j}$ we have

$\displaystyle \partial_{i+j-1} \partial_{i+j}(t'_i\otimes_R s''_j) = \partial_{... ...(t'_i)\otimes_R s''_j + (-1)^i\, t'_i\otimes_R \partial_{j} ''(s''_j) \right)=$

$\displaystyle = (-1)^{i-1}\, \partial_{i} '(t'_i)\otimes_R \partial_{j} ''(s''_j)+(-1)^i \partial_{i} '(t'_i)\otimes_R \partial_{j} ''(s''_j)=0,$

thus $C'\otimes_R C''$ is a chain complex.

"tensor product of chain complexes" is owned by Mazzu.

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Also defines:

tensor product of chain complexes

Keywords:

chain complex

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Cross-references: unity, commutative ring, chain complexes

This is version 10 of tensor product of chain complexes, born on 2006-09-07, modified 2007-01-08.
Object id is 8321, canonical name is TensorProductOfChainComplexes.
Accessed 1146 times total.

Classification:

AMS MSC:	16E05 (Associative rings and algebras :: Homological methods :: Syzygies, resolutions, complexes)
	18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)

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