tensor product of chain complexes

Let C={Cn,n} and C′′={Cn′′,n′′} be two chain complexesMathworldPlanetmath of R-modules, where R is a commutative ring with unity. Their tensor productPlanetmathPlanetmath CRC′′={(CRC′′)n,n} is the chain complex defined by


where CiRCj′′ denotes the tensor product (http://planetmath.org/TensorProduct) of R-modules Ci and Cj′′.

Indeed, this defines a chain complex, because for each tiRsj′′CiRCj′′(CRC′′)i+j we have


thus CRC′′ is a chain complex.

Title tensor product of chain complexes
Canonical name TensorProductOfChainComplexes
Date of creation 2013-03-22 16:13:21
Last modified on 2013-03-22 16:13:21
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 13
Author Mazzu (14365)
Entry type Definition
Classification msc 16E05
Classification msc 18G35
Defines tensor product of chain complexes