# tensor product of chain complexes

Let $C^{\prime}=\left\{C_{n}^{\prime},\partial_{n}^{\prime}\right\}$ and $C^{\prime\prime}=\left\{C_{n}^{\prime\prime},\partial_{n}^{\prime\prime}\right\}$ be two chain complexes of $R$-modules, where $R$ is a commutative ring with unity. Their tensor product $C^{\prime}\otimes_{R}C^{\prime\prime}=\left\{(C^{\prime}\otimes_{R}C^{\prime% \prime})_{n},\partial_{n}\right\}$ is the chain complex defined by

 $(C^{\prime}\otimes_{R}C^{\prime\prime})_{n}=\bigoplus_{i+j=n}(C_{i}^{\prime}% \otimes_{R}C_{j}^{\prime\prime}),$
 $\partial_{n}(t^{\prime}_{i}\otimes_{R}s^{\prime\prime}_{j})=\partial_{i}^{% \prime}(t^{\prime}_{i})\otimes_{R}s^{\prime\prime}_{j}+(-1)^{i}\,t^{\prime}_{i% }\otimes_{R}\partial_{j}^{\prime\prime}(s^{\prime\prime}_{j}),\ \ \ \forall t^% {\prime}_{i}\in C_{i}^{\prime},\ s^{\prime\prime}_{j}\in C_{j}^{\prime\prime},% \ (i+j=n),$

where $C_{i}^{\prime}\otimes_{R}C_{j}^{\prime\prime}$ denotes the tensor product (http://planetmath.org/TensorProduct) of $R$-modules $C_{i}^{\prime}$ and $C_{j}^{\prime\prime}$.

Indeed, this defines a chain complex, because for each $t^{\prime}_{i}\otimes_{R}s^{\prime\prime}_{j}\in C_{i}^{\prime}\otimes_{R}C_{j% }^{\prime\prime}\subseteq(C^{\prime}\otimes_{R}C^{\prime\prime})_{i+j}$ we have

 $\partial_{i+j-1}\partial_{i+j}(t^{\prime}_{i}\otimes_{R}s^{\prime\prime}_{j})=% \partial_{i+j-1}\left(\partial_{i}^{\prime}(t^{\prime}_{i})\otimes_{R}s^{% \prime\prime}_{j}+(-1)^{i}\,t^{\prime}_{i}\otimes_{R}\partial_{j}^{\prime% \prime}(s^{\prime\prime}_{j})\right)=$
 $=(-1)^{i-1}\,\partial_{i}^{\prime}(t^{\prime}_{i})\otimes_{R}\partial_{j}^{% \prime\prime}(s^{\prime\prime}_{j})+(-1)^{i}\partial_{i}^{\prime}(t^{\prime}_{% i})\otimes_{R}\partial_{j}^{\prime\prime}(s^{\prime\prime}_{j})=0,$

thus $C^{\prime}\otimes_{R}C^{\prime\prime}$ is a chain complex.

Title tensor product of chain complexes TensorProductOfChainComplexes 2013-03-22 16:13:21 2013-03-22 16:13:21 Mazzu (14365) Mazzu (14365) 13 Mazzu (14365) Definition msc 16E05 msc 18G35 tensor product of chain complexes