tensor product of chain complexes

Let $C^{\prime }=\{C_n^{\prime },{\partial }_n^{\prime }\}$ and $C^{\prime \prime }=\{C_n^{\prime \prime },{\partial }_n^{\prime \prime }\}$ be two chain complexes of $R$-modules, where $R$ is a commutative ring with unity. Their tensor product $C^{\prime }{\otimes }_R{C}^{\prime \prime }=\{{(C^{\prime }{\otimes }_R{C}^{\prime \prime })}_n,{\partial }_n\}$ is the chain complex defined by

$${(C^{\prime }{\otimes }_R{C}^{\prime \prime })}_n=\underset{i+j=n}{\oplus }(C_i^{\prime }{\otimes }_R{C}_j^{\prime \prime }),$$

$${\partial }_n(t_i^{\prime }{\otimes }_R{s}_j^{\prime \prime })={\partial }_i^{\prime }(t_i^{\prime }){\otimes }_R{s}_j^{\prime \prime }+{(-1)}^i{t}_i^{\prime }{\otimes }_R{\partial }_j^{\prime \prime }(s_j^{\prime \prime }),\forall t_i^{\prime }\in C_i^{\prime },s_j^{\prime \prime }\in C_j^{\prime \prime },(i+j=n),$$

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 $t_i^{\prime }{\otimes }_R{s}_j^{\prime \prime }\in C_i^{\prime }{\otimes }_R{C}_j^{\prime \prime }\subseteq {(C^{\prime }{\otimes }_R{C}^{\prime \prime })}_{i+j}$ we have

$${\partial }_{i+j-1}{\partial }_{i+j}(t_i^{\prime }{\otimes }_R{s}_j^{\prime \prime })={\partial }_{i+j-1}\left({\partial }_i^{\prime }(t_i^{\prime }){\otimes }_R{s}_j^{\prime \prime }+{(-1)}^i{t}_i^{\prime }{\otimes }_R{\partial }_j^{\prime \prime }(s_j^{\prime \prime })\right)=$$

$$={(-1)}^{i-1}{\partial }_i^{\prime }(t_i^{\prime }){\otimes }_R{\partial }_j^{\prime \prime }(s_j^{\prime \prime })+{(-1)}^i{\partial }_i^{\prime }(t_i^{\prime }){\otimes }_R{\partial }_j^{\prime \prime }(s_j^{\prime \prime })=0,$$

thus $C^{\prime }{\otimes }_R{C}^{\prime \prime }$ is a chain complex.