Exercise 3, § 4

Assume that the ring A is commutative. Let l be a finite set and $(C^{(i)}, d^{(i)})_{i \in I}$ a family of complexes of A-modules. (a) The tensor product $\otimes C^{(i)}$ is provided with its grading of type $\mathbf{Z}^l$ and with the A-endomorphisms

Now prove that one thus obtains an I-precomplex (X, exercise 13, p. 174); the associated I-complex is called the l-complex tensor product of the complexes C^{(i)}. The complex associated with this I-complex is called the tensor product complex of the C^{(i)}. (b) Now prove that the choice of a total order on I allows one to define an isomorphism of the tensor product complex of the C^{(i)} onto the complex defined in X, p. 64.