Complexity of tensor products of modules and a theorem of Huneke-Wiegand

This paper concerns the notion of complexity, a measure of the growth of the Betti numbers of a module. We show that over a complete intersection $R$ the complexity of the tensor product $M\otimes _{R} N$ of two finitely generated modules is the sum of the complexities of each if $\operatorname{Tor}_{i}^{R}(M,N)=0$ for $i\geq 1$. One of the applications is simplification of the proofs of central results in a paper of C. Huneke and R. Wiegand on the tensor product of modules and the rigidity of Tor.