Tor functor

Summary

The Tor functors $\operatorname{Tor}_n^R(M, N)$ are the left derived functors of the tensor product $M \otimes_R N$. They measure how far tensoring is from being exact. The vanishing of $\operatorname{Tor}_1^R(M, -)$ characterizes the flat modules, and Tor groups over $\Z$ appear in the universal coefficient and Künneth theorems of topology.