 Tor functor

The Tor functor is the homology of a tensor product of modules with a free resolution: ${\mathrm{Tor}}_i^R(M,N)=H_i(F_{\bullet }{\otimes }_{R}N)$ for a free resolution $F_{\bullet }$ of $M$. It measures the failure of tensoring to preserve exact sequences. For a commutative ring, one may resolve either argument; the double complex of the two resolutions proves that both computations agree.

In mathematics, particularly in the field of homological algebra and algebraic topology, the Tor functor is a significant construction related to the derived functors of the tensor product. The Tor functor, denoted as \(\text{Tor}_n^R(A, B)\), is used to study the properties of modules over a ring \(R\).