Tor functor
In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology.
Specifically, suppose R is a ring, and denote by R-Mod the category of left R-modules and by Mod-R the category of right R-modules (if R is commutative, the two categories coincide). Pick a fixed module B in R-Mod. For A in Mod-R, set T(A) = A ⊗RB. Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors LnT are defined. We set
i.e., we take a projective resolution
then remove the A term and tensor the projective resolution with B to get the complex
(note that A ⊗RB does not appear and the last arrow is just the zero map) and take the homology of this complex.
...
Wikipedia