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  • 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. Pick a fixed module B in R-Mod. For A in Mod-R, set T = A⊗RB. Then T is a right exact functor from Mod-R to the category of abelian groups Ab 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 and take the homology of this complex.