Tor functor

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.
In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book "Homological Algebra".
Definition.
Let "R" be a ring. Write "R"-Mod for the category of left "R"-modules and Mod-"R" for the category of right "R"-modules. (If "R" is commutative, the two categories can be identified.) For a fixed left "R"-module "B", let formula_1 for "A" in Mod-"R". This is a right exact functor from Mod-"R" to the category of abelian groups Ab, and so it has left derived functors formula_2. The Tor groups are the abelian groups defined by
for an integer "i". By definition, this means: take any projective resolution
remove the term "A", and form the chain complex:
For each integer "i", the group formula_6 is the homology of this complex at position "i". It is zero for "i" negative. Moreover, formula_7 is the cokernel of the map formula_8, which is isomorphic to formula_9.
Alternatively, one can define Tor by fixing "A" and taking the left derived functors of the right exact functor "G"("B") = "A" ⊗"R" "B". That is, tensor "A" with a projective resolution of "B" and take homology. Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups. Moreover, for a fixed ring "R", Tor is a functor in each variable (from "R"-modules to abelian groups).
For a commutative ring "R" and "R"-modules "A" and "B", Tor("A", "B") is an "R"-module (using that "A" ⊗"R" "B" is an "R"-module in this case). For a non-commutative ring "R", Tor("A", "B") is only an abelian group, in general. If "R" is an algebra over a ring "S" (which means in particular that "S" is commutative), then Tor("A", "B") is at least an "S"-module.
Properties.
Here are some of the basic properties and computations of Tor groups.