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Concept
Tor functor
Summary
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.
Let R be a ring. Write R-Mod for the 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 for A in Mod-R. This is a right exact functor from Mod-R to the Ab, and so it has left derived functors . The Tor groups are the abelian groups defined by
for an integer i. By definition, this means: take any projective resolution
and remove A, and form the chain complex:
For each integer i, the group is the homology of this complex at position i. It is zero for i negative. Moreover, is the cokernel of the map , which is isomorphic to .
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_R(A, B) is an R-module (using that A ⊗R B is an R-module in this case).
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