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 algebrass, 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 in 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, Ext was defined by Henri Cartan and Eilenberg in 1956.

Definition

be a ring for the category of R-modules is commutative. This is a right exact functor to the category of abelian groups, and so it has left derived functors for an integer. By definition, this means: take any projective resolution, and form the chain complex

is the homology is the cokernel, which is isomorphic.

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.-modules to abelian groups).

is an algebra over a ring-module.

Properties

Here are some of the basic properties and computations of Tor groups.

• .

• is flat (for example, free; this is more general than a projective (or free) resolution.

• is the torsion subgroup.

There are converses to the previous statement:

• ).

• ).

• By the general properties of derived functors, every short exact sequence-modules induces a long exact sequence of the form. The analogous exact sequence also holds for Tor with respect to the second variable.

• , there is a natural isomorphism.-modules.)

• is not a zero divisor for any finitely generated abelian group.

• Generalizing the previous example, one can compute Tor groups that involve the quotient of a commutative ring by any regular sequence, using the Koszul complex. is the polynomial ring is the exterior algebra.

• . The reason: every abelian group has a free resolution of length 1, since every subgroup of a free abelian group is free abelian.

• is a principal ideal domain over a PID has a free resolution of length 1, since every submodule of a free module over a PID is free.

• , Tor preserves direct sums (possibly infinite) and filtered colimits in each variable.

• , It follows that Tor commutes with localization. That is, for a multiplicatively closed set

• has the structure of a graded-commutative. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree.

Important special cases

• Group homology where G is a group, M is a representation of G is the group ring of G.

• For an algebra A over a field k and an A-bimodule M, Hochschild homology

• Lie algebra homology is a Lie algebra over a commutative ring R, M is the universal enveloping algebra.

• For a commutative ring R with a homomorphism onto a field k is a graded-commutative Hopf algebra over k. (If R is a Noetherian local ring with residue field k is Ext(k,k is the free graded-commutative divided power algebra on a graded vector space π*(R). When k has characteristic zero, π*(R) can be identified with the André-Quillen homology D*(k/R,k).

See also

• Flat morphism

• Serre's intersection formula

• Derived tensor product

• Eilenberg–Moore spectral sequence

Notes

References

External links