Tor functor

The Tor- functor is a mathematical term from the branch of homological algebra. It is a bi- functor, which occurs in the study of tensor product. He is next to the Ext- functor one of the most important constructions of homological algebra.

Motivation means of tensor products

We consider categories of modules over a ring. is

A short exact sequence of left - modules and module morphisms and is a right - module, the tensoring the above sequence from left leads to an exact sequence

Of abelian groups, which generally can not be with the zero- extended to the left object to an exact sequence, ie is not injective in general, or in short: the Tensorfunktor is quite accurate but generally not left exact.

As an example, consider the short exact sequence

Of -modules, where and are the natural mapping is the residue class group. Tensoriert to this sequence, it is not injective, since it is

Where the factor of 2 of the torsion group has been moved to the torsion group by Tensoroperation and has led there to a 0. This is the typical reason why the injectivity of the morphism is lost in the transition to tensorierten sequence. The lack of injectivity results in the appearance of a nucleus and gives rise to the following definition.

Definition

There were a right - and a left module - module. Next was

A short exact sequence with projective module. Then we define the abelian group

And one can show that this definition does not depend on the selected exact sequence with projective. This justifies the notation without reference to this sequence. Sometimes it adds to the ring and writes.

Is a morphism, then takes you to the commutative diagram

That the restriction defined by the core of after mapping and so a group homomorphism. In this way we obtain a functor from the category of right - modules to the category of abelian groups.

Next you can see the roles are reversed from and that is to go from the exact sequence of right - modules and shows that you get with the above definition naturally isomorphic to a group, which therefore also can be denoted by or. Overall, one obtains a bi- functor

Of the product of the category of right - modules over the category of left modules over. within the category of abelian groups

The Tor- functor is additive, that is, one has natural isomorphisms

For right - and left modules - modules.

Abelian groups

If you choose the base ring, as one moves into the category of abelian groups, because these are precisely the moduli, and one does not distinguish between left - and right -modules because of the commutativity of the base ring. In this category, certain simplifications arise and you find a relationship between the Tor functor and the eponymous for him torsion groups.

Alternative description of Tor (A, B)

Presented by generators and relations in the case of abelian groups, and may be as follows.

The set of generators is the set of all symbols, and, in which case the module operation only for practical reasons, it was left and right written once again, a distinction is, as mentioned above, not necessary. The set of relations containing all expressions of the form

Then one can show that is isomorphic to by presented group. To construct a mapping is a short exact sequence with projective module and a generator. Choose with. Then, and because of the accuracy there is exactly one. It can be shown that it is not dependent on the choice. because

Lies at the core of, and thus by definition in. Therefore, the presented construction defines a mapping, from which one can show that there is a Gruppenisomrphismus.

Characterization torsion-free groups

For an abelian group the following are equivalent:

• Is torsion-free, ie contains no elements of finite order except 0.
• For all abelian groups.
• For all injective group homomorphisms is also injective.
• Every exact sequence of abelian groups goes through tensoring with again in an exact sequence.

In particular, if one of the groups is the same.

Finitely generated abelian groups

Can be fully calculated for finitely generated abelian groups. After the main theorem on finitely generated abelian groups such groups are direct sums of cyclic groups, so that to determine because of the additivity of the Tor functor only for cyclic groups. If one of the groups are the same, then and there remains only the case of finite cyclic groups. Be the cyclic group of order. Then follows

And from this, if one the greatest common divisor of and may refer to:

But what you also can not derive directly from the definition of the resolution. This is intended for finitely generated abelian groups.

Tor as a derivative of the tensor functor

A more general definition is obtained by

As a -th leftmost derivation of Tensorfunktors. If the base ring is given by the context, so you let him continue in the designation and just write. This gives a sequence of bi- functors

If one uses projective resolutions to calculate, one sees that coincides with the above-defined functor.

Obtained from the general theory following long exact sequences that show how the Tor functor compensates for the lack of accuracy of the links Tensorfunktors.

Is a short exact sequence of right - modules and a left - modulus, one has a long exact sequence

Is a short exact sequence of left - and right modules - module, then one has a long exact sequence