Ext functor

Ext is a Bifunktor, which plays a central role in the homological algebra.

Definition

Be an abelian category, for example, the category of modules of a ring, which is the standard example of the embedding theorem of Mitchell. For two objects and is from the class of short exact sequences of the form

Now on an equivalence relation is defined. Two exact sequences and equivalent, if there is a morphism, so that the diagram

Commutes. This is the identical morphism.

From the Fünferlemma immediately follows that if there is such a morphism, this must be an isomorphism. The class modulo this equivalence relation is a set and is denoted by. In this set you can define a group structure.

Functoriality

Morphisms in the abelian category induce the following manner morphisms between the Ext- groups, so that becomes a double-digit functor.

For the sequence and can form the push -out:

Because of the universal property of the push -out there is an induced epimorphism from Y ' to Z, so that the following diagram commutated:

The bottom row is also exact and its residue class thus an element in.

If one forms the residue class of the residue class of ab, we obtain a well-defined group homomorphism.

Dual works with morphisms from Z ' to Z: to and the sequence can use the following pull-back form:

Because of the universal property of the pull -back, there is an induced Monomorphismus from X to Y ', so that the following diagram commutated:

The upper row is also exact and thus defines an element in.

If one forms the residue class of the residue class of ab, we obtain again a well-defined group homomorphism.

Ext as a derivation of Hom - functor

Another possibility for the definition uses the derived functors of Hom. The above- defined structure can be identified with the first derivative of the right Hom - functor.

More specifically it is an abelian category with enough projective objects ( ie each object is projective quotient of an object ) defines the contravariant functor and

That is, one is the -th rightmost derivation of and applies the so- formed functor on to.

More concretely, this means the following: Let and

A projective resolution of an epimorphism and a monomorphism. Next is the induced homomorphism

Then

The elements are thus of certain residue classes of elements.

Finally, it should be noted that one of the rollers, and may also exchange, obtained

Relationship between Ext and Ext1

In this section, we will explain how the above-defined constructs and related. We construct a mapping.

Be a short exact sequence defining an item. Next is a short exact sequence with projective. Using the projectivity of you can get a commutative diagram

. construct Then is a homomorphism, defines its residue class according to the above representation of an item.

If one forms the residue class of in the residue class of in from, one obtains a well-defined figure, from which one can show that there is a group isomorphism.

Therefore, one can identify with, that is, can be defined in this sense as the first derivative of the right - functor.

Long exact sequence

The Hom - functor is left exact, ie for a short exact sequence

And another object (module ) has an exact sequence

And this can not continue exactly 0 in general. For accuracy of the links does the 0-th derivative of the Hom functor with Hom match, that is, if one expands the above definition, so we have. Therefore, the long exact sequence of derived functors addititve provides the following exact sequence

Analogously, one obtains a long exact sequence

In this sense include the Ext- functors the gap left by the lack of exactness of the Hom - functor.