Let be an Abelian category , for example the category of the modules of a ring , which is the standard example according to Mitchell's embedding theorem . To two objects and from is the class of the short exact sequences of the form
commutes. It is the identical morphism.
From the five lemma it follows immediately that if there is such a morphism , it must be an isomorphism . The class modulo of this equivalence relation is a set and is denoted by. A group structure can be defined on this set.
Morphisms in the Abelian category induce morphisms between the Ext groups in the following way, so that becomes a two-digit functor.
You can push-out to and from the sequence :
Because of the universal property of the push-out there is an induced epimorphism from Y 'to Z, so that the following diagram commutes:
The bottom line is also exact and its equivalence class is therefore an element in .
If the equivalence class of is mapped to the equivalence class of , a well-defined group homomorphism is obtained .
This also works dual with morphisms from Z 'to Z. The following pull-back can be made to the sequence :
Because of the universal property of the pull-back, there is an induced monomorphism from X to Y ', so that the following diagram commutes:
The top line is also exact and thus defines an element in .
If the equivalence class of is mapped onto the equivalence class of , one again obtains a well-defined group homomorphism .
Ext as the derivative of the Hom functor
More precisely, one considers an Abelian category with a sufficient number of projective objects (i.e. each object is the quotient of a projective object) and defines the contravariant functor
that is, one forms the -th legal derivative of and applies the functor thus created .
More specifically, this means the following: Let it be and
The elements from are therefore certain equivalence classes of elements from .
Finally, it should be noted that one can swap the roles of and also one receives
Relationship between Ext and Ext 1
This section aims to explain how the constructs and defined above are related. We construct a picture .
Be a short exact sequence that defines an element from . Next is a short exact sequence with projective . Using the projectivity of , one can create a commutative diagram
to construct. Then there is a homomorphism whose equivalence class defines an element as described above .
If one forms the equivalence class of in the equivalence class of in starting, we obtain a well-defined figure , it can be shown from that it is a group isomorphism is.
Therefore one can identify with , that is , in this sense it can be defined as the first legal derivative of the -function.
Long exact sequence
The Hom functor is left exact, that is, for a short exact sequence
and another object (module) has an exact sequence
and in general this cannot be continued exactly with 0. Because of left-hand precision, the 0th derivative of the Hom functor agrees with Hom, that is, if one extends the above definition of to , one has . The long exact sequence for derived additive functors therefore gives the following exact sequence
A long exact sequence is obtained analogously
In this sense, the Ext functors close the gap created by the lack of accuracy of the Hom functor.
- Sergei I. Gelfand & Yuri Ivanovich Manin: Homological Algebra , Springer, Berlin, 1999, ISBN 978-3-540-65378-3
- Charles A. Weibel: An introduction to homological algebra , Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1999, ISBN 978-0-521-55987-4
- Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , sentence 3.13
- Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , Theorem 4.5
- Saunders Mac Lane : Homology , Springer Grundlehren der Mathematischen Wissenschaften Volume 114 (1967), chap. III, Theorem 3.4 and Theorem 9.1