Projective resolution
In the mathematical field of category theory and homological algebra , a projective resolution is a long exact sequence of projective objects that ends in a given object.
definition
Let there be an Abelian category (or the category Grp of the groups) and an object . Then a long exact sequence is called the form
projective resolution of when all are projective.
If all are even free , one speaks of a free dissolution .
existence
In the Abelian category, every object is quotient of a projective object, i. H. there are for each object an epimorphism , which is projective, it is said also possess sufficiently many projective objects .
Under these conditions there is also a projective resolution for every object . First of all there is an epimorphism , then an epimorphism on the core of this morphism and then further by induction .
The most important category with a sufficient number of projective objects is the category of (left) modules over a ring . If such a module is and is a generating system , then one has a surjective homomorphism by mapping the -th basic element of the free module to . Since free modules are projective, is the quotient of a projective module and thus has enough projective objects.
properties
Is
a projective resolution and
exactly, every homomorphism (not necessarily unique) can be converted into a commutative diagram
complete.
See also
- The dual term is that of injective dissolution .
- Projective resolutions are used in the computation of derived functors .
- Fundamental lemma of homological algebra
- Schanuel's Lemma
Individual evidence
- ^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , Chapter VII, Projective Resolutions
- ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , definition 2.5
- ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , sentence 2.7
- ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , Lemma 2.8 + following comment