Resolution (algebra)

In the mathematical field of category theory and homological algebra is a projective resolution is a long exact sequence of projective objects, which ends in a given object.

Definition

Let be an abelian category (or the category Grp of groups) and an object. Then is called a long exact sequence of the form

Projective resolution if all are projective.

Existence

Is in the abelian category of each object quotient of a projective object, that is, there is an epimorphism for each object in which is projective, so we can also say possess a sufficient number of projective objects.

Under these conditions, there are also about each object is a projective resolution. First, by assumption, namely, there is an epimorphism, then an epimorphism on the core of this morphism and then by induction on each.

The most important category with sufficiently many projective objects is the category of ( left ) modules over a ring. If such a module and is a generating set, so you have a surjective homomorphism by the - th basic element of the free module on maps. Since free modules are projective, the quotient is a projective module and thus has a sufficient number of projective objects.

Properties

Is

A projective resolution and

Exactly, so can any homomorphism (not necessarily unique ) to a commutative diagram

Complement.