Projective object

In the mathematical field of category theory projective objects are a generalization of the concept of freedom in algebra.

Definition

An object P in a category C is called projective if there exists for every epimorphism and every one, so that is. That is diagram on the left is commutative. So exactly is projective if for all epimorphisms, the induced map

Is surjective.

Examples

• Each initial object in a category is projective.
• In the category of sets Me every object is projective. This is a consequence of the axiom of choice.
• The coproduct of projective objects is projective.
• Projective groups are precisely the free groups.

Properties

Is in the category of each object quotient of a projective object, that is, for every object an epimorphism, where is projective, so we can also say possess enough projective objects. This property plays a role in the context of derived functors. For example, the category of groups has enough projective objects, because each group quotient of a free group is ( presentation by generators and relations ).

Projective module

In the category of modules, one can say more about what projective modules.

For a module, the following statements are equivalent.

• Is projective.
• There is for every epimorphism, so that applies. This means each Epimorphimus with target is a retraction.
• Each epimorphism decays. That is is a direct summand in.
• Is isomorphic to a direct summand of a free module.
• The functor is exact.

The direct sum of a family of modules is projective precisely when each is projective. In particular, every direct summand of a projective module is projective. The product of projective modules is not projective in general. For example, not projective.

Examples of projective modules

• Each ring is projective as a module. Every free module is therefore projective.
• Projective abelian groups are precisely the free abelian groups. Please note: free abelian groups are i.a. no free groups.
• More generally, over any principal ideal ring every projective module free.
• Broken ideals in a Dedekind ring is projective, but generally not free.
• A finitely generated module over a Noetherian ring is exactly projective module if the associated sheaf is locally free.

Dualbasislemma

A module will generated by. The module is projective precisely when there is a family of homomorphisms from the dual space with:

Consequences of the Dualbasislemma

• For each right module is a left module over the ring. This module is called to the dual module. The module is a right module again. One has the natural homomorphism. Is projective, so is injective.
• Is projective and finitely generated, so is an isomorphism. They say is reflexive.