projective R-module

Definition

A module is projective if (and only if) there is another module such that the direct sum is free (i.e. it is a direct summand of a free module).

Examples

• Any free module is projective

NOTE

Any projective module will be flat. This gives the following implications:

• Let be rings. Then is also a ring. The -module with module structure that is projective but not (usually) free. This is because is a free -module with direct summand .
• If is a PID or a local ring, then finitely generated projective modules agree with finitely generated free modules.
• Non-commutative example: Let be the ring of matrices with entries in a field . Then is a left -module. View as column -vectors. We have as -modules where is the ith column vector. Note that so is a projective -module.

Alternate characterizations

Let be an -module. The following are equivalent:

1. is projective.
2. For all surjective -module homomorphims and homomorphims , there exists a lift which satisfies . In pictures:

3. The functor is exact.

Proof

:

Suppose given projective, then the inclusion is split, that is there exists some module such that

Given a surjective homomorphism and homomorphism , which in diagram form would look like

We can define a map from using the index set . For each element of the basis we can chose and element of that it is sent to, then by the universal property of free modules this completely determines the map .

Therefore, for every , let be an element such that . This can always be chosen since is surjective. Then we can set . This ensures that the diagram below commutes

Then, we can use the composition to get the desired morphism.

:

The functor sends a short exact sequence

to

Exactness at : If , then because is injective.

Exactness at : Say . Then the image of is contained in , that is for some .

Exactness at : Let . This means that we have the diagram

By 2, there exists a lift such that .

is pretty clear.

: Let be a set of generators of . This gives the diagram

thus, by 2 there is a lift from such that thus the inclusion is split and is a direct summand of .