Projective Module - Projective Modules Over Commutative Rings

Projective Modules Over Commutative Rings

Projective modules over commutative rings have nice properties.

The localization of a projective module is a projective module over the localized ring. A projective module over a local ring is free. Thus a projective module is locally free.

The converse is true for finitely generated modules over Noetherian rings: a finitely generated module over a commutative noetherian ring is locally free if and only if it is projective.

However, there are examples of finitely generated modules over a non-Noetherian ring which are locally free and not projective. For instance, a Boolean ring has all of its localizations isomorphic to F₂, the field of two elements, so any module over a Boolean ring is locally free, but there are some non-projective modules over Boolean rings. One example is R/I where R is a direct product of countably many copies of F₂ and I is the direct sum of countably many copies of F₂ inside of R. The R-module R/I is locally free since R is Boolean (and it's finitely generated as an R-module too, with a spanning set of size 1), but R/I is not projective because I is not a principal ideal. (If a quotient module R/I, for any commutative ring R and ideal I, is a projective R-module then I is principal.)

However, it is true that for finitely presented modules M over a commutative ring R (in particular if M is a finitely generated R-algebra and R is noetherian), the following are equivalent.

1. M is flat.
2. is projective.
3. is free as -module for every maximal ideal of R.
4. is free as -module for every prime ideal of R.

Moreover, if R is a noetherian integral domain, then, by Nakayama's lemma, these conditions are equivalent to

• The dimension of the –vector space is the same for all prime ideals of R.

That is to say, M has constant rank ("rank" is defined in the section below).

The fourth condition can be restated as:

• is a locally free sheaf on .

Let A be a commutative ring. If B is a (possibly non-commutative) A-algebra that is a finitely generated projective A-module containing A as a subring, then A is a direct factor of B.