Editing Projective module (section)

== Projective modules over commutative rings ==
Projective modules over [[commutative ring]]s have nice properties.

The [[localization (commutative algebra)|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'' (in the sense that its localization at every [[prime ideal]] is free over the corresponding localization of the ring). The converse is true for [[finitely generated module]]s over [[Noetherian ring]]s: 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 that are locally free and not projective.  For instance, 
a [[Boolean ring]] has all of its localizations isomorphic to '''F'''<sub>2</sub>, 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 '''F'''<sub>2</sub> and ''I'' is the direct sum of countably many copies of '''F'''<sub>2</sub> inside of ''R''.
The ''R''-module ''R''/''I'' is locally free since ''R'' is Boolean (and it is 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 module]]s ''M'' over a commutative ring ''R'' (in particular if ''M'' is a finitely generated ''R''-module and ''R'' is Noetherian), the following are equivalent.<ref>Exercises 4.11 and 4.12 and Corollary 6.6 of David Eisenbud, ''Commutative Algebra with a view towards Algebraic Geometry'', GTM 150, Springer-Verlag, 1995. Also, {{harvnb|Milne|1980}}</ref>
#<math>M</math> is flat.
#<math>M</math> is projective.
#<math>M_\mathfrak{m}</math> is free as <math>R_\mathfrak{m}</math>-module for every [[maximal ideal]] <math>\mathfrak{m}</math> of ''R''.
#<math>M_\mathfrak{p}</math> is free as <math>R_\mathfrak{p}</math>-module for every prime ideal <math>\mathfrak{p}</math> of ''R''.
#There exist <math>f_1,\ldots,f_n \in R</math> generating the [[unit ideal]] such that <math>M[f_i^{-1}]</math> is free as <math>R[f_i^{-1}]</math>-module for each ''i''.  
#<math>\widetilde{M}</math> is a [[locally free sheaf]] on the [[affine scheme]] <math>\operatorname{Spec}R</math> (where <math>\widetilde{M}</math> is the [[sheaf associated to a module|sheaf associated to]] ''M''.)
Moreover, if ''R'' is a Noetherian [[integral domain]], then, by [[Nakayama's lemma]], these conditions are equivalent to
*The [[dimension (vector space)|dimension]] of the <math>k(\mathfrak{p})</math>-[[vector space]] <math>M \otimes_R k(\mathfrak{p})</math> is the same for all prime ideals <math>\mathfrak{p}</math> of ''R,'' where <math>k(\mathfrak{p})</math> is the residue field at <math>\mathfrak{p}</math>.<ref>That is, <math>k(\mathfrak{p})=R_\mathfrak{p}/\mathfrak{p}R_\mathfrak{p}</math> is the residue field of the local ring <math>R_\mathfrak{p}</math>.</ref>  That is to say, ''M'' has constant rank (as defined below).

Let ''A'' be a commutative ring. If ''B'' is a (possibly non-commutative) ''A''-[[algebra over a ring|algebra]] that is a finitely generated projective ''A''-module containing ''A'' as a [[subring]], then ''A'' is a direct factor of ''B''.<ref>{{harvnb|Bourbaki, Algèbre commutative|1989|loc=Ch II, §5, Exercise 4}}</ref>

=== Rank ===

Let ''P'' be a finitely generated projective module over a commutative ring ''R'' and ''X'' be the [[spectrum of a ring|spectrum]] of ''R''. The ''rank'' of ''P'' at a prime ideal <math>\mathfrak{p}</math> in ''X'' is the rank of the free <math>R_{\mathfrak{p}}</math>-module <math>P_{\mathfrak{p}}</math>.  It is a locally constant function on ''X''. In particular, if ''X'' is connected (that is if ''R'' has no other idempotents than 0 and 1), then ''P'' has constant rank.