Editing Projective module (section)

==Relation to other module-theoretic properties==

The relation of projective modules to free and [[flat module|flat]] modules is subsumed in the following diagram of module properties:

[[Image:Module properties in commutative algebra.svg|Module properties in commutative algebra]]

The left-to-right implications are true over any ring, although some authors define [[torsion-free module]]s only over a [[domain (ring theory)|domain]]. The right-to-left implications are true over the rings labeling them. There may be other rings over which they are true. For example, the implication labeled "[[local ring]] or PID" is also true for (multivariate) polynomial rings over a [[field (mathematics)|field]]: this is the [[Quillen–Suslin theorem]].

===Projective vs. free modules===
Any free module is projective. The converse is true in the following cases:
* if ''R'' is a field or [[skew field]]: ''any'' module is free in this case.
* if the ring ''R'' is a [[principal ideal domain]]. For example, this applies to {{nowrap|1=''R'' = '''Z'''}} (the [[integer]]s), so an [[abelian group]] is projective if and only if it is a [[free abelian group]]. The reason is that any [[submodule]] of a free module over a principal ideal domain is free.
* if the ring ''R'' is a [[local ring]]. This fact is the basis of the intuition of "locally free = projective". This fact is easy to [[mathematical proof|prove]] for [[finitely generated module|finitely generated]] projective modules. In general, it is due to {{harvtxt|Kaplansky|1958}}; see [[Kaplansky's theorem on projective modules]].

In general though, projective modules need not be free:
* Over a [[direct product of rings]] {{nowrap|''R'' × ''S''}} where ''R'' and ''S'' are [[zero ring|nonzero]] rings, both {{nowrap|''R'' × 0}} and {{nowrap|0 × ''S''}} are non-free projective modules.
* Over a [[Dedekind domain]] a non-[[principal ideal|principal]] [[ideal (ring theory)|ideal]] is always a projective module that is not a free module.
* Over a [[matrix ring]] M<sub>''n''</sub>(''R''), the natural module ''R''<sup>''n''</sup> is projective but is not free when ''n'' > 1.
* Over a [[semisimple ring]], ''every'' module is projective, but a nonzero proper left (or right) ideal is not a free module. Thus the only semisimple rings for which all projectives are free are [[division ring]]s.
The difference between free and projective modules is, in a sense, measured by the [[algebraic K-theory|algebraic ''K''-theory]] [[group (mathematics)|group]] ''K''<sub>0</sub>(''R''); see below.

===Projective vs. flat modules===
Every projective module is [[flat module|flat]].<ref>{{cite book|author=Hazewinkel |display-authors=etal |title=Algebras, Rings and Modules, Part 1|year=2004|contribution=Corollary 5.4.5|url={{Google books|plainurl=y|id=AibpdVNkFDYC|page=131|text=Every projective module is flat}}|page=131}}</ref> The converse is in general not true: the abelian group '''Q''' is a '''Z'''-module that is flat, but not projective.<ref>{{cite book|author=Hazewinkel |display-authors=etal |year=2004|contribution=Remark after Corollary 5.4.5|title=Algebras, Rings and Modules, Part 1|url={{Google books|plainurl=y|id=AibpdVNkFDYC|page=132|text=Q is flat but it is not projective}}|pages=131–132}}</ref>

Conversely, a [[finitely related module|finitely related]] flat module is projective.<ref>{{harvnb|Cohn|2003|loc=Corollary 4.6.4}}</ref>

{{harvtxt|Govorov|1965}} and {{harvtxt|Lazard|1969}} proved that a module ''M'' is flat if and only if it is a [[direct limit]] of [[finitely generated module|finitely-generated]] [[free module]]s.

In general, the precise relation between flatness and projectivity was established by {{harvtxt|Raynaud|Gruson|1971}} (see also {{harvtxt|Drinfeld|2006}} and {{harvtxt|Braunling|Groechenig|Wolfson|2016}}) who showed that a module ''M'' is projective if and only if it satisfies the following conditions:
*''M'' is flat,
*''M'' is a direct sum of [[countable set|countably]] generated modules,
*''M'' satisfies a certain [[Gösta Mittag-Leffler|Mittag-Leffler]]-type condition.
This characterization can be used to show that if <math>R \to S</math> is a [[Faithfully flat morphism|faithfully flat]] map of commutative rings and <math>M</math> is an <math>R</math>-module, then <math>M</math> is projective if and only if <math>M \otimes_R S</math> is projective.<ref>{{Cite web |title=Section 10.95 (05A4): Descending properties of modules—The Stacks project |url=https://stacks.math.columbia.edu/tag/05A4 |access-date=2022-11-03 |website=stacks.math.columbia.edu |language=en}}</ref> In other words, the property of being projective satisfies [[faithfully flat descent]].