Editing Projective module (section)

===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]].