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In mathematics, particularly in [[algebra]], the class of '''projective modules''' enlarges the class of [[free module]]s (that is, [[module (mathematics)|module]]s with [[basis vector]]s) over a [[ring (mathematics)|ring]], by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

A free module is a projective module, but the converse may not hold over some rings, such as [[Dedekind ring]]s. However, every projective module is a free module if the ring is a [[principal ideal domain]] such as the integers, or a [[polynomial ring]] (this is the [[Quillen–Suslin theorem]]).

Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by [[Henri Cartan]] and [[Samuel Eilenberg]].

== Definitions ==

=== Lifting property ===

The usual [[category theory]] definition is in terms of the property of [[lifting property|''lifting'']] that carries over from free to projective modules: a module ''P'' is projective if and only if for every surjective [[module homomorphism]] {{nowrap|''f'' : ''N'' ↠ ''M''}} and every module homomorphism {{nowrap|''g'' : ''P'' → ''M''}}, there exists a homomorphism {{nowrap|''h'' : ''P'' → ''N''}} such that {{nowrap|1=''fh'' = ''g''}}. (We don't require the lifting homomorphism ''h'' to be unique; this is not a [[universal property]].)

:[[Image:Projective-module-P.svg|120px]]

The advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to [[injective module]]s. The lifting property may also be rephrased as ''every morphism from <math>P</math> to <math>M</math> factors through every epimorphism to <math>M</math>''. Thus, by definition, projective modules are precisely the [[projective object]]s in the category of ''R''-modules.

=== Split-exact sequences ===
A module {{math|''P''}} is projective if and only if every [[short exact sequence]] of modules of the form

:<math>0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0</math>

is a [[split exact sequence]]. That is, for every surjective module homomorphism {{math|''f'' : ''B'' ↠ ''P''}} there exists a '''section map''', that is, a module homomorphism {{math|''h'' : ''P'' → ''B''}} such that {{math|1=''f'' ∘ ''h'' = id<sub>''P''</sub>}}. In that case, {{math|''h''(''P'')}} is a [[direct summand]] of {{math|''B''}}, {{mvar|h}} is an [[isomorphism]] from {{mvar|P}} to {{math|''h''(''P'')}}, and {{math|''h'' ∘ ''f''}} is a [[projection (linear algebra)|projection]] on the summand {{math|''h''(''P'')}}.  Equivalently,

:<math>B = \mathrm{Im}(h) \oplus \mathrm{Ker}(f) \ \ 
\text{where }\ \mathrm{Ker}(f) \cong A\ \text{ and }\ 
\mathrm{Im}(h) \cong P.</math>

=== Direct summands of free modules ===

A module ''P'' is projective if and only if there is another module ''Q'' such that the [[direct sum of modules|direct sum]] of ''P'' and ''Q'' is a free module.

=== Exactness ===

An ''R''-module ''P'' is projective if and only if the covariant functor {{nowrap|Hom(''P'',''-''): ''R''-Mod → AB}} is an [[exact functor]], where ''R''-Mod is the [[category (mathematics)|category]] of left ''R''-modules and AB the category of [[abelian group]]s. When the ring ''R'' is commutative, AB is advantageously replaced by ''R''-Mod in the preceding characterization. This functor is always left exact, but, when ''P'' is projective, it is also right exact. This means that ''P'' is projective if and only if this functor preserves epimorphisms (surjective homomorphisms), or if it preserves finite colimits.

===Dual basis===
A module ''P'' is projective if and only if there exists a set <math>\{a_i\in P \mid i\in I\}</math> and a set <math>\{f_i\in \mathrm{Hom}(P,R) \mid i\in I\}</math> such that for every ''x'' in ''P'', ''f''<sub>i</sub>(''x'') is only nonzero for finitely many ''i'', and <math>x=\sum f_i(x)a_i</math>.

== Elementary examples and properties ==
The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules:
* Direct sums and direct summands of projective modules are projective.
* If {{nowrap|1=''e'' = ''e''<sup>2</sup>}} is an [[idempotent element (ring theory)|idempotent]] in the ring ''R'', then ''Re'' is a projective left module over ''R''.

==Relation to other module-theoretic properties==

The relation of projective modules to free and 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. 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 polynomial rings over a 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 (mathematics)|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 prove for finitely generated projective modules. In general, it is due to {{harvtxt|Kaplansky|1958}}.

In general though, projective modules need not be free:
* Over a [[direct product of rings]] {{nowrap|''R'' × ''S''}} where ''R'' and ''S'' are nonzero rings, both {{nowrap|''R'' × 0}} and {{nowrap|0 × ''S''}} are non-free projective modules.
* Over a [[Dedekind domain]] a non-principal 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 not free.  More generally, over any [[semisimple ring]], ''every'' module is projective, but the [[zero ideal]] and the ring itself are the only free ideals.
The difference between free and projective modules is, in a sense, measured by the algebraic ''K''-theory 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 |booktitle=Algebras, Rings and Modules, Part 1|year=2004|title=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 which is flat, but not projective.<ref>{{cite book|author=Hazewinkel |display-authors=etal |year=2004|title=Remark after Corollary 5.4.5|booktitle=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 countably generated modules,
*''M'' satisfies a certain Mittag-Leffler type condition.

==The category of projective modules==
Submodules of projective modules need not be projective; a ring ''R'' for which every submodule of a projective left module is projective is called [[hereditary ring|left hereditary]].

Quotients of projective modules also need not be projective, for example '''Z'''/''n'' is a quotient of '''Z''', but not torsion free, hence not flat, and therefore not projective.

The category of finitely generated projective modules over a ring is an [[exact category]]. (See also [[algebraic K-theory]]).

== Projective resolutions ==
{{Main|Projective resolution}}
Given a module, ''M'', a '''projective [[resolution (algebra)|resolution]]''' of ''M'' is an infinite [[exact sequence]] of modules
:&middot;&middot;&middot; → ''P''<sub>''n''</sub> → &middot;&middot;&middot; → ''P''<sub>2</sub> → ''P''<sub>1</sub> → ''P''<sub>0</sub> → ''M'' → 0,
with all the ''P''<sub>''i''</sub>s projective. Every module possesses a projective resolution. In fact a '''free resolution''' (resolution by [[free module]]s) exists. The exact sequence of projective modules may sometimes be abbreviated to {{nowrap|''P''(''M'') → ''M'' → 0}} or {{nowrap|''P''<sub>•</sub> → ''M'' → 0}}. A classic example of a projective resolution is given by the [[Koszul complex]] of a [[regular sequence]], which is a free resolution of the [[ideal (ring theory)|ideal]] generated by the sequence.

The ''length'' of a finite resolution is the subscript ''n'' such that ''P''<sub>''n''</sub> is nonzero and {{nowrap|1=''P''<sub>''i''</sub> = 0}} for ''i'' greater than ''n''. If ''M'' admits a finite projective resolution, the minimal length among all finite projective resolutions of ''M'' is called its '''projective dimension''' and denoted pd(''M''). If ''M'' does not admit a finite projective resolution, then by convention the projective dimension is said to be infinite. As an example, consider a module ''M'' such that {{nowrap|1=pd(''M'') = 0}}. In this situation, the exactness of the sequence 0 → ''P''<sub>0</sub> → ''M'' → 0 indicates that the arrow in the center is an isomorphism, and hence ''M'' itself is projective.<ref>A module isomorphic to a projective module is of course projective.</ref>

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

The [[localization of a ring|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 which 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 <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 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}) = R/{\mathfrak{p}}</math>'' is the residue field.<ref>Here, <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 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.

== Vector bundles and locally free modules ==
{{more citations needed section|date=July 2008}}

A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of [[vector bundle]]s. This can be made precise for the ring of continuous real-valued functions on a [[compact space|compact]] [[Hausdorff space]], as well as for the ring of smooth functions on a [[manifold|smooth manifold]] (see [[Serre–Swan theorem]] that says a finitely generated projective module over the space of smooth functions on a compact manifold is the space of smooth sections of a smooth vector bundle).

Vector bundles are ''locally free''. If there is some notion of "localization" which can be carried over to modules, such as the usual [[localization of a ring]], one can define locally free modules, and the projective modules then typically coincide with the locally free modules.

== Projective modules over a polynomial ring ==
The [[Quillen–Suslin theorem]], which solves Serre's problem, is another [[deep result]]: if ''K'' is a [[field (mathematics)|field]], or more generally a [[principal ideal domain]], and {{nowrap|1=''R'' = ''K''[''X''<sub>1</sub>,...,''X''<sub>''n''</sub>]}} is a [[polynomial ring]] over ''K'', then every projective module over ''R'' is free.
This problem was first raised by Serre with ''K'' a field (and the modules being finitely generated).  Bass settled it for non-finitely generated modules and Quillen and Suslin independently and simultaneously treated the case of finitely generated modules.

Since every projective module over a principal ideal domain is free, one might ask this question: if ''R'' is a commutative ring such that every (finitely generated) projective ''R''-module is free, then is every (finitely generated) projective ''R''[''X'']-module free?  The answer is ''no''.  A counterexample occurs with ''R'' equal to the local ring of the curve {{nowrap|1=''y''<sup>2</sup> = ''x''<sup>3</sup>}} at the origin.  Thus the Quillen-Suslin theorem could never be proved by a simple induction on the number of variables.

== See also ==
{{Wikibooks|Commutative Algebra|Torsion-free, flat, projective and free modules}}

*[[Projective cover]]
*[[Schanuel's lemma]]
*[[Bass cancellation theorem]]

== Notes ==
<references/>

==References==
* {{cite book | author1=William A. Adkins |author2=Steven H. Weintraub |title=Algebra: An Approach via Module Theory | url=https://archive.org/details/springer_10.1007-978-1-4612-0923-2 |publisher=Springer |year=1992 |at=Sec 3.5}}
* {{cite book | author= Iain T. Adamson | title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 }}
* [[Nicolas Bourbaki]], Commutative algebra, Ch. II, §5
* {{citation|last1=Braunling|first1=Oliver |last2=Groechenig|first2=Michael|last3=Wolfson|first3=Jesse|title=Tate objects in exact categories|journal=Mosc. Math. J.|volume=16|year=2016|issue=3|mr=3510209|arxiv=1402.4969v4|doi=10.17323/1609-4514-2016-16-3-433-504}}
* {{cite book | author=Paul M. Cohn |authorlink=Paul Cohn | title=Further algebra and applications |year=2003 |publisher=Springer |isbn=1-85233-667-6}}
* {{citation|last=Drinfeld|first=Vladimir |editor=Pavel Etingof |editor2=Vladimir Retakh |editor3=I. M. Singer |chapter=Infinite-dimensional vector bundles in algebraic geometry: an introduction |title=The Unity of Mathematics |pages=263–304 |publisher=Birkhäuser Boston |year=2006 |mr=2181808 |doi=10.1007/0-8176-4467-9_7 |arxiv=math/0309155v4 |isbn=978-0-8176-4076-7}}
* {{citation|last=Govorov|first=V. E.|title=On flat modules (Russian)|journal=Siberian Math. J.|volume=6|year=1965|pages=300–304}}
* {{cite book |first1=Michiel |last1=Hazewinkel|authorlink1=Michiel Hazewinkel |first2=Nadiya|last2=Gubareni |authorlink2=Nadiya Gubareni|first3=Vladimir V.|last3=Kirichenko|authorlink3=Vladimir V. Kirichenko |year=2004 |title=Algebras, rings and modules |publisher=[[Springer Science]] |isbn=978-1-4020-2690-4 }}
* {{citation|last=Kaplansky|first=Irving|title=Projective modules|journal=Ann. of Math. |series= 2|volume=68|issue=2|year=1958|pages=372–377|mr=0100017|doi=10.2307/1970252|jstor=1970252|hdl=10338.dmlcz/101124}}
* {{cite book | last=Lang|first=Serge | authorlink=Serge Lang | title=Algebra | edition=3rd | publisher=[[Addison–Wesley]] | year=1993 | isbn=0-201-55540-9 }}
*{{citation|first=D.|last=Lazard |title=Autour de la platitude| journal=Bulletin de la Société Mathématique de France| year=1969| volume=97| pages=81–128| doi=10.24033/bsmf.1675}}
* {{cite book |first1=James |last1=Milne |year=1980 |title=Étale cohomology |publisher=Princeton Univ. Press |isbn=0-691-08238-3 |url-access=registration |url=https://archive.org/details/etalecohomology00miln }}
* Donald S. Passman (2004) ''A Course in Ring Theory'', especially chapter 2 Projective modules, pp 13&ndash;22, AMS Chelsea, {{isbn|0-8218-3680-3}} .
* {{citation |last1=Raynaud|first1=Michel |last2=Gruson|first2=Laurent |year=1971 |title=Critères de platitude et de projectivité. Techniques de "platification" d'un module |journal=Invent. Math. |volume=13 |pages=1–89 |mr=0308104|doi=10.1007/BF01390094|bibcode=1971InMat..13....1R}}
* [[Paulo Ribenboim]] (1969) ''Rings and Modules'', §1.6 Projective modules, pp 19–24, [[Interscience Publishers]].
* [[Charles Weibel]], [http://www.math.rutgers.edu/~weibel/Kbook.html The K-book: An introduction to algebraic K-theory]

[[Category:Homological algebra]]
[[Category:Module theory]]