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{{Short description|Direct summand of a free module (mathematics)}}
In [[mathematics]], particularly in [[algebra]], the [[Class (set theory)|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]], keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
Every free module is a projective module, but the [[converse (logic)|converse]] fails to hold over some rings, such as [[Dedekind ring]]s that are not [[principal ideal ___domain]]s. However, every projective module is a free module if the ring is a principal ideal ___domain such as the [[integer]]s, or a (multivariate) [[polynomial ring]] over a [[field (mathematics)|field]] (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|category theoretical]] 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 module 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 [[category (mathematics)|categories]] more general than [[module categories]]: we don't need a notion of "free object". It can also be [[dual (category theory)|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 modules|category of ''R''-modules]].
=== Split-exact sequences ===
A module ''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 {{nowrap|''f'' : ''B'' ↠ ''P''}} there exists a '''section map''', that is, a module homomorphism {{nowrap|''h'' : ''P'' → ''B''}} such that ''fh'' = id<sub>''P''</sub>. In that case, {{nowrap|''h''(''P'')}} is a [[direct summand]] of ''B'', ''h'' is an [[isomorphism]] from ''P'' to {{nowrap|''h''(''P'')}}, and {{nowrap|''hf''}} is a [[projection (linear algebra)|projection]] on the summand {{nowrap|''h''(''P'')}}. Equivalently,
:<math>B = \operatorname{Im}(h) \oplus \operatorname{Ker}(f) \ \
\text{ where } \operatorname{Ker}(f) \cong A\ \text{ and }
\operatorname{Im}(h) \cong P.</math>
=== Direct summands of free modules ===
===
An ''R''-module ''P'' is projective if and only if the covariant [[functor]] {{nowrap|Hom(''P'', -): ''R''-'''Mod''' → '''Ab'''}} is an [[exact functor]], where {{nowrap|''R''-'''Mod'''}} is the category of left ''R''-modules and '''Ab''' is the [[category of abelian groups]]. When the ring ''R'' is [[commutative ring|commutative]], '''Ab''' is advantageously replaced by {{nowrap|''R''-'''Mod'''}} in the preceding characterization. This functor is always [[left exact functor|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 [[epimorphism]]s (surjective homomorphisms), or if it preserves finite [[colimit]]s.
===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 {{math|1=''e'' = ''e''<sup>2</sup>}} is an [[idempotent (ring theory)|idempotent]] in the ring {{math|''R''}}, then {{math|''Re''}} is a projective left module over ''R''.
Let <math>R = R_1 \times R_2</math> be the [[direct product]] of two rings <math>R_1</math> and <math>R_2,</math> which is a ring with operations defined componentwise. Let <math>e_1=(1,0)</math> and <math>e_2=(0,1).</math> Then <math>e_1</math> and <math>e_2</math> are idempotents, and belong to the [[centre of a ring|centre]] of <math>R.</math> The [[two-sided ideal]]s <math>Re_1</math> and <math>Re_2</math> are projective modules, since their direct sum (as {{mvar|R}}-modules) equals the free {{mvar|R}}-module {{mvar|R}}. However, if <math>R_1</math> and <math>R_2</math> are nontrivial, then they are not free as modules over <math>R</math>. For instance <math>\mathbb{Z}/2\mathbb{Z}</math> is projective but not free over <math>\mathbb{Z}/6\mathbb{Z}</math>.
==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]].
==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]].
[[Quotient module|Quotients]] of projective modules also need not be projective, for example '''Z'''/''n'' is a quotient of '''Z''', but not [[torsion-free module|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
:⋅⋅⋅ → ''P''<sub>''n''</sub> → ⋅⋅⋅ → ''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 modules) 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 index ''n'' such that ''P''<sub>''n''</sub> is [[zero module|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.
== 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.
== 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 function (topology)|continuous]] [[real number|real]]-valued functions on a [[compact space|compact]] [[Hausdorff space]], as well as for the ring of [[smooth function]]s 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" that 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, 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). [[Hyman Bass|Bass]] settled it for non-finitely generated modules,<ref>{{cite journal|title=Big projective modules are free|last=Bass|first=Hyman|journal=[[Illinois Journal of Mathematics]]|volume=7|number=1|year=1963|publisher=Duke University Press|doi=10.1215/ijm/1255637479|at=Corollary 4.5|doi-access=free}}</ref> and [[Dan Quillen|Quillen]] and [[Andrei Suslin|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 [[mathematical induction|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]]
*[[Modular representation theory]]<!-- in this theory, it is important to understand/study projective modules - right? - so it makes sense to have some mention of a projective module in this theory -->
== Notes ==
{{Reflist}}
==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 |isbn=978-1-4612-0923-2 |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
* {{cite journal|last1=Braunling|first1=Oliver |last2=Groechenig|first2=Michael|last3=Wolfson|first3=Jesse|title= Tate Objects in Exact Categories (With an appendix by Jan Stovicek and Jan Trlifaj)|journal= Moscow Mathematical Journal|volume=16|year=2016|issue=3|pages=433–504 |mr=3510209|arxiv=1402.4969v4|doi=10.17323/1609-4514-2016-16-3-433-504|s2cid=118374422 }}
* {{cite book | author=Paul M. Cohn |author-link=Paul Cohn | title=Further algebra and applications |year=2003 |publisher=Springer |isbn=1-85233-667-6}}
* {{cite book|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}}
* {{cite journal|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|author-link1=Michiel Hazewinkel |first2=Nadiya|last2=Gubareni |author-link2=Nadiya Gubareni|first3=Vladimir V.|last3=Kirichenko|author-link3=Vladimir V. Kirichenko |year=2004 |title=Algebras, rings and modules |publisher=[[Springer Science]] |isbn=978-1-4020-2690-4 }}
* {{cite journal|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|hdl-access=free}}
* {{cite book | last=Lang|first=Serge | author-link=Serge Lang | title=Algebra | edition=3rd | publisher=[[Addison–Wesley]] | year=1993 | isbn=0-201-55540-9 }}
*{{cite journal|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| doi-access=free}}
* {{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 13–22, AMS Chelsea, {{isbn|0-8218-3680-3}} .
* {{cite journal |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|s2cid=117528099 }}
* [[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]
== Further reading ==
* https://mathoverflow.net/questions/272018/faithfully-flat-descent-of-projectivity-for-non-commutative-rings
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[[Category:Module theory]]
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