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| In [[mathematics]], particularly in [[abstract algebra]] and [[homological algebra]], the concept of '''projective module''' over a [[ring (mathematics)|ring]] R is a generalisation of the idea of a [[free module]] (that is, a [[module (mathematics)|module]] with [[basis vector]]s). Various equivalent characterizations of these modules appear below.
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| Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by [[Henri Cartan]] and [[Samuel Eilenberg]].
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| == Definitions ==
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| === Lifting property ===
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| The usual definition in line with [[category theory]] is the property of ''lifting'' that carries over from free to projective modules. We can summarize this lifting property as follows: a module ''P'' is projective if and only if for every surjective module homomorphism ''f'' : ''N'' ↠ ''M'' and every module homomorphism ''g'' : ''P'' → ''M'', there exists a homomorphism ''h'' : ''P'' → ''N'' such that ''fh'' = ''g''. (We don't require the lifting homomorphism ''h'' to be unique; this is not a [[universal property]].)
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| :[[Image:Projective module.png]]
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| 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.
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| === Split-exact sequences ===
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| A module ''P'' is projective if and only if for every surjective module homomorphism ''f'' : ''M'' ↠ ''P'' there exists a module homomorphism ''h'' : ''P'' → ''M'' such that ''fh'' = id<sub>''P''</sub>. The existence of such a '''section map''' ''h'' implies that ''P'' is a direct summand of ''M'' and that ''f'' is essentially a projection on the summand ''P''. More explicitly, ''M'' = im(''h'') ⊕ ker(''f''), and im(''h'') is isomorphic to ''P''.
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| The foregoing is a detailed description of the following statement: A module ''P'' is projective if every [[short exact sequence]] of modules of the form
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| :<math>0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0\,</math>
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| is a [[split exact sequence]].
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| === Direct summands of free modules ===
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| A module ''P'' is projective if and only if there is a free module ''F'' and another module ''Q'' such that the [[direct sum of modules|direct sum]] of ''P'' and ''Q'' is ''F''.
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| === Exactness ===
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| An ''R''-module ''P'' is projective if and only if the functor Hom(''P'',''-''): ''R''-Mod→ ''R''-Mod is an [[exact functor]]. This functor is always left exact, but when ''P'' is projective it is also right exact. Equivalently, we can demand that this functor preserves epimorphisms (surjective homomorphisms) or that it preserves finite colimits.
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| ===Dual basis===
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| 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>.
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| == Properties ==
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| * Direct sums and direct summands of projective modules are projective.
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| * If ''e'' = ''e''<sup>2</sup> is an [[idempotent element|idempotent]] in the ring ''R'', then ''Re'' is a projective left module over ''R''.
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| * 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]].
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| * The category of finitely generated projective modules over a ring is an [[exact category]]. (See also [[algebraic K-theory]]).
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| * Every module over a [[field (mathematics)|field]] or [[skew field]] is projective (even free). A ring over which every module is projective is called [[semisimple ring|semisimple]].
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| * An [[abelian group]] (i.e. a module over '''[[integer|Z]]''') is projective [[if and only if]] it is a [[free abelian group]]. The same is true for all [[principal ideal domain]]s; the reason is that for these rings, any submodule of a free module is free.
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| * Over a [[Dedekind domain]] a non-principal ideal is always a projective module that is not a free module.
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| * Over a [[direct product of rings]] ''R'' × ''S'' where ''R'' and ''S'' are nonzero rings, both ''R'' × 0 and 0 × ''S'' are non-free projective modules.
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| * 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.
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| * Every projective module is [[flat module|flat]].<ref>Hazewinkel, et. al. (2004), Corollary 5.4.5, {{Google books quote|id=AibpdVNkFDYC|page=131|text=Every projective module is flat|p. 131}}.</ref> The converse is in general not true: the abelian group '''Q''' is a '''Z'''-module which is flat, but not projective.<ref>Hazewinkel, et. al. (2004), Remark after Corollary 5.4.5, {{Google books quote|id=AibpdVNkFDYC|page=132|text=Q is flat but it is not projective|p. 131–132}}.</ref>
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| * In line with the above intuition of "locally free = projective" is the following theorem due to Kaplansky: over a [[local ring]], ''R'', every projective module is free. This is easy to prove for finitely generated projective modules, but the general case is difficult.
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| *A [[finitely related module]] is flat if and only if it is projective.<ref>{{harvnb|Cohn|2003|loc=Corollary 4.6.4}}</ref>
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| The relation of projective modules to free and flat modules is subsumed in the following diagram of module properties:
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| [[Image:Module properties in commutative algebra.svg|Module properties in commutative algebra]]
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| 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 [[Quillen-Suslin theorem]].
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| == Projective resolutions ==
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| {{Main|Projective resolution}}
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| Given a module, ''M'', a '''projective [[resolution (algebra)|resolution]]''' of ''M'' is an infinite [[exact sequence]] of modules
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| :· · · → ''P''<sub>''n''</sub> → · · · → ''P''<sub>2</sub> → ''P''<sub>1</sub> → ''P''<sub>0</sub> → ''M'' → 0,
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| 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 ''P''(''M'') → ''M'' → 0 or ''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.
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| The ''length'' of a finite resolution is the subscript ''n'' such that ''P''<sub>''n''</sub> is nonzero and ''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 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>
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| == Projective modules over commutative rings ==
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| Projective modules over [[commutative ring]]s have nice properties.
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| The [[localization of a ring|localization]] of a projective module is a projective module over the localized ring.
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| A projective module over a [[local ring]] is free. Thus a projective module is ''locally free''.
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| 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.
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| However, there are examples of finitely generated modules over a non-Noetherian ring which are locally free and not projective. For instance,
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| 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
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| there are some non-projective modules over Boolean rings. One example is ''R/I'' where
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| ''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''.
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| The ''R''-module ''R/I'' is locally free since ''R'' is Boolean (and it's finitely generated as an ''R''-module too, with a spanning set of size 1), but ''R/I'' is not projective because
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| ''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.)
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| 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>Eisenbud D.:''Commutative Algebra with a view towards Algebraic Geometry'', corollary 6.6, GTM 150, Springer-Verlag, 1995. Also, {{harvnb|Milne|1980}}</ref>
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| #''M'' is flat.
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| #<math>M</math> is projective.
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| #<math>M_\mathfrak{m}</math> is free as <math>R_\mathfrak{m}</math>-module for every maximal ideal <math>\mathfrak{m}</math> of ''R''.
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| #<math>M_\mathfrak{p}</math> is free as <math>R_\mathfrak{p}</math>-module for every prime ideal <math>\mathfrak{p}</math> of ''R''.
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| Moreover, if ''R'' is a noetherian integral domain, then, by [[Nakayama's lemma]], these conditions are equivalent to
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| *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''.<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>
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| That is to say, ''M'' has constant rank ("rank" is defined in the section below).
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| The fourth condition can be restated as:
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| *<math>\widetilde{M}</math> is a locally free sheaf on <math>\operatorname{Spec}R</math>.
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| 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>
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| === Rank ===
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| 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'' or its quotient by its [[nilradical of a ring|nilradical]] is an integral domain), then ''P'' has constant rank.
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| == Vector bundles and locally free modules ==
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| {{refimprove section|date=July 2008}}
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| 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).
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| Vector bundles are ''locally free''. If there is some notion of "localization" which can be carried over to modules, such as is given at [[localization of a ring]], one can define locally free modules, and the projective modules then typically coincide with the locally free ones.
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| == Projective modules over a polynomial ring ==
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| The [[Quillen–Suslin theorem]], which solves Serre's problem is another [[deep result]]; it states that if ''K'' is a [[field (mathematics)|field]], or more generally a [[principal ideal domain]], and ''R'' = ''K''[''X''<sub>1</sub>,...,''X''<sub>''n''</sub>] is a [[polynomial ring]] over ''K'', then every projective module over ''R'' is free.
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| 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.
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| 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 is free? The answer is ''no''. A counterexample occurs with ''R'' equal to the local ring of the curve ''y''<sup>2</sup> = ''x''<sup>3</sup> at the origin. So Serre's problem can not be proved by a simple induction on the number of variables.
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| == Notes ==
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| <references/>
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| == See also ==
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| *[[Projective cover]]
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| *[[Schanuel's lemma]]
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| *[[Bass cancellation theorem]]
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| ==References==
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| * {{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 }}
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| *[[Nicolas Bourbaki]], Commutative algebra, Ch. II, §5
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| * {{cite book | author=Paul M. Cohn |authorlink=Paul Cohn | title=Further algebra and applications |year=2003 |publisher=Springer |isbn=1-85233-667-6 }}
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| * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd | publisher=[[Addison–Wesley]] | year=1993 | isbn=0-201-55540-9 }}
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| * {{cite book |first1=James |last1=Milne |title=Étale cohomology |publisher=Princeton Univ. Press |year=1980 |isbn=0-691-08238-3}}
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| * {{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| title=Algebras, rings and modules | publisher=[[Springer Science]] | year=2004 | isbn=978-1-4020-2690-4 }}
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| * Donald S. Passman (2004) ''A Course in Ring Theory'', especially chapter 2 Projective modules, pp 13–22, AMS Chelsea, ISBN 0-8218-3680-3 .
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| * [[Paulo Ribenboim]] (1969) ''Rings and Modules'', §1.6 Projective modules, pp 19–24, [[Interscience Publishers]].
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| *[[Charles Weibel]], [http://www.math.rutgers.edu/~weibel/Kbook.html The K-book: An introduction to algebraic K-theory]
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| [[Category:Homological algebra]]
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| [[Category:Module theory]]
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