finitely generated projective module
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in algebra, the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
of projective modules enlarges the class of
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s (that is,
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
s with basis vectors) over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, by 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 fails to hold over some rings, such as Dedekind rings that are not
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
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 The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is ...
). Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by
Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...
and Samuel Eilenberg.


Definitions


Lifting property

The usual category theoretical definition is in terms of the property of ''lifting'' that carries over from free to projective modules: a module ''P'' is projective if and only if for every
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
module homomorphism and every module homomorphism , there exists a module homomorphism such that . (We don't require the lifting homomorphism ''h'' to be unique; this is not a universal property.) : The advantage of this definition of "projective" is that it can be carried out in categories more general than
module categories In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property may also be rephrased as ''every morphism from P to M factors through every epimorphism to M''. Thus, by definition, projective modules are precisely the projective objects in the 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 :0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0 is a
split exact sequence In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. Equivalent characterizations A short exact sequence of abelian groups or of modules over a ...
. That is, for every surjective module homomorphism there exists a section map, that is, a module homomorphism such that ''f'' ''h'' = id''P'' . In that case, is a direct summand of ''B'', ''h'' is an isomorphism from ''P'' to , and is a
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
on the summand . Equivalently, :B = \operatorname(h) \oplus \operatorname(f) \ \ \text \operatorname(f) \cong A\ \text \operatorname(h) \cong P.


Direct summands of free modules

A module ''P'' is projective if and only if there is another module ''Q'' such that the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of ''P'' and ''Q'' is a free module.


Exactness

An ''R''-module ''P'' is projective if and only if the covariant functor is an
exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
, where is the category of left ''R''-modules and Ab is the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...
. When the ring ''R'' is commutative, Ab is advantageously replaced by 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 \ and a set \ such that for every ''x'' in ''P'', ''f''''i''  (''x'') is only nonzero for finitely many ''i'', and x=\sum f_i(x)a_i.


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 is an 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 Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...
modules is subsumed in the following diagram of module properties: The left-to-right implications are true over any ring, although some authors define torsion-free modules only over a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
. 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 The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is ...
.


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 In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
. For example, this applies to (the integers), so an abelian group is projective if and only if it is a
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
. 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 Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...
for finitely generated projective modules. In general, it is due to ; see
Kaplansky's theorem on projective modules In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called ''local'' if for each element ''x'', eith ...
. In general though, projective modules need not be free: * Over a direct product of rings where ''R'' and ''S'' are nonzero rings, both and are non-free projective modules. * Over a Dedekind domain a non- principal
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
is always a projective module that is not a free module. * Over a
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
M''n''(''R''), the natural module ''R'' ''n'' 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''0(''R''); see below.


Projective vs. flat modules

Every projective module is
flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...
. The converse is in general not true: the abelian group Q is a Z-module which is flat, but not projective. Conversely, a finitely related flat module is projective. and proved that a module ''M'' is flat if and only if it is a
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of finitely-generated
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s. In general, the precise relation between flatness and projectivity was established by (see also and ) 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. This characterization can be used to show that if R \to S is a faithfully flat map of commutative rings and M is an R-module, then M is projective if and only if M \otimes_R S is projective. In other words, the property of being projective satisfies
faithfully flat descent Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open co ...
.


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 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 In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and co ...
. (See also algebraic K-theory).


Projective resolutions

Given a module, ''M'', a projective
resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual mak ...
of ''M'' is an infinite exact sequence of modules :··· → ''P''''n'' → ··· → ''P''2 → ''P''1 → ''P''0 → ''M'' → 0, with all the ''P''''i'' 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 or . 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 Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
generated by the sequence. The ''length'' of a finite resolution is the index ''n'' such that ''P''''n'' is nonzero and 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 . In this situation, the exactness of the sequence 0 → ''P''0 → ''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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s have nice properties. The
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
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 In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
is free over the corresponding localization of the ring). The converse is true for finitely generated modules over Noetherian rings: 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 F2, 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 F2 and ''I'' is the direct sum of countably many copies of F2 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 modules ''M'' over a commutative ring ''R'' (in particular if ''M'' is a finitely generated ''R''-module and ''R'' is Noetherian), the following are equivalent. #M is flat. #M is projective. #M_\mathfrak is free as R_\mathfrak-module for every maximal ideal \mathfrak of ''R''. #M_\mathfrak is free as R_\mathfrak-module for every prime ideal \mathfrak of ''R''. #There exist f_1,\ldots,f_n \in R generating the
unit ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
such that M _i^/math> is free as R _i^/math>-module for each ''i''. #\widetilde is a locally free sheaf on \operatornameR (where \widetilde is the sheaf associated to ''M''.) Moreover, if ''R'' is a Noetherian integral domain, then, by
Nakayama's lemma In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and ...
, these conditions are equivalent to *The dimension of the k(\mathfrak)- vector space M \otimes_R k(\mathfrak) is the same for all prime ideals \mathfrak of ''R,'' where k(\mathfrak) is the residue field at \mathfrak. 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 In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
, then ''A'' is a direct factor of ''B''.


Rank

Let ''P'' be a finitely generated projective module over a commutative ring ''R'' and ''X'' be the spectrum of ''R''. The ''rank'' of ''P'' at a prime ideal \mathfrak in ''X'' is the rank of the free R_-module P_. 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

A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of vector bundles. This can be made precise for the ring of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
real-valued functions on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Hausdorff space, as well as for the ring of smooth functions on a 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 The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is ...
, which solves Serre's problem, is another
deep result The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in l ...
: if ''K'' is a field, or more generally a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
, and 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 at the origin. Thus the Quillen-Suslin theorem could never be proved by a simple
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
on the number of variables.


See also

*
Projective cover In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes. Definition L ...
*
Schanuel's lemma In mathematics, especially in the area of abstract algebra, algebra known as module theory, Schanuel's lemma, named after Stephen Schanuel, allows one to compare how far modules depart from being projective module, projective. It is useful in defin ...
*
Bass cancellation theorem In mathematics, a stably free module is a module which is close to being free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be form ...
*
Modular representation theory Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as ...


Notes


References

* * *
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
, Commutative algebra, Ch. II, §5 * * * * * * * * * * Donald S. Passman (2004) ''A Course in Ring Theory'', especially chapter 2 Projective modules, pp 13–22, AMS Chelsea, . * *
Paulo Ribenboim Paulo Ribenboim (born March 13, 1928) is a Brazilian-Canadian mathematician who specializes in number theory. Biography Ribenboim was born into a Jewish family in Recife, Brazil. He received his BSc in mathematics from the University of São Pa ...
(1969) ''Rings and Modules'', §1.6 Projective modules, pp 19–24, Interscience Publishers. * Charles Weibel
The K-book: An introduction to algebraic K-theory
{{Authority control Homological algebra Module theory