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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, particularly in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

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 differently f ...
of projective modules enlarges the class of
free module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

free module
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 * Modula ...
s with
basis vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s) over a
ring Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
, 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 ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s that are not
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s. However, every projective module is a free module if the ring is a principal ideal domain such as the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, or a
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(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 a ...
). 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmeti ...

Henri Cartan
and
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph ...
.


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
module homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
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 In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
.) : 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 In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module (mathematics), module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if '' ...

injective module
s. 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 object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...

projective object
s in the category of ''R''-modules.


Split-exact sequences

A module ''P'' is projective if and only if every
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics ...
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 module (mathema ...

split exact sequence
. 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 The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of ''B'', ''h'' is an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
from ''P'' to , and is a projection 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 from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
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 of ...
, where ''R''-Mod is the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
of left ''R''-modules and Ab is the category of
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 \ 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 Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...
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: The left-to-right implications are true over any ring, although some authors define
torsion-free module In abstract algebra, algebra, a torsion-free module is a module (mathematics), module over a Ring (mathematics), ring such that zero is the only element Absorbing element, annihilated by a zero-divisor, regular element (non zero-divisor) of the ring ...
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 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 a ...
.


Projective vs. free modules

Any free module is projective. The converse is true in the following cases: * if ''R'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
or
skew field In algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, t ...
: ''any'' module is free in this case. * if the ring ''R'' is a
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. For example, this applies to (the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s), so an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
is projective
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it is a
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...
. 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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
. 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 ; see
Kaplansky's theorem on projective modulesIn abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring#General case, local ring is free module, free; where a not-necessary-commutative ring is called ''loca ...
. 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 In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...
a non-principal ideal is always a projective module that is not a free module. * Over a
matrix ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
M''n''(''R''), the natural module ''R''''n'' is projective but not free. More generally, over any
semisimple ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, ''every'' module is projective, but the
zero ideal In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive iden ...
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), a ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of finitely-generated
free module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

free module
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 The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
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 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 category, abelian categories without requiring that morphisms actually posse ...
. (See also
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of spac ...
).


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 make ...
of ''M'' is an infinite
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
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 module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

free module
s) 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of a
regular sequence The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular Moses Regular Jr. (born October 30, 1971) is a former American football linebacker who played one season with the New York Giants of the National ...
, which is a free resolution of the
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
generated by the sequence. The ''length'' of a finite resolution is the subscript ''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 ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s over
Noetherian ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
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 module In mathematics, a finitely generated module is a module (mathematics), module that has a Finite set, finite generating set. A finitely generated module over a Ring (mathematics), ring ''R'' may also be called a finite ''R''-module, finite over ''R'' ...
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. #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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, 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, 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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models expla ...
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 bundle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

vector bundle
s. This can be made precise for the ring of continuous real-valued functions on a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

Hausdorff space
, as well as for the ring of smooth functions on a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

smooth manifold
(see
Serre–Swan theorem In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population i ...
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 In commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with ...
, 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 a ...
, which solves Serre's problem, is another
deep result The language of mathematics has a vast vocabulary A vocabulary, also known as a wordstock or word-stock, is a set of familiar words within a person's language. A vocabulary, usually developed with age, serves as a useful and fundamental tool ...
: if ''K'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, or more generally a
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, and is a
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 on the number of variables.


See also

* Projective cover * Schanuel's lemma * Bass cancellation theorem *
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 (mathematics), field ''K'' of positive characteristic (algebra), characteristic ''p ...


Notes


References

* * *
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym A pseudonym () (originally: ψευδώνυμος in Greek) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true na ...
, 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 John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, Academic j ...
. *
Charles Weibel Charles Alexander Weibel (born October 28, 1950 in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra. Weibel studied physics and mathematics at the University of Michigan, ...

The K-book: An introduction to algebraic K-theory
{{Authority control Homological algebra Module theory