finitely generated projective module
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In mathematics, particularly in
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, the class of projective modules enlarges the class of free modules (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 * Mo ...
s with
basis vector In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s) over a ring, 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 Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
fails to hold over some rings, such as
Dedekind ring In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
s that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, or a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
(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 Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
.


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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
for every surjective
module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an '' ...
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 mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
.) : The advantage of this definition of "projective" is that it can be carried out in
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
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 ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule o ...
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, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective 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, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of modules of the form :0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0 is a 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 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 ''B'', ''h'' is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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 of ''P'' and ''Q'' is a free module.


Exactness

An ''R''-module ''P'' is projective if and only if the covariant
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
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 ...
, where is the category of left ''R''-modules and Ab is the category of abelian groups. When the ring ''R'' is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, 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
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
s (surjective homomorphisms), or if it preserves finite
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
s.


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 in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
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 algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to i ...
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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
or PID" is also true for polynomial rings over 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 grass ...
: 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 Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not l ...
: ''any'' module is free in this case. * if the ring ''R'' is a principal ideal domain. For example, this applies to (the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s), so an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is projective if and only if it is a free abelian group. The reason is that any
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
of a free module over a principal ideal domain is free. * if the ring ''R'' is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
. 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 modules. In general though, projective modules need not be free: * Over a
direct product of rings In mathematics, a product of rings or direct product of rings is a ring (mathematics), ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a di ...
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 which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
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 M''n''(''R''), the natural module ''R'' ''n'' is projective but not free. More generally, over any
semisimple ring In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
, ''every'' module is projective, but the
zero ideal In mathematics, a zero element is one of several generalizations of 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 identi ...
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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''K''0(''R''); see below.


Projective vs. flat modules

Every projective module is flat. 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 of finitely-generated free modules. 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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
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 c ...
.


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. (See also
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
).


Projective resolutions

Given a module, ''M'', a projective resolution of ''M'' is an infinite
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
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 In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ...
of a
regular sequence In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions Fo ...
, 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 rings have nice properties. The localization of a projective module is a projective module over the localized ring. A projective module over a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
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, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts in ...
s over
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
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, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
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 that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts incl ...
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 In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
\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 In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, then, by Nakayama's lemma, these conditions are equivalent to *The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the k(\mathfrak)-
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
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. The word was first used scientifically in optics to describe the rainbow of colors ...
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, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s. This can be made precise for the ring of continuous
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-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 In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
, 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 that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
(see
Serre–Swan theorem In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout ...
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 and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fraction ...
, 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 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 ...
: if ''K'' is a field, or more generally a principal ideal domain, and is a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
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 A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
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 In mathematics, a stably free module is a module which is close to being free. Definition A finitely generated module ''M'' over a ring ''R'' is ''stably free'' if there exist free finitely generated modules ''F'' and ''G'' over ''R'' such that : ...
*
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, 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 P ...
(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 publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...
. *
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