Algebraic K-theory
Encyclopedia
In mathematics
, algebraic K-theory is an important part of homological algebra
concerned with defining and applying a sequence
of functor
s from rings
to abelian group
s, for all integers n.
For historical reasons, the lower K-groups K0 and K1 are thought of in somewhat different terms from the higher algebraic K-groups Kn for n ≥ 2.
Indeed, the lower groups are more accessible, and have more applications, than the higher groups.
The theory of the higher K-groups is noticeably deeper, and certainly much harder to compute
(even when R is the ring of integer
s).
The group K0(R) generalises the construction of the ideal class group
of a ring,
using projective module
s. Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of Serre
on projective modules that now is the Quillen-Suslin theorem; numerous
other connections with classical algebraic problems were found in this era.
Similarly, K1(R) is a modification of the group of units
in a ring,
using elementary matrix theory. The group K1(R) is important in topology
,
especially when R is a group ring
, because its quotient
the Whitehead group contains the Whitehead torsion
used to study problems in simple homotopy theory
and surgery theory
; the group K0(R) also contains other invariants such as the finiteness invariant. Since the 1980s, algebraic K-theory has increasingly
had applications to algebraic geometry
. For example, motivic cohomology
is closely related to algebraic K-theory.
discovered K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann-Roch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah
and Hirzebruch
and is now known as topological K-theory
.
Applications of K-groups were found from 1960 onwards in surgery theory
for manifold
s, in particular; and numerous other connections with classical algebraic problems were brought out.
A little later a branch of the theory for operator algebra
s was fruitfully developed, resulting in operator K-theory
and KK-theory
. It also became clear that K-theory could play a role in algebraic cycle
theory in algebraic geometry
(Gersten's conjecture): here the higher K-groups become connected with the higher codimension phenomena, which are exactly those that are harder to access. The problem was that the definitions were lacking (or, too many and not obviously consistent). Using work
of Robert Steinberg on universal central extensions of classical algebraic groups, John Milnor
defined the group K2(A)
of a ring A as the center, isomorphic to H2(E(A),Z), of the universal central extension of the group E(A) of infinite
elementary matrices over A. (Definitions below.) There is a natural bilinear pairing from K1(A) × K1(A) to
K2(A). In the special case of a field k, with K1(k) isomorphic to the multiplicative group GL(1,k), computations of
Hideya Matsumoto showed that K2(k) is isomorphic to the group generated by K1(A) × K1(A) modulo an easily described set of relations.
Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by , who gave several definitions of Kn(A) for arbitrary non-negative n, via the +-construction
and the Q-construction.
Throughout, let A be a ring
.
of the set of isomorphism classes of its finitely generated projective module
s, regarded as a monoid under direct sum. Any ring homomorphism A → B gives a map K0(A) → K0(B) by mapping (the class of) a projective A-module M to M⊗AB, making K0 a covariant functor.
If the ring A is commutative, we can define a subgroup
of K0(A) as the set 
, where 
is the map sending every (class of a) finitely generated projective A-module M to the rank of the free 
-module 
(this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup 
is known as the reduced zeroth K-theory of A.
Examples: (Projective) modules over a field
k are vector spaces and K0(k) is isomorphic to Z, by dimension. For A a Dedekind ring,
where Pic(A) is the Picard group of A, and similarly the reduced K-theory is given by
An algebro-geometric variant of this construction is applied to the category of algebraic varieties
; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves (or coherent sheaves) on X. Given a compact topological space X, the topological K-theory
Ktop(X) of (real) vector bundle
s over X coincides with K0 of the ring of continuous
real-valued functions on X.
provided this definition, which generalizes the group of units of a ring: K1(A) is the abelianization of the infinite general linear group:
Here
is the direct limit
of the GLn, which embeds in GLn+1 as the upper left block matrix
, and the commutator subgroup
agrees with the group generated by elementary matrices
, by Whitehead's lemma. Indeed, the group 
was first defined and studied by Whitehead, and is called the Whitehead group of the ring A.
, one can define a determinant
to the group of units of A, which vanishes on 
and thus descends to a map 
.
As
, one can also define the special Whitehead group 
.
This map splits via the map
(unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence:
which is a quotient of the usual split short exact sequence defining the special linear group
, namely
Thus, since the groups in question are abelian,
splits as the direct sum of the group of units and the special Whitehead group: 
.
When A is a Euclidean domain (e.g. a field, or the integers) SK1(A) vanishes, and the determinant map is an isomorphism. In particular,
.
This is false in general for PIDs, thus providing one of the rare mathematical features of Euclidean domains that do not generalize to all PIDs. An explicit PID A such that SK1(A) is nonzero was given by Grayson in 1981. If A is a Dedekind domain whose quotient field is a finite extension of the rationals then shows that SK1(A) vanishes.
For a non-commutative ring, the determinant cannot be defined, but the map
generalizes the determinant.
found the right definition of K2 for fields: it is the center of the Steinberg group
of A.
It can also be defined as the kernel
of the map
or as the Schur multiplier
of the group of elementary matrices.
Matsumoto's theorem says that for a field k, the second K-group is given by
Matsumoto's original theorem is even more general: For any root system, it gives a presentation for the unstable K-theory.
This presentation is different from the one given here only for symplectic root systems.
For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for
.
Unstable second K-groups (in this context) are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system. This construction yields the kernel of the Steinberg extension for the root systems
(
) and, in the limit, stable second K-groups.
,
thus as graded parts of a quotient of the tensor algebra
of the multiplicative group
k× by the two-sided ideal, generated by the
for a ≠ 0,1. For n = 0,1,2 these coincide with those below, but for n≧3 they differ in general. For example, we have
for n≧3. Milnor K-theory modulo 2 is related to étale
(or Galois
) cohomology of the field by the Milnor conjecture
, proven by Voevodsky.
The analogous statement for odd primes is the Bloch-Kato conjecture, proved by Voevodsky, Rost, and others.
Here
is a homotopy group
, GL(R) is the direct limit
of the general linear group
s over R for the size of the matrix tending to infinity, B is the classifying space
construction of homotopy theory, and the + is Quillen's plus construction
.
This definition only holds for n>0 so one often defines the higher algebraic K-theory via
Since BGL(R)+ is path connected and K0(R) discrete, this definition doesn't differ in higher degrees and also holds for n=0.
Suppose P is an exact category
; associated to P a new category QP is defined, objects of which are those of P and morphisms from M′ to M″ are isomorphism classes of diagrams
where the first arrow is an admissible epimorphism
and the second arrow is an admissible monomorphism
.
The i-th K-group of P is then defined as
with a fixed zero-object 0, where BQ is the classifying space of Q, which is defined to be the geometric realisation of the nerve
of Q.
This definition coincides with the above definitions of K0.
The K-groups
of the ring A are then the K-groups 
where 
is the category of finitely generated projective A-modules
. More generally, for a scheme
X, the higher K-groups of X are by definition the K-groups of (the exact category of) locally free coherent sheaves
on X.
The following variant of this is also used: instead of finitely generated projective (=locally free) modules, take finitely generated modules. The resulting K-groups are usually called G-groups, or higher G-theory. When A is a noetherian
regular ring
, then G- and K-theory coincide. Indeed, the global dimension
of regular local rings is finite, i.e. any finitely generated module has a finite projective resolution, so the canonical map K0 → G0 is surjective. It is also injective, as can be shown. This isomorphism extends to the higher K-groups, too.
. It applies to categories with cofibrations (also called Waldhausen categories
). This is a more general concept than exact categories.
s:
If Fq is the finite field with q elements, then
, 
for
, and
for i≥1.
in an algebraic number field F (a finite extension of the rationals), then the algebraic K-groups of A are finitely generated. Borel
used this to calculate Ki(A) and Ki(F) modulo torsion. For example, for the integers Z, Borel proved that (modulo torsion)
for positive i unless 
with k positive
and (modulo torsion)
for positive k.
The torsion subgroups of K2i+1(Z), and the orders of the finite groups K4k+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K4k(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. See Quillen-Lichtenbaum conjecture for more details.
-groups are used in conjectures on special values of L-functions
and the formulation of an non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.
Another fundamental conjecture due to Hyman Bass
(Bass conjecture
) says that all G-groups G(A) (that is to say, K-groups of the category of finitely generated A-modules) are finitely generated when A is a finitely generated Z-algebra.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, algebraic K-theory is an important part of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
concerned with defining and applying a sequence
- Kn(R)
of functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s from rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
to abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
s, for all integers n.
For historical reasons, the lower K-groups K0 and K1 are thought of in somewhat different terms from the higher algebraic K-groups Kn for n ≥ 2.
Indeed, the lower groups are more accessible, and have more applications, than the higher groups.
The theory of the higher K-groups is noticeably deeper, and certainly much harder to compute
(even when R is the ring of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s).
The group K0(R) generalises the construction of the ideal class group
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
of a ring,
using projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s. Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
on projective modules that now is the Quillen-Suslin theorem; numerous
other connections with classical algebraic problems were found in this era.
Similarly, K1(R) is a modification of the group of units
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
in a ring,
using elementary matrix theory. The group K1(R) is important in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
,
especially when R is a group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
, because its quotient
the Whitehead group contains the Whitehead torsion
Whitehead torsion
In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...
used to study problems in simple homotopy theory
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
and surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
; the group K0(R) also contains other invariants such as the finiteness invariant. Since the 1980s, algebraic K-theory has increasingly
had applications to algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. For example, motivic cohomology
Motivic cohomology
Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...
is closely related to algebraic K-theory.
History
Alexander GrothendieckAlexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
discovered K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann-Roch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
and Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
and is now known as topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
.
Applications of K-groups were found from 1960 onwards in surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
for manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, in particular; and numerous other connections with classical algebraic problems were brought out.
A little later a branch of the theory for operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s was fruitfully developed, resulting in operator K-theory
Operator K-theory
In mathematics, operator K-theory is a variant of K-theory on the category of Banach algebras ....
and KK-theory
KK-theory
In mathematics, KK-theory is a common generalization both of K-homology and K-theory , as an additive bivariant functor on separable C*-algebras...
. It also became clear that K-theory could play a role in algebraic cycle
Algebraic cycle
In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V. Therefore the algebraic cycles on V are the part of the algebraic topology of V that is directly accessible in algebraic geometry...
theory in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
(Gersten's conjecture): here the higher K-groups become connected with the higher codimension phenomena, which are exactly those that are harder to access. The problem was that the definitions were lacking (or, too many and not obviously consistent). Using work
of Robert Steinberg on universal central extensions of classical algebraic groups, John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
defined the group K2(A)
of a ring A as the center, isomorphic to H2(E(A),Z), of the universal central extension of the group E(A) of infinite
elementary matrices over A. (Definitions below.) There is a natural bilinear pairing from K1(A) × K1(A) to
K2(A). In the special case of a field k, with K1(k) isomorphic to the multiplicative group GL(1,k), computations of
Hideya Matsumoto showed that K2(k) is isomorphic to the group generated by K1(A) × K1(A) modulo an easily described set of relations.
Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by , who gave several definitions of Kn(A) for arbitrary non-negative n, via the +-construction
Plus construction
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. It was introduced by Daniel Quillen. Given a perfect normal subgroup of the fundamental group of a connected CW complex X, attach two-cells along...
and the Q-construction.
Lower K-groups
The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful.Throughout, let A be a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
.
K0
The functor K0 takes a ring A to the Grothendieck groupGrothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way...
of the set of isomorphism classes of its finitely generated projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s, regarded as a monoid under direct sum. Any ring homomorphism A → B gives a map K0(A) → K0(B) by mapping (the class of) a projective A-module M to M⊗AB, making K0 a covariant functor.
If the ring A is commutative, we can define a subgroup
of K0(A) as the set , where is the map sending every (class of a) finitely generated projective A-module M to the rank of the free -module (this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup is known as the reduced zeroth K-theory of A.Examples: (Projective) modules over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k are vector spaces and K0(k) is isomorphic to Z, by dimension. For A a Dedekind ring,
- K0(A) = Pic(A) ⊕ Z,
where Pic(A) is the Picard group of A, and similarly the reduced K-theory is given by
An algebro-geometric variant of this construction is applied to the category of algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves (or coherent sheaves) on X. Given a compact topological space X, the topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
Ktop(X) of (real) vector bundle
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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s over X coincides with K0 of the ring of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
real-valued functions on X.
K1
Hyman BassHyman Bass
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
provided this definition, which generalizes the group of units of a ring: K1(A) is the abelianization of the infinite general linear group:
Here
is the direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
of the GLn, which embeds in GLn+1 as the upper left block matrix
Block matrix
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
, and the commutator subgroup
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group....
agrees with the group generated by elementary matrices
, by Whitehead's lemma. Indeed, the group was first defined and studied by Whitehead, and is called the Whitehead group of the ring A.Commutative rings and fields
For A a commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, one can define a determinant
to the group of units of A, which vanishes on and thus descends to a map .As
, one can also define the special Whitehead group .This map splits via the map
(unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence:which is a quotient of the usual split short exact sequence defining the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
, namely
Thus, since the groups in question are abelian,
splits as the direct sum of the group of units and the special Whitehead group: .When A is a Euclidean domain (e.g. a field, or the integers) SK1(A) vanishes, and the determinant map is an isomorphism. In particular,
.This is false in general for PIDs, thus providing one of the rare mathematical features of Euclidean domains that do not generalize to all PIDs. An explicit PID A such that SK1(A) is nonzero was given by Grayson in 1981. If A is a Dedekind domain whose quotient field is a finite extension of the rationals then shows that SK1(A) vanishes.
For a non-commutative ring, the determinant cannot be defined, but the map
generalizes the determinant.K2
John MilnorJohn Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
found the right definition of K2 for fields: it is the center of the Steinberg group
of A.It can also be defined as the kernel
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
of the map
or as the Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
of the group of elementary matrices.
Matsumoto's theorem says that for a field k, the second K-group is given by
Matsumoto's original theorem is even more general: For any root system, it gives a presentation for the unstable K-theory.
This presentation is different from the one given here only for symplectic root systems.
For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for
.Unstable second K-groups (in this context) are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system. This construction yields the kernel of the Steinberg extension for the root systems
() and, in the limit, stable second K-groups.Milnor K-theory
The above expression for K2 of a field k led Milnor to the following definition of "higher" K-groups by,thus as graded parts of a quotient of the tensor algebra
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
of the multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
k× by the two-sided ideal, generated by the
for a ≠ 0,1. For n = 0,1,2 these coincide with those below, but for n≧3 they differ in general. For example, we have
for n≧3. Milnor K-theory modulo 2 is related to étaleÉtale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
(or Galois
Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
) cohomology of the field by the Milnor conjecture
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by .-Statement of the theorem:Let F be a field of...
, proven by Voevodsky.
The analogous statement for odd primes is the Bloch-Kato conjecture, proved by Voevodsky, Rost, and others.
Higher K-theory
The definitive definitions of higher K-groups were given by , after a few years during which several incompatible definitions were suggested.The +-construction
One possible definition of higher algebraic K-theory of rings was given by QuillenHere
is a homotopy groupHomotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
, GL(R) is the direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
of the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
s over R for the size of the matrix tending to infinity, B is the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
construction of homotopy theory, and the + is Quillen's plus construction
Plus construction
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. It was introduced by Daniel Quillen. Given a perfect normal subgroup of the fundamental group of a connected CW complex X, attach two-cells along...
.
This definition only holds for n>0 so one often defines the higher algebraic K-theory via
Since BGL(R)+ is path connected and K0(R) discrete, this definition doesn't differ in higher degrees and also holds for n=0.
The Q-construction
The Q-construction gives the same results as the +-construction, but it applies in more general situations. Moreover, the definition is more direct in the sense that the K-groups, defined via the Q-construction are functorial by definition. This fact is not automatic in the +-construction.Suppose P is an exact category
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 cokernels, which is necessary for the usual definition...
; associated to P a new category QP is defined, objects of which are those of P and morphisms from M′ to M″ are isomorphism classes of diagrams
where the first arrow is an admissible epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
and the second arrow is an admissible monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
.
The i-th K-group of P is then defined as
with a fixed zero-object 0, where BQ is the classifying space of Q, which is defined to be the geometric realisation of the nerve
Nerve (category theory)
In category theory, the nerve N of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C...
of Q.
This definition coincides with the above definitions of K0.
The K-groups
of the ring A are then the K-groups where is the category of finitely generated projective A-modulesProjective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
. More generally, for a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
X, the higher K-groups of X are by definition the K-groups of (the exact category of) locally free coherent sheaves
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
on X.
The following variant of this is also used: instead of finitely generated projective (=locally free) modules, take finitely generated modules. The resulting K-groups are usually called G-groups, or higher G-theory. When A is a noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
regular ring
Regular ring
In commutative algebra, a regular ring is a commutative noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.Jean-Pierre...
, then G- and K-theory coincide. Indeed, the global dimension
Global dimension
In ring theory and homological algebra, the global dimension of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules...
of regular local rings is finite, i.e. any finitely generated module has a finite projective resolution, so the canonical map K0 → G0 is surjective. It is also injective, as can be shown. This isomorphism extends to the higher K-groups, too.
The S-construction
A third construction of K-theory groups is the S-construction, due to WaldhausenFriedhelm Waldhausen
Friedhelm Waldhausen is a German mathematician known for his work in algebraic topology.-Academic life:...
. It applies to categories with cofibrations (also called Waldhausen categories
Waldhausen category
In mathematics a Waldhausen category is a category C with a zero object equipped with cofibrations co and weak equivalences we, both containing all isomorphisms, both compatible with pushout, and co containing the unique morphisms\scriptstyle 0\,\rightarrowtail\, Afrom the zero-object to any...
). This is a more general concept than exact categories.
Examples
While the Quillen algebraic K-theory has provided deep insight into various aspects of algebraic geometry and topology, the K-groups have proved particularly difficult to compute except in a few isolated but interesting cases.Algebraic K-groups of finite fields
The first and one of the most important calculations of the higher algebraic K-groups of a ring were made by Quillen himself for the case of finite fieldFinite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s:
If Fq is the finite field with q elements, then
, for
, and for i≥1.Algebraic K-groups of rings of integers
Quillen proved that if A is the ring of algebraic integersRing of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
in an algebraic number field F (a finite extension of the rationals), then the algebraic K-groups of A are finitely generated. Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
used this to calculate Ki(A) and Ki(F) modulo torsion. For example, for the integers Z, Borel proved that (modulo torsion)
for positive i unless with k positiveand (modulo torsion)
for positive k.The torsion subgroups of K2i+1(Z), and the orders of the finite groups K4k+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K4k(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. See Quillen-Lichtenbaum conjecture for more details.
Applications and open questions
Algebraic-groups are used in conjectures on special values of L-functionsSpecial values of L-functions
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely...
and the formulation of an non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.
Another fundamental conjecture due to Hyman Bass
Hyman Bass
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
(Bass conjecture
Bass conjecture
In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated...
) says that all G-groups G(A) (that is to say, K-groups of the category of finitely generated A-modules) are finitely generated when A is a finitely generated Z-algebra.