ALGEBRAIC K-THEORY

In mathematics, 'algebraic K-theory' is an advanced part of homological algebra concerned with defining and applying a sequence
:''K_n''(''R'')
of functors from rings to abelian groups, 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' ''K_n'' 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 integers).
The group ''K₀(R)'' generalises the construction of the ideal class group of a ring,
using projective modules. Its development in the 1960's and 1970's 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 since as the finiteness invariant. Since the 1980's, algebraic K-theory has increasingly
had applications to algebraic geometry.
Alexander Grothendieck invented K-theory in the mid-1950's as a framework to state his
far-reaching generalization of the Riemann-Roch theorem. Within a few years, its topological counterpart
was considered by Atiyah and Hirzebruch and is now known as topological K-theory.
Applications of ''K''-groups were found from 1960 onwards in surgery theory for manifolds, in particular; and numerous other connections with classical algebraic problems were found.
A little later a branch of the theory for operator algebras was fruitfully developed. 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). A definition of ''K''₂ for fields by John Milnor, for example, gave an attractive theory that was too limited in scope, constructed as a quotient of the multiplicative group of the field tensored with itself, with some explicit relations imposed; and closely connected with central extensions.
Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by Daniel Quillen, who gave several definitions of higher algebraic ''K''-theory, via the +-construction and the ''Q''-construction.

Contents

Lower K-groups

''K''0

''K''1

''K''2

Milnor ''K''-theory

Higher ''K''-theory

The +-construction

The Q-construction

Examples

Algebraic K-groups of finite fields

Algebraic K-groups of rings of integers

References

External links

Lower K-groups

Let ''A'' be a ring.

''K''₀

The (covariant) functor ''K''₀ goes from the category of rings to the category of groups, taking ''A'' to the Grothendieck group of the isomorphism classes of its finitely generated projective modules.
(Projective) modules over a field ''k'' are vector spaces and ''K''₀(k) is isomorphic to ℤ. For ''A'' a Dedekind ring,
:$K\_0(A)=mathop\{mathrm\{Pic\}\}A\; imesmathbf\; Z$,
where Pic(''A'') is the Picard group of ''A''.

''K''₁

Hyman Bass provided this definition: ''K''₁(''A'') is the abelianisation
of the infinite general linear group:
:''K''₁(''A'') = GL(''A'')^ab = GL(''A'') / [GL(''A''),GL(''A'')]
Here
:GL(''A'') = colim GL_''n''(''A''),
the direct limit of the GL_''n'', which embeds in GL_n+1 as the upper left block matrix. See also the Whitehead torsion article.
For ''F'' a field this comes down to saying ''K''₁(''F'') is the group of units of ''F''. For ''A'' a commutative ring ''K''₁(''A'') splits as the direct sum of the group of units of ''A'' and a group ''SK''₁(''A''), called the 'special Whitehead group' of ''A''. When ''A'' is a Dedekind domain (e.g. the ring of algebraic integers in an algebraic number field), ''SK''₁(''A'') is zero.

''K''₂

John Milnor found the right definition of ''K''₂: it is the center of the Steinberg group St(''A'') (of Robert Steinberg) of ''A'', the universal central extension of the commutator subgroup of the general linear group. It can also described by generators and relations: generators
:''x''_''ij''(''r''),
for positive integer ''i'' ≠ ''j'' and ring elements ''r'', are subject to relations
# $mathrm\; x\_\{ij\}(r)mathrm\; x\_\{ij\}(r\text{'})\; =\; mathrm\; x\_\{ij\}(r+r\text{'})$
# $[mathrm\; x\_\{ij\}(r),mathrm\; x\_\{jk\}(r\text{'})]\; =\; mathrm\; x\_\{ik\}(rr\text{'})$ for $i$
ot=k
# $[mathrm\; x\_\{ij\}(r),mathrm\; x\_\{kl\}(r\text{'})]\; =\; 1$ for $i$
ot=l,j
ot=k
These relations hold also for elementary matrices; whence a group homomorphism
:
Now ''K''₂(''A'') is defined as the kernel of .
One can see that it is also the center of St(''A''). ''K''₁ and ''K''₂ are connected by the exact sequence
:$1longrightarrow\; K\_2(A)longrightarrowmathrm\{St\}(A)longrightarrowmathrm\{GL\}(A)longrightarrow\; K\_1(A)longrightarrow\; 1.$
For a field ''k'' one has
:$K\_2(k)\; =\; k^\; imesotimes\_\{mathbb\; Z\}\; k^\; imes/langle\; aotimes(1-a)mid\; a$
ot=0,1
angle.

Milnor ''K''-theory

The above expression for ''K₂'' of a field ''k'' led Milnor to the following definition of "higher" ''K''-groups by
:$K^M\_$
★ (k) := T^
★ k^ imes/(aotimes (1-a)) ,
thus as graded parts of a quotient of the tensor algebra of the multiplicative group ''k''^× by the two-sided ideal, generated by the
:$aotimes(1-a)$
for ''a'' ≠ 0,1. For ''n'' = 0,1,2 these coincide with those above, but for ''n''≧3 they differ in general.

Higher ''K''-theory

The master, definitive definitions of ''K''-theory were given by Daniel Quillen, after an extended period in which uncertainty had reigned.

The +-construction

One possible definition of higher algebraic ''K''-theory of rings was given by Quillen
:''K''_''n''(''R'') = π_''n''(''BGL''(''R'')⁺),
a very compressed piece of abstract mathematics. Here π_''k'' is a homotopy group, ''GL''(''R'') is the direct limit of the general linear groups 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.

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; associated to ''P'' a new category Q''P'' is defined, objects of which are those of ''P'' and morphisms from ''M''′ to ''M''″ are isomorphism classes of exact diagrams
:$M\text{'}longleftarrow\; Nlongrightarrow\; M\text{'}\text{'}.$
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
:$K\_i(P)=pi\_\{i+1\}(mathrm\{BQ\}P,0)$
with a fixed zero-object 0, where B''Q'' 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 ''K''₀, ''K''₁ and ''K''₂.
The ''K''-groups $K\_i(A)$ of the ring ''A'' are then the ''K''-groups $K\_i(P\_A)$ where $P\_A$ 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 ''K''₀ → ''G''₀ is surjective. It is also injective, as the relations in both groups are the same. This isomorphism promotes to the higher groups, too.

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 fields:
'Theorem'. Let ''F'' be a finite field with ''q'' elements. Then
:$K\_0(F)\; =\; Z$, $K\_\{2i\}(F)=0$
for $i$
eq 0, and
:$K\_\{2i-1\}(F)=\; mu\_\{q^i-1\}$ for $i=1,2,dots$
where $mu\_r$ denotes the cyclic group with ''r'' elements.

Algebraic K-groups of rings of integers

Quillen proved that if ''A'' is the ring of algebraic integers 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 K_''i''(''A'') and K_''i''(''F'') modulo torsion. For example, for the integers 'Z', Borel proved that (modulo torsion)
:$K\_i(Z)=0$ for positive ''i'' unless $i=4k+1$ with ''k'' positive
and (modulo torsion)
:$K\_\{4k+1\}(Z)=\; Z$ for positive ''k''.
The torsion subgroups of K_2''i''+1('Z'), and the orders of the finite groups K_4''k''+2('Z') have recently been determined, but whether the latter groups are cyclic, and whether the groups K_4''k''('Z') vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers.

References

★ J. Milnor: ''Algebraic K-theory and Quadratic Forms'', Inventiones math. 9 (1970), 318 - 344.

★ J. Milnor: ''Introduction to algebraic K-theory''. Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J., 1971 (lower K-groups)

★ D. Quillen: ''Higher algebraic K-theory: I''. In: H. Bass (ed.): ''Higher K-Theories''. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973. ISBN 3-540-06434-6 (Quillen's Q-construction)

★ D. Quillen: ''Higher K-theory for categories with exact sequences''. New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pp. 95–103. London Math. Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974. (relation of Q-construction to +-construction)

★

★ C. Weibel: ''Algebraic K-theory of rings of integers in local and global fields'' (survey article) PDF.

External links

★ C. Weibel "The K-book: An introduction to algebraic K-theory"