Algebraic K-theory

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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 n = 0,1,2, ... . Here for traditional reasons the cases of K₀ and K₁ are thought of in somewhat different terms from the higher algebraic K-groups K_n for n ≥ 2. In fact K₀ generalises the construction of the ideal class group, using projective modules; and K₁ as applied to a commutative ring is the unit group construction, which was generalised to all rings for the needs of topology (simple homotopy theory) by means of elementary matrix theory. Therefore the first two cases counted as relatively accessible; while after that the theory becomes quite noticeably deeper, and certainly quite hard to compute (even when R is the ring of integers).

It was Grothendieck who invented K-theory as an appropriate framework to state his farreaching generalization of Riemann-Roch theorem. Later on topological K-theory was considered by Atiyah and Hirzebruch as a topological counterpart. A supporting motivation is the conjecture of Serre that now is the Quillen-Suslin theorem. 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 K₂ 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 a definition of Daniel Quillen. Quillen defined

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. A variant on this construction is given below.

1 Detailed discussion
2 Literature

[edit] Detailed discussion

Let A be a ring.

[edit] Lower dimensions

[edit] K₀

The (covariant) functor K₀ goes from the category of rings to the category of groups, taking A to the Grothendieck group of the isomorphism classes of its projective modules. For A a Dedekind ring,

$K_0(A)=\mathop{\mathrm{Pic}}A\times\mathbf Z$ ,

where Pic(A) is the Picard group of A.

[edit] K₁

Hyman Bass provided this definition: K₁(A) is the abelianisation of the infinite general linear group:

K₁(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 K₁(F) is the group of units of F. For A a commutative ring K₁(A) splits as the direct sum of the group of units of A and a group SK₁(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), SK₁(A) is zero.

[edit] K₂

John Milnor found the right definition of K₂: it is the center of the Steinberg group St(A) (of Robert Steinberg) of A, defined by generators and relations. Generators

x_ij(r),

for positive integer i ≠ j and ring elements r, are subject to relations

$x i j (r)x i j (r') = x i j (r + r')$
$[x i j (r),x j k (r')] = x i k (r r')$ for $i\not=k$
$[x i j (r),x k l (r')] = 1$ for $i\not=l,j\not=k$

These relations hold also for elementary matrices; whence a group homomorphism

$\varphi\colon\mathrm{St}(A)\to\mathrm{GL}(A).$

Now K₂(A) is defined as the kernel of $\varphi$ .

One can see that it is also the center of St(A). K₁ and K₂ are connected by the exact sequence

$1\longrightarrow K_2(A)\longrightarrow\mathrm{St}(A)\longrightarrow\mathrm{GL}(A)\longrightarrow K_1(A)\longrightarrow 1.$

For a field k one has

$K_2(k) = k^\times\otimes_{\mathbb Z} k^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle.$

[edit] Milnor K-theory

Milnor further defined, for a field k, "higher" K-groups by

$K^M_*(k) := T^*k^\times/(a\otimes (1-a))$ ,

thus as graded parts of a quotient of the tensor algebra of the multiplicative group k^× by the two-sided ideal, generated by the

$a\otimes(1-a)$

for a ≠ 0,1. For n = 0,1,2 these coincide with those above..

[edit] Quillen's K-theory

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

[edit] Classifying spaces of categories

For a small category C, its nerve NC is defined as the semi-simplicial set, with as p-simplices the diagrams

$X_0\longrightarrow X_1\longrightarrow\ldots\longrightarrow X_p$ .

The geometric realisation BC of NC is the classifying space of C.

[edit] Quillen's Q-construction

Suppose P is an exact category, that is an additive category together with a class E of (short) "exact sequences"

$M'\longrightarrow M\longrightarrow M'',$

satisfying certain axioms, taken from the properties of short exact sequences in abelian categories:

E is closed under isomorphisms.
The canonical short exact sequences

$M'\longrightarrow M'\oplus M''\longrightarrow M''$

are in E.

The class of admissible epimorphisms - that is, those morphisms that occur as the second arrow of a sequence in E - is closed under composition and pullbacks (or "base changes").
The class of admissible monomorphisms - that is, those morphisms that occur as the first arrow of a sequence in E - is closed under composition and pushouts.

Associated to an exact category P a new category QP is defined, objects of which are those of P and morphisms from M′ to M″ are isomorphism classes of exact diagrams

$M'\longleftarrow N\longrightarrow M''.$

where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism.

[edit] The Quillen K-groups

The i-th K-group of P is then defined as

K i (P) = π i + 1 (BQ P,0)

with a fixed zero-object 0.

$K 0 (P)$ coincides with the Grothendieck group of $P$ .

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. When A is a noetherian ring, then $K i (A)$ is also isomorphic to $K i (M A)$ where $M A$ is the category of all finitely generated A-modules.

[edit] Calculations for special rings

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.

[edit] 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 2 i (F) = 0

for $i\neq 0$ , and

$K_{2i-1}(F)= \mu_{q^i-1}$ for $i=1,2,\dots$

where $μ r$ denotes the cyclic group with r elements.

[edit] 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 = 4 k + 1

with k positive

and (modulo torsion)

K 4 k + 1 (Z) = Z

for positive k.

The torsion subgroups of K_2i+1(Z), and the orders of the finite groups K_4k+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K_4k(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers.

[edit] Literature

Daniel 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
C. Weibel: Algebraic K-theory of rings of integers in local and global fields (survey article) [1].

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Categories: Abstract algebra | K-theory

Algebraic K-theory

From Wikipedia, the free encyclopedia

Contents

[edit] Detailed discussion

[edit] Lower dimensions

[edit] K₀

[edit] K₁

[edit] K₂

[edit] Milnor K-theory

[edit] Quillen's K-theory

[edit] Classifying spaces of categories

[edit] Quillen's Q-construction

[edit] The Quillen K-groups

[edit] Calculations for special rings

[edit] Algebraic K-groups of finite fields

[edit] Algebraic K-groups of rings of integers

[edit] Literature

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Algebraic K-theory

From Wikipedia, the free encyclopedia

Contents

[edit] Detailed discussion

[edit] Lower dimensions

[edit] K0

[edit] K1

[edit] K2

[edit] Milnor K-theory

[edit] Quillen's K-theory

[edit] Classifying spaces of categories

[edit] Quillen's Q-construction

[edit] The Quillen K-groups

[edit] Calculations for special rings

[edit] Algebraic K-groups of finite fields

[edit] Algebraic K-groups of rings of integers

[edit] Literature

Views

Navigation

interaction

Search

[edit] K₀

[edit] K₁

[edit] K₂