Algebraic K-theory

From Wikipedia, the free encyclopedia

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 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.

1 History
2 Lower K-groups
3 Milnor K-theory
4 Higher K-theory
- 4.1 The +-construction
- 4.2 The Q-construction
5 Examples
- 5.1 Algebraic K-groups of finite fields
- 5.2 Algebraic K-groups of rings of integers
6 References
7 External links

[edit] History

Alexander Grothendieck invented 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 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 brought out.

A little later a branch of the theory for operator algebras 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). 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 Daniel Quillen, who gave several definitions of higher algebraic K-theory, via the +-construction and the Q-construction.

[edit] 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.

[edit] K₀

The 0th K-group is related to dimension and the Picard group.

The (covariant) functor K₀ goes from the category of rings to the category of groups, taking A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum.

(Projective) modules over a field k are vector spaces and K₀(k) is isomorphic to $\mathbf{Z}$ , by dimension. For A a Dedekind ring,

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

where Pic(A) is the Picard group of A, and similarly the reduced K-theory is given by

$\tilde K_0(A)=\operatorname{Pic} A.$

[edit] K₁

Hyman Bass provided this definition, which generalizes the group of units of a field: K₁(A) is the abelianization of the infinite general linear group:

$K_1(A) = \operatorname{GL}(A)^{\mbox{ab}} = \operatorname{GL}(A) / [\operatorname{GL}(A),\operatorname{GL}(A)]$

Here

$\operatorname{GL}(A) = \operatorname{colim} \operatorname{GL}_n(A)$

is the direct limit of the GL_n, which embeds in GL_n+1 as the upper left block matrix, and the commutator subgroup agrees with the group generated by elementary matrices $\operatorname{E}(A)=[\operatorname{GL}(A),\operatorname{GL}(A)]$ , by Whitehead's lemma. Indeed, the group $\operatorname{GL}(A) / \operatorname{E}(A)$ was first defined and studied by Whitehead,^[1] and is called the Whitehead group of the ring^[2] A.

As $\operatorname{E}(A) \triangleleft \operatorname{SL}(A)$ , one can also define the special Whitehead group $SK_1(A) := \operatorname{SL}(A)/\operatorname{E}(A)$ .

[edit] Commutative rings and fields

For A a commutative ring, one can define a determinant $\det\colon \operatorname{GL}(A) \to A^*$ to the group of units of A, which vanishes on $\operatorname{E}(A)$ and thus descends to a map $\det\colon K_1(A) \to A^*$ . This map splits via the map $A^* \stackrel{\sim}{\to} GL_1(A) \to K_1(A)$ (unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence:

$1 \to SK_1(A) \to K_1(A) \to A^* \to 1,$

which is a quotient of the usual split short exact sequence defining the special linear group, namely

$1 \to \operatorname{SL}(A) \to \operatorname{GL}(A) \to A^* \to 1.$

Thus, since the groups in question are abelian, $K 1 (A)$ splits as the direct sum of the group of units and the special Whitehead group: $K_1(A) \approx A^* \oplus SK_1(A)$ .

When A is a Dedekind domain (e.g. a field, or the ring of algebraic integers in an algebraic number field), SK₁(A) vanishes, and the determinant map is an isomorphism. In particular, $\det\colon K_1(F) \stackrel{\sim}{\to} F^*$ .

For a non-commutative ring, the determinant cannot be defined, but the map $\operatorname{GL}(A) \to K_1(A)$ generalizes the determinant.

[edit] K₂

See also: Steinberg group (K-theory)

John Milnor found the right definition of K₂: it is the center of the Steinberg group $\operatorname{St}(A)$ of A.

It can also be defined as the kernel of the map

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

or as the Schur multiplier of the group of elementary matrices.

For a field k one has

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

[edit] Milnor K-theory

The above expression for K₂ of a field k led Milnor to the following definition of "higher" K-groups by

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

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, but for n≧3 they differ in general. For example, we have $K^M_n(\mathbb{F}_q) = 0$ for n≧3.

[edit] Higher K-theory

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

[edit] 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.

[edit] 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 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.

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, 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, K₁ and K₂.

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 K₀ → G₀ is surjective. It is also injective, as as can be shown. This isomorphism extends to the higher K-groups, too.

[edit] 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.

[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] References

^ J.H.C. Whitehead, Simple homotopy types Amer. J. Math. , 72 (1950) pp. 1–57
^ Not to be confused with the Whitehead group of a group.

Milnor, John Willard (1970), “Algebraic K-theory and quadratic forms”, Inventiones Mathematicae 9: 318–344, MR 0260844, ISSN 0020-9910
Milnor, John Willard (1971), Introduction to algebraic K-theory, Princeton, NJ: Princeton University Press, MR 0349811 (lower K-groups)
Quillen, Daniel (1975), “Higher algebraic K-theory”, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Montreal, Quebec: Canad. Math. Congress, pp. 171–176, MR 0422392 (Quillen's Q-construction)
Quillen, Daniel (1974), “Higher K-theory for categories with exact sequences”, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), vol. 11, London Math. Soc. Lecture Note Ser., Cambridge University Press, pp. 95–103, MR 0335604 (relation of Q-construction to +-construction)
Seiler, Wolfgang, “λ-Rings and Adams Operations in Algebraic K-Theory”, in Rapoport, M.; Schneider, P. & Schappacher, N., Beilinson's Conjectures on Special Values of L-Functions, Boston, MA: Academic Press, ISBN 978-0-12-581120-0
Weibel, Charles (2005), “Algebraic K-theory of rings of integers in local and global fields”, Handbook of K-theory, Berlin, New York: Springer-Verlag, pp. 139–190, MR 2181823, <http://www.math.uiuc.edu/K-theory/0691/KZsurvey.pdf> (survey article)

[edit] External links

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

Categories: Abstract algebra | K-theory

Algebraic K-theory

From Wikipedia, the free encyclopedia

Contents

[edit] History

[edit] Lower K-groups

[edit] K₀

[edit] K₁

[edit] Commutative rings and fields

[edit] K₂

[edit] Milnor K-theory

[edit] Higher K-theory

[edit] The +-construction

[edit] The Q-construction

[edit] Examples

[edit] Algebraic K-groups of finite fields

[edit] Algebraic K-groups of rings of integers

[edit] References

[edit] External links

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Algebraic K-theory

From Wikipedia, the free encyclopedia

Contents

[edit] History

[edit] Lower K-groups

[edit] K0

[edit] K1

[edit] Commutative rings and fields

[edit] K2

[edit] Milnor K-theory

[edit] Higher K-theory

[edit] The +-construction

[edit] The Q-construction

[edit] Examples

[edit] Algebraic K-groups of finite fields

[edit] Algebraic K-groups of rings of integers

[edit] References

[edit] External links

Views

Navigation

Interaction

Search

[edit] K₀

[edit] K₁

[edit] K₂