Algebraic Ktheory

From Exampleproblems

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).

Historically the roots of the theory were in topological K-theory (based on vector bundle theory); and its motivation 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+1(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

Contents 1 Detailed discussion 1.1 Lower dimensions 1.1.1 K0 1.1.2 K1 1.1.3 K2 1.2 Milnor K-theory 1.3 Quillen's K-theory 1.3.1 Classifying spaces of categories 1.3.2 Quillen's Q-construction 1.3.3 The Quillen K-groups 2 Literature

Detailed discussion

Let A be a ring.

Lower dimensions

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,

,

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

K₁

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

K₁(A) = GL(A)^ab

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].

For A a commutative ring this comes down to saying K₁(A) is its group of units.

K₂

John Milnor found the right definition of K₂: it is the Steinberg group St(A) (of R. 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

1. x_ij(r)x_ij(r') = x_ij(r + r')
2. [x_ij(r),x_jk(r')] = x_ik(rr')] für
3. [x_ij(r),x_kl(r')] = 1 für

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

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

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

verbunden.

For a field k one has

Milnor K-theory

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

,

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. Für n = 0,1,2 these coincide with those above..

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.

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

.

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

Quillen's Q-construction

Suppose P is an exact category, that is an additive category with a class E of "exact" diagrams

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

From P a category QP is defined, objects to be those of P, morphisms from M′ to M″ isomorphism classes of exact diagrams

The Quillen K-groups

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

K_i(P) = π_{i + 1}(BQP,0)

with a fixed zero-object 0.

K₀(P) coincides with the Grothendieck group of P.

The K-groups K_i(A) of the ring A are then the K-groups of the category its finitely-generated projective modules. When A is a noetherian ring, the K′_i(A) can alternatively be defined from the category of all finitely-generated A-modules.

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