Algebraic K-theory

From Academic Kids

In mathematics, algebraic K-theory is an advanced part of homological algebra concerned with defining and applying a sequence

Kn(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 Kn 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

Kn(R) = πn+1(BGL(R)+),

a very compressed piece of abstract mathematics. Here πk 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.

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