Home > 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 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 theoremThe Quillen-Suslin theorem is a theorem in abstract algebra about the relationship between free modules and projective modules. Projective modules are modules that are locally free. Not all projective modules are free, but in the mid- 1950s, Jean-Pierre S. Applications of K-groups were found from 1960 onwards in surgery theory for manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefors, in particular; and numerous other connections with classical algebraic problems were found. A little later a branch of the theory for operator algebraIn functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology. In particular, it is a set of operas was fruitfully developed. It also became clear that K-theory could play a role in algebraic cycle theory in algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations . When there is more than o ( 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 MilnorJohn Willard Milnor (b. February 20, 1931) is a mathematician known for his work in differential topology. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950. He continued on to graduate school at Princeton, wrote hi, 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 extensionIn group theory, a central extension of a group G is an exact sequence of groups : such that A is in Z ''E , the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A and setting E to be A 's.

Eventually the foundational difficulties were resolved (leaving a deep and difficult theory), by a definition of Daniel QuillenDaniel Quillen (born June 27, 1940) is an American mathematician, a Fields Medallist, and the current Waynflete Professor of Pure Mathematics. He is best known for his work on the definition of the higher groups in algebraic K-theory, by means of homotopy. 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.