|
In mathematics, K-theory is, roughly speaking, the study of certain kinds of invariants of large matrices. It originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include Bott periodicity, the Atiyah-Singer index theorem and the Adams operations. In high energy physics, K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes, Ramond–Ramond field strengths and also certain spinors on generalized complex manifolds. In condensed matter physics K-theory has been used to classify topological insulators, superconductors and stable Fermi surfaces. For more details, see K-theory (physics). ==Early history== The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck–Riemann–Roch theorem. It takes its name from the German ''Klasse'', meaning "class".〔Karoubi, 2006〕 Grothendieck needed to work with coherent sheaves on an algebraic variety ''X''. Rather than working directly with the sheaves, he defined a group using isomorphism classes of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called ''K(X)'' when only locally free sheaves are used, or ''G(X)'' when all are coherent sheaves. Either of these two constructions is referred to as the Grothendieck group; ''K(X)'' has cohomological behavior and ''G(X)'' has homological behavior. If ''X'' is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition. In topology, by applying the same construction to vector bundles, Michael Atiyah and Friedrich Hirzebruch defined ''K(X)'' for a topological space ''X'' in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Index Theorem (circa 1962). Furthermore this approach led to a noncommutative K-theory for C *-algebras. Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「K-theory」の詳細全文を読む スポンサード リンク
|