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The equivariant algebraic K-theory is an algebraic K-theory associated to the category of equivariant coherent sheaves on an algebraic scheme with action of a linear algebraic group, via Quillen's Q-construction; thus, by definition,
Tensor product of bundles gives K-theory a commutative ring structure. Without subscripts, () usually denotes complex K-theory whereas real K-theory is sometimes written as (). The remaining discussion is focused on complex K-theory. As a first example, note that the K-theory of a point is the integers. This is because vector bundles over a ...
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra.
The K-theory classification of D-branes has had numerous applications. For example, Hanany & Kol (2000) used it to argue that there are eight species of orientifold one-plane. Uranga (2001) applied the K-theory classification to derive new consistency conditions for flux compactifications.
In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups K i (C) associated to it.If C is an abelian category, there is no need for extra data, but in general it only makes sense to speak of K-theory after specifying on C a structure of an exact category, or of a Waldhausen category, or of a dg-category, or ...
The theory K(0) agrees with singular homology with rational coefficients, whereas K(1) is a summand of mod-p complex K-theory. The theory K(n) has coefficient ring F p [v n,v n −1] where v n has degree 2(p n − 1). In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2.
The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that the projection map : induces an isomorphism of prorings
The K-groups of finite fields are one of the few cases where the K-theory is known completely: [2] for , = (() +) {/ (), =,For n=2, this can be seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture.