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In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.In algebraic topology, it is a cohomology theory known as topological K-theory.
Wang's theorem states that if A has prime degree then SK 1 (A) is trivial, [59] and this may be extended to square-free degree. [60] Wang also showed that S K 1 ( A ) is trivial for any central simple algebra over a number field, [ 61 ] but Platonov has given examples of algebras of degree prime squared for which S K 1 ( A ) is non-trivial.
The Thom isomorphism theorem in topological K-theory is () ~ (()), where T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle. The Atiyah-Hirzebruch spectral sequence allows computation of K -groups from ordinary cohomology groups.
The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Alexander Merkurjev, Andrei Suslin, Markus Rost, Fabien Morel, Eric Friedlander, and others, including the newly minted theory of motivic cohomology (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra.
In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to [] or [,]. The theorem was first proved by Hyman Bass for K 0 , K 1 {\displaystyle K_{0},K_{1}} and was later extended to higher K -groups by Daniel Quillen .
Waldhausen Localization Theorem [2] — Let be the category with cofibrations, equipped with two categories of weak equivalences, () (), such that (,) and (,) are both Waldhausen categories. Assume ( A , w ) {\displaystyle (A,w)} has a cylinder functor satisfying the Cylinder Axiom, and that w ( A ) {\displaystyle w(A)} satisfies the Saturation ...
In this sense, the apparently ad hoc definition of Milnor K-theory becomes a theorem: certain motivic cohomology groups of a field can be explicitly computed by generators and relations. A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or étale ...
Theorem (Yau–Tian–Donaldson conjecture for Kähler–Einstein metrics): A Fano Manifold admits a Kähler–Einstein metric in the class of () if and only if the pair (,) is K-polystable. Chen, Donaldson, and Sun have alleged that Tian's claim to equal priority for the proof is incorrect, and they have accused him of academic misconduct.