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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,
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 Grothendieck–Riemann–Roch theorem says that these are equal. When Y is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the Grothendieck–Riemann–Roch theorem specializes to Hirzebruch's theorem. The group K(X) is now known as K 0 (X).
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.
A theorem of Ravindra Bapat, generalizing Sperner's lemma, [5]: chapter 16, pp. 257–261 implies the KKM lemma extends to connector-free coverings (he proved his theorem for =). The connector-free variant also has a permutation variant, so that both these generalizations can be used simultaneously.
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 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 .
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 ...