Hermitian Ktheory

General Introduction
See also Motivic homotopy theory
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Online References
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Paper References
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Definition
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Properties
arXiv:1101.2056 Periodicity of hermitian Kgroups from arXiv Front: math.KT by A. J. Berrick, M. Karoubi, P. A. Østvær Bott periodicity for the unitary, orthogonal and symplectic groups is fundamental to topological Ktheory. Analogous to unitary topological Ktheory, for algebraic Kgroups with finite coefficients similar periodicity results are consequences of the Milnor and BlochKato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic Kgroups for any ring implies periodicity for its hermitian Kgroups, analogous to orthogonal and symplectic topological Ktheory.
The proofs use in an essential way higher KSC theories extending those of Anderson and Green. They also provide an upper bound for the higher hermitian Kgroups in terms of the higher algebraic Kgroups.
We also relate periodicity to etale hermitian Kgroups by proving ahermitian version of Thomason's etale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Snaith: A descent theorem for Hermitian Ktheory link
Several papers of Hornbostel, not downloaded. See his webpage for list.
Hermitian Ktheory of the integers, by A. J. Berrick and M. Karoubi: K0649
Localization in Hermitian Ktheory of rings, by Jens Hornbostel and Marco Schlichting: K0653
Periodicity of Hermitian Ktheory and Milnor's Kgroups, by Max Karoubi K0659
Oriented Chow groups, Hermitian Ktheory and the Gersten conjecture, by Jens Hornbostel: K0835
arXiv:1011.4977 The homotopy fixed point theorem and the QuillenLichtenbaum conjecture in hermitian Ktheory from arXiv Front: math.AT by A. J. Berrick, M. Karoubi, M. Schlichting, P. A. Østvær Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian Ktheory of X to the homotopy fixed points of Ktheory under the natural Z/2action is a 2adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher GrothendieckWitt (hermitian K) theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the QuillenLichtenbaum conjecture in Ktheory. Applications compute higher GrothendieckWitt groups of complex algebraic varieties and rings of 2integers in number fields, and hence values of Dedekind zetafunctions.
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