Hornbostel on hermitian K-theory and Witt groups
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KT (K-theory), AG (Algebraic geometry)
Witt
See also Motivic homotopy theory
(A map) From algebraic K-theory to hermitian K-theory, by Max Karoubi: http://www.math.uiuc.edu/K-theory/0755
Cortinas constructs Stiefel-Whitney characters or Chern characters from Hermitian K-th of A to the dihedral homology of A, where A is an algebra with involution
arXiv:1101.2056 Periodicity of hermitian K-groups 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 K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with finite coefficients similar periodicity results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodicity for its hermitian K-groups, analogous to orthogonal and symplectic topological K-theory
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 K-groups in terms of the higher algebraic K-groups
We also relate periodicity to etale hermitian K-groups 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.
Snaith: A descent theorem for Hermitian K-theory link
Several papers of Hornbostel, not downloaded. See his webpage for list.
Hermitian K-theory of the integers, by A. J. Berrick and M. Karoubi: http://www.math.uiuc.edu/K-theory/0649
Localization in Hermitian K-theory of rings, by Jens Hornbostel and Marco Schlichting: http://www.math.uiuc.edu/K-theory/0653
Periodicity of Hermitian K-theory and Milnor’s K-groups, by Max Karoubi http://www.math.uiuc.edu/K-theory/0659
Oriented Chow groups, Hermitian K-theory and the Gersten conjecture, by Jens Hornbostel: http://www.math.uiuc.edu/K-theory/0835
arXiv:1011.4977 The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory 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 mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic 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 Grothendieck-Witt (hermitian K-) theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.
http://mathoverflow.net/questions/3701/stable-homology-of-arithmetic-groups
nLab page on Hermitian K-theory