Periodicity of hermitian K -groups
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In this article, it was shown that the periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-group, analogous to orthogonal and symplectic topological K-theory.Abstract:
Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic Kgroups with finite coefficients, similar 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 the 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 higher algebraic K-groups. We also relate periodicity to etale hermitian K-groups by proving a hermitian 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.read more
Citations
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The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory
TL;DR: In this paper, it was shown that the comparison map from the Hermitian K-theory of X to the homotopy fixed points of Ktheory under the natural Z/2 -action is a 2-adic equivalence.
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The homotopy limit problem and (etale) hermitian K-theory
TL;DR: In this article, it was shown that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its etale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory.
Journal ArticleDOI
Hermitian K -theory and 2-regularity for totally real number fields
TL;DR: In this paper, the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields were determined.
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The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory
TL;DR: In this article, it was shown that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its etale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory.
Posted Content
Hermitian K-theory and 2-regularity for totally real number fields
TL;DR: In this paper, the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in real 2-regular number fields were determined.
References
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Journal ArticleDOI
The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory
TL;DR: In this paper, it was shown that the comparison map from the Hermitian K-theory of X to the homotopy fixed points of Ktheory under the natural Z/2 -action is a 2-adic equivalence.
Journal ArticleDOI
Étale descent for real number fields
TL;DR: In this paper, the strong Quillen-Lichtenbaum conjecture for integers in real number fields at the prime two was shown to be weak equivalence on zero-connected covers.
Journal ArticleDOI
The Higher K-Theory of a Complex Surface
TL;DR: In this article, it was shown that the K-groups of F are divisible above the dimension of X, and that the k-group of X is divisible-by-finite.
Chain lemma for splitting fields of symbols
TL;DR: In this article, the Conner-Floyd theorem is used to compute the degree formula for s(X) and the chain-lemma construction (for general n) for n = 3.