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.
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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
Algebraic K-Theory Eventually Surjects onto Topological K-Theory
TL;DR: Theorem 1 shows that the existence of these non-algebraic topological bundles is reflected in the higher-dimensional algebraic K-groups of X as discussed by the authors, which essentially answers a question of Fulton, [9], w 5.
Journal ArticleDOI
The 2-torsion in the K-theory of the integers
TL;DR: The 2-torsion subgroups of the K-theory of the integers ℤ have been shown to be periodic with period 8, and there are no 2-Torsion elements except those known for over 20 years as discussed by the authors.
Book ChapterDOI
Norm Varieties and the Chain Lemma (After Markus Rost)
TL;DR: In this paper, the authors present proofs of two results of Markus Rost, the Chain Lemma 1 and the Norm Principle 3, that the norm residue maps are isomorphisms for every prime p, every n and every field k containing 1/p.
ComparisonBetweenAlgebraic and Topological K-Theory for Banach Algebras and C∗-Algebras*
TL;DR: Toeplitz Operators and K-Theory of Banach Algebras as mentioned in this paper, and K2 Smooth Extensions and K3 Smooth Extensions, K2, K3, K4, K5, K6, K7, K8, K9, K10, K11, K12, K13, K14, K15, K16, K17, K18, K19, K20, K21, K23, K24, K26, K28, K30, K31, K33, K34