Periodicity of hermitian K -groups
01 Jun 2011-Journal of K-theory: K-theory and Its Applications To Algebra, Geometry, and Topology (Cambridge University Press)-Vol. 7, Iss: 03, pp 429-493
TL;DR: 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.
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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.
21 citations
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23 Nov 2010
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.
Abstract: 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 etale 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.
12 citations
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.
Abstract: We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bokstedt square”, relating the K-theory of the 2-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8 in the orthogonal case. In both the orthogonal and symplectic cases, we prove a 2-primary hermitian homotopy limit conjecture for these rings.
7 citations
Posted Content•
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.
Abstract: 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 etale 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.
7 citations
Cites background from "Periodicity of hermitian K -groups"
...6 was conjectured in [7], where the map was shown to be split surjective in sufficiently high degrees....
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...See also [7] for another argument in that case....
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...Bott elements for higher GrothendieckWitt theory and arbitrary coefficients were first constructed in [7]....
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...We write GW (Xét,L) for the value at X of a globally fibrant replacement of GW ( ,L) on the small étale site Xét of X ; see [7], [14], or 2....
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...The second conjecture was explored in [7], where the étale comparison map was shown to be split surjective in sufficiently high degrees....
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Posted Content•
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.
Abstract: We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8. In both the orthogonal and symplectic cases, we prove the 2-primary hermitian Quillen-Lichtenbaum conjecture.
7 citations
References
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Book•
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01 Jan 1968
2,032 citations
Additional excerpts
...The regularity assumption on A can be dispensed with when working with negative K-theory [4], [30], [31], [61]....
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01 Jan 1990
TL;DR: In this article, a localization theorem for the K-theory of commutative rings and of schemes is presented, relating the k-groups of a scheme, of an open subscheme, and of those perfect complexes on the scheme which are acyclic on the open scheme.
Abstract: In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those perfect complexes on the scheme which are acyclic on the open subscheme. The localization theorem of Quillen [Q1] for K′- or G-theory is the main support of his many results on the G-theory of noetherian schemes. The previous lack of an adequate localization theorem for K-theory has obstructed development of this theory for the fifteen years since 1973. Hence our theorem unleashes a pack of new basic results hitherto known only under very restrictive hypotheses like regularity. These new results include the “Bass fundamental theorem” 6.6, the Zariski (Nisnevich) cohomolog-ical descent spectral sequence that reduces problems to the case of local (hensel local) rings 10.3 and 19.8, the Mayer-Vietoris theorem for open covers 8.1, invariance mod l under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating K-theory to cyclic cohomology 9.10, mod l Mayer-Vietoris for closed covers 9.8, and mod l comparison between algebraic and topological K-theory 11.5 and 11.9. Indeed most known results in K-theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses.
1,009 citations
Additional excerpts
...The regularity assumption on A can be dispensed with when working with negative K-theory [4], [30], [31], [61]....
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Book•
01 Jan 2000
TL;DR: Part I algebraic theory: Cohomology of Profinite groups as mentioned in this paper, some homological algebra, duality properties of profinite groups, free products of finite groups, Iwasawa Modules.
Abstract: Part I Algebraic Theory: Cohomology of Profinite Groups.- Some Homological Algebra.- Duality Properties of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa Modules.- Part II Arithmetic Theory: Galois Cohomology.- Cohomology of Local Fields.- Cohomology of Global Fields.- The Absolute Galois Group of a Global Field.- Restricted Ramification.- Iwasawa Theory of Number Fields.- Anabelian Geometry.- Literature.- Index.
948 citations
TL;DR: In this article, it was shown that the E*-localization functor of a spectrum E E Ho gives rise to a natural E*localisation functor ( )E: Ho" -+HoS and n : 1 +( )E.
760 citations
Additional excerpts
...Bousfield introduced this notion in [12]....
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