We settle two conjectures for computing higher Grothendieck-Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown 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. We also prove that the mod 2(nu) comparison map between the 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.

The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory / J. Berrick, M. Karoubi, M.&. Schlichting, P.A. Oestvaer. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 278(2015 Jun 25), pp. 34-55. [10.1016/j.aim.2015.01.018]

The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory

P.A. Oestvaer
2015-06-25

Abstract

We settle two conjectures for computing higher Grothendieck-Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown 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. We also prove that the mod 2(nu) comparison map between the 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.
Grothendieck-Witt groups of schemes; Hermitian Quillen-Lichtenbaum conjecture; Number fields; Algebraic varieties;
Settore MAT/03 - Geometria
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/860157
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