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The homotopy limit problem and (etale) hermitian K-theory
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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.read more
Citations
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Journal ArticleDOI
The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism
Po Hu,Igor Kriz,Kyle Ormsby +2 more
TL;DR: The homotopy limit problem for Karoubi's Hermitian K-theory was first posed by Thomason as discussed by the authors, who showed that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd 2 (F [ i ] ) ∞, but not in general.
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.
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Categorical Models for Equivariant Classifying Spaces
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Algebraic K-theory of Geometric Groups
TL;DR: In this article, the integral K-theoretic assembly map for the group ring $R[Gamma]$ is shown to be an isomorphism for groups of finite asymptotic dimension.
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The motivic Hopf map solves the homotopy limit problem for $K$-theory
TL;DR: In this article, the homotopy limit problem for $K$-theory over fields of finite virtual cohomological dimension was solved by employing the motivic slice filtration and the first motivic Hopf map.
References
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Book ChapterDOI
Higher Algebraic K-Theory of Schemes and of Derived Categories
R. W. Thomason,Thomas Trobaugh +1 more
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.
Journal ArticleDOI
Negative K -theory of derived categories
TL;DR: In this paper, negative K-groups are defined for exact categories and derived categories in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason.
Journal ArticleDOI
The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes
TL;DR: In this paper, the authors prove localization and Zariski-Mayer-Vietoris for higher Gro-thendieck-Witt groups, alias hermitian K-groups, of schemes admitting an ample family of line-bundles.
Journal ArticleDOI
Products and duality in Waldhausen categories
Michael Weiss,Bruce Williams +1 more
TL;DR: In this article, a Spanier-Whitehead product (SW product for short) is defined for the Tate cohomology of Z/2 acting on K-theory of any ring with inrvolu- tion, and it is 4-periodic.