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Geometric models for algebraic Ktheory
 Ktheory
"... Abstract. We show how algebraic Ktheory and related theories can be defined in terms of “algebraic mapping spaces ” to appropriate indschemes. Keywords: algebraic Ktheory, infinite loop space, Quotschemes, Γspace ..."
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Cited by 11 (7 self)
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Abstract. We show how algebraic Ktheory and related theories can be defined in terms of “algebraic mapping spaces ” to appropriate indschemes. Keywords: algebraic Ktheory, infinite loop space, Quotschemes, Γspace
SemiTopological KTheory of Real VARIETIES
"... . The semitopological Ktheory of real varieties, KR semi (), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bund ..."
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Cited by 8 (3 self)
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. The semitopological Ktheory of real varieties, KR semi (), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semitopological equivalence), consideration of Z/2equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic Ktheory of real varieties. The authors verify that the semitopological Ktheory of a real variety X interpolates between the algebraic Ktheory of X and Atiyah's Real Ktheory of the associated Real space of complex points, X R (C). The resulting natural maps of spectra K alg (X) # KR semi (X) # KRtop (X R (C)) satisfy numerous good properties: the first map is a modn equivalence for any projective real variety and any n > 0; the second map is an equivalence for smooth projective curves...
Techniques, computations, and conjectures for semitopological Ktheory
 MATH. ANN
, 2004
"... We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence tha ..."
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Cited by 6 (1 self)
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We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic Ktheory of varieties, and it is also compatible with the classical AtiyahHirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the BorelMoore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soulé — to compute the semitopological Ktheory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational threefolds, and related varieties, the semitopological Kgroups and topological Kgroups are isomorphic in all degrees permitted by cohomological considerations. We also
Rational Isomorphisms between KTheories and Cohomology Theories
"... The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character ma ..."
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Cited by 6 (3 self)
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The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced which establishes a useful general criterion for a natural transformation of functors on quasiprojective complex varieties to induce a homotopy equivalence of semitopological singular complexes. Since semitopological Ktheory and morphic cohomology can be formulated as the semitopological singular complexes associated to Ktheory and motivic cohomology, this criterion provides a rational isomorphism between the semitopological Ktheory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a RiemannRoch theorem for the Chern character on semitopological Ktheory and an interpretation of the "topological filtration" on singular cohomology groups in K theoretic terms.
Some remarks concerning modn Ktheory
 Invent. Math
"... The spectral sequence predicted by A. Beilinson relating motivic cohomology to algebraic Ktheory has been established for smooth quasiprojective varieties over a field (cf. [FS], [L1]). Among other properties verified, this spectral sequence has the expected multiplicative behavior (involving cup ..."
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Cited by 2 (0 self)
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The spectral sequence predicted by A. Beilinson relating motivic cohomology to algebraic Ktheory has been established for smooth quasiprojective varieties over a field (cf. [FS], [L1]). Among other properties verified, this spectral sequence has the expected multiplicative behavior (involving cup product in motivic cohomology and product in algebraic Ktheory) and a good multiplicative “modn reduction” relating modn motivic cohomology to modn algebraic Ktheory. The purpose of this short paper is to draw two conclusions concerning modn algebraic Ktheory of smooth, quasiprojective varieties over an algebraically closed field by comparing this spectral sequence to its “localization ” which we show converges to modn etale Ktheory. In Theorem 2.1, we verify that the map from modn algebraic Ktheory to modn etale Ktheory of a smooth quasiprojective variety X over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of X. This refines the surjectivity result of [DFST] and extends an argument by A. Suslin in [S1]. Then, in Example 3.4, we show that certain projective smooth 3folds X constructed
Semitopological Ktheory for certain projective varieties
, 2006
"... In this paper we compute Lawson homology groups and semitopological Ktheory for certain threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation m ..."
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Cited by 1 (1 self)
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In this paper we compute Lawson homology groups and semitopological Ktheory for certain threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of a technique of Bloch and Srinavas, of the BlochKato conjecture and of the spectral sequence relating morphic cohomology and semitopological Ktheory.
K^sst for certain . . .
, 2005
"... In this paper we compute Lawson homology groups and semitopological Ktheory for some threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation make ..."
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In this paper we compute Lawson homology groups and semitopological Ktheory for some threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of a technique of Bloch and Srinavas, of the BlochKato theorem and of the spectral sequence
SOME REMARKS CONCERNING MODn KTHEORY
"... The spectral sequence predicted by A. Beilinson relating motivic cohomology to algebraic Ktheory has been established for smooth quasiprojective varieties over a field (cf. [FS], [L1]). Among other properties verified, this spectral sequence has the expected multiplicative behavior (involving cup ..."
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The spectral sequence predicted by A. Beilinson relating motivic cohomology to algebraic Ktheory has been established for smooth quasiprojective varieties over a field (cf. [FS], [L1]). Among other properties verified, this spectral sequence has the expected multiplicative behavior (involving cup product in motivic cohomology and product in algebraic Ktheory) and a good multiplicative “modn reduction” relating modn motivic cohomology to modn algebraic Ktheory. The purpose of this short paper is to draw two simple conclusions concerning modn algebraic Ktheory of smooth, quasiprojective varieties over an algebraically closed field by comparing this spectral sequence to its “localization ” which we show converges to modn etale Ktheory. In Theorem 2.1, we verify that the map from modn algebraic Ktheory to modn etale Ktheory of a smooth quasiprojective variety X over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of X. This refines the surjectivity result of [DFST] and extends an argument by A. Suslin in [S1]. Then, in Example 3.4, we show that certain projective smooth 3folds X constructed
SEMITOPOLOGICAL KTHEORY OF REAL VARIETIES
"... Abstract. The semitopological Ktheory of real varieties, KRsemi (−), is an oriented multiplicative (generalized) cohomology theory which extends the authors ’ earlier theory, Ksemi (−), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles ..."
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Abstract. The semitopological Ktheory of real varieties, KRsemi (−), is an oriented multiplicative (generalized) cohomology theory which extends the authors ’ earlier theory, Ksemi (−), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semitopological equivalence), consideration of Z/2equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic Ktheory of real varieties. The authors verify that the semitopological Ktheory of a real variety X interpolates between the algebraic Ktheory of X and Atiyah’s Real Ktheory of the associated Real space of complex points, XR(C). The resulting natural maps of spectra K alg (X) → KR semi (X) → KRtop(XR(C)) satisfy numerous good properties: the first map is a modn equivalence for any projective real variety and any n> 0; the second map is an equivalence for smooth projective curves and flag varieties; the triple fits in a commutative diagram of spectra mapping via total Segre classes to a triple of cohomology theories. The authors also establish results for the semitopological Ktheory of real varieties, such as Nisnevich excision and a type of localization result, which were previously unknown even for complex varieties.
2.2. Twisted Chow Groups and Twisted Motivic Groups 4
"... Abstract. We define a total Chern class map for the Ktheory of a variety X twisted by a central simple algebra A. This includes defining a suitable notion of the motivic cohomology of X twisted by A to serve as the target for such a map. Our twisted motivic groups turn out to be different than thos ..."
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Abstract. We define a total Chern class map for the Ktheory of a variety X twisted by a central simple algebra A. This includes defining a suitable notion of the motivic cohomology of X twisted by A to serve as the target for such a map. Our twisted motivic groups turn out to be different than those defined