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Etale K-theory. II: Connections with algebraic K-theory. (English) Zbl 0537.14011
From the author’s introduction: ”We continue our study of etale K-theory begun in part I of this paper [Invent. Math. 60, 105-134 (1980; Zbl 0519.14010)]. We introduce a natural transformation from algebraic K- theory to \(\ell\)-adic etale K-theory which extends our previously defined natural transformation in degree 0 and which admits associated natural transformations between K-theories with coefficients. Our expectation is that etale K-theory may soon become a successful tool for deciding geometric questions, especially those involving galois actions. Consequently, we investigate various relationships between algebraic and etale K-theory of varieties quasi-projective over an algebraically closed field. Work in progress indicates that etale K-theory of more general schemes should prove to be a useful tool for certain number theoretic problems.”
Reviewer: M.Stoia

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
13D15 Grothendieck groups, \(K\)-theory and commutative rings
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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References:
[1] J. F. ADAMS , On Chern Characters and the Structure of the Unitary Group (Proc. Camb. Phil. Soc., Vol. 57, 1961 , pp. 198-199). MR 22 #12525 | Zbl 0103.16001 · Zbl 0103.16001
[2] S. ARAKI and H. TODA , Multiplicative Structures in Mod q Cohomology Theories, I and II , (Osaka J. Math., Vol. 2, 1965 , pp. 71-115 and Vol. 3, 1966 , pp. 81-120). Zbl 0129.15201 · Zbl 0129.15201
[3] M. ARTIN and B. MAZUR , Etale Homotopy (Lecture Notes in Math., 100, 1969 , Springer-Verlag). MR 39 #6883 | Zbl 0182.26001 · Zbl 0182.26001 · doi:10.1007/BFb0080957
[4] A. K. BOUSFIELD and D. M. KAN , Homotopy Limits, Completions, and Localizations (Lecture Notes in Math., Vol. 304, 1973 , Springer-Verlag). MR 51 #1825 | Zbl 0259.55004 · Zbl 0259.55004
[5] W. BROWDER , Algebraic K-Theory with Coefficients Z/p (Lecture Notes in Math., Vol. 651, 1978 , pp. 40-84, Springer-Verlag). MR 80b:18011 | Zbl 0386.18011 · Zbl 0386.18011
[6] K. S. BROWN and S. M. GERSTEN , Algebraic K-Theory as Generalized Cohomology (Lecture Notes in Math., Vol. 341, 1973 , pp. 266-292, Springer-Verlag). MR 50 #442 | Zbl 0291.18017 · Zbl 0291.18017
[7] P. DELIGNE , La Conjecture de Weil I (Publ. Math. I.H.E.S., Vol. 43, 1974 , pp. 273-307). Numdam | MR 49 #5013 | Zbl 0287.14001 · Zbl 0287.14001 · doi:10.1007/BF02684373 · numdam:PMIHES_1974__43__273_0 · eudml:103930
[8] P. DELIGNE , Cohomologie Etale (SGA 41/2) (Lecture Notes in Math., Vol. 569, 1977 , Springer-Verlag). MR 57 #3132 | Zbl 0345.00010 · Zbl 0345.00010
[9] E. FRIEDLANDER , Unstable K-Theories of the Algebraic Closure of a Finite Field (Comment. Math. Helvetici, Vol. 50, 1975 , pp. 145-154). MR 53 #14488 | Zbl 0307.18005 · Zbl 0307.18005 · doi:10.1007/BF02565742 · eudml:139616
[10] E. FRIEDLANDER , The Infinite Loop Adams Conjecture via Classification Theorems for F-Spaces (Proc. Camb. Phil. Soc., Vol. 87, 1980 , pp. 109-150). MR 81b:55023 | Zbl 0426.55010 · Zbl 0426.55010 · doi:10.1017/S0305004100056577
[11] E. FRIEDLANDER , Etale K-theory I: Connections with Etale Cohomology and Algebraic Vector Bundles (Inventiones Math., Vol. 60, 1980 , pp. 105-134). MR 82e:14029 | Zbl 0519.14010 · Zbl 0519.14010 · doi:10.1007/BF01405150 · eudml:142744
[12] E. FRIEDLANDER , Etale Homotopy of Simplicial Schemes . To appear in Princeton University Press. Zbl 0538.55001 · Zbl 0538.55001
[13] J.-P. JOUANOLOU , Une suite exact de Mayer-Vietoris en K-theorie algebrique (Lecture Notes in Math., Vol. 341, 1973 , pp. 293-316, Springer-Verlag). MR 53 #13231 | Zbl 0291.14006 · Zbl 0291.14006
[14] D. MACDUFF and G. SEGAL , Homology, Fibrations, and the ”Group Completion” Theorem (Inventiones Math., Vol. 31, 1976 , pp. 279-284). MR 53 #6547 | Zbl 0306.55020 · Zbl 0306.55020 · doi:10.1007/BF01403148 · eudml:142364
[15] J. P. MAY , The Spectra Associated to Permutative Categories (Topology, Vol. 17, 1978 , pp. 225-228). MR 80e:55015 | Zbl 0417.55011 · Zbl 0417.55011 · doi:10.1016/0040-9383(78)90027-7
[16] D. QUILLEN , Higher Algebraic K-Theory I (Lecture notes in Math., Vol. 341, 1973 , pp. 85-147, Springer-Verlag). MR 49 #2895 | Zbl 0292.18004 · Zbl 0292.18004
[17] G. SEGAL , Categories and Cohomology Theories (Topology, Vol. 13, 1974 , pp. 293-312). MR 50 #5782 | Zbl 0284.55016 · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6
[18] C. SOULÉ , K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale (Inventiones Math., Vol. 55, 1979 , pp. 251-295). MR 81i:12016 | Zbl 0437.12008 · Zbl 0437.12008 · doi:10.1007/BF01406843 · eudml:186136
[19] R. THOMASON , ALGEBRAIC K-THEORY AND ETALE COHOMOLOGY (Preprint). · Zbl 0714.14006
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