×

zbMATH — the first resource for mathematics

Algebraic \(K\)-theory and étale cohomology. (English) Zbl 0596.14012
This is a very long paper to which a short review like this could never do justice. Therefore I should firstly point out that this paper contains one of the most important discoveries concerning the role of algebraic \(K\)-theory in algebraic geometry.
Let \(\ell\) be a prime and \(X\) a separated, regular, noetherian scheme in with \(\ell\) is invertible. If \({\mathcal O}_ X\) contains \(\ell^{\nu}\)-th roots of unity there is a 2-dimensional “Bott element” in \(K_ 2({\mathcal O}_ X;{\mathbb Z}/\ell^{\nu})\) and one may, following the reviewer [Mem. Am. Math. Soc. 221 (1979; Zbl 0413.55004), Part IV, and ibid. 280 (1983; Zbl 0529.55015)], form the localisation \(K_*(X;{\mathbb Z}/\ell^{\nu})[\frac{1}{\beta}]\).
The author’s main theorem states that there is an isomorphism \[ K_*(X;{\mathbb Z}/\ell^{\nu})[\frac{1}{\beta}]\cong K_*^{top}(X;{\mathbb Z}/\ell^{\nu}) \] where the latter is topological or étale \(K\)-theory, which is constructed to be computable from étale cohomology.
When \(X\) is a smooth complex variety \(K_*^{top}(X;{\mathbb Z}/\ell^{\nu})\) is the usual mod \(\ell^{\nu} K\)-cohomology of \(X\) (with the classical topology). This connection permits the author to infer many fundamental consequences. For example, in § 4, he proves a purity conjecture of Grothendieck for \(\ell\)-adic étale cohomology by exploiting the relation with algebraic \(K\)-theory.
Reviewer: V. P. Snaith

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
19E20 Relations of \(K\)-theory with cohomology theories
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] J. F. ADAMS , Stable Homotopy and Generalized Homology , Univ. of Chicago Press, 1974 . MR 402720 | Zbl 0309.55016 · Zbl 0309.55016
[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 ; Vol. 3, 1966 , pp. 80-120). MR 182967 | Zbl 0129.15201 · Zbl 0129.15201
[3] M. ARTIN , Grothendieck Topologies (Lecture Notes, Harvard Univ., 1962 ). Zbl 0208.48701 · Zbl 0208.48701
[4] M. ARTIN , On the Joins of Hensel Rings (Advances in Math., Vol. 7, 1971 , pp. 282-296). MR 289501 | Zbl 0242.13021 · Zbl 0242.13021 · doi:10.1016/S0001-8708(71)80007-5
[5] M. ARTIN and B. MAZUR , Etale Homotopy (Springer Lecture Notes in Math., Vol. 100, 1969 ). MR 245577 | Zbl 0182.26001 · Zbl 0182.26001 · doi:10.1007/BFb0080957
[6] M. ARTIN and J.-L. VERDIER , Seminar on Etale Cohomology of Number Fields (Summer Institute on Algebraic Geometry, Woods Hole, 1964 , mineographed notes).
[7] M. BARR , Toposes without Points (J. Pure Appl. Alg., Vol. 5, 1974 , pp. 265-280). MR 409602 | Zbl 0294.18009 · Zbl 0294.18009 · doi:10.1016/0022-4049(74)90037-1
[8] A. A. BEILINSON , Higher Regulators and Values of L-Functions of Curves (Funct. Anal. and Appl., Vol. 14, No. 2, 1980 , pp. 116-118). MR 575206 | Zbl 0475.14015 · Zbl 0475.14015 · doi:10.1007/BF01086554
[9] A. A. BEILINSON , Visshi regulyatori i znacheniya L-funcni , (Sovremennge Problemy Matematiki, Vol. 24, 1984 , pp. 181-238). MR 760999
[10] J. M. BOARDMAN , Conditionally Convergent Spectral Sequences , preprint, 1981 . MR 1718076
[11] A. K. BOUSFIELD , The Localization of Spaces with Respect to Homology (Topology, Vol. 14, 1975 , pp. 133-150). MR 380779 | Zbl 0309.55013 · Zbl 0309.55013 · doi:10.1016/0040-9383(75)90023-3
[12] A. K. BOUSFIELD , The Localization of Spectra with Respect to Homology (Topology, Vol. 18, 1979 , pp. 257-281). MR 551009 | Zbl 0417.55007 · Zbl 0417.55007 · doi:10.1016/0040-9383(79)90018-1
[13] A. K. BOUSFIELD , K-Localizations and K-Equivalences of Infinite Loop Spaces (Proc. London Math. Soc., Ser. 3, Vol. 44, 1982 , pp. 291-311). MR 647434 | Zbl 0511.55007 · Zbl 0511.55007 · doi:10.1112/plms/s3-44.2.291
[14] A. K. BOUSFIELD and E. M. FRIEDLANDER , Homotopy Theory of \?-Spaces, Spectra, and Bisimplicial Sets (Geometric Applications of Homotopy Theory II, Springer Lecture Notes in Math., Vol. 658, 1978 , pp. 80-130). MR 513569 | Zbl 0405.55021 · Zbl 0405.55021
[15] A. K. BOUSFIELD and D. M. KAN , The Homotopy Spectral Sequence of a Space with Coefficients in a Ring (Topology, Vol. 11, 1972 , pp. 79-106). MR 283801 | Zbl 0202.22803 · Zbl 0202.22803 · doi:10.1016/0040-9383(72)90024-9
[16] A. K. BOUSFIELD and D. M. KAN , Homotopy Limits, Completions, and Localizations (Springer Lecture Notes in Math., Vol. 304, 1972 ). MR 365573 | Zbl 0259.55004 · Zbl 0259.55004
[17] W. BROWDER , Algebraic K-Theory with Coefficients \Bbb Z/p (Geometric Applications of Homotopy Theory I, Springer Lecture Notes in Math., Vol. 657, pp. 40-84). MR 513541 | Zbl 0386.18011 · Zbl 0386.18011
[18] K. S. BROWN , Abstract Homotopy Theory and Generalized Sheaf Cohomology (Trans. Amer. Math. Soc., Vol. 186, 1973 , pp. 419-458). MR 341469 | Zbl 0245.55007 · Zbl 0245.55007 · doi:10.1090/S0002-9947-1973-0341469-9
[19] K. S. BROWN , Cohomology of Groups (Graduate Texts in Math., Vol. 87, Springer, 1982 ). MR 672956 | Zbl 0584.20036 · Zbl 0584.20036
[20] K. S. BROWN and S. M. GERSTEN , Algebraic K-Theory as Generalized Sheaf Cohomology (Higher K-Theories, Springer Lecture Notes in Math., Vol. 341, 1973 , pp. 266-292). MR 347943 | Zbl 0291.18017 · Zbl 0291.18017
[21] H. CARTAN and S. EILENBERG , Homological Algebra , Princeton University Press, 1956 . MR 77480 | Zbl 0075.24305 · Zbl 0075.24305
[22] E. B. CURTIS , Simplicial Homotopy Theory (Advances in Math., Vol. 6, 1971 , pp. 107-209). MR 279808 | Zbl 0225.55002 · Zbl 0225.55002 · doi:10.1016/0001-8708(71)90015-6
[23] B. H. DAYTON and C. A. WEIBEL , A Spectral Sequence for the K-Theory of Affine Glued Schemes , (Algebraic K-Theory : Evanston 1980 , Springer Lecture Notes in Math., Vol. 854, 1981 , pp. 24-92). MR 618299 | Zbl 0462.18006 · Zbl 0462.18006
[24] P. DELIGNE , Théorie de Hodge III (Publ. Math. I.H.E.S., Vol. 44, 1974 , pp. 5-77). Numdam | MR 498552 | Zbl 0237.14003 · Zbl 0237.14003 · doi:10.1007/BF02685881 · numdam:PMIHES_1974__44__5_0 · eudml:103935
[25] T. TOM DIECK , Transformation Groups and Representation Theory (Springer Lecture Notes in Math., Vol. 766, 1979 ). MR 551743 | Zbl 0445.57023 · Zbl 0445.57023
[26] A. DOLD , Homology of Symmetric Products and Other Functors of Complexes (Ann. Math., Vol. 68, 1958 , pp. 54-80). MR 97057 | Zbl 0082.37701 · Zbl 0082.37701 · doi:10.2307/1970043
[27] A. DOLD and D. PUPPE , Homologie Nich-Additive-Funktoren. Anwendungen (Ann. Inst. Fourier, Vol. 11, 1961 , pp. 201-312). Numdam | MR 150183 | Zbl 0098.36005 · Zbl 0098.36005 · doi:10.5802/aif.114 · numdam:AIF_1961__11__201_0 · eudml:73776
[28] A. DRESS , Contributions to the Theory of Induced Representations (Classical Algebraic K-theory, and Connections with Arithmetic, Springer Lecture Notes in Math., Vol. 342, 1973 , pp. 183-240). MR 384917 | Zbl 0331.18016 · Zbl 0331.18016
[29] E. DUBUC , Kan Extensions in Enriched Category Theory (Springer Lectures Notes in Math., Vol. 145, 1970 ). MR 280560 | Zbl 0228.18002 · Zbl 0228.18002
[30] W. DWYER and E. FRIEDLANDER , Etale K-Theory and Arithmetic , preprint, 1983 . MR 648533
[31] W. DWYER , E. FRIEDLANDER , V. SNAITH and R. THOMASON , Algebraic K-Theory Eventually Surjects onto Topological K-Theory (Invent. Math., Vol. 66, 1982 , pp. 481-491). MR 662604 | Zbl 0501.14013 · Zbl 0501.14013 · doi:10.1007/BF01389225 · eudml:142894
[32] E. DYER , Cohomology Theories , Benjamin, 1969 . MR 268883 | Zbl 0182.57002 · Zbl 0182.57002
[33] E. FRIEDLANDER , Computations of K-Theories of Finite Fields (Topology, Vol. 15, 1976 , pp. 87-109). MR 394660 | Zbl 0401.18007 · Zbl 0401.18007 · doi:10.1016/0040-9383(76)90054-9
[34] E. FRIEDLANDER , Etale K-Theory I : Connections with Etale Cohomology and Algebraic Vector Bundles (Invent. Math., Vol. 60, 1980 , pp. 105-134). MR 586424 | Zbl 0519.14010 · Zbl 0519.14010 · doi:10.1007/BF01405150 · eudml:142744
[35] E. FRIEDLANDER , Etale K-Theory II : Connections with Algebraic K-Theory (Ann. scient. Ec. Norm. Sup., Vol. 15, 1982 , pp. 231-256). Numdam | MR 683636 | Zbl 0537.14011 · Zbl 0537.14011 · numdam:ASENS_1982_4_15_2_231_0 · eudml:82096
[36] E. FRIEDLANDER , Etale Homotopy of Simplicial Schemes , Princeton Univ. Press, 1982 . MR 676809 | Zbl 0538.55001 · Zbl 0538.55001
[37] E. FRIEDLANDER and B. PARSHALL , Etale Cohomology of Reductive Groups (Algebraic K-Theory : Evanston. 1980 , Springer Lecture Notes in Math., Vol. 854, 1981 , pp. 127-140). MR 618302 | Zbl 0495.14029 · Zbl 0495.14029
[38] P. GABRIEL and M. ZISMAN , Calculus of Fractions and Homotopy Theory (Springer Ergebnisse, Vol. 35, 1967 ). MR 210125 | Zbl 0186.56802 · Zbl 0186.56802
[39] S. M. GERSTEN , Mayer-Vietoris Functors and Algebraic K-Theory (J. Alg., Vol. 18, 1971 , pp. 55-58). MR 280570 | Zbl 0215.09801 · Zbl 0215.09801 · doi:10.1016/0021-8693(71)90127-X
[40] S. M. GERSTEN , Problems about Higher K-Functors (Higher K-Theories, Springer Lecture Notes in Math., Vol. 341, 1973 , pp. 43-56). MR 338125 | Zbl 0285.18011 · Zbl 0285.18011
[41] S. M. GERSTEN , Higher K-Theory of Rings (Higher K-Theories, Springer Lecture Notes in Math., Vol. 341, 1973 , pp. 3-42). MR 382398 | Zbl 0285.18010 · Zbl 0285.18010
[42] H. GILLET , Riemann-Roch Theorems for Higher Algebraic K-Theory (Advances in Math., Vol. 40, 1981 , pp. 203-289). MR 624666 | Zbl 0478.14010 · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0
[43] H. GILLET , Comparison of K-Theory Spectral Sequences , with Applications (Algebraic K-Theory : Evanston. 1980 , Springer Lecture Notes in Math., Vol. 854, 1981 , pp. 141-167). MR 618303 | Zbl 0478.14011 · Zbl 0478.14011
[44] H. GILLET , On the K-Theory of Surfaces with Multiple Curves , (Duke Math., Journal, Vol. 51, 1984 , pp. 195-233). Article | MR 744295 | Zbl 0557.14003 · Zbl 0557.14003 · doi:10.1215/S0012-7094-84-05111-1 · minidml.mathdoc.fr
[45] H. GILLET and R. THOMASON , The K-Theory of Strict Hensel Local Rings and a Theorem of Suslin (J. Pure Applied Algebra, Vol. 34, 1984 , pp. 241-254). MR 772059 | Zbl 0577.13009 · Zbl 0577.13009 · doi:10.1016/0022-4049(84)90037-9
[46] R. GODEMENT , Topologie algébrique et Théorie des Faisceaux , Herman, 1958 . MR 102797 | Zbl 0080.16201 · Zbl 0080.16201
[47] J. W. GRAY , Fibred and Cofibred Categories (Proceedings of the Conference on Categorical Algebra, La Jolla, 1965 , Springer, 1966 , pp. 21-83). MR 213413 | Zbl 0192.10701 · Zbl 0192.10701
[48] J. W. GRAY , Closed Categories, Lax Limits, and Homotopy Limits (J. Pure Appl. Alg., Vol. 19, 1980 , pp. 127-158). MR 593251 | Zbl 0462.55008 · Zbl 0462.55008 · doi:10.1016/0022-4049(80)90098-5
[49] A. GROTHENDIECK , Sur quelques points d’algèbre homologique (Tohoku Math. J., Vol. 9, 1957 , pp. 119-221). Article | MR 102537 | Zbl 0118.26104 · Zbl 0118.26104 · minidml.mathdoc.fr
[50] R. HARTSHORNE , Residues and Duality (Springer Lecture Notes in Math., Vol. 20, 1966 ). MR 222093 | Zbl 0212.26101 · Zbl 0212.26101 · doi:10.1007/BFb0080482 · eudml:203789
[51] R. HARTSHORNE , Algebraic Geometry (Graduate Texts in Math., Vol. 52, Springer, 1977 ). MR 463157 | Zbl 0367.14001 · Zbl 0367.14001
[52] A. HELLER , Stable Homotopy Categories (Bull. Amer. Math. Soc., Vol. 74, 1968 , pp. 28-64). Article | MR 224090 | Zbl 0177.25605 · Zbl 0177.25605 · doi:10.1090/S0002-9904-1968-11871-3 · minidml.mathdoc.fr
[53] P. J. HILTON and U. STAMMBACH , A Course in Homological Algebra (Graduate Texts in Math., Vol. 4, Springer, 1971 ). MR 346025 | Zbl 0238.18006 · Zbl 0238.18006
[54] F. HIRZEBRUCH ( SCHWARZENBERGER , BOREL ), Topological Methods in Algebraic Geometry , 3rd Ed., Grundlehren 131, Springer, 1978 . MR 1335917 | Zbl 0376.14001 · Zbl 0376.14001
[55] L. HODGKIN , On the K-Theory of Lie Groups (Topology, Vol. 6, 1967 , pp. 1-36). MR 214099 | Zbl 0186.57103 · Zbl 0186.57103 · doi:10.1016/0040-9383(67)90010-9
[56] L. ILLUSIE , Complexe cotangent et déformations I, II (Springer Lecture Notes in Math., Vol. 239, 1971 and 283, 1972 ). Zbl 0238.13017 · Zbl 0238.13017 · doi:10.1007/BFb0059573
[57] P. JOHNSTONE , Topos Theory , Academic Press, 1977 . MR 470019 | Zbl 0368.18001 · Zbl 0368.18001
[58] J. P. JOUANOLOU , Une suite exacte de Mayer-Vietoris en K-théorie algébrique (Higher K-Theories, Springer Lecture Notes in Math., Vol. 341, 1973 , pp. 293-316). MR 409476 | Zbl 0291.14006 · Zbl 0291.14006
[59] M. KAROUBI and O. VILLAMAYOR , Foncteurs Kn en algèbre et en topologie (C. R. Acad. Sc., Paris, T. 269, série A, 1969 , pp. 416-419). MR 251717 | Zbl 0182.57001 · Zbl 0182.57001
[60] M. KAROUBI and O. VILLAMAYOR , K-théorie algébrique et K-théorie topologique (Math. Scand., Vol. 28, 1971 , pp. 265-307). MR 313360 | Zbl 0231.18018 · Zbl 0231.18018 · eudml:166183
[61] D. M. KAN , Functors Involving c.s.s. Complexes (Trans. Amer. Math. Soc., Vol. 87, 1958 , pp. 330-346). MR 131873 | Zbl 0090.39001 · Zbl 0090.39001 · doi:10.2307/1993103
[62] D. M. KAN , Semisimplicial Spectra (Ill. J. Math., Vol. 7, 1963 , pp. 463-478). MR 153017 | Zbl 0115.40401 · Zbl 0115.40401
[63] D. M. LATCH , A Fibred Homotopy Equivalence and Homology Theories for the Category of Small Categories (J. Pure Appl. Alg., Vol. 15, 1979 , pp. 247-269). MR 537499 | Zbl 0407.55006 · Zbl 0407.55006 · doi:10.1016/0022-4049(79)90020-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.