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The spectral sequence relating algebraic \(K\)-theory to motivic cohomology. (English) Zbl 1047.14011
Suppose that \(X\) is a smooth variety over a field \(F\). The main result of this paper asserts that there is a spectral sequence \[ E_{2}^{p,q} = H^{p-q}(X,\mathbb{Z}(-q)) = \text{CH}^{-q}(X,-p-q) \Rightarrow K_{p-q}(X) \] from the motivic cohomology of \(X\) to the algebraic \(K\)-theory of \(X\). This result, as stated and proved in this paper, extends and depends on a theorem of Bloch and Lichtenbaum, which asserts the existence of the same spectral sequence in the case where \(X\) is the field \(F\).
The motivic cohomology spectral sequence is now a fundamental construction for algebraic \(K\)-theory. In particular, a torsion coefficients analogue is used to show that the Bloch-Kato conjecture identifying Milnor K-theory with Galois cohomology (both with torsion coefficients) implies the Lichtenbaum-Quillen conjecture. The Lichtenbaum-Quillen conjecture says, roughly, that \(K\)-theory with torsion coefficients can be recovered from étale cohomology in all but finitely many degrees.
The method of proof of the main result in the paper under review is to show that the objects in a tower of cofibrations arising from multirelative \(K\)-theory spectra have an adequate theory of transfers and satisfy the homotopy property. Techniques of V. Voevodsky [in: Motives, polylogarithms and Hodge theory, I: Motives and polylogarithms. Int. Press. Lect. Ser. 3, 3–34 (2002; see the following review Zbl 1047.14012)] are then used to show that the fibre sequences making up the Bloch-Lichtenbaum spectral sequence [S. Bloch and S. Lichtenbaum, “A spectral sequence for motivic cohomology” (preprint 1995)] extend to semi-local rings, and then the desired spectral sequence is recovered from a Zariski (i.e., Brown-Gersten) descent argument.
Since the appearance of this paper, A. Suslin [Proc. Steklov Inst. Math. 241, 202–237 (2003)] has removed the dependence on the Bloch-Lichtenbaum result by showing that a variant of motivic cohomology developed by D. Grayson [\(K\)-Theory 9, 139–172 (1995; Zbl 0826.19003)] – which has a cohomological spectral sequence converging to \(K\)-theory for smooth semi-local schemes – coincides with motivic cohomology.
Most recently, M. Levine [“\(K\)-theory and motivic cohomology of schemes” (preprint 1999) and “The homotopy coniveau filtration” (preprint 2003)] has shown that the multirelative \(K\)-theory tower is a special case of Voevodsky’s slice filtration, which is a construction that exists quite generally in the motivic stable category (loc. cit.). The layers of the slice filtration for \(K\)-theory are shifted motivic cohomology spectra, and so one obtains another description of the motivic cohomology spectral sequence.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
14C25 Algebraic cycles
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
19E08 \(K\)-theory of schemes
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References:
[1] Bass H. , Algebraic K-Theory , W.A. Benjamin , 1968 . MR 249491 | Zbl 0174.30302 · Zbl 0174.30302
[2] Beilinson A., Letter to C. Soulé (1982).
[3] Bloch S. , Algebraic cycles and higher K -theory , Adv. in Math. 61 ( 1986 ) 267 - 304 . MR 852815 | Zbl 0608.14004 · Zbl 0608.14004 · doi:10.1016/0001-8708(86)90081-2
[4] Bloch S. , The moving lemma for higher Chow groups , J. Algebraic Geom. 3 ( 1994 ) 537 - 568 . MR 1269719 | Zbl 0830.14003 · Zbl 0830.14003
[5] Bloch S., Lichtenbaum S., A spectral sequence for motivic cohomology , Preprint.
[6] Bourbaki N. , General Topology , Springer-Verlag , 1989 . · Zbl 0145.19302
[7] Bousfield A. , Friedlander E. , Homotopy theory of \Gamma -spaces, spectra, and bisimplicial sets , in: Geometric Applications of Homotopy Theory II , Lecture Notes in Math. , 658 , Springer-Verlag , 1978 , pp. 80 - 130 . Zbl 0405.55021 · Zbl 0405.55021
[8] Browder W. , Algebraic K-theory with coefficients Z/p , in: Geometric Applications of Homotopy Theory , Lecture Notes in Math. , 657 , Springer-Verlag , 1978 , pp. 40 - 84 . MR 513541 | Zbl 0386.18011 · Zbl 0386.18011
[9] Brown K. , Gersten S. , Algebraic K-theory as generalized sheaf cohomology , in: Algebraic K-theory, I: Higher K -theories , Lecture Notes in Math. , 341 , Springer-Verlag , 1973 , pp. 266 - 292 . MR 347943 | Zbl 0291.18017 · Zbl 0291.18017
[10] Friedlander E. , Voevodsky V. , Bivariant cycle cohomology , in: Voevodsky V. , Suslin A. , Friedlander E. (Eds.), Cycles, Transfers and Motivic Homology Theories , Annals of Math. Studies , 143 , 2000 , pp. 138 - 187 . MR 1764201 | Zbl 1019.14011 · Zbl 1019.14011
[11] Godement R. , Topologie algébrique et théorie des faisceaux , Hermann , Paris , 1958 . MR 102797 | Zbl 0080.16201 · Zbl 0080.16201
[12] Grayson D. , Weight filtrations via commuting automorphisms , K-Theory 9 ( 1995 ) 139 - 172 . MR 1340843 | Zbl 0826.19003 · Zbl 0826.19003 · doi:10.1007/BF00961457
[13] Jardine J. , Generalized étale cohomology theories , Birkhäuser Verlag , Basel , 1997 . MR 1437604 | Zbl 0868.19003 · Zbl 0868.19003
[14] Kieboom R. , A pullback theorem for cofibrations , Manuscripta Math. 58 ( 1987 ) 381 - 384 . Article | MR 893162 | Zbl 0617.55002 · Zbl 0617.55002 · doi:10.1007/BF01165895 · eudml:155228
[15] Landsburg S. , Some filtrations on higher K-theory and related invariants , K-theory 6 ( 1992 ) 431 - 457 . MR 1194843 | Zbl 0774.14006 · Zbl 0774.14006 · doi:10.1007/BF00961338
[16] Levine M. , Techniques of localization in the theory of algebraic cycles , J. Algebraic Geom. 10 ( 2001 ) 299 - 363 . MR 1811558 | Zbl 1077.14509 · Zbl 1077.14509
[17] Lillig J. , A union theorem for cofibrations , Arch. Math. 24 ( 1973 ) 410 - 415 . MR 334193 | Zbl 0274.55008 · Zbl 0274.55008 · doi:10.1007/BF01228231
[18] Massey W. , Products in exact couples , Ann. Math. 59 ( 1954 ) 558 - 569 . MR 60829 | Zbl 0057.15204 · Zbl 0057.15204 · doi:10.2307/1969719
[19] MacLane S. , Homology , Springer-Verlag , 1963 . MR 156879 | Zbl 0133.26502 · Zbl 0133.26502
[20] May P. , Simplicial Objects in Algebraic Topology , University of Chicago Press , 1967 . Zbl 0769.55001 · Zbl 0769.55001
[21] Morel F. , Voevodsky V. , A 1 -homotopy theory of schemes , Publ. Math. IHES 90 ( 2001 ) 45 - 143 . Numdam | MR 1813224 | Zbl 0983.14007 · Zbl 0983.14007 · doi:10.1007/BF02698831 · numdam:PMIHES_1999__90__45_0 · eudml:104163
[22] Oka S. , Multiplications on the Moore spectrum , Mem. Fac. Sci Kyushu, Ser. A 38 ( 1984 ) 257 - 276 . MR 760188 | Zbl 0552.55009 · Zbl 0552.55009 · doi:10.2206/kyushumfs.38.257
[23] Puppe D. , Bemerkungen über die Erweiterung von Homotopien , Arch. Math. 18 ( 1967 ) 81 - 88 . MR 206954 | Zbl 0149.20101 · Zbl 0149.20101 · doi:10.1007/BF01899475
[24] Quillen D. , Homotopical Algebra , Lecture Notes in Math. , 43 , Springer-Verlag , 1967 . MR 223432 | Zbl 0168.20903 · Zbl 0168.20903 · doi:10.1007/BFb0097438
[25] Quillen D. , Higher algebraic K-theory: I , in: Algebraic K -theory, I: Higher K -theories , Lecture Notes in Math. , 341 , Springer-Verlag , 1973 , pp. 85 - 147 . MR 338129 | Zbl 0292.18004 · Zbl 0292.18004
[26] Segal G. , Categories and cohomology theories , Topology 13 ( 1974 ) 293 - 312 . MR 353298 | Zbl 0284.55016 · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6
[27] Serre J.-P. , Algèbre locale multiplicités , Lecture Notes in Math. , 341 , Springer-Verlag , 1975 . MR 201468 | Zbl 0296.13018 · Zbl 0296.13018
[28] Strom A. , The homotopy category is a homotopy category , Arch. Math. 23 ( 1972 ) 435 - 441 . MR 321082 | Zbl 0261.18015 · Zbl 0261.18015 · doi:10.1007/BF01304912
[29] Suslin A. , Higher Chow groups and étale cohomology , in: Voevodsky V. , Suslin A. , Friedlander E. (Eds.), Cycles, Transfers and Motivic Homology Theories , Annals of Math. Studies , 143 , 2000 , pp. 237 - 252 . MR 1764203 | Zbl 1019.14001 · Zbl 1019.14001
[30] Suslin A. , Voevodsky V. , Singular homology of abstract algebraic varieties , Invent. Math. 123 ( 1996 ) 61 - 94 . MR 1376246 | Zbl 0896.55002 · Zbl 0896.55002 · doi:10.1007/BF01232367 · eudml:144336
[31] Suslin A. , Voevodsky V. , Bloch-Kato conjecture and motivic cohomology with finite coefficients , in: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998) , NATO Sci. Ser. C Math. Phys. Sci. , 548 , 1998 , pp. 117 - 189 . MR 1744945 | Zbl 1005.19001 · Zbl 1005.19001
[32] Thomason R., Trobaugh T., Higher algebraic K -theory of schemes and of derived categories, in: The Grothendieck Festschrift, Vol. III , in: Progr. Math., Vol. 88 , pp. 247-435. MR 1106918
[33] Voevodsky V. , Cohomological theory of presheaves with transfers , in: Voevodsky V. , Suslin A. , Friedlander E. (Eds.), Cycles, Transfers and Motivic Homology Theories , Annals of Math. Studies , 143 , 2000 , pp. 87 - 137 . MR 1764200 | Zbl 1019.14010 · Zbl 1019.14010
[34] Voevodsky V. , Triangulated category of motives over a field , in: Voevodsky V. , Suslin A. , Friedlander E. (Eds.), Cycles, Transfers and Motivic Homology Theories , Annals of Math. Studies , 143 , 2000 , pp. 188 - 238 . MR 1764202 | Zbl 1019.14009 · Zbl 1019.14009
[35] Voevodsky V., Motivic cohomology are isomorphic to higher Chow groups , Preprint. · Zbl 1057.14026 · doi:10.1155/S107379280210403X
[36] Voevodsky V., Mazza C., Weibel C., Lectures on motivic cohomology , Preprint. · Zbl 1115.14010
[37] Waldhausen F. , Algebraic K-theory of spaces , in: Algebraic and Geometric Topology (New Brunswick, NJ, 1983) , Lecture Notes in Math. , 1126 , Springer-Verlag , 1985 , pp. 318 - 419 . MR 802796 | Zbl 0579.18006 · Zbl 0579.18006
[38] Weibel C. , An Introduction to Homological Algebra , Cambridge University Press , 1994 . MR 1269324 | Zbl 0797.18001 · Zbl 0797.18001
[39] Weibel C. , Products in higher Chow groups and motivic cohomology , in: Algebraic K -theory (Seattle, WA, 1997) , Proceedings of Symposia in Pure Math. , 67 , 1999 , pp. 305 - 315 . MR 1743246 | Zbl 0942.19003 · Zbl 0942.19003
[40] Whitehead G. , Elements of Homotopy Theory , Springer-Verlag , 1978 . MR 516508 | Zbl 0406.55001 · Zbl 0406.55001
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