Motivic cohomology with \(\mathbb Z/2\)-coefficients. (English) Zbl 1057.14028

Voevodsky, Vladimir

Publ. Math., Inst. Hautes Étud. Sci. 98, 59-104 (2003).

Let \(k\) be a field and \(\ell\) a prime number different from \(\text{char}(k)\). For a fixed separable closure \(k_{\text{sep}}\) of \(k\), the group \(\mu_\ell\) of \(\ell\)th roots of unity in \(k_{\text{sep}}\) is a \(\text{Gal}(k_{\text{sep}}/k)\)-module occuring in the short exact Kummer sequence \[ 1\to \mu_\ell\to k^*_{\text{sep}}@> z^\ell>> k^*_{\text{sep}}\to 1. \] By the classical work of J. Milnor, H. Bass and J. Tate there is then a natural homomorphism \[ K^M_*(k)\to H^*(k, \mu^{\otimes_*}_\ell) \] from the \(K\)-theoretic Milnor ring to the group cohomology ring, which factors through a map \[ K^M_*(k)/\ell\to H^*(k, \mu^{\otimes_*}_\ell) \] called the norm residue homomorphism.
There is a recent general conjecture about this particular homomorphism, the so-called Bloch-Kato conjecture, stating that the norm residue homomorphism is an isomorphism of graded rings for any field \(k\) with \(\text{char}(k)\neq\ell\). This conjecture generalizes the famous Milnor conjecture [J. Milnor, Invent. Math. 9, 318–344 (1970; Zbl 0199.55501)] predicting that this homomorphism is at least injective in degree two. The general Bloch-Kato conjecture was formulated in 1980 [K. Kato, J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 603–683 (1980; Zbl 0463.12006)], and partial results in confirming it have been obtained by A. Merkurjev [Sov. Math. Dokl. 24, 546–551 (1981; Zbl 0496.16020)] for \(\ell= 2\) and degree 2, by A. Merkurjev and A. Suslin [Math. USSR. Izv. 21, 307–340 (1983; Zbl 0525.18008)] for all \(\ell\) and degree 2, by M. Rost [On Hilbert Satz 90 for \(K_3\) for degree-two extensions, Preprint, http://www.math.uni-bielefeld.de/~rost/K3-86.html] for \(\ell= 2\) and degree 3, and independently by A. Merkurjev and A. Suslin [Math. USSR Izv. 36, 541–565 (1991; Zbl 0725.19003)] for the same case of \(\ell= 2\) and degree 3.
One of the main objectives of the paper under review is to present a proof of the Bloch-Kato conjecture for \(\ell= 2\) in general. The author’s approach is based on earlier ideas introduced by S. Lichtenbaum [in: Number theory, Lect. Notes Math. 1068, 127–138 (1984; Zbl 0591.14014)] and A. Beilinson [in: \(K\)-theory, arithmetic and geometry, Semin. Moscow Univ. 1984–86, Lect. Notes Math. 1289, 1–26 (1987; Zbl 0651.14002)] in the 1980s. Back then they formulated several conjectures describing properties of certain motivic complexes of sheaves, which became known as the Beilinson-Lichtenbaum conjectures. One of these conjectures can be shown to imply the Bloch-Kato conjecture, and that is the approach which the author elaborates in the present paper. To this end, he developes a refined analysis of the motivic cohomology with coefficients in \(\mathbb{Z}/\ell\) to a great extent, relates then his results to the Beilinson-Lichtenbaum conjectures, and finally places the Bloch-Kato conjecture for \(\ell= 2\) into this motivic context. The whole treatise culminates, in the last section, in the main result (Theorem 7.4.) establishing the 2-local version of the Bloch-Kato conjecture, together with a series of applications and corollaries.
The author’s first version of his main result appeared in 1995 [V. Voevodsky, Bloch-Kato conjecture for \(\mathbb{Z}/\ell\)-coefficients and algebraic Morava \(K\)-theories, Electronic paper,
http://www.math.uiuc.edu/K-theory/76], and a second version was published in 1996 [V. Voevodsky, The Milnor conjecture, Electronic paper, http://www.math.uiuc.edu/K-theory/170], where an approach via motivic cohomology operations was carried out. A full account of the theory of motivic cohomology operations used in the latter version, which also transpired a proof of the long-standing Milnor conjecture, was recently provided by the author in the preceding paper [V. Voevodsky, Publ.Math., Inst. Hautes Étud. Sci. 98, 1–57 (2003; Zbl 1057.14027)]. The third version for \(\ell= 2\), presented in the paper under review, is close to the second one, but it is technically much simpler and more elegant. The most important simplification is due to the fact that the use of motivic stable homotopy and the topological realization functor could now be completely eliminated. In addition to a new, simplified proof of the Bloch-Kato conjecture for \(\ell= 2\), this important paper contains several new results on motivic cohomology which are interesting in their own right.

Reviewer: Werner Kleinert (Berlin)

MSC:

14F42	Motivic cohomology; motivic homotopy theory
19E15	Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
55S10	Steenrod algebra

Keywords:

norm residue homomorphism; Bloch-Kato conjecture; Milnor conjecture; motivic cohomology operations; Beilinson-Lichtenbaum conjectures

Citations:

Zbl 1057.14027; Zbl 0725.19003; Zbl 0496.16020; Zbl 0525.18008; Zbl 0463.12006; Zbl 0591.14014; Zbl 0651.14002; Zbl 0199.55501

References:

[1]	H. Bass and J. Tate, The Milnor ring of a global field, In K-theory II, Lecture Notes in Math. 342 (1973), pp. 349–446, Springer. · Zbl 0299.12013
[2]	A. Beilinson, Height pairing between algebraic cycles, In K-theory, Arithmetic and Geometry, Lecture Notes in Math. 1289 (1987), pp. 1–26, Springer. · Zbl 0651.14002
[3]	S. Bloch, Lectures on algebraic cycles, Duke Univ. Press, 1980. · Zbl 0436.14003
[4]	S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, www.math.uiuc.edu/K-theory/062, 1994.
[5]	S. Borghesi, Algebraic Morava K-theories, Invent. Math., 151 (2) (2003), 381–413. · Zbl 1030.55003
[6]	E. M. Friedlander and A. Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4), 35 (6) (2002), 773–875. · Zbl 1047.14011
[7]	T. Geisser and M. Levine, The K-theory of fields in characteristic p, Invent. Math., 139 (3) (2000), 459–493. · Zbl 0957.19003
[8]	T. Geisser and M. Levine, The Bloch-Kato conjecture and a theorem of Suslin–Voevodsky, J. Reine Angew. Math., 530 (2001), 55–103. · Zbl 1023.14003
[9]	R. Hartshorne, Algebraic Geometry, Heidelberg: Springer, 1971.
[10]	J.-P. Jouanolou, Une suite exacte de Mayer-Vietoris en K-theorie algebrique, Lecture Notes in Math. 341 (1973), pp. 293–317.
[11]	B. Kahn, La conjecture de Milnor (d’après V. Voevodsky), Astérisque, (245): Exp. No. 834, 5 (1997), 379–418. Séminaire Bourbaki, Vol. 1996/97.
[12]	N. Karpenko, Characterization of minimal Pfister neighbors via Rost projectors, J. Pure Appl. Algebra, 160 (2001), 195–227. · Zbl 0998.11016
[13]	K. Kato, A generalization of local class field theory by using K-groups, II, J. Fac. Sci., Univ Tokyo, 27 (1980), 603–683. · Zbl 0463.12006
[14]	T. Y. Lam, The algebraic theory of quadratic forms, Reading, MA: The Benjamin/Cummings Publ., 1973. · Zbl 0259.10019
[15]	S. Lichtenbaum, Values of zeta-functions at non-negative integers, In Number theory, Lecture Notes in Math. 1068 (1983), pp. 127–138, Springer.
[16]	H. R. Margolis, Spectra and Steenrod algebra, North-Holland, 1983.
[17]	V. Voevodsky, C. Mazza and C. Weibel, Lectures on motivic cohomology, I, www.math.uiuc.edu/K-theory/486, 2002.
[18]	A. Merkurjev, On the norm residue symbol of degree 2, Sov. Math. Dokl., (1981), 546–551. · Zbl 0496.16020
[19]	A. Merkurjev and A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR Izvestiya, 21 (1983), 307–340. · Zbl 0525.18008
[20]	A. Merkurjev and A. Suslin, The norm residue homomorphism of degree three, Math. USSR Izvestiya, 36(2) (1991), 349–367. · Zbl 0716.19002
[21]	J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math., 9 (1970), 318–344. · Zbl 0199.55501
[22]	J. Milnor, Introduction to Algebraic K-theory, Princeton, N.J.: Princeton Univ. Press, 1971. · Zbl 0237.18005
[23]	F. Morel and V. Voevodsky, A 1-homotopy theory of schemes, Publ. Math. IHES, (90) (1999), 45–143. · Zbl 0983.14007
[24]	Y. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, In Algebraic K-theory: connections with geometry and topology, pp. 241–342, Dordrecht: Kluwer Acad. Publ., 1989. · Zbl 0715.14009
[25]	D. Orlov, A. Vishik and V. Voevodsky, An exact sequence for Milnor’s K-theory with applications to quadratic forms, www.math.uiuc.edu/K-theory/0454, 2000.
[26]	D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Ann. of Math. Studies 128. Princeton, 1992. · Zbl 0774.55001
[27]	M. Rost, Hilbert 90 for K3 for degree-two extensions, www.math.ohio-state.edu/ost/K3-86.html, 1986.
[28]	M. Rost, On the spinor norm and A0(X,K1) for quadrics, www.math.ohio-state.edu/ost/spinor.html, 1988.
[29]	M. Rost, Some new results on the Chowgroups of quadrics, www.math.ohio-state.edu/ost/chowqudr.html, 1990.
[30]	M. Rost, The motive of a Pfister form, www.math.ohio-state.edu/ost/motive.html, 1998.
[31]	A. Suslin, Algebraic K-theory and the norm residue homomorphism, J. Soviet Math., 30 (1985), 2556–2611. · Zbl 0566.12016
[32]	A. Suslin, Higher Chow groups and etale cohomology, In Cycles, transfers and motivic homology theories, pp. 239–254, Princeton: Princeton Univ. Press, 2000. · Zbl 1019.14001
[33]	A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, In The arithmetic and geometry of algebraic cycles, pp. 117–189, Kluwer, 2000. · Zbl 1005.19001
[34]	J. Tate, Relations between K2 and Galois cohomology, Invent. Math., 36 (1976), 257–274. · Zbl 0359.12011
[35]	V. Voevodsky, Bloch-Kato conjecture for Z/2-coefficients and algebraic Morava K-theories, www.math.uiuc.edu/K-theory/76, 1995.
[36]	V. Voevodsky, The Milnor Conjecture, www.math.uiuc.edu/K-theory/170, 1996.
[37]	V. Voevodsky, The A 1-homotopy theory, In Proceedings of the international congress of mathematicians, 1 (1998), pp. 579–604, Berlin. · Zbl 0907.19002
[38]	V. Voevodsky, Cohomological theory of presheaves with transfers, In Cycles, transfers and motivic homology theories, Annals of Math Studies, pp. 87–137, Princeton: Princeton Univ. Press, 2000. · Zbl 1019.14010
[39]	V. Voevodsky, Triangulated categories of motives over a field, In Cycles, transfers and motivic homology theories, Annals of Math Studies, pp. 188–238, Princeton: Princeton Univ. Press, 2000. · Zbl 1019.14009
[40]	V. Voevodsky, Lectures on motivic cohomology 2000/2001 (written by Pierre Deligne), www.math.uiuc.edu/ K-theory /527, 2000/2001. · Zbl 1005.19001
[41]	V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not., (7) (2002), 351–355. · Zbl 1057.14026
[42]	V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. IHES (this volume), 2003. · Zbl 1057.14027
[43]	V. Voevodsky, E. M. Friedlander and A. Suslin, Cycles, transfers and motivic homology theories, Princeton: Princeton University Press, 2000. · Zbl 1021.14006