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

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.

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
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\textit{V. Voevodsky}, Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)

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