## On motivic cohomology with $$\mathbb{Z}/l$$-coefficients.(English)Zbl 1236.14026

This paper is a fundamental one which provides the final step to the proof of a long standing conjecture, the Bloch-Kato conjecture, which was announced several years ago.
Theorem 1. Let $$k$$ be a field of characteristic $$\neq l$$, then the norm residue homomorphism $K^M_n(k)/l\to H^n_{et}(k, \mu^{\otimes n}_l)$ is an isomorphism for all $$n$$.
This result is a generalization of the particular case $$l=2$$, whose proof appeared in [the author, Publ. Math., Inst. Hautes Étud. Sci. 98, 59–104 (2003; Zbl 1057.14028)]. Here one of the main new results is a theorem which relates two types of cohomological operations in motivic cohomology: one is defined in terms of symmetric power functors in the categories of relative Tate motives and another one in terms of the reduced power operation. Let $$R$$ be a ring such that all primes but $$l$$ are invertible in $$R$$, e.g $$R= \mathbb{Z}_{(l)}$$ or $$R= \mathbb{Z}/l$$. Using several results about the structure of the symmetric powers $$S^i(M)$$, for $$i<l$$, when $$M$$ is a Tate motive of the form $R(p)[2q]\to M\to R(p)[2q+ 1]$ one defines a cohomological operation $\Phi_{l-1}: H^{2q+ 1,p}(+,R)\to H^{2q+l,pl}(-, R).$ Then the author proves the following
Theorem 2. For any $$n\geq 0$$ there exists $$c\in(\mathbb{Z}/l)^*$$ such that for any $$\alpha\in\widetilde H^{2n+1,n}({\mathcal X},\mathbb{Z}/l)$$ one has $\Phi_{l-1}(\alpha)= c\beta P^n(\alpha),$ where $$\beta$$ is the Bockstein homomorphism, $$P^n$$ is the motivic reduced power operation and $$\widetilde H^{2n+1,n}({\mathcal X},\mathbb{Z}/l)$$ are the motivic cohomology groups on pointed simplicial schemes.
The proof of the above theorem has a long history: a proof of it appeared in the first version of this paper in 2003 and was based on a lemma, the validity of which is under serious doubt. Then in 2001 C. Weibel suggested another approach of the same result. In the present paper the author uses a modified version of Weibel’s approach to prove a new lemma which finally gave a complete proof of Theorem 2.
Another fundamental ingredient in the proof of Theorem 1 is the notion of generalized Rost motives, which unify the previously known families of motives, the Rost motives, which appear in the proof for $$l=2$$, and the motives of cyclic field extensions of prime degree. Every embedded symplicial scheme $${\mathcal X}$$ which has a non-trivial cohomology class of certain bidegree and such that the corresponding class of varieties contains a $$\nu_n$$-variety defines a generalized Rost motive. A smooth projective variety $$X$$ is a $$\nu_n$$-variety if $$\dim X= l^n- 1$$ and $$\deg(s_{l^n-1}(X)\neq 0\pmod{l^2}$$. Here $$s_d(X)$$ denotes the $$d$$-th Milnor class of $$X$$ in $$H^{2d,d}(X,\mathbb{Z})$$, for $$\dim X= d$$.

### MSC:

 14F42 Motivic cohomology; motivic homotopy theory

Zbl 1057.14028
Full Text:

### References:

 [1] T. Geisser and M. Levine, ”The $$K$$-theory of fields in characteristic $$p$$,” Invent. Math., vol. 139, iss. 3, pp. 459-493, 2000. · Zbl 0957.19003 · doi:10.1007/s002220050014 [2] T. Geisser and M. Levine, ”The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky,” J. Reine Angew. Math., vol. 530, pp. 55-103, 2001. · Zbl 1023.14003 · doi:10.1515/crll.2001.006 [3] D. Kraines, ”Massey higher products,” Trans. Amer. Math. Soc., vol. 124, pp. 431-449, 1966. · Zbl 0146.19201 · doi:10.2307/1994385 [4] M. Lazard, ”Lois de groupes et analyseurs,” Ann. Sci. École Norm. Sup., vol. 72, pp. 299-400, 1955. · Zbl 0068.02702 [5] J. P. May, ”The additivity of traces in triangulated categories,” Adv. Math., vol. 163, iss. 1, pp. 34-73, 2001. · Zbl 1007.18012 · doi:10.1006/aima.2001.1995 [6] J. Milnor, ”The Steenrod algebra and its dual,” Ann. of Math., vol. 67, pp. 150-171, 1958. · Zbl 0080.38003 · doi:10.2307/1969932 [7] A. Suslin and S. Joukhovitski, ”Norm varieties,” J. Pure Appl. Algebra, vol. 206, iss. 1-2, pp. 245-276, 2006. · Zbl 1091.19002 · doi:10.1016/j.jpaa.2005.12.012 [8] V. Voevodsky, ”Triangulated categories of motives over a field,” in Cycles, Transfers, and Motivic Homology Theories, Princeton, NJ: Princeton Univ. Press, 2000, vol. 143, pp. 188-238. · Zbl 1019.14009 [9] V. Voevodsky, ”Reduced power operations in motivic cohomology,” Publ. Math. Inst. Hautes Études Sci., vol. 98, pp. 1-57, 2003. · Zbl 1057.14027 · doi:10.1007/s10240-003-0009-z [10] V. Voevodsky, ”Motivic cohomology with $${\mathbf Z}/2$$-coefficients,” Publ. Math. Inst. Hautes Études Sci., vol. 98, pp. 59-104, 2003. · Zbl 1057.14028 · doi:10.1007/s10240-003-0010-6 [11] V. Voevodsky, On motivic cohomology with $${\mathbf Z}/l$$-coefficients, 2003. · Zbl 1057.14028 · doi:10.1007/s10240-003-0010-6 [12] V. Voevodsky, ”Simplicial radditive functors,” J. K-Theory, vol. 5, iss. 2, pp. 201-244, 2010. · Zbl 1194.55021 · doi:10.1017/is010003026jkt097 [13] V. Voevodsky, ”Motives over simplicial schemes,” J. K-Theory, vol. 5, iss. 1, pp. 1-38, 2010. · Zbl 1194.14029 · doi:10.1017/is010001030jkt107 [14] V. Voevodsky, ”Motivic Eilenberg-Maclane spaces,” Publ. IHES, vol. 112, 2010. · Zbl 1227.14025 · doi:10.1007/s10240-010-0024-9 [15] C. Weibel, Patching the norm residue isomorphism theorem, 2007. · Zbl 1214.14018
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.