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**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 \(\dim X= d\).

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 \(\dim X= d\).

Reviewer: Claudio Pedrini (Genova)

### MSC:

14F42 | Motivic cohomology; motivic homotopy theory |

### Citations:

Zbl 1057.14028### References:

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