# zbMATH — the first resource for mathematics

Torsion cycles of codimension 2 and $$\ell$$-adic realizations of motivic cohomology. (English) Zbl 0833.14004
David, Sinnou (ed.), Séminaire de théorie des nombres, Paris, France, 1991-92. Boston, MA: Birkhäuser. Prog. Math. 116, 247-277 (1994).
Let $$X$$ be a smooth projective variety defined over a number field and $$\text{Ch}^r (X)$$ be the Chow group of cycles of codimension $$r$$ on $$X$$ modulo linear equivalence. Then $$\text{Ch}^1 (X)$$ is finitely generated by the theorems of Mordell-Weil and Néron-Séveri. One expects similar results when $$r > 1$$, but no one seems to have any idea how to prove that these groups are of finite rank except in trivial cases. – The situation is somewhat better for the torsion part $$\text{Ch}^2 (X)_{\text{tors}}$$ of $$\text{Ch}^2 (X)$$. J.-L. Colliot-Thélène [Invent. Math. 71, 1-20 (1983; Zbl 0527.14011)] proved that $$\text{Ch}^2 (X)_{\text{tors}}$$ is finite for arbitrary rational surfaces over number fields. I found in the end of 1987 another proof of his theorem based on the following sequence due to Bloch and Merkurjev-Suslin: $0 \to H^1_{Zar} (X, {\mathcal K}_2) \otimes \mathbb{Q}_\ell/ \mathbb{Z}_\ell \to NH^3_{\text{ét}} (X, \mathbb{Q}_\ell/ \mathbb{Z}_\ell (2)) \to\text{Ch}^2 (X) \{\ell \} \to 0. \tag{0.1}$ Here Ch$$^2 (X) \{\ell\}$$ is the $$\ell$$-primary part of $$\text{Ch}^2 (X)_{\text{tors}}$$ for some prime $$\ell$$, $${\mathcal K}_2$$ the Zariski sheaf of Milnor’s $$K_2$$-groups and $$NH^3_{\text{ét}} (X, \mathbb{Q}_\ell/ \mathbb{Z}_\ell (2))$$ the kernel of the restriction map $H^3_{\text{ét}} (X, \mathbb{Q}_\ell/ \mathbb{Z}_\ell (2)) \to H^3_{\text{ét}} (k(X), \mathbb{Q}_\ell/ \mathbb{Z}_\ell (2)).$ The aim of this paper is to extend this proof in order to prove the following main theorem (0.2):
Let $$X$$ be a smooth, projective, geometrically irreducible variety defined over a number field. Suppose that $$H^2_{\text{Zar}} (X, {\mathcal O}_X) = 0$$. Then $$\text{Ch}^2 (X)_{\text{tors}}$$ is finite. In a seminar at Paris VII in October 1989, Colliot-Thélène announced the following result of him and Raskind and described the main ideas of the proof.
Theorem (0.3) (Colliot-Thélène and Raskind): Let $$X$$ be a smooth, projective, geometrically irreducible variety defined over a number field such that $$H^2_{\text{Zar}} (X, {\mathcal O}_X) = 0$$. Then the $$n$$- torsion subgroup $$_n\text{Ch}^2 (X)$$ of $$\text{Ch}^2 (X)$$ is finite for any positive integer $$n$$.
Theorem (0.4) (Salberger): Let $$X$$ be a smooth, projective, geometrically irreducible variety defined over a number field such that $$H^2_{\text{Zar}} (X,O_X) = 0$$. Then there exists an integer $$n$$ such that $$\text{Ch}^2 (X)_{\text{tors}}$$ is contained in $$_n\text{Ch}^2 (X)$$.
The combination of (0.3) and (0.4) clearly led to a proof of (0.2). – The first published proof of (0.2) gave J.-L. Colliot-Thélène and W. Raskind [Invent. Math. 105, No. 2, 221-245 (1991; Zbl 0752.14004)]. In that paper, they describe their proof of (0.3) as well as a variant of the proof of (0.4) which I communicated to Colliot- Thélène.
The aim of the present paper is to record my original proofs for (0.3) and (0.4), hence for (0.2).
For the entire collection see [Zbl 0807.00013].

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14C25 Algebraic cycles 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14A20 Generalizations (algebraic spaces, stacks) 14C20 Divisors, linear systems, invertible sheaves