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].

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 |