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Torsion algebraic cycles and algebraic \(K\)-theory. (Cycles algébriques de torsion et \(K\)-théorie algébrique.) (French) Zbl 0806.14002
Colliot-Thélène, Jean-Louis et al., Arithmetic algebraic geometry. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Trento, Italy, June 24 - July 2, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1553, 1-49 (1993).
Let \(X\) be a smooth irreducible algebraic variety over the field \(k\). Then one can define its Picard group \(\text{Pic} (X)\). \(\text{Pic} (X)\) is a quotient of the divisor group \(\text{Div} (X)\), i.e. it is defined by the codimension one subvarieties of \(X\). For absolutely irreducible and complete \(X\) and the field extension \(k \subset F\) the natural map \(\text{Pic} (X) \to \text{Pic} (X \times_ kF)\) is injective. Also, in this case there exists an abelian variety \(J = \underline {\text{Pic}}^ 0_{X/k}\) defined over \(k\) which ‘represents’ \(\text{Pic} (X)\) in the sense that there exists a \(\text{Gal} (\overline k/k)\)- equivariant short exact sequence \[ 0 \to J (\overline k) \to \text{Pic} (\overline X) \to NS(\overline X\to 0, \] where \(\overline X = X \times_ k \overline k\) and \(NS (\overline X)\) is an abelian group of finite type, the Néron-Severi group of \(X\). One also has various finiteness properties of the torsion part \(\text{Pic} (X)_{\text{tors}}\) of \(\text{Pic} (X)\), eventually depending on the nature of \(k\).
In higher codimension one tries to generalize the notion of the Picard group and one is led to define the Chow groups \(CH_ i (X)\), or, in case \(X\) is equidimensional of dimension \(d\), \(CH^ j(X) = CH_{d-j} (X)\). One has \(CH^ 1(X) = \text{Pic} (X)\). However, in codimension \(\geq 2\) one has no injectivity or representability results as for the Picard group. The main theme of the paper is to study torsion phenomena for \(CH^ 2\) and to give an overview of the author’s results, his joint work with W. Raskind, J.-J. Sansuc and C. Soulé, work of P. Salberger, S. Saito and of N. Suwa. As one might expect, important ingredients such as the Merkurev-Suslin theorem and ideas of S. Bloch on algebraic cycles and algebraic \(K\)-theory are at the basis of these results. A basic theorem, following from the work of these people (and A. Ogus), says that for smooth \(X/k\) and an integer \(n\) invertible in \(k\), the subgroup \(_ nCH^ 2(X)\) of elements in \(CH^ 2(X)\) killed by multiplication by \(n\) is a subquotient of the étale cohomology group \(H^ 3_{\acute e t} (X, \mu^{\otimes 2}_ n)\), and for \(\ell\) prime to \(\text{char} (k)\), the group \(CH^ 2(X)_{\ell \text{ - tors}}\) is a subquotient of \(H^ 3_{\acute e t} (X, \mathbb{Q}_ \ell/ \mathbb{Z}_ \ell (2))\). This result (and the ideas behind it) is used at various places in the sequel where the torsion of \(CH^ i(X)\) \((i=2\) most of the time) is discussed for (i) \(k\) a separately closed field: (ii) \(k\) a finite field; (iii) \(k\) a number field; and (iv) \(k\) a local field, respectively.
In case (i) it is shown, among other things, that for smooth connected \(X\) of dimension \(d\) and \(n\) invertible in \(k\), \(_ nCH^ d(X)\) is finite. Also, Roitman’s theorem on torsion zero-cycles is shown without reducing to the case of surfaces. The main result in case (ii) for smooth projective and geometrically connected \(X\) says that \(CH^ 2(X)_{\text{tors}}\) is finite. Here one uses a Hochschild-Serre spectral sequence and Deligne’s theorem on the Weil conjecture with twisted coefficients. Also, a sketch of the proof of the theorem of Kato and Saito which says that for such \(X\), the group of degree zero 0-cycles modulo rational equivalence is finite and isomorphic to the geometric abelian fundamental group \(\pi_ 1^{ab, \text{geom}} (X)\) of \(X\), is given. In case (iii) the most general known result is proved in various steps. It is due to the author and Raskind, and Salberger. It says that for smooth projective geometrically connected \(X\) over the number field \(k\) with \(H^ 2(X, {\mathcal O}_ X) = 0\), \(CH^ 2 (X)_{\text{tors}}\) is finite. Saito has a different approach, also discussed in the underlying paper, to obtain this result. In case (iv), \(k\) a finite extension of \(\mathbb{Q}_ p\), one has the general result, due to the author, Sansuc and Soulé, establishing the finiteness of \(_ nCH^ 2(X)\) for any integer \(n>0\), where \(X\) is a smooth \(k\)-variety. Several other finiteness results, assuming that \(H^ 2 (X, {\mathcal O}_ X) = 0\), are discussed and proved. In the final section, again on varieties over number fields, Salberger’s ideas on the finiteness of the exponent of \(CH^ 2 (X)_{\text{tors}}\) for \(X\) with \(H^ 2 (X, {\mathcal O}_ X) = 0\), used in the proof of the main result in case (iii), are explained.
For the entire collection see [Zbl 0780.00022].

14C25 Algebraic cycles
14C22 Picard groups
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry