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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].

##### MSC:
 14C25 Algebraic cycles 14C22 Picard groups 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry