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Zbl 1073.14034
Kahn, Bruno
Rational and numerical equivalences on certain abelian type varieties over a finite field. (Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 36, No. 6, 977-1002 (2003). ISSN 0012-9593

Summary: We prove that if $X$ is a smooth projective variety $X$ ``of abelian type" over a finite field $k$ for which the Tate conjecture holds (e.g. a product of elliptic curves [cf. {\it M. Spiess}, Math. Ann. 314, 285--290 (1999; Zbl 0941.11026)]), rational and numerical equivalences agree on $X$. The proof uses {\it C. Soulé}~'s ideas [Math. Ann. 268, 317--345 (1984; Zbl 0573.14001)], {\it U. Jannsen}'s semi-simplicity theorem [Invent. Math. 107, 447--452 (1992; Zbl 0762.14003)], and a result of {\it Y. André} and the author [Rend. Sem. Math. Univ. Padova 108, 107--291 (2002)] inspired by {\it S.I. Kimura}'s results on finite-dimensional Chow motives [J. Alg. Geom., to appear]. We give some consequences, among which: the conjectures of {\it S. Lichtenbaum} [in: Number theory, Lect. Notes Math. 1068, 127--138 (1984; Zbl 0591.14014)] hold true for $X$, the second Chow group of $X$ is finitely generated, the Beilinson--Soulé conjecture holds in weight $n$ for the function field of $X$ provided $n \leqslant 2$ or dim $X \leqslant 2$, Gersten's conjecture holds for discrete valuation rings with residue field such a function field if dim $X \leqslant 2$.

MSC 2000:: *14C35 Appl. of methods of algebraic K-theory
11G25 Varieties over finite and local fields
14C25 Algebraic cycles
14F20 Grothendieck cohomology and topology
14F42 Motivic cohomology
19E15 Algebraic cycles (K-theory)

Citations: Zbl 0941.11026; Zbl 0573.14001; Zbl 0762.14003; Zbl 0591.14014

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