×

zbMATH — the first resource for mathematics

Groupes de Chow et K-théorie de variétés sur un corps fini. (French) Zbl 0573.14001
The paper is concerned with relations between the algebraic K-theory and the Chow groups of projective varieties over finite fields. The notion of motif due to Grothendieck and Manin is used.
Reviewer: L.Vaserstein

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G15 Finite ground fields in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bass, H.:K 2 des corps globaux (d’après Tate, Garland, ...), Séminaire Bourbaki, No. 394 (1971). Lecture Notes in Mathematics, Vol. 244. Berlin, Heidelberg, New York: Springer 1972
[2] Colliot-Thélène, J.L., Sansuc, J.J., Soulé, C.: Torsion dans le groupe de Chow de codimension deux. Duke Math. J.50, 763-801 (1983) · Zbl 0574.14004 · doi:10.1215/S0012-7094-83-05038-X
[3] Deligne, P.: La conjecture de Weil. I. Publ. Math. IHES43, 273-308 (1974)
[4] Deligne, P.: Cohomologie étale. In: SGAIV 1/2., Lecture Notes in Mathematics, Vol. 569. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0349.14008
[5] Dwyer, W., Friedlander, E.: EtaleK-theory and arithmetic. Preprint (1983)
[6] Fulton, W.: Rational equivalence on singular varieties. Publ. Math. IHES45, 147-167 (1975) · Zbl 0332.14002
[7] Gillet, H.: Riemann-Roch theorems for higher algebraicK-theory. Adv. Math.40, 203-289 (1981) · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0
[8] Grayson, D.: Products inK-theory and intersecting algebraic cycles. Invent. Math.47, 71-84 (1978) · Zbl 0394.14004 · doi:10.1007/BF01609480
[9] Grothendieck, A.: Théorie des intersections et théorème de Riemann-Roch, SGA VI. In: Lecture Notes in Mathematics, Vol. 225. Berlin, Heidelberg, New York: Springer 1971
[10] Harder, G.: Die KomohologieS-arithmetischer Gruppen über Funktionenkörper. Invent. Math.42, 135-175 (1977) · Zbl 0391.20036 · doi:10.1007/BF01389786
[11] Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, Vol. 52. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[12] Jouanolou, J.P.: Riemann-Roch sans dénominateurs. Invent. Math.11, 15-26 (1970) · Zbl 0199.55901 · doi:10.1007/BF01389802
[13] Kato, K., Saito, S.: Unramified class field theory of arithmetical sur faces. Preprint (1982)
[14] Katsura, T., Shioda, T.: On Fermat varieties. Tôhoku Math. J.31, 97-115 (1979) · Zbl 0415.14022 · doi:10.2748/tmj/1178229881
[15] Kleiman, S.L.: Motives, dans ?algebraic geometry?. pp. 53-82. Oslo: Wolters-Noordhoff 1970
[16] Kratzer, C.: ?-structure enK-théorie algébrique. Commun. Math. Helv.55, 233-254 (1980) · Zbl 0444.18008 · doi:10.1007/BF02566684
[17] Lang, S.: Sur les sériesL d’une variété algébrique. Bull. Soc. Math. France84, 385-407 (1956) · Zbl 0089.26301
[18] Lichtenbaum, S.: Values of zeta functions, etale cohomology and algebraicK-theory. In: Lecture Notes in Mathematics, Vol. 342, pp. 489-499. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0284.12005
[19] Manin, Y.I.: Correspondences, motives and monoïdal transformations. Mat. Sborn.77, 475-507 (1970), AMS Transl.
[20] Mumford, D.: Abelian varieties. Oxford: Oxford Univ. Press 1970 · Zbl 0223.14022
[21] Nisnevic, L.: Number of points of algebraic varieties over finite fields (en russe) Dokl. Akad. Nauk99, 17-20 (1954)
[22] Quillen, D.: On the cohomology andK-theory of the general linear groups over a finite field. Ann. Math.96, 552-486 (1972) · Zbl 0249.18022 · doi:10.2307/1970825
[23] Quillen, D.: AlgebraicK-theory I. In: Lecture Notes in Mathematics, Vol 341, pp. 85-147. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0292.18004
[24] Quillen, D.: Finite generation of the groupsK i of rings algebraic integers. In: Lecture Notes in Mathematics, Vol. 341, pp. 179-210. Berlin, Heidelberg, New York: Springer 1973; Finite generation ofK-groups of a curve over a finite field, rédigé par D. Grayson. In: Lecture Notes in Mathematics, Vol. 966, pp. 69-90. Berlin, Heidelberg, New York: Springer 1982
[25] Quillen, D.: Higher algebraicK-theory. Actes ICM, pp. 171-176. Vancouver (1974)
[26] Shermenev, A.M.: The motif of an abelian variety. Functional Analysis8, 55-61 (1974)
[27] Soulé, C.:K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math.55, 251-295 (1979) · Zbl 0437.12008 · doi:10.1007/BF01406843
[28] Soulé, C.: Opérations enK-théorie algébrique. Prépublication (1983)
[29] Tate, J.: Algebraic cycles and poles of zeta functions. Dans: Arithmetical algebraic geometry, pp. 93-110. New York: Harper and Row 1965 · Zbl 0213.22804
[30] Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math.2, 134-144 (1966) · Zbl 0147.20303 · doi:10.1007/BF01404549
[31] Weil, A.: Variétés abéliennes et courbes algébriques. Paris: Hermann 1948
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.