Karpenko, Nikita A. Weil transfer of algebraic cycles. (English) Zbl 1047.14004 Indag. Math., New Ser. 11, No. 1, 73-86 (2000). Summary: Let \(L/F\) be a finite separable field extension of degree \(n\), \(X\) a smooth quasi-projective \(L\)-scheme, and \(R(X)\) the Weil transfer of \(X\) with respect to \(L/F\). The map \(Z\to R(Z\)) of the set of simple cycles \(Z\subset X\) extends in a natural way to a map \(Z(X)\to Z(R(X))\) on the whole group of algebraic cycles \(Z(X)\). This map factors through the rational equivalence of cycles and induces this way a map of the Chow groups \(\text{CH}(X)\to \text{CH}(R(X))\), which, in its turn, produces a natural functor of the categories of Chow correspondences \(\mathcal{CV}(L)\to \mathcal{CV}(F)\). Restricting to the graded components, one has a map \(Z^\ast (X)\to Z_n^\ast (R(X))\), which produces a functor of the categories of degree 0 Chow correspondences \(\mathcal{CV}^0(L)\to\mathcal{CV}^0(F)\), a functor of the categories of the Grothendieck-Chow motives \(\mathcal M(L)\to\mathcal M(F)\), as well as functors of several other classical motivic categories. Cited in 15 Documents MSC: 14C25 Algebraic cycles 14C15 (Equivariant) Chow groups and rings; motives PDFBibTeX XMLCite \textit{N. A. Karpenko}, Indag. Math., New Ser. 11, No. 1, 73--86 (2000; Zbl 1047.14004) Full Text: DOI References: [1] Borel, A.; Serre, J.-P., Théorèmes de finitude en cohomology galoisienne, Comment. Math. Helv., 39, 111-164 (1964-1965) · Zbl 0143.05901 [2] Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron Models, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21 (1990), Springer: Springer Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong Kong), 3. Folge · Zbl 0705.14001 [3] Demazure, M.; Gabriel, P., Groupes Algébriques I, (Masson & Cie (1970), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam), Paris [4] Eilenberg, S.; MacLane, S., On the groups \(H(Π,n)\), II. Methods of computations, Ann. of Math., 60, 49-139 (1954) · Zbl 0055.41704 [5] Fulton, W., Intersection Theory (1984), Springer: Springer Berlin-Heidelberg-New York-Tokyo · Zbl 0541.14005 [6] Jannsen, U., Motives, numerical equivalence, and semi-simplicity, Invent. Math., 107, 447-452 (1992) · Zbl 0762.14003 [7] Joukhovitski, V., \(K\)-theory of the Weil restriction functor, Prépublications de l’Equipe de Mathématiques de Besançon, 99/01 (1999), Preprint [8] Karpenko, N.A. — Cohomology of relative cellular spaces and isotropic flag varieties. St. Petersburg Math. J. (to appear).; Karpenko, N.A. — Cohomology of relative cellular spaces and isotropic flag varieties. St. Petersburg Math. J. (to appear). · Zbl 1003.14016 [9] Köck, B., Chow motif and higher Chow theory of \(GP\), Manuscripta Math., 70, 363-372 (1991) · Zbl 0735.14001 [10] Manin, Yu. I., Correspondences, motives, and monoidal transformations, Math. USSR-Sb., 6, 439-470 (1968) [11] Merkurjev, A. S.; Panin, I. A., \(K\)-Theory of algebraic tory and toric varieties, \(K\)-Theory, 12, 101-143 (1997) · Zbl 0882.19002 [12] Panin, I. A., On the algebraic \(K\)-theory of twisted flag varieties, \(K\)-Theory, 8, 541-585 (1994) · Zbl 0854.19002 [13] Rost, M., The motive of a Pfister form (1998), Preprint [14] Scheiderer, C., Real and Étale Cohomology, (Lect. Notes Math., 1588 (1994), Springer: Springer Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong Kong-Barcelona-Budapest) · Zbl 0852.14003 [15] Scholl, A. J., Classical motives, (Proc. Symp. Pure Math., 55 (1994)), 163-187, Part 1 · Zbl 0814.14001 [16] Serre, J.-P., Algèbre Locale — Multiplicités, (Lect. Notes Math., 11 (1965), Springer: Springer Heidelberg), (redigé par P. Gabriel) · Zbl 0091.03701 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.