Lifting of Coefficients for Chow Motives of Quadrics

Abstract

Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally attaches an ideal H X (k) of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of X. This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of H, such as the Chow group. When H is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When H is the algebraic cobordism, we give a new proof of a theorem of Levine and Morel. Finally we discuss some splitting properties of geometrically unirational field extensions of small transcendence degree.

Équivalence motivique des groupes algébriques semisimples

“…Sous ces conditions les motifs des quadriques projectives de et sont isomorphes à coefficients dans . Cet isomorphisme se relève en un isomorphisme des motifs de ces quadriques à coefficients entiers par [Hau10]. Le critère de Vishik [Kar00, Criterion 0.1] assure donc que les indices de Witt de et coincident sur toutes les extensions de leur corps de définition.◻…”

Morava K-theory of orthogonal groups and motives of projective quadrics

“…It is easy to deduce now, that decompositions of K(n) * (−; Z (2) )and K(n) * (−; F 2 )-motives of a quadric coincide. For the Chow theory it was shown in [Vi04,Hau].…”