A counter-example by Yagita

Abstract

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between [Formula: see text]- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group [Formula: see text], due to the author, fails for [Formula: see text] the spinor group [Formula: see text]. Yagita’s tools include a Brown–Peterson version of algebraic cobordism, ordinary and connective Morava [Formula: see text]-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic [Formula: see text]. We provide a proof using only the [Formula: see text]- and Chow theories themselves and extend the (slightly modified) example to arbitrary characteristic.

Maximal orthogonal grassmannians of quadratic forms of dimensions up to $22$

“…In this section we shall collect some basic results on the Chow and the Grothendieck rings of maximal orthogonal grassmannians. For details and the general theory we refer the reader to [1], [2] and [9].…”

“…Recently, Yagita showed that G = Spin(17) provides a counter-example on the conjecture by Karpenko [10]. Later, Karpenko extended Yagita's counter-example to base fields of arbitrary characteristic [2]. However, as mentioned in [2, §1.…”

“…For the proof of our main theorem, we provide certain elements which are not divisible by 2 in the Chow groups of the maximal orthogonal grassmannians, whereas their images under φ are divisible by 2 in the associated graded rings, following the approach of [2].…”

A remark on connective K-theory

“…On the other hand, the counter-example of [10] (see also [4]) provides by Theorem 3.1 a generic flag variety X (given by the spinor group Spin(17)) with non-trivial torsion in CK(X).…”

“…where CH(X) is the Chow ring, K(X) is the Grothendieck ring of X, and ChowK(X) is the ring associated with the Chow filtration (i.e., the filtration by codimension of supports of coherent sheaves) on K(X). Being recently disproved for G = Spin(17) by Yagita in [10] (see also [4]), Conjecture 1.1 has been confirmed for many other G. (For instance, the 2-local version for G of type E 7 is proved in the very [4].) An overview of some positive cases is given in [5].…”