A remark on connective K-theory

Abstract

Let X be a smooth algebraic variety over an arbitrary field. Let φ be the canonical surjective homomorphism of the Chow ring of X onto the ring associated with the Chow filtration on the Grothendieck ring K(X). We remark that φ is injective if and only if the connective K-theory CK(X) coincides with the terms of the Chow filtration on K(X). As a consequence, CK(X) turns out to be computed for numerous flag varieties (under semisimple algebraic groups) for which the injectivity of φ had already been established. This especially applies to the so-called generic flag varieties X of many different types, identifying for them CK(X) with the terms of the explicit Chern filtration on K(X). Besides, for arbitrary X, we compare CK(X) with the fibered product of the Chow ring of X and the graded ring formed by the terms of the Chow filtration on K(X).