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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).