Some remarks concerning mod-nK-theory

Abstract

The spectral sequence predicted by A. Beilinson relating motivic cohomology to algebraic K-theory has been established for smooth quasi-projective varieties over a field (cf. [FS], [L1]). Among other properties verified, this spectral sequence has the expected multiplicative behavior (involving cup product in motivic cohomology and product in algebraic K-theory) and a good multiplicative “mod-n reduction” relating mod-n motivic cohomology to mod-n algebraic K-theory. The purpose of this short paper is to draw two simple conclusions concerning mod-n algebraicK-theory of smooth, quasi-projective varieties over an algebraically closed field by comparing this spectral sequence to its “localization” which we show converges to mod-n etale K-theory. In Theorem 2.1, we verify that the map from mod-n algebraic K-theory to mod-n etale K-theory of a smooth quasi-projective variety X over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of X. This refines the surjectivity result of [DFST] and extends an argument by A. Suslin in [S1]. Then, in Example 3.4, we show that certain projective smooth 3-folds X constructed by S. Bloch and H. Esnault have non-zero elements in K0(X,Z/n) which are killed by multiplication by a sufficiently high power of the Bott element β ∈ K2(X,Z/n). Although the existence of such examples is perhaps not surprising, there have been attempts to prove that multiplication by β is always injective. The authors gratefully acknowledge useful conversations with H. Esnault and A. Suslin.