On the structure of étale motivic cohomology

Abstract

We discuss the structure of integralétale motivic cohomology groups of smooth and projective schemes over algebraically closed fields, finite fields, local fields, and arithmetic schemes.

Zeta-values of one-dimensional arithmetic schemes at strictly negative integers

“…-see, for example, [14,Example 4.2]. This is related to Quillen's calculation of the K-theory of finite fields [35].…”

“…Suppose that X/F q is a smooth curve. The groups H i (X ét , Z c (n)) are finitely generated by [14,Proposition 4.3], so that the duality ( 14) holds. The Q/Z-dual groups…”

“…We have H i c (Spec O F [1/p], µ ⊗n p r ) = 0 for i ≥ 3 by Artin-Verdier duality [32, Chapter II, Corollary 3.3], or by [37, p. 268]. Therefore, it follows that H i c (X ét , Z(n)) = 0 for i ≥ 4, and hence by duality (14),…”

Duality for integral motivic cohomology

“…) modulo its divisible subgroup is isomorphic to the dual of Tor NS X. Hence the fact that (A ♯ /m) ♯ = m A for a finite group A imply that (7) gives a short exact sequence…”

On Suslin homology with integral coefficients in characteristic zero (with an appendix by Bruno Kahn)

“…For the precise statements are theorems 4.7 and 4.8. For X smooth over k, where k is an algebraically closed field of arbitrary characteristic, a decomposition of the form (1) with a slight modification on the p-primary part, is shown in [Gei17], and in this case Z i = 0. In general for singular X, for example certain singular curves, Z i is nonzero.…”