# Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.

STATEMENT:

Let $X$ be a smooth and projective scheme over a finite field $\mathbb{F}_{p}$.

Then, Bloch's higher chow group $CH_{0}(X,i,\mathbb{Z}/n)$ vanishes for $n$ satisfying (n,p) =1 and $i>2dim(X)$.

Please give me any advice.

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$n\in\mathbf{F}_q^\times$ ? – Chandan Singh Dalawat Dec 16 '12 at 7:28
Sorry, I meant n is an integer prime to p. – Hiro Dec 16 '12 at 8:02
Why don't you just say that $(p,n) = 1$, or (perhaps more usefully) that $p$ is invertible in the coefficient ring $\mathbb{Z}/n$? – S. Carnahan Dec 16 '12 at 11:34
This is probably true; it should be an easy consequence of the Beilinson-Lichtenbaum conjecture (that was recently proved by Voevodsky and others). – Mikhail Bondarko Dec 16 '12 at 17:59
See this paper math.uiuc.edu/K-theory/handbook/1-351-428.pdf (in particular, page 379 in it). – Mikhail Bondarko Dec 17 '12 at 9:18