Zero-cycles on an arithmetic surface

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Could anyone give a reference for the following statement, which I believe is true.

"Let X be a regular scheme, flat over $Spec( \mathbb{Z})$, with fiber dimension $1$. Then the Chow group $CH^2(X)$ is finite."

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1 Answer

The finiteness is known; see Szamuely's Seminaire Bourbaki expose and Remarque 3.4 (5) on page 11 is a precise reference.

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A similar result is valid for any regular scheme $X$ of finite type over $\mathop{\rm Spec}(\mathbb Z)$: the Chow group of zero cycles is finite, with the exceptions of schemes which are proper over a finite field~$\mathbb F_p$ for which the Chow group of zero cycles has the form $\mathbb Z\oplus G$, for some finite group $G$.  This is a part of geometric class field theory,  due to Rosenlicht, Lang (50s), Bloch (1981), Kato-Saito (1986)...  Szamuely's Seminaire Bourbaki explains the new point of view given by work of Wiesend. – ACL Apr 15 2011 at 6:43
Yes, indeed! Thanks! – SGP Apr 15 2011 at 21:06
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