The finiteness is knownfor any scheme $Y$ which is regular, flat and proper over Spec Z. This is due to the work of Bloch, Kato-Saito among others; see Szamuely's Seminaire Bourbaki expose for and Remarque 3.4 (5) on page 11 is a modern accountprecise reference. Your $X$ may be an open subscheme of a $Y$; since the fibers of $X$ (and $Y$) over Spec Z are one-dimensional, the finiteness for $Y$ implies that for $X$.
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The finiteness is known for any scheme $Y$ which is regular, flat and proper over Spec Z. This is due to the work of Bloch, Kato-Saito among others; see Szamuely's Seminaire Bourbaki expose for a modern account. Your $X$ may be an open subscheme of a $Y$; since the fibers of $X$ (and $Y$) over Spec Z are one-dimensional, the finiteness for $Y$ implies that for $X$. |
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