Lemma 62.6.10. Let $f : X \to S$ be a finite type morphism of schemes with $S$ the spectrum of a discrete valuation ring. Let $r \geq 0$. Then (62.6.8.1) is surjective.
Proof. This of course follows from Lemma 62.6.9 but we can also see it directly as follows. Say $\alpha $ is a relative $r$-cycle on $X/S$. Write $\alpha _\eta = \sum n_ i[Z_ i]$ (the sum is finite). Denote $\overline{Z}_ i \subset X$ the closure of $Z_ i$ as in Section 62.4. Then $\alpha = \sum n_ i[\overline{Z}_ i/X/S]$. $\square$
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