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Given an integral scheme $X$, let $K(X)=\mathrm{Frac}(R)$ be its function field, where $\mathrm{Spec}(R)$ is some non-empty open affine subscheme of $X$. Take the maximal ideal $P$ of some DVR of $K(X)$ (i.e. a place $P$). How can one associate a closed point of $X$? Does it exist a bijection between closed points of $X$ and places of $K(X)$?

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    I$n$ fact, any valuation determines uniquely a point of the scheme (not a closed point, though) as its center if and only if the scheme is separated.2012-12-14

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