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Take an integral, proper variety $X$ over $k$ with function field $k(X)$. Let $A$ be a DVR containing $k$ having field of fractions $k(X)$. Take $P \in X$. Does there always exist an injection $\mathcal{O}_{X,P} \to A$?

EDIT: I changed the question a very little bit (replaced $\subseteq$ by the existence of an injection).

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I would say the answer is no. Suppose we take a smooth irreducible curve over $k$. Then the local rings are DVR's containing $k$, but they don't contain each other.

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    @Georges, Thanks! Now I see where my confusion was: I was somehow thinking interchangeably $Spec O_{X,P}$ and $X$ in valuative criterion, without noticing that $Spec O_{X,P}$ is not of finite type over $k$, so in fact I couldn't replace $X$ with $Spec O_{X,P}$. Thanks!2011-10-05