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In §8. Specialization in the Red Book of Varieties and Schemes by Mumford, one considers a valuation ring $(R, \mathfrak{m})$ with algebraically closed field of fractions $k = Q(R)$. This is a well-behaved situation, for example the quotient field $L = R / \mathfrak m$ is algebraically closed as well.

I wonder, what are examples of such rings? And where do they appear in algebraic geometry?

The typical example of a (discrete) valuation ring I have in mind is the local ring $R = \mathcal{O}_{X,x}$ of a point $x$ of codimension one on a smooth variety $X$ (let's say over $\mathbb{C}$). Then the fraction field of such $R$ is just the function field of the variety, and by Noether normalization, that is a finite field extension of an transcendental extension of $\mathbb{C}$. Though I don't have a proof for this, I don't believe such a thing can be algebraically closed.

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Regarding the first part of your question, that is, giving an example of a valuation in an algebraically closed field, recall that any valuation $\nu$ on a field $K$ has an extension to an overfield $L$. If $L$ is the algebraic closure of $K$, then the value group of $L$ is the divisible closure of the value group of $K$ and the residue field of $L$ is the algebraic closure of the residue field of $K$.

So for example consider the $X$-adic valuation on the power series field $k((X))$ and let us denote it by $\nu$. Then the value group $\nu k((X)) = \mathbb{Z}$ and the residue field $k((X)) \nu = k$. Extending $\nu$ to the algebraic closure $\overline{k((X))}$, we have $\nu \overline{k((X))} = \mathbb{Q}$ and $\overline{k((X))} \nu = \overline{k}$. As a bonus we obtain that when char $k = 0$, then $\overline{k((X))}$ equals the Puiseux series field $P(k)$ if and only if $k = \overline{k}$.

Edit : I do not have a concrete example regarding the second part of your question. However one reason I think we talk of valuations in algebraically closed fields is to make sense of ramification theory of subextensions without ambiguity.

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  • $\begingroup$ This sounds very interesting. Is there a textbook or a reference that contains the proof of the statements of your answer? $\endgroup$ Commented Mar 17, 2020 at 20:07
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    $\begingroup$ Dear @youngsu, you can check out the paper "Valuation theoretic and model theoretic aspects of local uniformization" by Kuhlmann. Basically, $k((X))$ is maximally valued, hence henselian. $P(k)$ being an algebraic extension of $k((X))$ is hence also henselian. And char $k = 0$ implies that $P(k)$ is a defectless field. The result now follows from the Lemma of Ostrowski. $\endgroup$ Commented Mar 17, 2020 at 20:42

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