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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 $\mathbb{C}$. thoughThough I don't have a proveproof for this, I don't believe such a thing can be algebraically closed.

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 prove for this, I don't believe such a thing can be algebraically closed.

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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Example of a valuation ring in an algebraically closed field

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 prove for this, I don't believe such a thing can be algebraically closed.