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4 Fixed an issue pointed out by nicoj

1) Here's another example. $k[y, xy, y/x, y/x^2, y/x^3, \dots]$. The localization of this at the origin is a valuation ring (and this idea can be used to construct many other examples).

2+3) If you are constructing examples of this type, many are constructed by gluing. In other words, as pushouts of diagrams of affine schemes $$\{ X \leftarrow Z \rightarrow W \}.$$ where $Z \to X$ is a closed immersion and $Z \rightarrow W$ is arbitrary. The condition you then want in (2) is for $Z \rightarrow W$ to be a finite map. Some relevant references include Ferrand, "Conducteur, descente et pincement", MR2044495 (2005a:13016) and Artin, "Algebraization of Formal Moduli II: Existence of Modifications", MR0260747 (41 #5370)

For example, your the ring $k[x, xy, xy^2, \dots]$ is the pushout of $$\{ \mathbb{A}^2 \leftarrow \text{coordinate-axis} \rightarrow \text{point} \}.$$ This gives a nice geometric interpretation, you just contracted a coordinate axis to a point, you can contract other schemes and get new examples. Note the $Z \to W$ in this example is not finite.

My example in 1) is the pushout of

$$\{ \mathbb{A}^2 \setminus{V(x)} \leftarrow \text{Spec } k[x,y,x^{-1}]/(y) \rightarrow \text{Spec } k[x] \}.$$

Where the maps are the obvious ones. The $Z \rightarrow W$ map is not finite in this example either.

3 Typo fixed

1) Here's another example. $k[y, xy, y/x, y/x^2, y/x^3, \dots]$. The localization of this at the origin is a valuation ring (and this idea can be used to construct many other examples).

2+3) If you are constructing examples of this type, many are constructed by gluing. In other words, as pushouts of diagrams of affine schemes $$\{ X \leftarrow Z \rightarrow W \}.$$ where $Z \to X$ is a closed immersion and $Z \rightarrow W$ is arbitrary. The condition you then want in (2) is for $Z \rightarrow W$ to be a finite map. Some relevant references include Ferrand, "Conducteur, descente et pincement", MR2044495 (2005a:13016) and Artin, "Algebraization of Formal Moduli II: Existence of Modifications", MR0260747 (41 #5370)

For example, your ring is the pushout of $$\{ \mathbb{A}^2 \leftarrow \text{coordinate-axis} \rightarrow \text{point} \}.$$ This gives a nice geometric interpretation, you just contracted a coordinate axis to a point, you can contract other schemes and get new examples. Note the $Z \to W$ in this example is not finite.

My example in 1) is the pushout of

$$\{ \mathbb{A}^2 \setminus{V(x)} \leftarrow \text{Spec } k[x,y,x^{-1}]/(y) \rightarrow \text{Spec } k[x] \}.$$

Where the maps are the obvious ones. The $Z \rightarrow W$ map is not finite in this example either.

1) Here's another example. $k[y, xy, y/x, y/x^2, y/x^3, \dots]$. The localization of this at the origin is a valuation ring (and this idea can be used to construct many other examples).

2+3) If you are constructing examples of this type, many are constructed by gluing. In other words, as pushouts of diagrams of affine schemes $$\{ X \leftarrow Z \rightarrow W \}.$$ where $Z \to X$ is a closed immersion and $Z \rightarrow W$ is arbitrary. The condition you then want in (2) is for $Z \rightarrow W$ to be a finite map. Some relevant references include Ferrand, "Conducteur, descente et pincement", MR2044495 (2005a:13016) and Artin, "Algebraization of Formal Moduli II: Existence of Modifications", MR0260747 (41 #5370)

For example, your ring is the pushout of $$\{ \mathbb{A}^2 \leftarrow \text{coordinate-axis} \rightarrow \text{point} \}.$$ This gives a nice geometric interpretation, you just contracted a coordinate axis to a point, you can contract other schemes and get new examples. Note the $Z \to W$ in this example is finite.

My example in 1) is the pushout of

$$\{ \mathbb{A}^2 \setminus{V(x)} \leftarrow \text{Spec } k[x,y,x^{-1}]/(y) \rightarrow \text{Spec } k[x] \}.$$

Where the maps are the obvious ones. The $Z \rightarrow W$ map is not finite in this example either.