Example 27.8.13. Let $R = \mathbf{C}[t]$ and

$S = R[X_1, X_2, X_3, \ldots ]/(X_ i^2 - tX_{i + 1}).$

The grading is such that $R = S_0$ and $\deg (X_ i) = 2^{i - 1}$. Note that if $\mathfrak p \in \text{Proj}(S)$ then $t \not\in \mathfrak p$ (otherwise $\mathfrak p$ has to contain all of the $X_ i$ which is not allowed for an element of the homogeneous spectrum). Thus we see that $D_{+}(X_ i) = D_{+}(X_{i + 1})$ for all $i$. Hence $\text{Proj}(S)$ is quasi-compact; in fact it is affine since it is equal to $D_{+}(X_1)$. It is easy to see that the image of $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(R)$ is $D(t)$. Hence the morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(R)$ is not closed. Thus the valuative criterion cannot apply because it would imply that the morphism is closed (see Schemes, Proposition 26.20.6 ).

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