The Stacks project

Example 27.8.12. Let $\mathbf{C}[X_1, X_2, X_3, \ldots ]$ be the graded $\mathbf{C}$-algebra with each $X_ i$ in degree $1$. Consider the ring map

which maps $X_ i$ to $t^{1/i}$. The right hand side becomes a valuation ring $A$ upon localization at the ideal $\mathfrak m = (t^\alpha ; \alpha > 0)$. Let $K$ be the fraction field of $A$. The above gives a morphism $\mathop{\mathrm{Spec}}(K) \to \text{Proj}(\mathbf{C}[X_1, X_2, X_3, \ldots ])$ which does not extend to a morphism defined on all of $\mathop{\mathrm{Spec}}(A)$. The reason is that the image of $\mathop{\mathrm{Spec}}(A)$ would be contained in one of the $D_{+}(X_ i)$ but then $X_{i + 1}/X_ i$ would map to an element of $A$ which it doesn't since it maps to $t^{1/(i + 1) - 1/i}$.