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

$\mathbf{C}[X_1, X_2, X_3, \ldots ] \longrightarrow \mathbf{C}[t^\alpha ; \alpha \in \mathbf{Q}_{\geq 0}]$

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 $t^{1/(i + 1) - 1/i}$.

Comment #5004 by Laurent Moret-Bailly on

I guess the degree of each $X_i$ should be $1$.

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• 10 comment(s) on Section 27.8: Proj of a graded ring

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