Example 27.8.15. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme, and let $U \subset X$ be an open subscheme. Grade $A[T]$ by setting $\deg T = 1$. Define $S$ to be the subring of $A[T]$ generated by $A$ and all $fT^ i$, where $i \ge 0$ and where $f \in A$ is such that $D(f) \subset U$. We claim that $S$ is a graded ring with $S_0 = A$ such that $\text{Proj}(S) \cong U$, and this isomorphism identifies the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(A)$ of Lemma 27.8.10 with the inclusion $U \subset X$.

Suppose $\mathfrak p \in \text{Proj}(S)$ is such that every $fT \in S_1$ is in $\mathfrak p$. Then every generator $fT^ i$ with $i \ge 1$ is in $\mathfrak p$ because $(fT^ i)^2 = (fT)(fT^{2i-1}) \in \mathfrak p$ and $\mathfrak p$ is radical. But then $\mathfrak p \supset S_+$, which is impossible. Consequently $\text{Proj}(S)$ is covered by the standard open affine subsets $\{ D_+(fT)\} _{fT \in S_1}$.

Observe that, if $fT \in S_1$, then the inclusion $S \subset A[T]$ induces a graded isomorphism of $S[(fT)^{-1}]$ with $A[T, T^{-1}, f^{-1}]$. Hence the standard open subset $D_+(fT) \cong \mathop{\mathrm{Spec}}(S_{(fT)})$ is isomorphic to $\mathop{\mathrm{Spec}}(A[T, T^{-1}, f^{-1}]_0) = \mathop{\mathrm{Spec}}(A[f^{-1}])$. It is clear that this isomorphism is a restriction of the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(A)$. If in addition $gT \in S_1$, then $S[(fT)^{-1}, (gT)^{-1}] \cong A[T, T^{-1}, f^{-1}, g^{-1}]$ as graded rings, so $D_+(fT) \cap D_+(gT) \cong \mathop{\mathrm{Spec}}(A[f^{-1}, g^{-1}])$. Therefore $\text{Proj}(S)$ is the union of open subschemes $D_+(fT)$ which are isomorphic to the open subschemes $D(f) \subset X$ under the canonical morphism, and these open subschemes intersect in $\text{Proj}(S)$ in the same way they do in $X$. We conclude that the canonical morphism is an isomorphism of $\text{Proj}(S)$ with the union of all $D(f) \subset U$, which is $U$.

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