The Stacks project

Proof. The uniqueness part follows from the fact that $\text{Proj}(S)$ is separated (Lemma 27.8.8 and Schemes, Lemma 26.22.1). Choose $x_ i \in S_{+}$ homogeneous, $i = 1, \ldots , n$ which generate $S$ over $S_0$. Let $d_ i = \deg (x_ i)$ and set $d = \text{lcm}\{ d_ i\}$. Suppose we are given a diagram

as in Schemes, Definition 26.20.3. Denote $v : K^* \to \Gamma$ the valuation of $A$, see Algebra, Definition 10.50.13. We may choose an $f \in S_{+}$ homogeneous such that $\mathop{\mathrm{Spec}}(K)$ maps into $D_{+}(f)$. Then we get a commutative diagram of ring maps

After renumbering we may assume that $\varphi (x_ i^{\deg (f)}/f^{d_ i})$ is nonzero for $i = 1, \ldots , r$ and zero for $i = r + 1, \ldots , n$. Since the open sets $D_{+}(x_ i)$ cover $\text{Proj}(S)$ we see that $r \geq 1$. Let $i_0 \in \{ 1, \ldots , r\}$ be an index minimizing $\gamma _ i = (d/d_ i)v(\varphi (x_ i^{\deg (f)}/f^{d_ i}))$ in $\Gamma$. For convenience set $x_0 = x_{i_0}$ and $d_0 = d_{i_0}$. The ring map $\varphi$ factors though a map $\varphi ' : S_{(fx_0)} \to K$ which gives a ring map $S_{(x_0)} \to S_{(fx_0)} \to K$. The algebra $S_{(x_0)}$ is generated over $S_0$ by the elements $x_1^{e_1} \ldots x_ n^{e_ n}/x_0^{e_0}$, where $\sum e_ i d_ i = e_0 d_0$. If $e_ i > 0$ for some $i > r$, then $\varphi '(x_1^{e_1} \ldots x_ n^{e_ n}/x_0^{e_0}) = 0$. If $e_ i = 0$ for $i > r$, then we have

because $\gamma _0$ is minimal among the $\gamma _ i$. This implies that $S_{(x_0)}$ maps into $A$ via $\varphi '$. The corresponding morphism of schemes $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(S_{(x_0)}) = D_{+}(x_0) \subset \text{Proj}(S)$ provides the morphism fitting into the first commutative diagram of this proof. $\square$