The Stacks project

Lemma 27.8.8. Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is separated.

Proof. We have to show that the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated. We will use Schemes, Lemma 26.21.7. Thus it suffices to show given any pair of standard opens $D_{+}(f)$ and $D_{+}(g)$ that $D_{+}(f) \cap D_{+}(g) = D_{+}(fg)$ is affine (clear) and that the ring map

\[ S_{(f)} \otimes _{\mathbf{Z}} S_{(g)} \longrightarrow S_{(fg)} \]

is surjective. Any element $s$ in $S_{(fg)}$ is of the form $s = h/(f^ ng^ m)$ with $h \in S$ homogeneous of degree $n\deg (f) + m\deg (g)$. We may multiply $h$ by a suitable monomial $f^ ig^ j$ and assume that $n = n' \deg (g)$, and $m = m' \deg (f)$. Then we can rewrite $s$ as $s = h/f^{(n' + m')\deg (g)} \cdot f^{m'\deg (g)}/g^{m'\deg (f)}$. So $s$ is indeed in the image of the displayed arrow. $\square$

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