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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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