Lemma 27.8.6 (01MA)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 27: Constructions of Schemes
Section 27.8: Proj of a graded ring
Lemma 27.8.6 (cite)

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Lemma 27.8.6. Let $S$ be a graded ring. Let $f \in S$ be homogeneous of positive degree. Suppose that $D(g) \subset \mathop{\mathrm{Spec}}(S_{(f)})$ is a standard open. Then there exists an $h \in S$ homogeneous of positive degree such that $D(g)$ corresponds to $D_{+}(h) \subset D_{+}(f)$ via the homeomorphism of Algebra, Lemma 10.57.3. In fact we can take $h$ such that $g = h/f^ n$ for some $n$ .

Proof. Write $g = h/f^ n$ for some $h$ homogeneous of positive degree and some $n \geq 1$ . If $D_{+}(h)$ is not contained in $D_{+}(f)$ then we replace $h$ by $hf$ and $n$ by $n + 1$ . Then $h$ has the required shape and $D_{+}(h) \subset D_{+}(f)$ corresponds to $D(g) \subset \mathop{\mathrm{Spec}}(S_{(f)})$ . $\square$

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