The Stacks project

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 a $h \in S$ 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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