The Stacks project

Lemma 27.10.5. Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. Let $W = W_ d \subset X$ be the open subscheme defined in Lemma 27.10.4. Let $n \geq 1$ and $f \in S_{nd}$. Denote $s \in \Gamma (W, \mathcal{O}_ W(nd))$ the section which is the image of $f$ via ( restricted to $W$. Then

\[ W_ s = D_{+}(f) \cap W. \]

Proof. Let $D_{+}(ab) \subset W$ be a standard affine open with $a, b \in S$ homogeneous and $\deg (a) = \deg (b) + d$. Note that $D_{+}(ab) \cap D_{+}(f) = D_{+}(abf)$. On the other hand the restriction of $s$ to $D_{+}(ab)$ corresponds to the element $f/1 = b^ nf/a^ n (a/b)^ n \in (S_{ab})_{nd}$. We have seen in the proof of Lemma 27.10.4 that $(a/b)^ n$ is a generator for $\mathcal{O}_ W(nd)$ over $D_{+}(ab)$. We conclude that $W_ s \cap D_{+}(ab)$ is the principal open associated to $b^ nf/a^ n \in \mathcal{O}_ X(D_{+}(ab))$. Thus the result of the lemma is clear. $\square$

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