The Stacks project

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. $\square$