Lemma 27.10.4. Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. The following open subsets of $X$ are equal:

1. The largest open subset $W = W_ d \subset X$ such that each $\mathcal{O}_ X(dn)|_ W$ is invertible and all the multiplication maps $\mathcal{O}_ X(nd)|_ W \otimes _{\mathcal{O}_ W} \mathcal{O}_ X(md)|_ W \to \mathcal{O}_ X(nd + md)|_ W$ (see 27.10.1.1) are isomorphisms.

2. The union of the open subsets $D_{+}(fg)$ with $f, g \in S$ homogeneous and $\deg (f) = \deg (g) + d$.

Moreover, all the maps $\widetilde M(nd)|_ W = \widetilde M|_ W \otimes _{\mathcal{O}_ W} \mathcal{O}_ X(nd)|_ W \to \widetilde{M(nd)}|_ W$ (see 27.10.1.5) are isomorphisms.

Proof. If $x \in D_{+}(fg)$ with $\deg (f) = \deg (g) + d$ then on $D_{+}(fg)$ the sheaves $\mathcal{O}_ X(dn)$ are generated by the element $(f/g)^ n = f^{2n}/(fg)^ n$. This implies $x$ is in the open subset $W$ defined in (1) by arguing as in the proof of Lemma 27.10.2.

Conversely, suppose that $\mathcal{O}_ X(d)$ is free of rank 1 in an open neighbourhood $V$ of $x \in X$ and all the multiplication maps $\mathcal{O}_ X(nd)|_ V \otimes _{\mathcal{O}_ V} \mathcal{O}_ X(md)|_ V \to \mathcal{O}_ X(nd + md)|_ V$ are isomorphisms. We may choose $h \in S_{+}$ homogeneous such that $D_{+}(h) \subset V$. By the definition of the twists of the structure sheaf we conclude there exists an element $s$ of $(S_ h)_ d$ such that $s^ n$ is a basis of $(S_ h)_{nd}$ as a module over $S_{(h)}$ for all $n \in \mathbf{Z}$. We may write $s = f/h^ m$ for some $m \geq 1$ and $f \in S_{d + m \deg (h)}$. Set $g = h^ m$ so $s = f/g$. Note that $x \in D(g)$ by construction. Note that $g^ d \in (S_ h)_{-d\deg (g)}$. By assumption we can write this as a multiple of $s^{\deg (g)} = f^{\deg (g)}/g^{\deg (g)}$, say $g^ d = a/g^ e \cdot f^{\deg (g)}/g^{\deg (g)}$. Then we conclude that $g^{d + e + \deg (g)} = a f^{\deg (g)}$ and hence also $x \in D_{+}(f)$. So $x$ is an element of the set defined in (2).

The existence of the generating section $s = f/g$ over the affine open $D_{+}(fg)$ whose powers freely generate the sheaves of modules $\mathcal{O}_ X(nd)$ easily implies that the multiplication maps $\widetilde M(nd)|_ W = \widetilde M|_ W \otimes _{\mathcal{O}_ W} \mathcal{O}_ X(nd)|_ W \to \widetilde{M(nd)}|_ W$ (see 27.10.1.5) are isomorphisms. Compare with the proof of Lemma 27.10.2. $\square$

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