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The Stacks project

Lemma 27.10.6. Let S be a graded ring. Let X = \text{Proj}(S). Let Y \subset X be a quasi-compact open subscheme. Denote \mathcal{O}_ Y(n) the restriction of \mathcal{O}_ X(n) to Y. There exists an integer d \geq 1 such that

  1. the subscheme Y is contained in the open W_ d defined in Lemma 27.10.4,

  2. the sheaf \mathcal{O}_ Y(dn) is invertible for all n \in \mathbf{Z},

  3. all the maps \mathcal{O}_ Y(nd) \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(m) \longrightarrow \mathcal{O}_ Y(nd + m) of Equation (27.10.1.1) are isomorphisms,

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

  5. given f \in S_{nd} denote s \in \Gamma (Y, \mathcal{O}_ Y(nd)) the image of f via (27.10.1.3) restricted to Y, then D_{+}(f) \cap Y = Y_ s,

  6. a basis for the topology on Y is given by the collection of opens Y_ s, where s \in \Gamma (Y, \mathcal{O}_ Y(nd)), n \geq 1, and

  7. a basis for the topology of Y is given by those opens Y_ s \subset Y, for s \in \Gamma (Y, \mathcal{O}_ Y(nd)), n \geq 1 which are affine.

Proof. Since Y is quasi-compact there exist finitely many homogeneous f_ i \in S_{+}, i = 1, \ldots , n such that the standard opens D_{+}(f_ i) give an open covering of Y. Let d_ i = \deg (f_ i) and set d = d_1 \ldots d_ n. Note that D_{+}(f_ i) = D_{+}(f_ i^{d/d_ i}) and hence we see immediately that Y \subset W_ d, by characterization (2) in Lemma 27.10.4 or by (1) using Lemma 27.10.2. Note that (1) implies (2), (3) and (4) by Lemma 27.10.4. (Note that (3) is a special case of (4).) Assertion (5) follows from Lemma 27.10.5. Assertions (6) and (7) follow because the open subsets D_{+}(f) form a basis for the topology of X and are affine. \square


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