Lemma 27.10.6 (01MW)—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

the subscheme $Y$ is contained in the open $W_ d$ defined in Lemma 27.10.4,
the sheaf $\mathcal{O}_ Y(dn)$ is invertible for all $n \in \mathbf{Z}$ ,
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,
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,
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 $D_{+}(f) \cap Y = Y_ s$ ,
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
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. $\square$

The Stacks project

Comments (0)

Post a comment