The Stacks project

Lemma 36.36.6. Let $X$ be a scheme. If $X$ has an ample family of invertible modules (Morphisms, Definition 29.12.1), then $X$ has the resolution property.

Proof. Since $X$ is quasi-compact, there exists $n$ and pairs $(\mathcal{L}_ i, s_ i)$, $i = 1, \ldots , n$ where $\mathcal{L}_ i$ is an invertible $\mathcal{O}_ X$-module and $s_ i \in \Gamma (X, \mathcal{L}_ i)$ is a section such that the set of points $U_ i \subset X$ where $s_ i$ is nonvanishing is affine and $X = U_1 \cup \ldots \cup U_ n$. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be the image of $s_ i : \mathcal{L}_ i^{\otimes -1} \to \mathcal{O}_ X$. Applying Lemma 36.36.5 we find that $X$ has the resolution property. $\square$


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