Definition 27.21.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$-module^{1}. We denote

and we call it the *projective bundle associated to $\mathcal{E}$*. The symbol $\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n)$ indicates the invertible $\mathcal{O}_{\mathbf{P}(\mathcal{E})}$-module of Lemma 27.16.11 and is called the $n$th *twist of the structure sheaf*.

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