The Stacks project

Definition 27.21.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$-module1. We denote

\[ \pi : \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ S(\text{Sym}(\mathcal{E})) \longrightarrow S \]

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.

[1] The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with [II, Definition 4.1.1, EGA].

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