Definition 27.10.1. Let $S$ be a graded ring. Let $X = \text{Proj}(S)$.

1. We define $\mathcal{O}_ X(n) = \widetilde{S(n)}$. This is called the $n$th twist of the structure sheaf of $\text{Proj}(S)$.

2. For any sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we set $\mathcal{F}(n) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(n)$.

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