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In Hartshorne's Algebraic Geometry, the twisting sheaf of Serre $\mathscr{O}(n)$ is defined to be $S(n)^\sim$, where $S$ is an $\mathbb{N}$-graded ring. But I couldn't find the definition of $S(n)$ anywhere in the book (if any one knows where it is, please let me know). It is presumably the regrading of $S$ by $n$, but I wanted to make sure there is absolutely no room for misunderstanding. For example, regrading in which direction? It is probably $S(n)_i=S_{n+i}$, but did Hartshorne use this convention? And presumably $S(n)_i=0\quad\forall i ?

2 Answers 2

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Indeed, he uses this definition. Have a look at II.7, he describes $\mathcal{O}_X(1)$ there, which implies the rest.

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See Chapter I, Sect. 7, p. 50.