I have tried looking at my sheaves notes but couldn't find anything.
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1It's one of Serre's twisting sheafs. In particular, it is the hyperplane bundle of $\Bbb{P}^n$. – A.P. Jun 19 '15 at 14:15
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I'm far from being an expert, but I think I recall that line bundles on projective spaces are classified by integers (Chern class). So this would be the generator of the Picard group. – Najib Idrissi Jun 19 '15 at 14:15
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For details a reference is the book of Hartshorne "Algebraic Geometry", page 116. If $S$ is a graded ring, then a graded $S$-module $M$ gives a sheaf $\tilde{M}$ on $Proj(S)$. On the basic open, $\tilde{M}(D(f)) \cong \tilde{M_f}$.
Now, it's only pure algebra.
There is a general construction with $S$ a graded ring, and $M$ a graded $S-$module. We can define the "twisted module" $M(k)$ as $M$ with modified grading : $M(k)_n = M_{n+k}$.
Now, you can define $\mathcal O_{\mathbb P^n}(1) = \tilde{S(1)}$.
For more details about the sheaves $\mathcal O(n)$, you can take a look this other question.
