Serre's twisting sheaf

Summary

Let $S$ be a graded ring. Serre's twisting sheaf $\mathcal{O}(1)$ on $\operatorname{Proj} S$ is the sheaf attached to the graded module $S$ shifted by one degree. On projective space $\mathbb{P}^n_k$ it is the line bundle whose global sections are the linear forms in the homogeneous coordinates. Its powers $\mathcal{O}(m)$ turn graded modules into sheaves and sheaves back into graded modules, and they control embeddings into projective space.