Definition 27.8.3. Let $S$ be a graded ring.

1. The structure sheaf $\mathcal{O}_{\text{Proj}(S)}$ of the homogeneous spectrum of $S$ is the unique sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ which agrees with $\widetilde S$ on the basis of standard opens.

2. The locally ringed space $(\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ is called the homogeneous spectrum of $S$ and denoted $\text{Proj}(S)$.

3. The sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules extending $\widetilde M$ to all opens of $\text{Proj}(S)$ is called the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$. This sheaf is denoted $\widetilde M$ as well.

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