The Stacks project

Definition 26.5.3. Let $R$ be a ring.

  1. The structure sheaf $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ of the spectrum of $R$ is the unique sheaf of rings $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ which agrees with $\widetilde R$ on the basis of standard opens.

  2. The locally ringed space $(\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ is called the spectrum of $R$ and denoted $\mathop{\mathrm{Spec}}(R)$.

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

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