Spectrum of a ring

If $$R$$ is a ring, its spectrum is a locally ringed space denoted $$\operatorname{Spec} R$$.


 * The underlying set of $$\operatorname{Spec} R$$ is the set of prime ideals of $$R$$.
 * The topology of $$\operatorname{Spec} R$$ is the topology generated by the sets $$D(f) := \{ \mathfrak{p} < R \text{ prime } \; | \; f \not \in \mathfrak{p} \}$$, known as the distinguished open sets of $$\operatorname{Spec} R$$.
 * The structure sheaf of $$\operatorname{Spec} R$$ is the sheaf $$\mathcal{O}_{\operatorname{Spec} R}$$ for which $$\mathcal{O}_{\operatorname{Spec} R}(D(f)) := R_f$$, the localization of $$R$$ with respect to $$f$$.