The localization construction extends to the structure sheaf on Spec A

The preceding lemma gives a sheaf of rings $\tilde{A}$ on the distinguished-open basis.

For an open $U$, let $\mathcal{O}(U)$ be the ring of families $(s_x{)}_{x\in U}$ of germs of $\tilde{A}$ such that every $x\in U$ has a distinguished neighborhood $D(f)\subseteq U$ on which the family is represented by one section of $\tilde{A}(D(f))$. Restriction discards the germs outside the smaller open, and ring operations are pointwise. These maps make $\mathcal{O}$ a presheaf of rings; for $U=\mathrm{\varnothing }$, the empty germ family is its unique element.

Local representability is itself local, so compatible families in the rings $\mathcal{O}(U_i)$ glue uniquely by taking their pointwise germs. Hence $\mathcal{O}$ is a sheaf of rings. If $U=D(f)$ is distinguished, the map from $\tilde{A}(D(f))$ to its family of germs is bijective by locality and gluing for the basis sheaf in step 1.1. Thus $\mathcal{O}$ agrees with the localization assignment on every distinguished open.

If $\mathcal{G}$ is any other sheaf of rings with the same basis restriction, its sections on each open $U$ map to the locally representable germ families of step 2.1. The sheaf axiom makes this map bijective and compatible with restrictions, so $\mathcal{G}\cong \mathcal{O}$ uniquely through the prescribed basis identifications.