Definition:Structure Sheaf of Spectrum of Ring

Definition
Let $A$ be a commutative ring with unity.

Let $\struct {\Spec A, \tau}$ be its spectrum with Zariski topology $\tau$

Definition 1
Note that Principal Open Subsets form Basis of Zariski Topology on Prime Spectrum.

We define the structure sheaf of $\Spec A$ to be the sheaf induced by a sheaf on this basis defined as follows:
 * For $f \in A$, $\map \OO {\map X f}$ is the localization of $A$ at $f$
 * For $f, g \in A$ with $\map X f \supset \map X g$, the restriction is the induced homomorphism of $A$-algebras $A_f \to A_g$.

Also see

 * Equivalence of Definitions of Structure Sheaf on Prime Spectrum
 * Definition:Affine Scheme