Definition:Structure Sheaf of Spectrum of Ring

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

Let $(\operatorname{Spec}(A), \tau)$ 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 $\operatorname{Spec}(A)$ to be the sheaf induced by a sheaf on this basis defined as follows:
 * For $f\in A$, $\mathcal O(X(f))$ is the localization of $A$ at $f$
 * For $f, g \in A$ with $X(f)\supset X(g)$, 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