Definition:Structure Sheaf of Spectrum of Ring

From ProofWiki
Jump to navigation Jump to search


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 $A_f \to A_g$.


Definition 2


Definition 3


Also see