Definition:Structure Sheaf of Spectrum of Ring

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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

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Definition 3

This definition needs to be completed. In particular: using explicit formulas You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

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