# Definition:Structure Sheaf of Spectrum of Ring

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring of basic complexity.In particular: complete definitions, then subpagesUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.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 `{{Refactor}}` from the code. |

## 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

This definition needs to be completed.In particular: the étalé space point of viewYou 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. |

### Definition 3

This definition needs to be completed.In particular: using explicit formulasYou 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. |