# Affine scheme

## Contents

## Definition

For a commutative ring , the set (called the *prime spectrum of *) denotes the set of prime ideals of *A*. This set is endowed with a topology of closed sets, where closed subsets are defined to be of the form

for any subset . This topology of closed sets is called the *Zariski topology* on . It is easy to check that , where
is the ideal of generated by .

## The functor V and the Zariski topology

The Zariski topology on satisfies some properties: it is quasi-compact and , but is rarely Hausdorff. is not, in general, a Noetherian topological space (in fact, it is a Noetherian topological space if and only if is a Noetherian ring.

## The Structural Sheaf

has a natural sheaf of rings, denoted by and called the *structural sheaf* of *X*. The pair is called an *affine scheme*. The important properties of this sheaf are that

- The stalk is isomorphic to the local ring , where is the prime ideal corresponding to .
- For all , , where is the localization of by the multiplicative set . In particular, .

*locally constant*if for every , there is an open neighborhood contained in and elements such that for all , (in particular, is required to not be an element of any ). This description is phrased in a common way of thinking of sheaves, and in fact captures their local nature. One construction of the sheafification functor makes use of such a perspective.

## The Category of Affine Schemes

Regarding as a contravariant functor between the category of commutative rings and the category of affine schemes, one can show that it is in fact an anti-equivalence of categories.