# Affine scheme

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## 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

1. The stalk  is isomorphic to the local ring , where  is the prime ideal corresponding to .
2. For all , , where  is the localization of  by the multiplicative set . In particular, .
Explicitly, the structural sheaf  may be constructed as follows. To each open set , associate the set of functions

; that is,  is 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.