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.

Curves