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Affine scheme

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This is a draft article, under development. These unapproved articles are subject to a disclaimer.

Contents

Definition

For a commutative ring A, the set Spec(A) (called the prime spectrum of A) 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

V(E)=\{p\in Spec(A)| p\supseteq E\}

for any subset E\subseteq A. This topology of closed sets is called the Zariski topology on Spec(A). It is easy to check that V(E)=V\left((E)\right)=V(\sqrt{(E)}), where (E) is the ideal of A generated by E.

The functor V and the Zariski topology

The Zariski topology on Spec(A) satisfies some properties: it is quasi-compact and T0, but is rarely Hausdorff. Spec(A) is not, in general, a Noetherian topological space (in fact, it is a Noetherian topological space if and only if A is a Noetherian ring.

The Structural Sheaf

X = Spec(A) has a natural sheaf of rings, denoted by OX and called the structural sheaf of X. The pair (Spec(A),OX) is called an affine scheme. The important properties of this sheaf are that

  1. The stalk OX,x is isomorphic to the local ring A_{\mathfrak{p}}, where \mathfrak{p} is the prime ideal corresponding to x\in X.
  2. For all f\in A, \Gamma(D(f),O_X)\simeq A_f, where Af is the localization of A by the multiplicative set S=\{1,f,f^2,\ldots\}. In particular, \Gamma(X,O_X)\simeq A.
Explicitly, the structural sheaf OX = may be constructed as follows. To each open set U, associate the set of functions
O_X(U):=\{s:U\to \coprod_{p\in U} A_p|s(p)\in A_p, \text{ and }s\text{ is locally constant}\}
; that is, s is locally constant if for every p\in U, there is an open neighborhood V contained in U and elements a,f\in A such that for all q\in V, s(q)=a/f\in A_q (in particular, f is required to not be an element of any q\in V). 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 Spec(\cdot) 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

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