Affine scheme

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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 T_0, 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 O_X and called the structural sheaf of X. The pair (Spec(A),O_X) is called an affine scheme. The important properties of this sheaf are that

  1. The stalk O_{X,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 A_f 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 O_X= 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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