Affine scheme

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} .
2. For all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\in A} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(D(f),O_X)\simeq A_f} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_f} is the localization of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} by the multiplicative set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\{1,f,f^2,\ldots\}} . In particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(X,O_X)\simeq A} .

Explicitly, the structural sheaf Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O_X=} may be constructed as follows. To each open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , associate the set of functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} is locally constant if for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in U} , there is an open neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,f\in A} such that for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\in V} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(q)=a/f\in A_q} (in particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is required to not be an element of any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.