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 $x\in X$.
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.