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 $E$.

Some Topological Properties

$Spec(A)$ is quasi-compact and $T_{0}$, but is rarely Hausdorff.

The Structural Sheaf

$X=Spec(A)$ has a natural sheaf of rings, denoted $O_{X}=$, called the structural sheaf of X. 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, $\Gamma (X,O_{X})\simeq A$.

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.