# Affine scheme

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## Definition

For a commutative ring ${\displaystyle A}$, the set ${\displaystyle Spec(A)}$ (called the prime spectrum of ${\displaystyle A}$) 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

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

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

## The functor V and the Zariski topology

The Zariski topology on ${\displaystyle Spec(A)}$ satisfies some properties: it is quasi-compact and ${\displaystyle T_{0}}$, but is rarely Hausdorff. ${\displaystyle Spec(A)}$ is not, in general, a Noetherian topological space (in fact, it is a Noetherian topological space if and only if ${\displaystyle A}$ is a Noetherian ring.

## The Structural Sheaf

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

1. The stalk ${\displaystyle O_{X,x}}$ is isomorphic to the local ring ${\displaystyle A_{\mathfrak {p}}}$, where ${\displaystyle {\mathfrak {p}}}$ is the prime ideal corresponding to ${\displaystyle x\in X}$.
2. For all ${\displaystyle f\in A}$, ${\displaystyle \Gamma (D(f),O_{X})\simeq A_{f}}$, where ${\displaystyle A_{f}}$ is the localization of ${\displaystyle A}$ by the multiplicative set ${\displaystyle S=\{1,f,f^{2},\ldots \}}$. In particular, ${\displaystyle \Gamma (X,O_{X})\simeq A}$.

Explicitly, the structural sheaf ${\displaystyle O_{X}=}$ may be constructed as follows. To each open set ${\displaystyle U}$, associate the set of functions

${\displaystyle O_{X}(U):=\{s:U\to \coprod _{p\in U}A_{p}|s(p)\in A_{p},{\text{ and }}s{\text{ is locally constant}}\}}$

; that is, ${\displaystyle s}$ is locally constant if for every ${\displaystyle p\in U}$, there is an open neighborhood ${\displaystyle V}$ contained in ${\displaystyle U}$ and elements ${\displaystyle a,f\in A}$ such that for all ${\displaystyle q\in V}$, ${\displaystyle s(q)=a/f\in A_{q}}$ (in particular, ${\displaystyle f}$ is required to not be an element of any ${\displaystyle q\in V}$). 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 ${\displaystyle Spec(\cdot )}$ 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.