# prime spectrum

## 1 Spec as a set

Let $R$ be any commutative ring with identity     . The prime spectrum $\operatorname{Spec}(R)$ of $R$ is defined to be the set

 $\{P\subsetneq R\mid P\text{ is a prime ideal of }R\}.$

For any subset $A$ of $R$, we define the of $A$ to be the set

 $V(A):=\{P\in\operatorname{Spec}(R)\mid A\subset P\}\subset\operatorname{Spec}(R)$

It is enough to restrict attention to subsets of $R$ which are ideals, since, for any subset $A$ of $R$, we have $V(A)=V(I)$ where $I$ is the ideal generated by  $A$. In fact, even more is true: $V(I)=V(\sqrt{I})$ where $\sqrt{I}$ denotes the radical   of the ideal $I$.

## 2 Spec as a topological space

We impose a topology  on $\operatorname{Spec}(R)$ by defining the sets $V(A)$ to be the collection  of closed subsets of $\operatorname{Spec}(R)$ (that is, a subset of $\operatorname{Spec}(R)$ is open if and only if it equals the complement  of $V(A)$ for some subset $A$). The equations

 $\displaystyle\bigcap_{\alpha}V(I_{\alpha})$ $\displaystyle=$ $\displaystyle V\left(\bigcup_{\alpha}I_{\alpha}\right)$ $\displaystyle\bigcup_{i=1}^{n}V(I_{i})$ $\displaystyle=$ $\displaystyle V\left(\bigcap_{i=1}^{n}I_{i}\right),$

for any ideals $I_{\alpha}$, $I_{i}$ of $R$, establish that this collection does constitute a topology on $\operatorname{Spec}(R)$. This topology is called the in light of its relationship to the Zariski topology on an algebraic variety (see Section        4 below). Note that a point $P\in\operatorname{Spec}(R)$ is closed if and only if $P\subset R$ is a maximal ideal  .

A distinguished open set of $\operatorname{Spec}(R)$ is defined to be an open set of the form

 $\operatorname{Spec}(R)_{f}:=\{P\in\operatorname{Spec}(R)\mid f\notin P\}=% \operatorname{Spec}(R)\setminus V(\{f\}),$

for any element $f\in R$. The collection of distinguished open sets forms a topological basis for the open sets of $\operatorname{Spec}(R)$. In fact, we have

 $\operatorname{Spec}(R)\setminus V(A)=\bigcup_{f\in A}\operatorname{Spec}(R)_{f}.$

The topological space $\operatorname{Spec}(R)$ has the following additional properties:

## 3 Spec as a sheaf

For convenience, we adopt the usual convention of writing $X$ for $\operatorname{Spec}(R)$. For any $f\in R$ and $P\in X_{f}$, let $\iota_{f,P}:R_{f}\longrightarrow R_{P}$ be the natural inclusion map  . Define a presheaf    of rings $\mathcal{O}_{X}$ on $X$ by setting

 $\mathcal{O}_{X}(U):=\left\{\left.(s_{P})\in\prod_{P\in U}R_{P}\ \right|\mbox{ \begin{tabular}[]{l}U has an open cover \{X_{f_{\alpha}}\} with elements s_{\alpha}\in R_{f_{\alpha}}\\ such that s_{P}=\iota_{f_{\alpha},P}(s_{\alpha}) whenever P\in X_{f_{\alpha}}\end{tabular} }\right\},$

for each open set $U\subset X$. The restriction  map $\operatorname{res}_{U,V}:\mathcal{O}_{X}(U)\longrightarrow\mathcal{O}_{X}(V)$ is the map induced by the projection map

 $\prod_{P\in U}R_{P}\longrightarrow\prod_{P\in V}R_{P},$

for each open subset $V\subset U$. The presheaf $\mathcal{O}_{X}$ satisfies the following properties:

1. 1.

$\mathcal{O}_{X}$ is a sheaf.

2. 2.

$\mathcal{O}_{X}(X_{f})=R_{f}$ for every $f\in R$.

3. 3.

The stalk $(\mathcal{O}_{X})_{P}$ is equal to $R_{P}$ for every $P\in X$. (In particular, $X$ is a locally ringed space.)

4. 4.

The restriction sheaf of $\mathcal{O}_{X}$ to $X_{f}$ is isomorphic   as a sheaf to $\mathcal{O}_{\operatorname{Spec}(R_{f})}$.

## 4 Relationship to algebraic varieties

Let $k$ be a field and write as usual $\mathbb{A}_{k}^{n}$ for the vector space $k^{n}$. Recall that an affine variety $V$ in $\mathbb{A}_{k}^{n}$ is the set of common zeros of some prime ideal $I\subset k[X_{1},\ldots,X_{n}]$. The coordinate ring of $V$ is defined to be the ring $R:=k[X_{1},\ldots,X_{n}]/I$, and there is an embedding   $i:V\hookrightarrow\operatorname{Spec}(R)$ given by

 $i(a_{1},\ldots,a_{n}):=(X_{1}-a_{1},\ldots,X_{n}-a_{n})\in\operatorname{Spec}(% R).$

The function $i$ is not a homeomorphism, because it is not a bijection (its image is contained inside the set of maximal ideals of $R$). However, the map $i$ does define an order preserving bijection between the open sets of $V$ and the open sets of $\operatorname{Spec}(R)$ in the Zariski topology. This isomorphism    between these two lattices of open sets can be used to equate the sheaf $\operatorname{Spec}(R)$ with the structure sheaf of the variety $V$, showing that the two objects are identical in every respect except for the minor detail of $\operatorname{Spec}(R)$ having more points than $V$.

The additional points of $\operatorname{Spec}(R)$ are valuable in many situations and a systematic study of them leads to the general notion of schemes. As just one example, the classical Bezout’s theorem is only valid for algebraically closed fields, but admits a scheme–theoretic generalization  which holds over non–algebraically closed fields as well. We will not attempt to explain the theory of schemes in detail, instead referring the interested reader to the references below.

Remark. The spectrum $\operatorname{Spec}(R)$ of a ring $R$ may be generalized to the case when $R$ is not commutative   , as long as $R$ contains the multiplicative identity  . For a ring $R$ with $1$, the $\operatorname{Spec}(R)$, like above, is the set of all proper prime ideals of $R$. This definition is used to develop the noncommutative version of Hilbert’s Nullstellensatz.

## References

• 1
• 2 David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, Springer–Verlag, 1999 (LNM 1358).
• 3 Louis H. Rowen, Ring Theory, Vol. 1, Academic Press, New York, 1988.
Title prime spectrum PrimeSpectrum 2013-03-22 12:38:07 2013-03-22 12:38:07 CWoo (3771) CWoo (3771) 16 CWoo (3771) Definition msc 14A15 Scheme Sheaf ZariskiTopology LocallyRingedSpace distinguished open set