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prime spectrum

(Definition)

Spec as a set

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

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

For any subset

, we define the variety of

to be the set

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

It is enough to restrict attention to subsets of

which are ideals, since, for any subset

, we have

where

is the ideal generated by

. In fact, even more is true: $V(I) = V(\sqrt{I})$ where $\sqrt{I}$ denotes the radical of the ideal

Spec as a topological space

We impose a topology on $\operatorname{Spec}(R)$ by defining the sets 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 for some subset ). 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 $I_\alpha$ ,

, establish that this collection does constitute a topology on $\operatorname{Spec}(R)$ . This topology is called the Zariski topology 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

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

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

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

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

$\operatorname{Spec}(R)$ is compact (but almost never Hausdorff).
A subset of $\operatorname{Spec}(R)$ is an irreducible closed set if and only if it equals for some prime ideal of .
For $f \in R$ , let denote the localization of at . Then the topological spaces $\operatorname{Spec}(R)_f$ and $\operatorname{Spec}(R_f)$ are naturally homeomorphic, via the correspondence sending a prime ideal of not containing to the induced prime ideal in .
For $P \in \operatorname{Spec}(R)$ , let denote the localization of at the prime ideal . Then the topological spaces $V(P) \subset \operatorname{Spec}(R)$ and $\operatorname{Spec}(R_P)$ are naturally homeomorphic, via the correspondence sending a prime ideal of contained in to the induced prime ideal in .

Spec as a sheaf

For convenience, we adopt the usual convention of writing 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 $\O _X$ on by setting

$\displaystyle \O _X(U) := \left\{ \left.(s_P) \in \prod_{P \in U} R_P \ \right\... ...{f_\alpha,P}(s_\alpha)$ whenever $P \in X_{f_\alpha}$ \end{tabular}} \right\},$

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

$\displaystyle \prod_{P \in U} R_P \longrightarrow \prod_{P \in V} R_P,$

for each open subset $V \subset U$ . The presheaf $\O _X$ satisfies the following properties:

$\O _X$ is a sheaf.
$\O _X(X_f) = R_f$ for every $f \in R$ .
The stalk $(\O _X)_P$ is equal to for every $P \in X$ . (In particular, is a locally ringed space.)
The restriction sheaf of $\O _X$ to is isomorphic as a sheaf to $\O _{\operatorname{Spec}(R_f)}$ .

Relationship to algebraic varieties

$\operatorname{Spec}(R)$ is sometimes called an affine scheme because of the close relationship between affine varieties in $\mathbb{A}_k^n$ and the $\operatorname{Spec}$ of their corresponding coordinate rings. In fact, the correspondence between the two is an equivalence of categories, although a complete statement of this equivalence requires the notion of morphisms of schemes and will not be given here. Nevertheless, we explain what we can of this correspondence below.

Let be a field and write as usual $\mathbb{A}_k^n$ for the vector space . Recall that an affine variety 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 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

$\displaystyle i(a_1, \ldots, a_n) := (X_1 - a_1, \ldots, X_n - a_n) \in \operatorname{Spec}(R).$

The function

is not a homeomorphism, because it is not a bijection (its image is contained inside the set of maximal ideals of

). However, the map

does define an order preserving bijection between the open sets of

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

, showing that the two objects are identical in every respect except for the minor detail of $\operatorname{Spec}(R)$ having more points than

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.

Bibliography

1: Robin Hartshorne, Algebraic Geometry, Springer-Verlag New York, Inc., 1977 (GTM 52).
2: David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, Springer-Verlag, 1999 (LNM 1358).

"prime spectrum" is owned by djao.

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Also defines:

distinguished open set

Attachments:: proof that $\operatorname{Spec}(R)$ is quasi-compact (Proof) by Wkbj79
$V(I)=\emptyset$ implies (Theorem) by Wkbj79

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Cross-references: references, theory, algebraically closed, Bezout's theorem, schemes, minor, objects, structure sheaf, equate, isomorphism, order, image, bijection, homeomorphism, function, embedding, vector space, field, morphisms of schemes, equivalence, complete, equivalence of categories, coordinate, affine varieties, affine scheme, isomorphic, locally ringed space, stalk, sheaf, projection map, map, restriction, rings, presheaf, inclusion map, contained, induced, homeomorphic, localization, prime ideal, irreducible, Hausdorff, compact, properties, basis, open set, maximal ideal, closed, point, section, algebraic, Zariski topology, equations, complement, open, closed subsets, collection, topology, radical, even, ideal generated by, ideals, variety, subset, identity, commutative ring
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This is version 11 of prime spectrum, born on 2002-05-09, modified 2003-08-20.
Object id is 2898, canonical name is PrimeSpectrum.
Accessed 8046 times total.

Classification:

AMS MSC:

14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)

Pending Errata and Addenda

None.[ View all 1 ]

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