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prime spectrum
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(Definition)
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Let be any commutative ring with identity. The prime spectrum
of is defined to be the set
 is a prime ideal of 
For any subset of , we define the variety of to be the set
It is enough to restrict attention to subsets of which are ideals, since, for any subset of , we have
where is the ideal generated by . In fact, even more is true:
where denotes the radical of the ideal .
We impose a topology on
by defining the sets to be the collection of closed subsets of
(that is, a subset of
is open if and only if it equals the complement of for some subset ). The equations
for any ideals , of , establish that this collection does constitute a topology on
. 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
is closed if and only if
is a maximal ideal.
A distinguished open set of
is defined to be an open set of the form
for any element . The collection of distinguished open sets forms a topological basis for the open sets of
. In fact, we have
The topological space
has the following additional properties:
-
is compact (but almost never Hausdorff).
- A subset of
is an irreducible closed set if and only if it equals for some prime ideal of .
- For
, let denote the localization of at . Then the topological spaces
and
are naturally homeomorphic, via the correspondence sending a prime ideal of not containing to the induced prime ideal in .
- For
, let denote the localization of at the prime ideal . Then the topological spaces
and
are naturally homeomorphic, via the correspondence sending a prime ideal of contained in to the induced prime ideal in .
For convenience, we adopt the usual convention of writing for
. For any and , let
be the natural inclusion map. Define a presheaf of rings on by setting
for each open set
. The restriction map
is the map induced by the projection map
for each open subset
. The presheaf satisfies the following properties:
is a sheaf.
-
for every .
- The stalk
is equal to for every . (In particular, is a locally ringed space.)
- The restriction sheaf of
to is isomorphic as a sheaf to
.
Relationship to algebraic varieties
is sometimes called an affine scheme because of the close relationship between affine varieties in
and the
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
for the vector space . Recall that an affine variety in
is the set of common zeros of some prime ideal
. The coordinate ring of is defined to be the ring
, and there is an embedding
given by
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
in the Zariski topology. This isomorphism between these two lattices of open sets can be used to equate the sheaf
with the structure sheaf of the variety , showing that the two objects are identical in every respect except for the minor detail of
having more points than .
The additional points of
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.
- 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).
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"prime spectrum" is owned by djao.
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(view preamble)
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
There are 15 references to this entry.
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) |
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Pending Errata and Addenda
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