In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, the prime spectrum (or simply the spectrum) of a
ring ''R'' is the set of all
prime ideals of ''R'', and is usually denoted by
; in
algebraic geometry it is simultaneously a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
equipped with the
sheaf of rings .
Zariski topology
For any
ideal ''I'' of ''R'', define
to be the set of prime ideals containing ''I''. We can put a topology on
by defining the
collection of closed sets to be
:
This topology is called the
Zariski topology.
A
basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''
''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''
''f'' is an open subset of
, and
is a basis for the Zariski topology.
is a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, but almost never
Hausdorff: in fact, the
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
s in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a
T1 space. However,
is always a
Kolmogorov space (satisfies the T
0 axiom); it is also a
spectral space.
Sheaves and schemes
Given the space
with the Zariski topology, the structure sheaf ''O''
''X'' is defined on the distinguished open subsets ''D''
''f'' by setting Γ(''D''
''f'', ''O''
''X'') = ''R''
''f'', the
localization of ''R'' by the powers of ''f''. It can be shown that this defines a
B-sheaf and therefore that it defines a sheaf. In more detail, the distinguished open subsets are a
basis of the Zariski topology, so for an arbitrary open set ''U'', written as the union of
''i''∈''I'', we set Γ(''U'',''O''
''X'') = lim
''i''∈''I'' ''R''
''fi''. One may check that this presheaf is a sheaf, so
is a
ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
. Any ringed space isomorphic to one of this form is called an affine scheme. General
schemes are obtained by gluing affine schemes together.
Similarly, for a module ''M'' over the ring ''R'', we may define a sheaf
on
. On the distinguished open subsets set Γ(''D''
''f'',
) = ''M''
''f'', using the
localization of a module. As above, this construction extends to a presheaf on all open subsets of
and satisfies gluing axioms. A sheaf of this form is called a
quasicoherent sheaf.
If ''P'' is a point in
, that is, a prime ideal, then the stalk of the structure sheaf at ''P'' equals the
localization of ''R'' at the ideal ''P'', and this is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
. Consequently,
is a
locally ringed space.
If ''R'' is an integral domain, with field of fractions ''K'', then we can describe the ring Γ(''U'',''O''
''X'') more concretely as follows. We say that an element ''f'' in ''K'' is regular at a point ''P'' in ''X'' if it can be represented as a fraction ''f'' = ''a''/''b'' with ''b'' not in ''P''. Note that this agrees with the notion of a
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regul ...
in algebraic geometry. Using this definition, we can describe Γ(''U'',''O''
''X'') as precisely the set of elements of ''K'' which are regular at every point ''P'' in ''U''.
Functorial perspective
It is useful to use the language of
category theory and observe that
is a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
. Every
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition prese ...
induces a
continuous map
(since the preimage of any prime ideal in
is a prime ideal in
). In this way,
can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime
the homomorphism
descends to homomorphisms
:
of local rings. Thus
even defines a contravariant functor from the category of commutative rings to the category of
locally ringed spaces. In fact it is the universal such functor hence can be used to define the functor
up to natural isomorphism.
The functor
yields a contravariant equivalence between the
category of commutative rings and the category of affine schemes; each of these categories is often thought of as the
opposite category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yield ...
of the other.
Motivation from algebraic geometry
Following on from the example, in
algebraic geometry one studies ''algebraic sets'', i.e. subsets of ''K''
''n'' (where ''K'' is an
algebraically closed field) that are defined as the common zeros of a set of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s in ''n'' variables. If ''A'' is such an algebraic set, one considers the commutative ring ''R'' of all polynomial functions ''A'' → ''K''. The ''maximal ideals'' of ''R'' correspond to the points of ''A'' (because ''K'' is algebraically closed), and the ''prime ideals'' of ''R'' correspond to the ''subvarieties'' of ''A'' (an algebraic set is called
irreducible or a variety if it cannot be written as the union of two proper algebraic subsets).
The spectrum of ''R'' therefore consists of the points of ''A'' together with elements for all subvarieties of ''A''. The points of ''A'' are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of ''A'', i.e. the maximal ideals in ''R'', then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in ''R'', i.e.
, together with the Zariski topology, is homeomorphic to ''A'' also with the Zariski topology.
One can thus view the topological space
as an "enrichment" of the topological space ''A'' (with Zariski topology): for every subvariety of ''A'', one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the
generic point for the subvariety. Furthermore, the sheaf on
and the sheaf of polynomial functions on ''A'' are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of
schemes.
Examples
* The affine scheme
is the final object in the category of affine schemes since
is the initial object in the category of commutative rings.
* The affine scheme
is scheme theoretic analogue of
. From the functor of points perspective, a point
can be identified with the evaluation morphism
. This fundamental observation allows us to give meaning to other affine schemes.
*
looks topologically like the transverse intersection of two complex planes at a point, although typically this is depicted as a
since the only well defined morphisms to
are the evaluation morphisms associated with the points
.
* The prime spectrum of a
Boolean ring (e.g., a
power set ring) is a (Hausdorff)
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
.
* (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a
spectral space) if and only if it is quasi-compact,
quasi-separated In algebraic geometry, a morphism of schemes from to is called quasi-separated if the diagonal map from to is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme is called quasi-separated if ...
and
sober.
Non-affine examples
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.
* The Projective
-Space