Spectrum of a ring explained
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative 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 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
\{VI\colonIisanidealofR\}.
This topology is called the
Zariski topology.
A basis for the Zariski topology can be constructed as follows. For f ∈ R, define Df to be the set of prime ideals of R not containing f. Then each Df is an open subset of
, and
is a basis for the Zariski topology.
is a
compact space, but almost never
Hausdorff: in fact, the
maximal ideals in
R are precisely the closed points in this topology. By the same reasoning,
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
is defined on the distinguished open subsets
by setting
the
localization of by the powers of . 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, written as the union of
, we set
where
denotes the
inverse limit with respect to the natural ring homomorphisms
One may check that this presheaf is a sheaf, so
is a
ringed space. 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 over the ring, we may define a sheaf
on
. On the distinguished open subsets set
using the localization of a module. As above, this construction extends to a presheaf on all open subsets of
and satisfies the
gluing axiom. A sheaf of this form is called a
quasicoherent sheaf.
If is a point in
, that is, a prime ideal, then the
stalk of the structure sheaf at equals the
localization of at the ideal, and this is a
local ring. Consequently,
is a
locally ringed space.
If is an integral domain, with field of fractions, then we can describe the ring
more concretely as follows. We say that an element in is regular at a point in if it can be represented as a fraction with not in . Note that this agrees with the notion of a
regular function in algebraic geometry. Using this definition, we can describe
as precisely the set of elements of that are regular at every point in .
Functorial perspective
It is useful to use the language of category theory and observe that
is a
functor. Every
ring homomorphism
induces a continuous map
\operatorname{Spec}(f):\operatorname{Spec}(S)\to\operatorname{Spec}(R)
(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, and 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 of the other.
Motivation from algebraic geometry
Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kn (where K is an algebraically closed field) that are defined as the common zeros of a set of polynomials 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.
\operatorname{MaxSpec}(R)
, 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 spectrum of integers: 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 scheme-theoretic analogue of
: The affine scheme
=\operatorname{Spec}(C[x1,\ldots,xn])
. From the functor of points perspective, a point
(\alpha1,\ldots,\alphan)\inCn
can be identified with the evaluation morphism
C[x1,\ldots,xn]
\xrightarrow{ev | |
| (\alpha1,\ldots,\alphan) |
} \mathbb. This fundamental observation allows us to give meaning to other affine schemes.
\operatorname{Spec}(C[x,y]/(xy))
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
\{(\alpha1,0),(0,\alpha2):\alpha1,\alpha2\inC\}
.
Non-affine examples
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.
=\operatorname{Proj}k[x0,\ldots,xn]
over a field
. This can be easily generalized to any base ring, see
Proj construction (in fact, we can define projective space for any base scheme). The projective
-space for
is not affine as the global section of
is
.
- Affine plane minus the origin. Inside
=\operatorname{Spec}k[x,y]
are distinguished open affine subschemes
. Their union
is the affine plane with the origin taken out. The global sections of
are pairs of polynomials on
that restrict to the same polynomial on
, which can be shown to be
, the global section of
.
is not affine as
in
.
Non-Zariski topologies on a prime spectrum
Some authors (notably M. Hochster) consider topologies on prime spectra other than the Zariski topology.
First, there is the notion of constructible topology: given a ring A, the subsets of
of the form
\varphi*(\operatorname{Spec}B),\varphi:A\toB
satisfy the axioms for closed sets in a topological space. This topology on
is called the constructible topology.
In, Hochster considers what he calls the patch topology on a prime spectrum. By definition, the patch topology is the smallest topology in which the sets of the forms
and
\operatorname{Spec}(A)-V(f)
are closed.
Global or relative Spec
There is a relative version of the functor
called global
, or relative
. If
is a scheme, then relative
is denoted by
\underline{\operatorname{Spec}}S
or
. If
is clear from the context, then relative Spec may be denoted by
\underline{\operatorname{Spec}}
or
. For a scheme
and a
quasi-coherent sheaf of
-algebras
, there is a scheme
\underline{\operatorname{Spec}}S(l{A})
and a morphism
f:\underline{\operatorname{Spec}}S(l{A})\toS
such that for every open affine
, there is an isomorphism
f-1(U)\cong\operatorname{Spec}(l{A}(U))
, and such that for open affines
, the inclusion
is induced by the restriction map
. That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the
Spec of the sheaf.
Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative
-algebras and schemes over
. In formulas,
\operatorname{Hom}l{OS-alg
}(\mathcal, \pi_*\mathcal_X) \cong \operatorname_(X, \mathbf(\mathcal)),where
is a morphism of schemes.
Example of a relative Spec
The relative spec is the correct tool for parameterizing the family of lines through the origin of
over
Consider the sheaf of algebras
and let
be a sheaf of ideals of
Then the relative spec
| 1 |
\underline{\operatorname{Spec}} | |
| a,b |
parameterizes the desired family. In fact, the fiber over
is the line through the origin of
containing the point
Assuming
the fiber can be computed by looking at the composition of pullback diagrams
\begin{matrix}
\operatorname{Spec}\left(
| C[x,y] |
\left(y- | \beta | x\right) | \alpha |
|
\right)&\to&\operatorname{Spec}\left(
\right)&\to&\underline{\operatorname{Spec}}X\left(
| l{O |
X[x,y]}{\left(ay-bx\right)} |
\right)\\
\downarrow&&\downarrow&&\downarrow\\
\operatorname{Spec}(C)&\to&\operatorname{Spec}\left(C\left[
\right]\right)=Ua&\to&
\end{matrix}
where the composition of the bottom arrows
\operatorname{Spec}(C)\xrightarrow{[\alpha:\beta]}
gives the line containing the point
and the origin. This example can be generalized to parameterize the family of lines through the origin of
over
by letting
and
l{I}=\left(2 x 2minorsof\begin{pmatrix}a0& … &an\\
x0& … &xn\end{pmatrix}\right).
Representation theory perspective
From the perspective of representation theory, a prime ideal I corresponds to a module R/I, and the spectrum of a ring corresponds to irreducible cyclic representations of R, while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is the study of modules over its group algebra.
or, without a basis,
As the latter formulation makes clear, a polynomial ring is the group algebra over a
vector space, and writing in terms of
corresponds to choosing a basis for the vector space. Then an ideal
I, or equivalently a module
is a cyclic representation of
R (cyclic meaning generated by 1 element as an
R-module; this generalizes 1-dimensional representations).
In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in n-space, by the Nullstellensatz (the maximal ideal generated by
(x1-a1),(x2-a2),\ldots,(xn-an)
corresponds to the point
). These representations of
are then parametrized by the dual space
the covector being given by sending each
to the corresponding
. Thus a representation of
(
K-linear maps
) is given by a set of
n numbers, or equivalently a covector
Thus, points in n-space, thought of as the max spec of
correspond precisely to 1-dimensional representations of
R, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to
infinite-dimensional representations.
Functional analysis perspective
See main article: Spectrum (functional analysis).
The term "spectrum" comes from the use in operator theory.Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[''T''], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[''T''] (as a ring) equals the spectrum of T (as an operator).
Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:
the 2×2 zero matrix has module
showing geometric multiplicity 2 for the zero
eigenvalue,while a non-trivial 2×2 nilpotent matrix has module
showing algebraic multiplicity 2 but geometric multiplicity 1.
In more detail:
- the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
- the primary decomposition of the module corresponds to the unreduced points of the variety;
- a diagonalizable (semisimple) operator corresponds to a reduced variety;
- a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space);
- the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial.
Generalizations
The spectrum can be generalized from rings to C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra. Notably, for a Hausdorff space, the algebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) is a commutative C*-algebra, with the space being recovered as a topological space from
of the algebra of scalars, indeed functorially so; this is the content of the
Banach–Stone theorem. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to
non-commutative C*-algebras yields
noncommutative topology.
See also
References
- Book: Atiyah . Michael Francis . Michael Atiyah . Macdonald . I.G. . Ian G. Macdonald . Introduction to Commutative Algebra . Westview Press . 978-0-201-40751-8 . 1969.
- Book: Arkhangel’skii. Alexander Arhangelskii . A. V.. Pontryagin. Lev Pontryagin . L. S.. 10.1007/978-3-642-61265-7. General Topology I . Encyclopaedia of Mathematical Sciences . 1990 . 17 . 978-3-642-64767-3. .
- Book: Brandal . Willy . 10.1007/BFb0069021. Commutative Rings whose Finitely Generated Modules Decompose . Lecture Notes in Mathematics . 1979 . 723 . 978-3-540-09507-1 .
- Web site: Kock . Joachim . 2007 . Remarks on spectra, supports, and Hochster duality. 54501563 .
- Book: Sharp, Rodney Y. . Steps in Commutative Algebra . 2001 . 2nd . . 978-0-511-62368-4.
- 10.1080/00927872.2018.1469637 . Flat topology and its dual aspects . 2019 . Tarizadeh . Abolfazl . Communications in Algebra . 47 . 195–205 . 1503.04299 . 119574163 .
- Web site: Vakil . Ravi . Ravi Vakil . n.d. . Foundations Of Algebraic Geometry . math.stanford.edu.
Further reading
- https://mathoverflow.net/questions/441029/intrinsic-topology-on-the-zariski-spectrum
External links
Notes and References
- see https://www.math.ias.edu/~lurie/261ynotes/lecture14.pdf