prime spectrum

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In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s of ''R'', and is usually denoted by $\operatorname$; in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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 point ...
equipped with the sheaf of rings $\mathcal$.

# Zariski topology

For any
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring consider ...
''I'' of ''R'', define $V_I$ to be the set of prime ideals containing ''I''. We can put a topology on $\operatorname\left(R\right)$ by defining the collection of closed sets to be :$\.$ This topology is called the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
. A
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...
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 $\operatorname\left(R\right)$, and $\$ is a basis for the Zariski topology. $\operatorname\left(R\right)$ 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 co ...
s in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a T1 space. However, $\operatorname\left(R\right)$ is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.

# Sheaves and schemes

Given the space $X = \operatorname\left(R\right)$ 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 Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is ...
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 Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...
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 $\operatorname\left(R\right)$ 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 o ...
. 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 $\tilde$ on $\operatorname\left(R\right)$. On the distinguished open subsets set Γ(''D''''f'', $\tilde$) = ''M''''f'', using the localization of a module. As above, this construction extends to a presheaf on all open subsets of $\operatorname\left(R\right)$ and satisfies gluing axioms. A sheaf of this form is called a quasicoherent sheaf. If ''P'' is a point in $\operatorname\left(R\right)$, that is, a prime ideal, then the stalk of the structure sheaf at ''P'' equals the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is ...
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, $\operatorname\left(R\right)$ 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. 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
and observe that $\operatorname$ 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, and ...
. 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 preserv ...
$f: R \to S$ induces a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
map $\operatorname\left(f\right): \operatorname\left(S\right) \to \operatorname\left(R\right)$ (since the preimage of any prime ideal in $S$ is a prime ideal in $R$). In this way, $\operatorname$ can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime $\mathfrak$ the homomorphism $f$ descends to homomorphisms :$\mathcal_ \to \mathcal_\mathfrak$ of local rings. Thus $\operatorname$ 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 $\operatorname$ up to natural isomorphism. The functor $\operatorname$ 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 yields ...
of the other.

# Motivation from algebraic geometry

Following on from the example, in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
one studies ''algebraic sets'', i.e. subsets of ''K''''n'' (where ''K'' is an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, becaus ...
) 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 examp ...
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. $\operatorname\left(R\right)$, together with the Zariski topology, is homeomorphic to ''A'' also with the Zariski topology. One can thus view the topological space $\operatorname\left(R\right)$ 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 $\operatorname\left(R\right)$ 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 $\operatorname\left(\mathbb\right)$ is the final object in the category of affine schemes since $\mathbb$ is the initial object in the category of commutative rings. * The affine scheme is scheme theoretic analogue of $\mathbb^n$. From the functor of points perspective, a point $\left(\alpha_1,\ldots,\alpha_n\right) \in \mathbb^n$ 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 $\mathbb$ are the evaluation morphisms associated with the points $\$. * The prime spectrum of a
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
(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 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 $n$-Space
Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not funct ...
(in fact, we can define Projective Space for any base scheme). The Projective $n$-Space for $n \geq 1$ is not affine as the global section of $\mathbb^n_k$ is $k$. * Affine plane minus the origin. Inside

# Non-Zariski topologies on a prime spectrum

Some authors (notably M. Hochster) consider topologies on prime spectra other than Zariski topology. First, there is the notion of constructible topology: given a ring ''A'', the subsets of $\operatorname\left(A\right)$ of the form $\varphi^*\left(\operatorname B\right), \varphi: A \to B$ satisfy the axioms for closed sets in a topological space. This topology on $\operatorname\left(A\right)$ is called the constructible topology. In , Hochster considers what he calls the patch topology on a prime spectrum.Willy Brandal, Commutative Rings whose Finitely Generated Modules Decompose By definition, the patch topology is the smallest topology in which the sets of the forms $V\left(I\right)$ and $\operatorname\left(A\right) - V\left(f\right)$ are closed.

# Global or relative Spec

There is a relative version of the functor $\operatorname$ called global $\operatorname$, or relative $\operatorname$. If $S$ is a scheme, then relative $\operatorname$ is denoted by $\underline_S$ or $\mathbf_S$. If $S$ is clear from the context, then relative Spec may be denoted by $\underline$ or $\mathbf$. For a scheme $S$ and a quasi-coherent sheaf of $\mathcal_S$-algebras $\mathcal$, there is a scheme $\underline_S\left(\mathcal\right)$ and a morphism $f : \underline_S\left(\mathcal\right) \to S$ such that for every open affine $U \subseteq S$, there is an isomorphism $f^\left(U\right) \cong \operatorname\left(\mathcal\left(U\right)\right)$, and such that for open affines $V \subseteq U$, the inclusion $f^\left(V\right) \to f^\left(U\right)$ is induced by the restriction map $\mathcal\left(U\right) \to \mathcal\left(V\right)$. 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 $\mathcal_S$-algebras and schemes over $S$. In formulas, :$\operatorname_\left(\mathcal, \pi_*\mathcal_X\right) \cong \operatorname_\left(X, \mathbf\left(\mathcal\right)\right),$ where $\pi \colon X \to S$ 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 $\mathbb^2_\mathbb$ over $X = \mathbb^1_.$ Consider the sheaf of algebras and let $\mathcal = \left(ay-bx\right)$ be a sheaf of ideals of $\mathcal.$ Then the relative spec $\underline_X\left(\mathcal/\mathcal\right) \to \mathbb^1_$ parameterizes the desired family. In fact, the fiber over

# Representation theory perspective

From the perspective of
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, 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. The connection to representation theory is clearer if one considers the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...
, and writing in terms of $x_i$ corresponds to choosing a basis for the vector space. Then an ideal ''I,'' or equivalently a module $R/I,$ 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 $\left(x_1-a_1\right), \left(x_2-a_2\right),\ldots,\left(x_n-a_n\right)$ corresponds to the point $\left(a_1,\ldots,a_n\right)$). These representations of

# Functional analysis perspective

The term "spectrum" comes from the use in
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
. 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 The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then :M\cong R^r\o ...
of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
.

# Generalizations

The spectrum can be generalized from rings to
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
s in
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
, 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 $\operatorname$ of the algebra of scalars, indeed functorially so; this is the content of the
Banach–Stone theorem In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recov ...
. 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.

*
Scheme (mathematics) In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different s ...
* Projective scheme * Spectrum of a matrix *
Serre's theorem on affineness In the mathematical discipline of algebraic geometry, Serre's theorem on affineness (also called Serre's cohomological characterization of affineness or Serre's criterion on affineness) is a theorem due to Jean-Pierre Serre which gives sufficient ...
* Étale spectrum * Ziegler spectrum * Primitive spectrum

* * * * * *