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| {{for|the concept of ring spectrum in homotopy theory|Ring spectrum}}
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| In [[abstract algebra]] and [[algebraic geometry]], the '''spectrum''' of a [[commutative ring]] ''R'', denoted by Spec(''R''), is the set of all proper [[prime ideal]]s of ''R''. It is commonly augmented with the [[Zariski topology]] and with a structure [[sheaf (mathematics)|sheaf]], turning it into a [[locally ringed space]].
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| ==Zariski topology==
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| For any [[ideal (ring theory)|ideal]] ''I'' of ''R'', define <math>V_I</math> to be the set of prime ideals containing ''I''. We can put a topology on Spec(''R'') by defining the [[Characterizations of the category of topological spaces#Definition via closed sets|collection of closed sets]] to be
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| :<math>\{ V_I \colon I \text{ is an ideal of } R \}.</math>
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| This topology is called the [[Zariski topology]].
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| A [[Base (topology)|basis]] for the Zariski topology can be constructed as follows. For ''f''∈''R'', define ''D''<sub>''f''</sub> to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''<sub>''f''</sub> is an open subset of Spec(''R''), and <math>\{D_f:f\in R\}</math> is a basis for the Zariski topology.
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| Spec(''R'') is a [[compact space]], but almost never [[Hausdorff space|Hausdorff]]: in fact, the [[maximal ideal]]s in ''R'' are precisely the closed points in this topology. However, Spec(''R'') is always a [[Kolmogorov space]]. It is also a [[spectral space]].
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| ==Sheaves and schemes==
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| Given the space ''X''=Spec(''R'') with the Zariski topology, the structure sheaf ''O''<sub>''X''</sub> is defined on the ''D''<sub>''f''</sub> by setting Γ(''D''<sub>''f''</sub>, ''O''<sub>''X''</sub>) = ''R''<sub>''f''</sub>, the [[localization of a ring|localization]] of ''R'' at the multiplicative system {1,''f'',''f''<sup>2</sup>,''f''<sup>3</sup>,...}. It can be shown that this satisfies the necessary axioms to be a [[B-Sheaf#Sheaves on a basis of open sets|B-Sheaf]]. Next, if ''U'' is the union of {''D''<sub>''fi''</sub>}<sub>''i''∈''I''</sub>, we let Γ(''U'',''O''<sub>''X''</sub>) = lim<sub>''i''∈''I''</sub> ''R''<sub>''fi''</sub>, and this produces a sheaf; see the [[Gluing axiom#Sheaves on a basis of open sets|Gluing axiom]] article for more detail.
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| If ''R'' is an integral domain, with field of fractions ''K'', then we can describe the ring Γ(''U'',''O''<sub>''X''</sub>) 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''<sub>''X''</sub>) as precisely the set of elements of ''K'' which are regular at every point ''P'' in ''U''.
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| If ''P'' is a point in Spec(''R''), that is, a prime ideal, then the stalk at ''P'' equals the [[localization of a ring|localization]] of ''R'' at ''P'', and this is a [[local ring]]. Consequently, Spec(''R'') is a [[locally ringed space]].
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| Every locally ringed space isomorphic to one of this form is called an ''affine scheme''.
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| General [[scheme (mathematics)|schemes]] are obtained by "gluing together" several affine schemes.
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| ==Functoriality==
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| It is useful to use the language of [[category theory]] and observe that Spec is a [[functor]].
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| Every [[ring homomorphism]] ''f'' : ''R'' → ''S'' induces a [[continuous function (topology)|continuous]] map Spec(''f'') : Spec(''S'') → Spec(''R'') (since the preimage of any prime ideal in ''S'' is a prime ideal in ''R''). In this way, Spec can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover for every prime ''P'' the homomorphism ''f'' descends to homomorphisms
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| :''O''<sub>''f''<sup> -1</sup>(''P'')</sub> → ''O''<sub>''P''</sub>
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| of local rings. Thus Spec even defines a contravariant functor from the category of commutative rings to the category of [[locally ringed space]]s. In fact it is the universal such functor and this can be used to define the functor Spec up to natural isomorphism.
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| The functor Spec 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.
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| ==Motivation from algebraic geometry==
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| Following on from the example, in [[algebraic geometry]] one studies ''algebraic sets'', i.e. subsets of ''K''<sup>''n''</sup> (where ''K'' is an [[algebraically closed field]]) which are defined as the common zeros of a set of [[polynomial]]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 component|irreducible]] or a variety if it cannot be written as the union of two proper algebraic subsets).
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| 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).
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| One can thus view the topological space Spec(''R'') 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 Spec(''R'') 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 [[scheme (mathematics)|scheme]]s.
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| ==Global Spec==
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| There is a relative version of the functor Spec called global Spec, or relative Spec, and denoted by '''Spec'''. For a scheme ''Y'', and a quasi-coherent sheaf of ''O<sub>Y</sub>''-algebras ''A'', there is a unique scheme '''Spec'''''A'', and a morphism <math>f \colon \bold{Spec} \ A \to Y</math> such that for every open affine <math>U \subseteq Y</math>, there is an isomorphism induced by ''f'': <math>f^{-1}(U) \cong \mathrm{Spec} \ A(U)</math>, and such that for open affines <math>U \subseteq V</math>, the inclusion <math>f^{-1}(U) \to f^{-1}(V)</math> induces the restriction map <math>A(V) \to A(U).</math> 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.
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| ==Representation theory perspective==
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| 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]].
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| The connection to representation theory is clearer if one considers the [[polynomial ring]] <math>R=K[x_1,\dots,x_n]</math> or, without a basis, <math>R=K[V].</math> As the latter formulation makes clear, a polynomial ring is the group algebra over a [[vector space]], and writing in terms of <math>x_i</math> corresponds to choosing a basis for the vector space. Then an ideal ''I,'' or equivalently a module <math>R/I,</math> is a cyclic representation of ''R'' (cyclic meaning generated by 1 element as an ''R''-module; this generalizes 1-dimensional representations).
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| In the case that the field is closed (say, the complex numbers) and one uses a maximal ideal, which corresponds (by the [[nullstellensatz]]) to a point in ''n''-space (the maximal ideal generated by <math>(x_1-a_1), (x_2-a_2),\ldots,(x_n-a_n)</math> corresponds to the point <math>(a_1,\ldots,a_n)</math>), these representations are parametrized by the dual space <math>V^*,</math> (the covector is given by the <math>a_i</math>). This is precisely [[Fourier theory]]: the representations the additive group <math>K</math> are given by the [[dual group]]{{disambiguation needed|date=July 2011}} (simply, maps <math>K \to K</math> are multiplication by a scalar), and thus the representations of <math>K^n</math> (''K''-linear maps <math>K^n \to K</math>) are given by a set of ''n''-numbers, or equivalently a covector <math>K^n \to K.</math>
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| Thus, points in ''n''-space, thought of as the max spec of <math>R=K[x_1,\dots,x_n],</math> 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.
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| ==Functional analysis perspective==
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| {{main|Spectrum (functional analysis)}}
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| The term "spectrum" comes from the use in [[operator theory]].
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| 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).
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| 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:
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| :<math>K[T]/(T-1) \oplus K[T]/(T-1)</math>
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| the 2×2 zero matrix has module
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| :<math>K[T]/(T-0) \oplus K[T]/(T-0),</math>
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| showing geometric multiplicity 2 for the zero eigenvalue,
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| while a non-trivial 2×2 nilpotent matrix has module
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| :<math>K[T]/T^2,</math>
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| showing algebraic multiplicity 2 but geometric multiplicity 1.
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| In more detail:
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| * the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
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| * the primary decomposition of the module corresponds to the unreduced points of the variety;
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| * a diagonalizable (semisimple) operator corresponds to a reduced variety;
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| * a cyclic module (one generator) corresponds to the operator having a [[cyclic vector]] (a vector whose orbit under ''T'' spans the space);
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| * the first [[invariant factor]] of the module equals the [[Minimal polynomial (linear algebra)|minimal polynomial]] of the operator, and the last invariant factor equals the [[characteristic polynomial]].
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| ==Generalizations==
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| The spectrum can be generalized from rings to [[C*-algebra]]s 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) are a ''commutative'' C*-algebra, with the space being recovered as a topological space from MSpec 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]].
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| ==See also==
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| *[[Spectrum of a matrix]]
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| *[[Constructible topology]]
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| ==References==
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| * {{Citation | last1=Cox | first1=David | author1-link=David Cox (mathematician) | last2=O'Shea | first2=Donal | last3=Little | first3=John | title=Ideals, Varieties, and Algorithms | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94680-1 | year=1997}}
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| * {{Citation | last1=Eisenbud | first1=David | author1-link = David Eisenbud | last2=Harris | first2=Joe | author2-link = Joe Harris (mathematician) | title=The geometry of schemes | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98638-8; 978-0-387-98637-1 | mr=1730819 | year=2000 | volume=197}}
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| * {{Citation | last1=Hartshorne | first1=Robin | author1-link = Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | year=1977}}
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| ==External links==
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| * Kevin R. Coombes: [http://odin.mdacc.tmc.edu/~krcoombes/agathos/spec.html ''The Spectrum of a Ring'']
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| [[Category:Commutative algebra]]
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| [[Category:Scheme theory]]
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| [[Category:Prime ideals]]
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