Incommutative algebra, theprime spectrum (or simply thespectrum) of acommutative ring${\displaystyle R}$ is the set of allprime ideals of${\displaystyle R}$, and is usually denoted by${\displaystyle \mathrm{Spec}R}$;^[1] inalgebraic geometry it is simultaneously atopological space equipped with asheaf of rings.^[2]

For anyideal${\displaystyle I}$ of${\displaystyle R}$, define${\displaystyle V_I}$ to be the set ofprime ideals containing${\displaystyle I}$. We can put atopology on${\displaystyle \mathrm{Spec}(R)}$ by defining thecollection of closed sets to be

${\displaystyle \{V_I:I\text{ is an ideal of }R\}.}$

This topology is called theZariski topology.

Abasis for the Zariski topology can be constructed as follows: For${\displaystyle f\in R}$, define${\displaystyle D_f}$ to be the set of prime ideals of${\displaystyle R}$ not containing${\displaystyle f}$. Then each${\displaystyle D_f}$ is an open subset of${\displaystyle \mathrm{Spec}(R)}$, and${\displaystyle \{D_f:f\in R\}}$ is a basis for the Zariski topology.

${\displaystyle \mathrm{Spec}(R)}$ is acompact space, but almost neverHausdorff: In fact, themaximal ideals in${\displaystyle R}$ are precisely the closed points in this topology. By the same reasoning,${\displaystyle \mathrm{Spec}(R)}$ is not, in general, aT₁ space.^[3] However,${\displaystyle \mathrm{Spec}(R)}$ is always aKolmogorov space (satisfies the T₀ axiom); it is also aspectral space.

Given the space${\displaystyle X=\mathrm{Spec}(R)}$ with the Zariski topology, thestructure sheaf${\displaystyle {\mathcal{O}}_X}$ is defined on the distinguished open subsets${\displaystyle D_f}$ by setting${\displaystyle \mathrm{\Gamma }(D_f,{\mathcal{O}}_X)=R_f,}$ thelocalization of${\displaystyle R}$ by the powers of${\displaystyle f}$. It can be shown that this defines aB-sheaf and therefore that it defines asheaf. In more detail, the distinguished open subsets are abasis of the Zariski topology, so for an arbitrary open set${\displaystyle U}$, written as the union of$U=\bigcup _{i\in I}{D}_{f_i}$, we set$\mathrm{\Gamma }(U,{\mathcal{O}}_X)={\underleftarrow{\lim }}_{i\in I}{R}_{f_i},$ where${\displaystyle \underleftarrow{\lim }}$ denotes theinverse limit with respect to the natural ring homomorphisms${\displaystyle R_f\to R_{fg}.}$ One may check that thispresheaf is a sheaf, so${\displaystyle \mathrm{Spec}(R)}$ is aringed space. Any ringed space isomorphic to one of this form is called anaffine scheme. Generalschemes are obtained by gluing affine schemes together.

Similarly, for amodule${\displaystyle M}$ over the ring${\displaystyle R}$, we may define a sheaf${\displaystyle \tilde{M}}$ on${\displaystyle \mathrm{Spec}(R)}$. On the distinguished open subsets set${\displaystyle \mathrm{\Gamma }(D_f,\tilde{M})=M_f,}$ using thelocalization of a module. As above, this construction extends to a presheaf on all open subsets of${\displaystyle \mathrm{Spec}(R)}$ and satisfies thegluing axiom. A sheaf of this form is called aquasicoherent sheaf.

If${\displaystyle \mathfrak{p}}$ is a point in${\displaystyle \mathrm{Spec}(R)}$, that is, a prime ideal, then thestalk of the structure sheaf at${\displaystyle \mathfrak{p}}$ equals thelocalization of${\displaystyle R}$ at the ideal${\displaystyle \mathfrak{p}}$, which is generally denoted${\displaystyle R_{\mathfrak{p}}}$, and this is alocal ring. Consequently,${\displaystyle \mathrm{Spec}(R)}$ is alocally ringed space.

If${\displaystyle R}$ is anintegral domain, withfield of fractions${\displaystyle K}$, then we can describe the ring${\displaystyle \mathrm{\Gamma }(U,{\mathcal{O}}_X)}$ more concretely as follows. We say that an element${\displaystyle f}$ in${\displaystyle K}$ is regular at a point${\displaystyle \mathfrak{p}}$ in${\displaystyle X=\mathrm{Spec}R}$ if it can be represented as a fraction${\displaystyle f=a/b}$ with${\displaystyle b\notin \mathfrak{p}}$. Note that this agrees with the notion of aregular function in algebraic geometry. Using this definition, we can describe${\displaystyle \mathrm{\Gamma }(U,{\mathcal{O}}_X)}$ as precisely the set of elements of${\displaystyle K}$ that are regular at every point${\displaystyle \mathfrak{p}}$ in${\displaystyle U}$.

It is useful to use the language ofcategory theory and observe that${\displaystyle \mathrm{Spec}}$ is afunctor. Everyring homomorphism${\displaystyle f:R\to S}$ induces acontinuous map${\displaystyle \mathrm{Spec}(f):\mathrm{Spec}(S)\to \mathrm{Spec}(R)}$ (since the preimage of any prime ideal in${\displaystyle S}$ is a prime ideal in${\displaystyle R}$). In this way,${\displaystyle \mathrm{Spec}}$ can be seen as a contravariant functor from thecategory of commutative rings to thecategory of topological spaces. Moreover, for every prime${\displaystyle \mathfrak{p}}$ the homomorphism${\displaystyle f}$ descends to homomorphisms

${\displaystyle {\mathcal{O}}_{f^{-1}(\mathfrak{p})}\to {\mathcal{O}}_{\mathfrak{p}}}$

of local rings. Thus${\displaystyle \mathrm{Spec}}$ even defines a contravariant functor from the category of commutative rings to the category oflocally ringed spaces. In fact it is the universal such functor, and hence can be used to define the functor${\displaystyle \mathrm{Spec}}$ up tonatural isomorphism.^{[citation needed]}

The functor${\displaystyle \mathrm{Spec}}$ yields a contravariantequivalence between thecategory of commutative rings and thecategory of affine schemes; each of these categories is often thought of as theopposite category of the other.

Following on from the example, inalgebraic geometry one studiesalgebraic sets, i.e. subsets of${\displaystyle K^n}$ (where${\displaystyle K}$ is analgebraically closed field) that are defined as the common zeros of a set ofpolynomials in${\displaystyle n}$ variables. If${\displaystyle A}$ is such an algebraic set, one considers the commutative ring${\displaystyle R}$ of allpolynomial functions${\displaystyle A\to K}$. Themaximal ideals of${\displaystyle R}$ correspond to the points of${\displaystyle A}$ (because${\displaystyle K}$ is algebraically closed), and theprime ideals of${\displaystyle R}$ correspond to theirreducible subvarieties of${\displaystyle A}$ (an algebraic set is calledirreducible if it cannot be written as the union of two proper algebraic subsets).

The spectrum of${\displaystyle R}$ therefore consists of the points of${\displaystyle A}$ together with elements for all irreducible subvarieties of${\displaystyle A}$. The points of${\displaystyle A}$ 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${\displaystyle A}$, i.e. the maximal ideals in${\displaystyle R}$, 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${\displaystyle R}$, i.e.${\displaystyle \mathrm{MaxSpec}(R)}$, together with the Zariski topology, ishomeomorphic to${\displaystyle A}$ also with the Zariski topology.

One can thus view the topological space${\displaystyle \mathrm{Spec}(R)}$ as an "enrichment" of the topological space${\displaystyle A}$ (with Zariski topology): for every irreducible subvariety of${\displaystyle A}$, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding irreducible subvariety. One thinks of this point as thegeneric point for the irreducible subvariety. Furthermore, the structure sheaf on${\displaystyle \mathrm{Spec}(R)}$ and the sheaf of polynomial functions on${\displaystyle A}$ are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with the Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language ofschemes.

• The spectrum of integers: The affine scheme${\displaystyle \mathrm{Spec}(\mathbb{Z})}$ is thefinal object in the category of affine schemes since${\displaystyle \mathbb{Z}}$ is theinitial object in the category of commutative rings.
• The scheme-theoretic analogue of${\displaystyle {\mathbb{C}}^n}$: The affine scheme${\displaystyle {\mathbb{A}}_{\mathbb{C}}^n=\mathrm{Spec}(\mathbb{C}[x_1,\dots ,x_n])}$. From thefunctor of points perspective, a point${\displaystyle ({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{C}}^n}$ can be identified with the evaluation morphism${\displaystyle \mathbb{C}[x_1,\dots ,x_n]\underset{e{v}_{({\alpha }_1,\dots ,{\alpha }_n)}}{\overset{}{\to }}\mathbb{C}}$. This fundamental observation allows us to give meaning to other affine schemes.
• The cross:${\displaystyle \mathrm{Spec}(\mathbb{C}[x,y]/(xy))}$ looks topologically like the transverse intersection of two complex planes at a point (in particular, this scheme is not irreducible), although typically this is depicted as a${\displaystyle +}$, since the only well defined morphisms to${\displaystyle \mathbb{C}}$ are the evaluation morphisms associated with the points${\displaystyle \{({\alpha }_1,0),(0,{\alpha }_2):{\alpha }_1,{\alpha }_2\in \mathbb{C}\}}$.
• The prime spectrum of aBoolean ring (e.g., apower set ring) is a compacttotally disconnected Hausdorff space (that is, aStone space).^[4]
• (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., aspectral space) if and only if it is compact,quasi-separated andsober.^[5]

Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.

• Theprojective${\displaystyle n}$-space${\displaystyle {\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]}$ over a field${\displaystyle k}$. This can be easily generalized to any base ring, seeProj construction (in fact, we can define projective space for any base scheme). The projective${\displaystyle n}$-space for${\displaystyle n\ge 1}$ is not affine as the ring of global sections of${\displaystyle {\mathbb{P}}_k^n}$ is${\displaystyle k}$.
• Affine plane minus the origin.^[6] Inside${\displaystyle {\mathbb{A}}_k^2=\mathrm{Spec}k[x,y]}$ are distinguished open affine subschemes${\displaystyle D_x,D_y}$. Their union${\displaystyle D_x\cup D_y=U}$ is the affine plane with the origin taken out. The global sections of${\displaystyle U}$ are pairs of polynomials on${\displaystyle D_x,D_y}$ that restrict to the same polynomial on${\displaystyle D_{xy}}$, which can be shown to be${\displaystyle k[x,y]}$, the global sections of${\displaystyle {\mathbb{A}}_k^2}$.${\displaystyle U}$ is not affine as${\displaystyle V_{(x)}\cap V_{(y)}=\varnothing }$ in${\displaystyle U}$.

This sectionneeds expansion. You can help byadding to it.(June 2020)

Some authors (notably M. Hochster) consider topologies on prime spectra other than the Zariski topology.

First, there is the notion ofconstructible topology: given a ringA, the subsets of${\displaystyle \mathrm{Spec}(A)}$ of the form${\displaystyle {\phi }^{\ast }(\mathrm{Spec}B),\phi :A\to B}$ satisfy the axioms for closed sets in a topological space. This topology on${\displaystyle \mathrm{Spec}(A)}$ is called the constructible topology.^[7][8]

InHochster (1969), Hochster considers what he calls the patch topology on a prime spectrum.^[9][10][11] By definition, the patch topology is the smallest topology in which the sets of the forms${\displaystyle V(I)}$ and${\displaystyle \mathrm{Spec}(A)-V(f)}$ are closed.

There is a relative version of the functor${\displaystyle \mathrm{Spec}}$ called global${\displaystyle \mathrm{Spec}}$, or relative${\displaystyle \mathrm{Spec}}$. If${\displaystyle S}$ is a scheme, then relative${\displaystyle \mathrm{Spec}}$ is denoted by${\displaystyle {\underset{\_}{\mathrm{Spec}}}_S}$ or${\displaystyle {\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}}_S}$. If${\displaystyle S}$ is clear from the context, then relative Spec may be denoted by${\displaystyle \underset{\_}{\mathrm{Spec}}}$ or${\displaystyle \mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}}$. For a scheme${\displaystyle S}$ and aquasi-coherentsheaf of${\displaystyle {\mathcal{O}}_S}$-algebras${\displaystyle \mathcal{A}}$, there is a scheme${\displaystyle {\underset{\_}{\mathrm{Spec}}}_S(\mathcal{A})}$ and a morphism${\displaystyle f:{\underset{\_}{\mathrm{Spec}}}_S(\mathcal{A})\to S}$ such that for every open affine${\displaystyle U\subseteq S}$, there is an isomorphism${\displaystyle f^{-1}(U)\cong \mathrm{Spec}(\mathcal{A}(U))}$, and such that for open affines${\displaystyle V\subseteq U}$, the inclusion${\displaystyle f^{-1}(V)\to f^{-1}(U)}$ is induced by the restriction map${\displaystyle \mathcal{A}(U)\to \mathcal{A}(V)}$. 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 theSpec 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${\displaystyle {\mathcal{O}}_S}$-algebras and schemes over${\displaystyle S}$.^{[dubious –discuss]} In formulas,

${\displaystyle {\mathrm{Hom}}_{{\mathcal{O}}_S\text{-alg}}(\mathcal{A},{\pi }_{\ast }{\mathcal{O}}_X)\cong {\mathrm{Hom}}_{\text{Sch}/S}(X,\mathbf{S}\mathbf{p}\mathbf{e}\mathbf{c}(\mathcal{A})),}$

where${\displaystyle \pi :X\to S}$ is a morphism of schemes.

The relative spec is the correct tool for parameterizing the family of lines through the origin of${\displaystyle {\mathbb{A}}_{\mathbb{C}}^2}$ over${\displaystyle X={\mathbb{P}}_{a,b}^1.}$ Consider the sheaf of algebras${\displaystyle \mathcal{A}={\mathcal{O}}_X[x,y],}$ and let${\displaystyle \mathcal{I}=(ay-bx)}$ be a sheaf of ideals of${\displaystyle \mathcal{A}.}$ Then the relative spec${\displaystyle {\underset{\_}{\mathrm{Spec}}}_X(\mathcal{A}/\mathcal{I})\to {\mathbb{P}}_{a,b}^1}$ parameterizes the desired family. In fact, the fiber over${\displaystyle [\alpha :\beta ]}$ is the line through the origin of${\displaystyle {\mathbb{A}}^2}$ containing the point${\displaystyle (\alpha ,\beta ).}$ Assuming${\displaystyle \alpha \ne 0,}$ the fiber can be computed by looking at the composition of pullback diagrams

where the composition of the bottom arrows

${\displaystyle \mathrm{Spec}(\mathbb{C})\stackrel{[\alpha :\beta ]}{\to }{\mathbb{P}}_{a,b}^1}$

gives the line containing the point${\displaystyle (\alpha ,\beta )}$ and the origin. This example can be generalized to parameterize the family of lines through the origin of${\displaystyle {\mathbb{A}}_{\mathbb{C}}^{n+1}}$ over${\displaystyle X={\mathbb{P}}_{a_0,...,a_n}^n}$ by letting${\displaystyle \mathcal{A}={\mathcal{O}}_X[x_0,...,x_n]}$ and

From the perspective ofrepresentation theory, a prime idealI corresponds to a moduleR/I, and the spectrum of a ring corresponds toirreducible cyclic representations ofR, while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of agroup is the study of modules over itsgroup algebra.

The connection to representation theory is clearer if one considers thepolynomial ring${\displaystyle R=K[x_1,\dots ,x_n]}$ or, without a basis,${\displaystyle R=K[V].}$ As the latter formulation makes clear, a polynomial ring is the group algebra over avector space, and writing in terms of${\displaystyle x_i}$ corresponds to choosing a basis for the vector space. Then an idealI, or equivalently a module${\displaystyle R/I,}$ is a cyclic representation ofR (cyclic meaning generated by 1 element as anR-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 inn-space, by theNullstellensatz (the maximal ideal generated by${\displaystyle (x_1-a_1),(x_2-a_2),\dots ,(x_n-a_n)}$ corresponds to the point${\displaystyle (a_1,\dots ,a_n)}$). These representations of${\displaystyle K[V]}$ are then parametrized by thedual space${\displaystyle V^{\ast },}$ the covector being given by sending each${\displaystyle x_i}$ to the corresponding${\displaystyle a_i}$. Thus a representation of${\displaystyle K^n}$ (K-linear maps${\displaystyle K^n\to K}$) is given by a set ofn numbers, or equivalently a covector${\displaystyle K^n\to K.}$

Thus, points inn-space, thought of as the max spec of${\displaystyle R=K[x_1,\dots ,x_n],}$ correspond precisely to 1-dimensional representations ofR, 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 toinfinite-dimensional representations.

The term "spectrum" comes from the use inoperator theory.Given alinear operatorT on afinite-dimensional vector spaceV, one can consider the vector space with operator as a module over the polynomial ring in one variableR =K[T], as in thestructure theorem for finitely generated modules over a principal ideal domain. Then the spectrum ofK[T] (as a ring) equals the spectrum ofT (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:

${\displaystyle K[T]/(T-1)\oplus K[T]/(T-1)}$

the 2×2 zero matrix has module

${\displaystyle K[T]/(T-0)\oplus K[T]/(T-0),}$

showing geometric multiplicity 2 for the zeroeigenvalue,while a non-trivial 2×2 nilpotent matrix has module

${\displaystyle K[T]/T^2,}$

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 acyclic vector (a vector whose orbit underT spans the space);
• the lastinvariant factor of the module equals theminimal polynomial of the operator, and the product of the invariant factors equals thecharacteristic polynomial.

The spectrum can be generalized from rings toC*-algebras inoperator theory, yielding the notion of thespectrum of a C*-algebra. Notably, for aHausdorff space, thealgebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) is acommutative C*-algebra, with the space being recovered as a topological space from${\displaystyle \mathrm{MaxSpec}}$ of the algebra of scalars, indeed functorially so; this is the content of theBanach–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 tonon-commutative C*-algebras yieldsnoncommutative topology.

• Scheme (mathematics)
• Projective scheme
• Spectrum of a matrix
• Serre's theorem on affineness
• Étale spectrum
• Ziegler spectrum
• Primitive spectrum
• Stone duality^[12]

• https://mathoverflow.net/questions/441029/intrinsic-topology-on-the-zariski-spectrum

• Commutative algebra
• Scheme theory
• Prime ideals
• Functional analysis

• Articles with short description
• Short description matches Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from August 2015
• Articles to be expanded from June 2020
• All articles to be expanded
• All accuracy disputes
• Articles with disputed statements from January 2018
• CS1 maint: year