In

_{''f''} to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''_{''f''} is an open subset of $\backslash operatorname(R)$, and $\backslash $ is a basis for the Zariski topology.
$\backslash operatorname(R)$ is a compact space, but almost never Hausdorff space, Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a T1 space, T_{1} space. However, $\backslash operatorname(R)$ is always a Kolmogorov space (satisfies the T_{0} axiom); it is also a spectral space.

_{''X''} is defined on the distinguished open subsets ''D''_{''f''} by setting Γ(''D''_{''f''}, ''O''_{''X''}) = ''R''_{''f''}, the localization of a ring, 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 base (topology), 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 $\backslash operatorname(R)$ is a ringed space. Any ringed space isomorphic to one of this form is called an affine scheme. General scheme (mathematics), schemes are obtained by gluing affine schemes together.
Similarly, for a module ''M'' over the ring ''R'', we may define a sheaf $\backslash tilde$ on $\backslash operatorname(R)$. On the distinguished open subsets set Γ(''D''_{''f''}, $\backslash tilde$) = ''M''_{''f''}, using the localization of a module. As above, this construction extends to a presheaf on all open subsets of $\backslash operatorname(R)$ and satisfies gluing axioms. A sheaf of this form is called a quasicoherent sheaf.
If ''P'' is a point in $\backslash operatorname(R)$, that is, a prime ideal, then the stalk of the structure sheaf at ''P'' equals the localization of a ring, localization of ''R'' at the ideal ''P'', and this is a local ring. Consequently, $\backslash operatorname(R)$ 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''.

^{''n''} (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 component, 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. $\backslash operatorname(R)$, together with the Zariski topology, is homeomorphic to ''A'' also with the Zariski topology.
One can thus view the topological space $\backslash operatorname(R)$ 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 $\backslash operatorname(R)$ 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), schemes.

''The Spectrum of a Ring''

* http://stacks.math.columbia.edu/tag/01LL, relative spec * {{cite web, author=Miles Reid, url=http://dmat.cfm.cl/library/ac.pdf, title=Undergraduate Commutative Algebra, page=22, archive-url=https://web.archive.org/web/20160414151327/http://dmat.cfm.cl/library/ac.pdf, archive-date=14 April 2016, url-status=dead Commutative algebra Scheme theory Prime ideals Functional analysis

commutative algebra
Commutative algebra is the branch of algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...

, 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 $\backslash operatorname$; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings $\backslash mathcal$.
Zariski topology

For any ideal (ring theory), ideal ''I'' of ''R'', define $V\_I$ to be the set of prime ideals containing ''I''. We can put a topology on $\backslash operatorname(R)$ by defining the Characterizations of the category of topological spaces#Definition via closed sets, collection of closed sets to be :$\backslash .$ This topology is called the Zariski topology. A Base (topology), basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''Sheaves and schemes

Given the space $X\; =\; \backslash operatorname(R)$ with the Zariski topology, the structure sheaf ''O''Functorial perspective

It is useful to use the language of category theory and observe that $\backslash operatorname$ is a functor. Every ring homomorphism $f:\; R\; \backslash to\; S$ induces a continuous function (topology), continuous map $\backslash operatorname(f):\; \backslash operatorname(S)\; \backslash to\; \backslash operatorname(R)$ (since the preimage of any prime ideal in $S$ is a prime ideal in $R$). In this way, $\backslash operatorname$ can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime $\backslash mathfrak$ the homomorphism $f$ descends to homomorphisms :$\backslash mathcal\_\; \backslash to\; \backslash mathcal\_\backslash mathfrak$ of local rings. Thus $\backslash operatorname$ 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 $\backslash operatorname$ up to natural isomorphism. The functor $\backslash operatorname$ 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 of the other.Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies ''algebraic sets'', i.e. subsets of ''K''Examples

* The affine scheme $\backslash operatorname(\backslash mathbb)$ is the final object in the category of affine schemes since $\backslash mathbb$ is the initial object in the category of commutative rings. * The affine scheme $\backslash mathbb^n\_\backslash mathbb\; =\; \backslash operatorname(\backslash mathbb[x\_1,\backslash ldots,\; x\_n])$ is scheme theoretic analogue of $\backslash mathbb^n$. From the functor of points perspective, a point $(\backslash alpha\_1,\backslash ldots,\backslash alpha\_n)\; \backslash in\; \backslash mathbb^n$ can be identified with the evaluation morphism $\backslash mathbb[x\_1,\backslash ldots,\; x\_n]\; \backslash xrightarrow\; \backslash mathbb$. This fundamental observation allows us to give meaning to other affine schemes. * $\backslash operatorname(\backslash mathbb[x,y]/(xy))$ 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 $\backslash mathbb$ are the evaluation morphisms associated with the points $\backslash $. *The prime spectrum of a Boolean ring (e.g., a power set ring) is a (Hausdorff) compact space. *(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 space, quasi-separated and sober space, 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 $\backslash mathbb^n\_k\; =\; \backslash operatornamek[x\_0,\backslash ldots,\; x\_n]$ over a field $k$ . 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 $n$-Space for $n\; \backslash geq\; 1$ is not affine as the global section of $\backslash mathbb^n\_k$ is $k$. *Affine plane minus the origin. Inside $\backslash mathbb^2\_k\; =\; \backslash operatorname\backslash ,\; k[x,y]$ are distinguished open affine subschemes $D\_x\; ,\; D\_y$. Their union $D\_x\; \backslash cup\; D\_y\; =\; U$ is the affine plane with the origin taken out. The global sections of $U$ are pairs of polynomials on $D\_x,D\_y$ that restrict to the same polynomial on $D\_$, which can be shown to be $k[x,y]$, the global section of $\backslash mathbb^2\_k$. $U$ is not affine as $V\_\; \backslash cap\; V\_\; =\; \backslash varnothing$ in $U$.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 $\backslash operatorname(A)$ of the form $\backslash varphi^*(\backslash operatorname\; B),\; \backslash varphi:\; A\; \backslash to\; B$ satisfy the axioms for closed sets in a topological space. This topology on $\backslash operatorname(A)$ 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 $V(I)$ and $\backslash operatorname(A)\; -\; V(f)$ are closed.Global or relative Spec

There is a relative version of the functor $\backslash operatorname$ called global $\backslash operatorname$, or relative $\backslash operatorname$. If $S$ is a scheme, then relative $\backslash operatorname$ is denoted by $\backslash underline\_S$ or $\backslash mathbf\_S$. If $S$ is clear from the context, then relative Spec may be denoted by $\backslash underline$ or $\backslash mathbf$. For a scheme $S$ and a quasi-coherent sheaf, quasi-coherent sheaf of algebras, sheaf of $\backslash mathcal\_S$-algebras $\backslash mathcal$, there is a scheme $\backslash underline\_S(\backslash mathcal)$ and a morphism $f\; :\; \backslash underline\_S(\backslash mathcal)\; \backslash to\; S$ such that for every open affine $U\; \backslash subseteq\; S$, there is an isomorphism $f^(U)\; \backslash cong\; \backslash operatorname(\backslash mathcal(U))$, and such that for open affines $V\; \backslash subseteq\; U$, the inclusion $f^(V)\; \backslash to\; f^(U)$ is induced by the restriction map $\backslash mathcal(U)\; \backslash to\; \backslash mathcal(V)$. 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 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 $\backslash mathcal\_S$-algebras and schemes over $S$. In formulas, :$\backslash operatorname\_(\backslash mathcal,\; \backslash pi\_*\backslash mathcal\_X)\; \backslash cong\; \backslash operatorname\_(X,\; \backslash mathbf(\backslash mathcal)),$ where $\backslash pi\; \backslash colon\; X\; \backslash to\; S$ 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 $\backslash mathbb^2\_\backslash mathbb$ over $X\; =\; \backslash mathbb^1\_.$ Consider the sheaf of algebras $\backslash mathcal\; =\; \backslash mathcal\_X[x,y],$ and let $\backslash mathcal\; =\; (ay-bx)$ be a sheaf of ideals of $\backslash mathcal.$ Then the relative spec $\backslash underline\_X(\backslash mathcal/\backslash mathcal)\; \backslash to\; \backslash mathbb^1\_$ parameterizes the desired family. In fact, the fiber over $[\backslash alpha:\backslash beta]$ is the line through the origin of $\backslash mathbb^2$ containing the point $(\backslash alpha,\backslash beta).$ Assuming $\backslash alpha\; \backslash neq\; 0,$ the fiber can be computed by looking at the composition of pullback diagrams :$\backslash begin\; \backslash operatorname\backslash left(\; \backslash frac\; \backslash right)\; \&\; \backslash to\; \&\; \backslash operatorname\backslash left(\; \backslash frac\; \backslash right)\; \&\; \backslash to\; \&\; \backslash underline\_X\backslash left(\; \backslash frac\; \backslash right)\backslash \backslash \; \backslash downarrow\; \&\; \&\; \backslash downarrow\; \&\; \&\; \backslash downarrow\; \backslash \backslash \; \backslash operatorname(\backslash mathbb)\&\; \backslash to\; \&\; \backslash operatorname\backslash left(\backslash mathbb\backslash left[\backslash frac\backslash right]\backslash right)=U\_a\; \&\; \backslash to\; \&\; \backslash mathbb^1\_\; \backslash end$ where the composition of the bottom arrows :$\backslash operatorname(\backslash mathbb)\backslash xrightarrow\; \backslash mathbb^1\_$ gives the line containing the point $(\backslash alpha,\backslash beta)$ and the origin. This example can be generalized to parameterize the family of lines through the origin of $\backslash mathbb^\_\backslash mathbb$ over $X\; =\; \backslash mathbb^n\_$ by letting $\backslash mathcal\; =\; \backslash mathcal\_X[x\_0,...,x\_n]$ and $\backslash mathcal\; =\; \backslash left(\; 2\backslash times\; 2\; \backslash text\; \backslash begina\_0\; \&\; \backslash cdots\; \&\; a\_n\; \backslash \backslash \; x\_0\; \&\; \backslash cdots\; \&\; x\_n\backslash end\; \backslash right).$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 ring, group algebra. The connection to representation theory is clearer if one considers the polynomial ring $R=K[x\_1,\backslash dots,x\_n]$ or, without a basis, $R=K[V].$ As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, 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 $(x\_1-a\_1),\; (x\_2-a\_2),\backslash ldots,(x\_n-a\_n)$ corresponds to the point $(a\_1,\backslash ldots,a\_n)$). These representations of $K[V]$ are then parametrized by the dual space $V^*,$ the covector being given by sending each $x\_i$ to the corresponding $a\_i$. Thus a representation of $K^n$ (''K''-linear maps $K^n\; \backslash to\; K$) is given by a set of ''n'' numbers, or equivalently a covector $K^n\; \backslash to\; K.$ Thus, points in ''n''-space, thought of as the max spec of $R=K[x\_1,\backslash dots,x\_n],$ 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

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: :$K[T]/(T-1)\; \backslash oplus\; K[T]/(T-1)$ the 2×2 zero matrix has module :$K[T]/(T-0)\; \backslash oplus\; K[T]/(T-0),$ showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module :$K[T]/T^2,$ 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 a cyclic vector (a vector whose orbit under ''T'' spans the space); * the last invariant factor of the module equals the Minimal polynomial (linear algebra), 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 $\backslash operatorname$ 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

*Scheme (mathematics) *Projective scheme *Spectrum of a matrix *Serre's theorem on affineness *Étale spectrum *Ziegler spectrum *Primitive spectrumCitations

References

* * * * *External links

* Kevin R. Coombes''The Spectrum of a Ring''

* http://stacks.math.columbia.edu/tag/01LL, relative spec * {{cite web, author=Miles Reid, url=http://dmat.cfm.cl/library/ac.pdf, title=Undergraduate Commutative Algebra, page=22, archive-url=https://web.archive.org/web/20160414151327/http://dmat.cfm.cl/library/ac.pdf, archive-date=14 April 2016, url-status=dead Commutative algebra Scheme theory Prime ideals Functional analysis