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Set of a ring's prime ideals

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by $operatorname Spec(R)$ , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.

Zariski topology

For any ideal I of R, define $V_I$ to be the set of prime ideals containing I. We can put a topology on $operatorname Spec(R)$ by defining the collection of closed sets to be

V_Icolon Itext is an ideal of R.

This topology is called the Zariski topology.

A basis 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 $operatorname Spec(R)$ , and $D_f:fin R$ is a basis for the Zariski topology.

$operatorname Spec(R)$ 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, it is not, in general, a T₁ space.^[1] However, $operatorname Spec(R)$ is always a Kolmogorov space (satisfies the T₀ axiom); it is also a spectral space.

Sheaves and schemes

Given the space $X=operatorname Spec(R)$ 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 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 of the Zariski topology, so for an arbitrary open set U, written as the union of D_fi_i∈I, we set Γ(U,O_X) = lim_i∈IR_fi. One may check that this presheaf is a sheaf, so $operatorname Spec(R)$ 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 M over the ring R, we may define a sheaf $tilde M$ on $operatorname Spec(R)$ . On the distinguished open subsets set Γ(D_f, $tilde M$ ) = M_f, using the localization of a module. As above, this construction extends to a presheaf on all open subsets of $operatorname Spec(R)$ and satisfies gluing axioms. A sheaf of this form is called a quasicoherent sheaf.

If P is a point in $operatorname Spec(R)$ , that is, a prime ideal, then the stalk of the structure sheaf at P equals the localization of R at the ideal P, and this is a local ring. Consequently, $operatorname Spec(R)$ 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.

Functorial perspective

It is useful to use the language of category theory and observe that $operatorname Spec$ is a functor. Every ring homomorphism $f:Rto S$ induces a continuous map $displaystyle operatorname Spec (f):operatorname Spec (S)to operatorname Spec (R)$ (since the preimage of any prime ideal in $S$ is a prime ideal in $R$ ). In this way, $operatorname Spec$ can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime $mathfrak p$ the homomorphism $f$ descends to homomorphisms

displaystyle mathcal O_f^-1(mathfrak p)to mathcal O_mathfrak p

of local rings. Thus $operatorname Spec$ 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 $operatorname Spec$ up to natural isomorphism.^{[citation needed]}

The functor $operatorname Spec$ 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.

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. $displaystyle operatorname MaxSpec (R)$ , together with the Zariski topology, is homeomorphic to A also with the Zariski topology.

One can thus view the topological space $operatorname Spec(R)$ 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 $operatorname Spec(R)$ 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 schemes.

Examples

The affine scheme $displaystyle operatorname Spec (mathbb Z )$ is the final object in the category of affine schemes since $mathbb Z$ is the initial object in the category of commutative rings.

The affine scheme $displaystyle mathbb A _mathbb C ^n=operatorname Spec (mathbb C [x_1,ldots ,x_n])$ is scheme theoretic analogue of $mathbb C ^n$ . From the functor of points perspective, a point $displaystyle (alpha _1,ldots ,alpha _n)in mathbb C ^n$ can be identified with the evaluation morphism $displaystyle mathbb C [x_1,ldots ,x_n]xrightarrow ev_(alpha _1,ldots ,alpha _n)mathbb C$ . This fundamental observation allows us to give meaning to other affine schemes.

$displaystyle operatorname Spec (mathbb C [x,y]/(xy))$ 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 $mathbb C$ are the evaluation morphisms associated from the points $displaystyle (alpha _1,0),(0,alpha _2):alpha _1,alpha _2in mathbb C$ .

Not-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 $displaystyle mathbb P _k^n=operatorname Proj k[x_0,ldots ,x_n]$ 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 $n geq 1$ is not affine as the global section of $mathbb P_k^n$ is $k$ .

Affine plane minus the origin.^[2] Inside $displaystyle mathbb A _k^2=operatorname Spec ,k[x,y]$ are distinguished open affine subschemes $displaystyle D_x,D_y$ . Their union $displaystyle D_xcup D_y=U$ is the affine plane with the origin taken out. The global sections of $U$ 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 section of $displaystyle mathbb A _k^2$ . $U$ is not affine as $displaystyle V_(x)cap V_(y)=varnothing$ in $U$ .

Global or relative Spec

There is a relative version of the functor $operatorname Spec$ called global $operatorname Spec$ , or relative $operatorname Spec$ . If $S$ is a scheme, then relative $operatorname Spec$ is denoted by $displaystyle underline operatorname Spec _S$ or $displaystyle mathbf Spec _S$ . If $S$ is clear from the context, then relative Spec may be denoted by $displaystyle underline operatorname Spec$ or $displaystyle mathbf Spec$ . For a scheme $S$ and a quasi-coherent sheaf of $mathcal O_S$ -algebras $mathcal A$ , there is a scheme $displaystyle underline operatorname Spec _S(mathcal A)$ and a morphism $displaystyle f:underline operatorname Spec _S(mathcal A)to S$ such that for every open affine $displaystyle Usubseteq S$ , there is an isomorphism $displaystyle f^-1(U)cong operatorname Spec (mathcal A(U))$ , and such that for open affines $displaystyle Vsubseteq U$ , 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 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 $mathcal O_S$ -algebras and schemes over $S$ .^{[dubious – discuss]} In formulas,

displaystyle operatorname Hom _mathcal O_Stext-alg(mathcal A,pi _*mathcal O_X)cong operatorname Hom _textSch/S(X,mathbf Spec (mathcal A)),

where $displaystyle pi colon Xto S$ is a morphism of schemes.

Example of relative Spec

Suppose we want to parameterize a family of affine hyperelliptic curves over some projective space, then relative spec gives the correct tools for this. Let $displaystyle X=mathbb P _a,b,c^2$ , consider the sheaf of algebras $displaystyle mathcal A=mathcal O_X[x,y]$ , and let $displaystyle mathcal I=(y^2-(x-a)(x-b)(x-c))$ be a sheaf of ideals of $mathcal A$ . Then, the relative spec, $displaystyle underline operatorname Spec _X(mathcal A/mathcal I)to mathbb P _a,b,c^2$ , parameterizes the desired family. For instance, we can check that the fibers are indeed affine hyperelliptic curves. If we let $displaystyle alpha =[alpha _0:alpha _1:alpha _2]$ be a point in $displaystyle mathbb P _a,b,c^2$ , and assume that $displaystyle alpha _0neq 0$ , then we can find the curve in the fiber by looking at the composition of pullback diagrams

$displaystyle beginmatrixoperatorname Spec left(frac mathbb C [x,y](y^2-(x-1)(x-(alpha _1/alpha _0))(x-(alpha _2/alpha _0)))right)&to &operatorname Spec left(frac mathbb C [b/a,c/a][x,y](y^2-(x-1)(x-(b/a))(x-(c/a)))right)&to &underline operatorname Spec _Xleft(frac mathcal O_X[x,y](y^2-(x-a)(x-b)(x-c))right)\downarrow &&downarrow &&downarrow \operatorname Spec (mathbb C )&to &operatorname Spec (mathbb C [b/a,c/a])=U_a&to &mathbb P _a,b,c^2endmatrix$

where the composition of the bottom arrows is

$displaystyle operatorname Spec (mathbb C )xrightarrow [alpha _0:alpha _1:alpha _2]mathbb P _a,b,c^2$

gives us the corresponding hyperelliptic curve over our point $alpha$ .

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.

The connection to representation theory is clearer if one considers the polynomial ring $R=K[x_1,dots ,x_n]$ 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 $(x_1-a_1),(x_2-a_2),ldots ,(x_n-a_n)$ corresponds to the point $(a_1,ldots ,a_n)$ ). 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 $K^nto K$ ) is given by a set of n numbers, or equivalently a covector $K^nto K.$

Thus, points in n-space, thought of as the max spec of $R=K[x_1,dots ,x_n],$ 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)oplus K[T]/(T-1)

the 2×2 zero matrix has module

K[T]/(T-0)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 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) are a commutative C*-algebra, with the space being recovered as a topological space from $displaystyle operatorname MaxSpec$ 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.

References

^ A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag .mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
ISBN 3-540-18178-4 (See example 21, section 2.6.)

^ R.Vakil, Foundations of Algebraic Geometry (see Chapter 4, example 4.4.1)

Cox, David; O'Shea, Donal; Little, John (1997), Ideals, Varieties, and Algorithms, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94680-1

Eisenbud, David; Harris, Joe (2000), The geometry of schemes, Graduate Texts in Mathematics, 197, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98637-1, MR 1730819

Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

External links

Kevin R. Coombes: The Spectrum of a Ring

http://stacks.math.columbia.edu/tag/01LL, relative spec

Miles Reid. "Undergraduate Commutative Algebra" (PDF). p. 22. Archived from the original (PDF) on 14 April 2016.

[1] A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) General Topology I (1990) Springer-Verlag .mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
ISBN 3-540-18178-4 (See example 21, section 2.6.)

[2] R.Vakil, Foundations of Algebraic Geometry (see Chapter 4, example 4.4.1)

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Spectrum of a ring

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