Spectrum of a ring


Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring "R", denoted by Spec("R"), is defined to be the set of all proper prime ideals of "R". It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

Zariski topology

Spec("R") can be turned into a topological space as follows: a subset "V" of Spec("R") is "closed" if and only if there exists a subset "I" of "R" such that "V" consists of all those prime ideals in "R" that contain "I". This is called the Zariski topology on Spec("R").

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. However, Spec("R") is always a Kolmogorov space. It is also a spectral space.

heaves and schemes

To define a structure sheaf on Spec("R"), first let "D""f" be the set of all prime ideals "P" in Spec("R") such that "f" is not in "P". The sets {"Df"}"f"∈"R" form a basis for the topology on Spec("R"). Define a sheaf on the "D""f" by setting Γ("D""f", "O""X") = "R""f", the localization of "R" at the multiplicative system {1,"f","f"2,"f"3,...}. It can be shown that this satisfies the necessary axioms to be a B-Sheaf. Next, if "U" is the union of {"D""fi"}"i"∈"I", we let Γ("U","O""X") = lim"i"∈"I" "R""fi", and this produces a sheaf; see the sheaf article for more detail.

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".

If "P" is a point in Spec("R"), that is, a prime ideal, then the stalk at "P" equals the localization of "R" at "P", and this is a local ring. Consequently, Spec("R") is a locally ringed space.

Every locally ringed space isomorphic to one of this form is called an "affine scheme". General schemes are obtained by "gluing together" several affine schemes.

Functoriality

It is useful to use the language of category theory and observe that Spec is a functor.Every ring homomorphism "f" : "R" → "S" induces a 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 :"O""f" -1("P") → "O""P",of local rings. Thus 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 and this can be used to define the functor Spec up to natural isomorphism.

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.

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies "algebraic sets", i.e. subsets of "K""n" (where "K" is an algebraically closed field) which 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).

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 schemes.

Global Spec

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 "OY"-algebras "A", there is a unique scheme "X", called Spec "A", and a morphism f colon X o Y such that for every open affine U subseteq Y, there is an isomorphism induced by "f": f^{-1}(U) cong mathrm{Spec} A(U), and such that for an inclusion of open affines U subseteq V, the restriction map f^{-1}(U) o f^{-1}(V) is the restriction map A(V) o A(U).

References

* | year=2000 | volume=197
* | year=1977

External links

* Kevin R. Coombes: [http://odin.mdacc.tmc.edu/~krc/agathos/spec.html "The Spectrum of a Ring"]


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