|
|
| word looked up : | home / archive |
Spectrum of a ringIn abstract algebra and algebraic geometry, the spectrum of a commutative ring R is defined to be the set of all prime ideals of R. It is denoted by Spec(R).
| |||
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. Spec(R) is always a T0 space, however.
To every open set U of Spec(R), one may assign a commutative ring RU in the following way: let S be the complement of the union of all the prime ideals in U. Then S is a multiplicative set and we define RU as the ring of quotients[?] of R with respect to S. This endows Spec(R) with a sheaf of rings O. If P is an element of Spec(R), then the stalk OP at P of this sheaf is equal to the localization of R at P, which is a local ring. Thus Spec(R) is a locally ringed space.
Every sheaf of rings of this form is called an affine scheme. The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes.
General schemes are obtained by "glueing together" several affine schemes.
Note that 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 (see category theory). Moreover for every prime P the homomorphism f descends to homomorphisms
In algebraic geometry, one studies algebraic sets, i.e. subsets of Kn (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 or 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 therefore 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. 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 eventually to schemes.
External links:
reminding them that he, also, was to take part in the play.html">play,
the stage door and bought railway passage to the West.
Presently the curtain rolled up, and the audience applauded
made their first bow before the footlights.
I have said that Will did not know a line of his part,
It had been faithfully memorized, but oozed from his mind like the
who was on the stage with him, "he needs time to recover."
In glancing over the audience, he had recognized in one of
guided on a big hunt near McPherson. The expedition had been
were well known.
"I've been out on a hunt with Milligan," replied Will,
but had been the butt of innumerable jokes because of his
that greeted the sally stocked the scout with confidence,
It became manifest to the playwright-actor that he would have
to perform, and that he must prepare it on the spot.
"Tell us about it, Bill," said he, and the prompter groaned.
One of the pleasures of frontier life consists in telling stories
pretty sure to be a good raconteur. Will was at ease immediately,
That it was amusing was attested by the frequent rounds of applause.
tried again and again to give Will his cue, but even cues had been forgotten.
The dialogue of that performance must have been delightfully absurd.
during the entire evening. In the Indian scenes, however, they scored
and the mock red men were slain at an astonishing rate.
Financially the play proved all that its projectors could
with the hapless dramatist. The professionals in the company
were let down gently in the criticisms; but the critics had no
.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|