Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Spectrum of a ring
Summary
In commutative algebra, 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 ; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .
For any ideal I of R, define to be the set of prime ideals containing I. We can put a topology on by defining the to be
This topology is called the Zariski topology.
A basis for the Zariski topology can be constructed as follows. For f ∈ R, define Df to be the set of prime ideals of R not containing f. Then each Df is an open subset of , and is a basis for the Zariski topology.
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. However, is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.
Given the space with the Zariski topology, the structure sheaf OX is defined on the distinguished open subsets Df by setting Γ(Df, OX) = Rf, 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 {Dfi}i∈I, we set Γ(U,OX) = limi∈I Rfi. One may check that this presheaf is a sheaf, so 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 on . On the distinguished open subsets set Γ(Df, ) = Mf, using the localization of a module. As above, this construction extends to a presheaf on all open subsets of and satisfies gluing axioms. A sheaf of this form is called a quasicoherent sheaf.
If P is a point in , 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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (5)
MATH-726: Working group in Topology I
The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, and topological algebraic geometry.
MATH-510: Algebraic geometry II - schemes and sheaves
The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.
MATH-726(2): Working group in Topology II
The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, topological algebraic geometry and t
Related publications (44)
Related concepts (26)
Commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third.
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.