|
Serre's Theorem On Affineness
In the mathematical discipline of algebraic geometry, Serre's theorem on affineness (also called Serre's cohomological characterization of affineness or Serre's criterion on affineness) is a theorem due to Jean-Pierre Serre which gives sufficient conditions for a scheme to be affine. The theorem was first published by Serre in 1957. Statement Let be a scheme with structure sheaf If: :(1) is quasi-compact, and :(2) for every quasi-coherent ideal sheaf of -modules, , then is affine. Related results * A special case of this theorem arises when is an algebraic variety, in which case the conditions of the theorem imply that is an affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea .... * A similar result has stricter conditions on but looser conditions on the cohomology: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ''x''2 = 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). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of A Ring
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 \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. 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(R) by defining the collection of closed sets to be :\. 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(R), and \ is a basis for the Zariski topology. \operatorname(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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Sheaf
In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces. Definition Let ''X'' be a topological space and ''A'' a sheaf of rings on ''X''. (In other words, (''X'', ''A'') is a ringed space.) An ideal sheaf ''J'' in ''A'' is a subobject of ''A'' in the category of sheaves of ''A''-modules, i.e., a subsheaf of ''A'' viewed as a sheaf of abelian groups such that : Γ(''U'', ''A'') · Γ(''U'', ''J'') ⊆ Γ(''U'', ''J'') for all open subsets ''U'' of ''X''. In other words, ''J'' is a sheaf of ''A''-submodules of ''A''. General properties * If ''f'': ''A'' → ''B'' is a homomorphism between two sheaves of rings on the same space ''X'', the kernel of ''f'' is an ideal sheaf in ''A''. * Conversely, for any ideal sheaf ''J'' in a sheaf of rings ''A'', there is a natural structure of a sheaf of rings on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety. Some texts do not require a prime ideal, and call ''irreducible'' an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets. In some contexts, it is useful to distinguish the field in which the coefficients are considered, from the algebraically closed field (containing ) over which the zero-locus is considered (that is, the points of the affine variety are in ). In this case, the variety is said ''defined over'' , and the points of the variety that belong to are said '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
