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Quillen–Suslin Theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space. The theorem states that every finitely generated projective module over a polynomial ring is free. History Background Geometrically, finitely generated projective modules over the ring R _1,\dots,x_n/math> correspond to vector bundles over affine space \mathbb^n_R, where free modules correspond to trivial vector bundles. This correspondence (from modules to (algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending M\to \widetilde (cite Hartshorne II.5, page 110). Affine space is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence and the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Sheaf Sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let ''M'' be a complex manifold, and write ''O''''M'' for the sheaf of holomorphic functions on ''M''. Let ''O''''M''* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism :\exp : \mathcal O_M \to \mathcal O_M^*, because for a holomorphic function ''f'', exp(''f'') is a non-vanishing holomorphic function, and exp(''f'' + ''g'') = exp(''f'')exp(''g''). Its kernel is the sheaf 2π''i''Z of locally constant functions on ''M'' taking the values 2π''in'', with ''n'' an integer. The exponential sheaf sequence is therefore :0\to 2\pi i\,\mathbb Z \to \mathcal O_M\to\mathcal O_M^*\to 0. The exponential mapping here is not always a surjective map on sections; this can be seen for example when ''M'' is a punctured disk in the complex plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (russian: Доклады Академии Наук СССР, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), french: Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (russian: Доклады Академии Наук), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bass–Quillen Conjecture
In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring ''A'' and over the polynomial ring A _1, \dots, t_n/math>. The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture. Statement of the conjecture The conjecture is a statement about finitely generated projective modules. Such modules are also referred to as vector bundles. For a ring ''A'', the set of isomorphism classes of vector bundles over ''A'' of rank ''r'' is denoted by \operatorname_r(A). The conjecture asserts that for a regular Noetherian ring ''A'' the assignment :M \mapsto M \otimes_A A _1, \dots, t_n/math> yields a bijection :\operatorname_r(A) \stackrel \sim \to \operatorname_r(A _1, \dots, t_n. Known cases If ''A'' = ''k'' is a field, the Bass–Quillen conjecture asserts that any projective module over k _1, \dots, t_n/math> is free. This question was raised by Jean-Pierre Serre and was later proved by Quillen and Susli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonid Vaseršteĭn
Leonid Nisonovich Vaserstein ( rus, Леонид Нисонович Васерштейн) is a Russian-American mathematician, currently Professor of Mathematics at Penn State University. His research is focused on algebra and dynamical systems. He is well known for providing a simple proof of the Quillen–Suslin theorem, a result in commutative algebra, first conjectured by Jean-Pierre Serre in 1955, and then proved by Daniel Quillen and Andrei Suslin in 1976. Leonid Vaserstein got his Master's degree and doctorate in Moscow State University, where he was until 1978. He then moved to Europe and United States. Alternate forms of the last name: Vaseršteĭn, Vasershtein, Wasserstein. The Wasserstein metric was named after him by R.L. Dobrushin in 1970. Biography Leonid Vaserstein grew up in the Soviet Union. In secondary school he won the second prize in the All-Russian High School Mathematical Olympiad. Vaserstein got his undergraduate, masters (1966), and doctoral degrees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields. The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics. The prize includes a monetary award which, since 2006, has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stably Free
In mathematics, a stably free module is a module which is close to being free. Definition A finitely generated module ''M'' over a ring ''R'' is ''stably free'' if there exist free finitely generated modules ''F'' and ''G'' over ''R'' such that : M \oplus F = G . \, Properties * A projective module is stably free if and only if it possesses a finite free resolution. * An infinitely generated module is stably free if and only if it is free. See also * Free object * Eilenberg–Mazur swindle * Hermite ring In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to (p. 465), a ring is right Hermite if, for every two elements ''a'' and ''b'' of the ring, there is an element ''d'' of the ring and ... References {{Reflist Module theory Free algebraic structures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Important Publications In Mathematics
This is a list of important publications in mathematics, organized by field. Some reasons why a particular publication might be regarded as important: *Topic creator – A publication that created a new topic *Breakthrough – A publication that changed scientific knowledge significantly *Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of mathematics. Among published compilations of important publications in mathematics are ''Landmark writings in Western mathematics 1640–1940'' by Ivor Grattan-Guinness and ''A Source Book in Mathematics'' by David Eugene Smith. Algebra Theory of equations ''Baudhayana Sulba Sutra'' * Baudhayana (8th century BCE) Believed to have been written around the 8th century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia and its surrounding regions, and Indian mathematics#Charges of Eu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...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] |
