|
Suslin's Theorem (other)
{{mathdab ...
In mathematics, Suslin's theorem may refer to: *The Quillen–Suslin theorem (formerly the Serre conjecture), due to Andrei Suslin. *Any of several theorems about analytic sets due to Mikhail Yakovlevich Suslin; in particular: **There is an analytic subset of the reals that is not Borel **An analytic set whose complement is also analytic is a Borel set, a special case of the Lusin separation theorem **Any analytic set in R''n'' is the projection of a Borel set in R''n''+1 **Analytic sets can be constructed using the Suslin operation In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers. The Suslin operation was introduced by and . In Russia it is sometimes called the A-operati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (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-bar P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Suslin
Andrei Suslin (, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University. He was born on 27 December 1950 in St. Petersburg, Russia. As a youth, he was an "all Leningrad" gymnast. He received his PhD from Leningrad University in 1974; his thesis was titled ''Projective modules over polynomial rings''. In 1976 he and Daniel Quillen independently proved Serre's conjecture about the triviality of algebraic vector bundles on affine space. In 1982 he and Alexander Merkurjev proved the Merkurjev–Suslin theorem on the norm residue homomorphism in Milnor K2-theory, with applications to the Brauer group. Suslin was an invited speaker at the International Congress of Mathematicians in 1978 and 1994, and he gave a plenary invited address at the Congress in 1986. He was awarded the Frank Nelson Cole Prize in Alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Set
In the mathematical field of descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student . Definition There are several equivalent definitions of analytic set. The following conditions on a subspace ''A'' of a Polish space ''X'' are equivalent: *''A'' is analytic. *''A'' is empty or a continuous image of the Baire space ωω. *''A'' is a Suslin space, in other words ''A'' is the image of a Polish space under a continuous mapping. *''A'' is the continuous image of a Borel set in a Polish space. *''A'' is a Suslin set, the image of the Suslin operation. *There is a Polish space Y and a Borel set B\subseteq X\times Y such that A is the projection of B onto X; that is, : A=\. *''A'' is the projection of a closed set in the cartesian product of ''X'' with the Baire space. *''A'' is the projection of a Gδ set in the cartesian product of ''X'' with the Cantor space 2� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Yakovlevich Suslin
Mikhail Yakovlevich Suslin (; November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory. Biography Mikhail Suslin was born on November 15, 1894, in the village of Krasavka, the only child of poor peasants Yakov Gavrilovich and Matrena Vasil'evna Suslin. From a young age, Suslin showed a keen interest in mathematics and was encouraged to continue his education by his primary school teacher, Vera Andreevna Teplogorskaya-Smirnova. From 1905 to 1913 he attended Balashov boys' grammar school. In 1913, Suslin enrolled at the Imperial Moscow University and studied under the tutelage of Nikolai Luzin. He graduated with a degree in mathematics in 1917 and immediately began working at the Ivanovo-Voznesensk Polytechnic Institute. Suslin died of typhus in the 1919 Moscow epidemic following the Russian Civil War, at the age of 24. Work His name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lusin Separation Theorem
In descriptive set theory and mathematical logic, Lusin's separation theorem states that if ''A'' and ''B'' are disjoint analytic subsets of Polish space, then there is a Borel set ''C'' in the space such that ''A'' ⊆ ''C'' and ''B'' ∩ ''C'' = ∅.. It is named after Nikolai Luzin Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Sov ..., who proved it in 1927.. The theorem can be generalized to show that for each sequence (''A''''n'') of disjoint analytic sets there is a sequence (''B''''n'') of disjoint Borel sets such that ''A''''n'' ⊆ ''B''''n'' for each ''n''. An immediate consequence is Suslin's theorem, which states that if a set and its complement are both analytic, then the set is Borel. Notes References * ( for the Euro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |