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Torsor (algebraic Geometry)
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra. Though other notions of torsors are known in more general context (e.g. over stacks) this article will focus on torsors over schemes, the original setting where torsors have been thought for. The word ''torsor'' comes from the French ''torseur''. They are indeed widely discussed, for instance, in Michel Demazure's and Pierre Gabriel's famous book ''Groupes algébriques, Tome I''. Definition Let \mathcal be a Grothendieck topology and X a scheme. Moreover let G be a group scheme over X, a G-torsor (or principal G-bundle) over X for the topology \mathcal (or simply a G-torsor when the topology is clear from the context) is the data of a scheme P and a morphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Descent
In mathematics, given a ''G''-torsor ''X'' → ''Y'' and a stack ''F'', the descent along torsors says there is a canonical equivalence between ''F''(''Y''), the category of ''Y''-points and ''F''(''X'')''G'', the category of ''G''-equivariant ''X''-points. It is a basic example of descent, since it says the "equivariant data" (which is an additional data) allows one to "descend" from ''X'' to ''Y''. When ''G'' is the Galois group of a finite Galois extension ''L''/''K'', for the ''G''-torsor \operatorname L \to \operatorname K, this generalizes classical Galois descent (cf. field of definition In mathematics, the field of definition of an algebraic variety ''V'' is essentially the smallest field to which the coefficients of the polynomials defining ''V'' can belong. Given polynomials, with coefficients in a field ''K'', it may not be ob ...). For example, one can take ''F'' to be the stack of quasi-coherent sheaves (in an appropriate topology). Then ''F''(''X'')''G'' consists of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlos Simpson
Carlos Tschudi Simpson (born 30 June 1962) is an American mathematician, specializing in algebraic geometry. Simpson received his Ph.D. in 1987 from Harvard University, where he was supervised by Wilfried Schmid; his thesis was titled ''Systems of Hodge Bundles and Uniformization''. He became a professor at the University of Toulouse III (Paul Sabatier University) and then at the University of Nice. He is research director of the Centre national de la recherche scientifique. He works on moduli spaces of vector bundles, higher non-abelian de Rham cohomology (Hodge theory), the theory of higher categories and computer verification of mathematical proofs (''e.g.'' verification of proofs within Zermelo–Fraenkel set theory using Coq). In his Ph.D. dissertation, Simpson studied the notion of system of Hodge bundles, which can be seen as a special case of the higher dimensional generalization of Higgs bundles introduced earlier by Nigel Hitchin. The Simpson correspondence (or th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld (; born February 14, 1954), surname also romanized as Drinfel'd, is a mathematician from Ukraine, who immigrated to the United States and works at the University of Chicago. Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory of solitons. He was awarded the Fields Medal in 1990. In 2016, he was elected to the National Academy of Sciences. In 2018 he received the Wolf Prize in Mathematics. In 2023 he was awarded the Shaw Prize in Mathematical Sciences. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Of Structure Group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(''n'')-structure defines a Riemannian metric, and for the special linear group an SL(''n'',R)-structure is the same as a volume form. For the trivial group, an -structure consists of an absolute parallelism of the manifold. Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal G-bundle over a group G "comes from" a subgroup H of G. This is called reduction of the structure group (to H). Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are ''G''-structures with an additional integrability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marco Antei
Marco Antei (born 1978, Sanremo) is an Italian mathematician and LGBT+ activist. Career Antei was awarded his PhD in mathematics in 2008 from the University of Lille under the supervision of Michel Emsalem. He later worked at the Max Planck Institute for Mathematics in Bonn, the KAIST in Daejeon, the Ben-Gurion University of the Negev in Beersheba, the Côte d'Azur University in Nice before joining the University of Costa Rica. He has been lecturer at Lucerne University of Applied Sciences and Arts since 2022. Antei studies within the field of geometry. His areas of interest in research focus on algebraic and arithmetic geometry, and applications. He particularly studies the fundamental group scheme, torsors and their connections. The fundamental group scheme The existence of the fundamental group scheme was conjectured by Alexander Grothendieck, while the first proof of its existence is due, for schemes defined over fields, to Madhav Nori. Antei, Michel Emsalem and Carlo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Madhav V
Mādhava means Krishna. It may also refer to: *a Sanskrit patronymic, "descendant of Madhu (a man of the Yadu tribe)". ** especially of Krishna, see Madhava (Vishnu) *** an icon of Krishna ** Madhava of Sangamagrama, fourteenth-century Indian mathematician ** Madhvacharya, philosopher in the Vaishnavism tradition ** Madhava Vidyaranya, Advaita saint and brother of Sayana ** Venkata Madhava, 10th to 12th century commentator of the Rigveda ** Madhavdeva, 16th-century proponent of Ekasarana dharma, neo-Vaishnavism of Assam *relating to springtime; the first month of spring, see Chaitra *Madhava or Madhava-kara, an Indian physician of the 7th or early 8th century *Madhava, titular protagonist of the ancient Indian drama Mālatīmādhava by Bhavabhuti *Madhava, a character in the 11th-century Indian story collection '' Shringara-manjari-katha'' See also * * * Madhavan (other) * Madhavi (other) * Madhab (other) * Madhavaram (other) * ''Madhav'' ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called Grothendieck's relative point of view, "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des Hautes Études Scientifiques, Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He receive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group Scheme
In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental group. Although its existence was conjectured by Alexander Grothendieck, the first proof of its existence is due, for schemes defined over fields, to Madhav Nori. A proof of its existence for schemes defined over Dedekind schemes is due to Marco Antei, Michel Emsalem and Carlo Gasbarri. History The (topological) fundamental group associated with a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. Although it is still being studied for the classification of algebraic varieties even in algebraic geometry, for many applications the fundamental group has been found to be inadequate for the classification of objects, such as schemes, that are more than just topological spaces. The sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Morphism
In mathematics, in particular in algebraic geometry, a flat morphism ''f'' from a scheme (mathematics), scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every Stalk (sheaf), stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \to \mathcal_ is a flat map for all ''P'' in ''X''. A map of rings A\to B is called flat if it is a homomorphism that makes ''B'' a flat module, flat ''A''-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: *flatness is a generic property; and *the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some finiteness condition on a morphism of schemes, finiteness conditions on ''f'', it can be shown that there is a non-empty open subscheme Y' of ''Y'', such that ''f'' restricted to Y' is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Čech Cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topological space, and let \mathcal be an open cover of ''X''. Let N(\mathcal) denote the nerve of a covering, nerve of the covering. The idea of Čech cohomology is that, for an open cover \mathcal consisting of sufficiently small open sets, the resulting simplicial complex N(\mathcal) should be a good combinatorial model for the space ''X''. For such a cover, the Čech cohomology of ''X'' is defined to be the simplicial homology, simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of ''X'', ordered by Open cover#Refinement, refinement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Torsor
Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group where the commutator subgroup is abelian * Abelianisation Topology and number theory * Abelian variety, a complex torus that can be embedded into projective space * Abelian surface, a two-dimensional abelian variety * Abelian function, a meromorphic function on an abelian variety * Abelian integral, a function related to the indefinite integral of a differential of the first kind Other mathematics * Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel * Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series * Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian * Abelia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
