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List Of Things Named After Alexander Grothendieck
The mathematician Alexander Grothendieck (1928–2014) is the eponym An eponym is a person, a place, or a thing after whom or which someone or something is, or is believed to be, named. The adjectives which are derived from the word eponym include ''eponymous'' and ''eponymic''. Usage of the word The term ''epon ... of many things. Mathematics {{DEFAULTSORT:List Of Topics Named After Alexander Grothendieck Grothendieck ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give models for every homotopy type. It is conjectured that there are many different "equivalent" models for ∞-groupoids all which can be realized as homotopy types. See also *''Pursuing Stacks'' *N-group (category theory) In mathematics, an ''n''-group, or ''n''-dimensional higher group, is a special kind of ''n''-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander ... References *John BaezThe Homotopy Hypothesis* * External links *What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in honour of Erik Ivar Fredholm. Much of the abstract theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955. Definition Let ''B'' be an arbitrary Banach space, and let ''B''* be its dual, that is, the space of bounded linear functionals on ''B''. The tensor product B^*\otimes B has a completion under the norm :\Vert X \Vert_\pi = \inf \sum_ \Vert e^*_i\Vert \Vert e_i \Vert where the infimum is taken over all finite representations :X=\sum_ e^*_i \otimes e_i \in B^*\otimes B The completion, under this norm, is often denoted as :B^* \widehat_\pi B and is called the projective topological tensor product. The elements of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlessinger%27s Theorem
In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck. Definitions Λ is a complete Noetherian local ring with residue field ''k'', and ''C'' is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field ''k''. A small extension in ''C'' is a morphism ''Y''→''Z'' in ''C'' that is surjective with kernel a 1-dimensional vector space over ''k''. A functor is called representable if it is of the form ''h''''X'' where ''h''''X''(''Y'')=hom(''X'',''Y'') for some ''X'', and is called pro-representable if it is of the form ''Y''→lim hom(''X''''i'',''Y'') for a filtered direct limit over ''i'' in some filtered ordered set. A morphism of functors ''F''→''G'' from ''C'' to sets is called smooth if whenever ''Y''→''Z'' is an epimorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck's Relative Point Of View
Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck, who made extensive use of it in treating foundational aspects of algebraic geometry. Outside that field, it has been influential particularly on category theory and categorical logic. In the usual formulation, the language of category theory is applied, to describe the point of view as treating, not objects ''X'' of a given category ''C'' as such, but morphisms :''f'': ''X'' → ''S'' where ''S'' is a fixed object. This idea is made formal in the idea of the slice category of objects of ''C'' 'above' ''S''. To move from one slice to another requires a base change; from a technical point of view base change becomes a major issue for the whole approach (see for exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-ring
In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after Alexander Grothendieck. A ring that is a both G-ring and a J-2 ring is called a quasi-excellent ring, and if in addition it is universally catenary it is called an excellent ring. Definitions *A (Noetherian) ring ''R'' containing a field ''k'' is called geometrically regular over ''k'' if for any finite extension ''K'' of ''k'' the ring ''R'' ⊗''k'' ''K'' is a regular ring. *A homomorphism of rings from ''R'' to ''S'' is called regular if it is flat and for every ''p'' ∈ Spec(''R'') the fiber ''S'' ⊗''R'' ''k''(''p'') is geometrically regular over the residue field ''k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Prime
57 (fifty-seven) is the natural number following 56 and preceding 58. In mathematics Fifty-seven is the sixteenth discrete semiprime, and the fourth discrete bi-prime pair with 58. It is a Blum integer since its two prime factors are both Gaussian primes. It is also an icosagonal (20-gonal) number and a repdigit in base-7 (111). 57 is the fourth Leyland number, as it can be written in the form: :5^ + 2^ = 57 57 is the number of compositions of 10 into distinct parts. With an aliquot sum of 23, fifty-seven is the first composite member of the 23-aliquot tree. 57 is the seventh fine number, equivalently the number of ordered rooted trees with ''seven'' nodes having root of even degree. In geometry, there are: *57 uniform star polyhedra in the third dimension, including four Kepler-Poinsot star polyhedra that are regular. *57 vertices and hemi-dodecahedral facets in the 57-cell, a 4-dimensional abstract regular polytope. *57 uniform prismatic 5-polytopes in the fifth d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mumford%E2%80%93Tate Group
In algebraic geometry, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a Hodge structure ''F'' is a certain algebraic group ''G''. When ''F'' is given by a rational representation of an algebraic torus, the definition of ''G'' is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the ''p''-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of on p-divisible groups, and named them Mumford–Tate groups. Formulation The algebraic torus ''T'' used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of ''a''+''bi'' on the basis of the complex numbers C over R: :\begin a & b \\ -b & a \end. The circle group inside this group of matrices is the unitary group ''U''(1). Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck–Ogg–Shafarevich Formula
In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, in terms of local data involving the Swan conductor In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function. Local Artin conductors .... and proved the formula for abelian varieties with tame ramification over curves, and extended the formula to constructible sheaves over a curve . Statement Suppose that ''F'' is a constructible sheaf over a genus ''g'' smooth projective curve ''C'', of rank ''n'' outside a finite set ''X'' of points where it has stalk 0. Then :\chi(C,F) = n(2-2g) -\sum_(n+Sw_x(F)) where ''Sw'' is the Swan conductor at a point. References * * * * {{DEFAULTSORT:Grothendieck-Ogg-Shafarevich formula Elliptic curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Hodge Theory
In mathematics, ''p''-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic ''p'' (such as Q''p''). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of ''p''-adic cohomology theories analogous to the Hodge decomposition, hence the name ''p''-adic Hodge theory. Further developments were inspired by properties of ''p''-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field. General classification of ''p''-adic representations Let ''K'' be a local field with residue field ''k'' of characteristic ''p''. In this article, a ''p-adic representation'' of ''K'' (or of ''GK'', the absolute Galois group of ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Local Duality
In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Statement Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ''d'' with maximal ideal ''m'' and residue field ''k'' = ''R''/''m''. Let ''E''(''k'') be a Matlis module, an injective hull of ''k'', and let be the completion of its dualizing module. Then for any ''R''-module ''M'' there is an isomorphism of modules over the completion of ''R'': : \operatorname_R^i(M,\overline\Omega) \cong \operatorname_R(H_m^(M),E(k)) where ''H''''m'' is a local cohomology group. There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |