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Group Functor
In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no scheme theory. Because of this feature, some authors, notably Waterhouse and Milne (who followed Waterhouse), develop the theory of group schemes based on the notion of group functor instead of scheme theory. A formal group is usually defined as a particular kind of a group functor. Group functor as a generalization of a group scheme A scheme may be thought of as a contravariant functor from the category \mathsf_S of ''S''-schemes to the category of sets satisfying the gluing axiom; the perspective known as the rational points#Definition, functor of points. Under this perspective, a group scheme is a contravariant functor from \mathsf_S to the category of groups that is a Zariski sheaf (i.e., satisfying the gluing axiom for the Zariski topology). For example, if Γ is a finite group, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The Category (mathematics), category of group schemes is somewhat better behaved than that of Group variety, group varieties, since all homomorphisms have Kernel (category theory), kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The ini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme Theory
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 commu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Group
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. Definitions A one-dimensional formal group law over a commutative ring ''R'' is a power series ''F''(''x'',''y'') with coefficients in ''R'', such that # ''F''(''x'',''y'') = ''x'' + ''y'' + terms of higher degree # ''F''(''x'', ''F''(''y'',''z'')) = ''F''(''F''(''x'',''y''), ''z'') (associativity). The simplest example is the additive formal group law ''F''(''x'', ''y'') = ''x'' + ''y''. The idea of the definition is that ''F'' should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gluing Axiom
In mathematics, the gluing axiom is introduced to define what a sheaf \mathcal F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor ::(X) \rightarrow C to a category C which initially one takes to be the category of sets. Here (X) is the partial order of open sets of X ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism :U \rightarrow V if U is a subset of V, and none otherwise. As phrased in the sheaf article, there is a certain axiom that F must satisfy, for any open cover of an open set of X. For example, given open sets U and V with union X and intersection W, the required condition is that :(X) is the subset of (U) \times (V) With equal image in (W) In less formal language, a section s of F over X is equally well given by a pair of sections :(s', s'') on U and V respectively, which 'agree' in the sense that s' and s'' have a common image in (W) under the respec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Points
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field ''k'', and an algebraically closed extension ''K'' of ''k'', an affine variety ''X'' over ''k'' is the set of common zeros in K^n of a collection of polynomials with coefficients in ''k'': :f_1(x_1,\ldots,x_n)=0,\ldots, f_r(x_1,\dots,x_n)=0. These common zeros are called the ''points'' of ''X''. A ''k''-rational point (or ''k''-point) of ''X'' is a point of ''X'' that belongs to ''k''''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsor (algebraic Geometry)
In algebraic geometry, a torsor or a principal bundle is an analog 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. The category of torsors over a fixed base forms a stack. Conversely, a prestack can be stackified by taking the category of torsors (over the prestack). Definition Given a smooth algebraic group ''G'', a ''G''-torsor (or a principal ''G''-bundle) ''P'' over a scheme ''X'' is a scheme (or even algebraic space) with an action of ''G'' that is locally trivial in the given Grothendieck topology in the sense that the base change Y \times_X P along some covering map Y \to X is isomorphic to the trivial torsor Y \times G \to Y (''G'' acts only on the second factor). Equivalently, a ''G''-torsor ''P'' on ''X'' is a principal homogeneous space for the group scheme G_X = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group Functor
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional structur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |