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Verschiebung Operator
In mathematics, the Verschiebung or Verschiebung operator ''V'' is a homomorphism between affine commutative group schemes over a field of nonzero characteristic ''p''. For finite group schemes it is the Cartier dual In mathematics, Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by . Definition using characters Given any finite flat commutative group scheme ''G'' over ''S'', its Cartier dual is the group o ... of the Frobenius homomorphism. It was introduced by as the shift operator on Witt vectors taking (''a''0, ''a''1, ''a''2, ...) to (0, ''a''0, ''a''1, ...). ("Verschiebung" is German for "shift", but the term "Verschiebung" is often used for this operator even in other languages.) The Verschiebung operator ''V'' and the Frobenius operator ''F'' are related by ''FV'' = ''VF'' = 'p'' where 'p''is the ''p''th power homomorphism of an abelian group scheme. Examples *If ''G'' is the discrete group with ''n'' elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of 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] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartier Dual
In mathematics, Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by . Definition using characters Given any finite flat commutative group scheme ''G'' over ''S'', its Cartier dual is the group of characters, defined as the functor that takes any ''S''-scheme ''T'' to the abelian group of group scheme homomorphisms from the base change G_T to \mathbf_ and any map of ''S''-schemes to the canonical map of character groups. This functor is representable by a finite flat ''S''-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative ''S''-group schemes to itself. If ''G'' is a constant commutative group scheme, then its Cartier dual is the diagonalizable group ''D''(''G''), and vice versa. If ''S'' is affine, then the duality functor is given by the duality of the Hopf algebras of functions. Definition using Hopf algebras A finite commutative group scheme over a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
