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Dual Module
In mathematics, the dual module of a left (respectively right) module ''M'' over a ring ''R'' is the set of left (respectively right) ''R''-module homomorphisms from ''M'' to ''R'' with the pointwise right (respectively left) module structure. The dual module is typically denoted ''M''∗ or . If the base ring ''R'' is a field, then a dual module is a dual vector space. Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = \operatorname(A) is a finite commutative group scheme represented by a Hopf algebra ''A'' over a commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...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 a scheme ''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 sche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of 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 of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have 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 initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Rayn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Function
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsionless Module
In abstract algebra, a module (mathematics), module ''M'' over a ring (mathematics), ring ''R'' is called torsionless if it can be embedded into some direct product ''R''''I''. Equivalently, ''M'' is torsionless if each non-zero element of ''M'' has non-zero image under some ''R''-linear functional ''f'': : f\in M^=\operatorname_R(M,R),\quad f(m)\ne 0. This notion was introduced by Hyman Bass. Properties and examples A module is torsionless if and only if the canonical map into its double dual module, dual, : M\to M^=\operatorname_R(M^,R), \quad m\mapsto (f\mapsto f(m)), m\in M, f\in M^, is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive. * A unital free module is torsionless. More generally, a direct sum of torsionless modules is torsionless. * A free module is reflexive if it is finitely generated module, finitely generated, and for some rings there are also infinitely genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Module
Reflexive, or the property reflexivity, may refer to: Fiction *Metafiction Grammar * Reflexivity (grammar): **Reflexive pronoun, a pronoun with a reflexive relationship with its self-identical antecedent **Reflexive verb, where a semantic agent and patient are the same Mathematics and computer science *Reflexive relation, a relation where elements of a set are self-related *Reflexive user interface, an interface that permits its own command verbs and sometimes underlying code to be edited *Reflexive operator algebra, an operator algebra that has enough invariant subspaces to characterize it *Reflexive space, a subset of Banach spaces *Reflexive bilinear form, a bilinear form for which the order of a pair of vectors does not affect whether it evaluates to zero. Biology *Reflexive antagonism, the phenomenon by which muscles with opposing functions tend to antagonistically inhibit each other. * Self-reflexivity (see Self-reference) Other * Reflexive Entertainment, a video game deve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Homomorphism
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention). A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps. A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of ''choices'' of canonical maps or canonical isomorphisms; for a typical example, see prestack. For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference. Examples *If is a normal subgroup of a group , then there is a canonical surjective group homomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Vector Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 192 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
