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Localizing Subcategory
In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category. Serre subcategories Let \mathcal be an abelian category. A non-empty full subcategory \mathcal is called a ''Serre subcategory'' (or also a ''dense subcategory''), if for every short exact sequence 0\rightarrow A' \rightarrow A\rightarrow A''\rightarrow 0 in \mathcal the object A is in \mathcal if and only if the objects A' and A'' belong to \mathcal. In words: \mathcal is closed under subobjects, quotient objects and extensions. Each Serre subcategory \mathcal of \mathcal is itself an abelian category, and the inclusion functor \mathcal\to\mathcal is exact. The importance of this notion stems from the fact that kernels of exact functors between abelian categories are Serre subcategories, and that one can build (for locally small \mathcal) the q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Full And Faithful Functors
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor. Formal definitions Explicitly, let ''C'' and ''D'' be (locally small) categories and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function :F_\colon\mathrm_(X,Y)\rightarrow\mathrm_(F(X),F(Y)) for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be *faithful if ''F''''X'',''Y'' is injectiveJacobson (2009), p. 22 *full if ''F''''X'',''Y'' is surjectiveMac Lane (1971), p. 14 *fully faithful (= full and faithful) if ''F''''X'',''Y'' is bijective for each ''X'' and ''Y'' in ''C''. A mnemonic for remembering the term "full" is that the image of the function fills the codomain; a mnemonic for remembering the term "faithful" is that you can trust (have faith) that F(X)=F(Y) implies X=Y. Properties A faithful functor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolae Popescu
Nicolae Popescu (; 22 September 1937 – 29 July 2010) was a Romanian mathematician and professor at the University of Bucharest. He also held a research position at the Institute of Mathematics of the Romanian Academy, and was elected corresponding Member of the Romanian Academy in 1997. He is best known for his contributions to algebra and the theory of abelian categories. From 1964 to 2007 he collaborated with Pierre Gabriel on the characterization of abelian categories; their best-known result is the Gabriel–Popescu theorem, published in 1964. His areas of expertise were category theory, abelian categories with applications to rings and modules, adjoint functors, limits and colimits, the theory of sheaves, the theory of rings, fields and polynomials, and valuation theory. He also had interests and published in algebraic topology, algebraic geometry, commutative algebra, K-theory, class field theory, and algebraic function theory. Biography Popescu was born on September ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giraud Subcategory
In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud. Definition Let \mathcal be a Grothendieck category. A full subcategory \mathcal is called ''reflective'', if the inclusion functor i\colon\mathcal\rightarrow\mathcal has a left adjoint. If this left adjoint of i also preserves kernels, then \mathcal is called a ''Giraud subcategory''. Properties Let \mathcal be Giraud in the Grothendieck category \mathcal and i\colon\mathcal\rightarrow\mathcal the inclusion functor. * \mathcal is again a Grothendieck category. * An object X in \mathcal is injective if and only if i(X) is injective in \mathcal. * The left adjoint a\colon\mathcal\rightarrow\mathcal of i is exact. * Let \mathcal be a localizing subcategory of \mathcal and \mathcal/\mathcal be the associated quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Modules
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. Properties The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules. Projective limits and inductive limits exist in the categories of left and right modules. Over a commutative ring, together with the tensor product of modules ⊗, the category of mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Theory
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by , based on a suggestion of . In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by and . Definitions In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have a multiplicative identity. In particular, every ideal in a ring is also a ring. A radical class (also called radical property or just radical) is a class σ of rings possibly without identities, such that: # the homomorphic image of a ring in σ is also in σ # every ring ''R'' contains an ideal ''S''(''R'') in σ that contains every other ideal of ''R'' that is in σ # ''S''(''R''/''S''(''R'')) = 0. The ideal ''S''(''R'') is called the radical, or σ-radi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Category
In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962. To every algebraic variety V one can associate a Grothendieck category \operatorname(V), consisting of the quasi-coherent sheaves on V. This category encodes all the relevant geometric information about V, and V can be recovered from \operatorname(V) (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories. Definition By definition, a Grothendieck category \mathcal is an AB5 category with a generator. Spelled out, this means that * \mathcal is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Hull
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . Definition A module ''E'' is called the injective hull of a module ''M'', if ''E'' is an essential extension of ''M'', and ''E'' is injective. Here, the base ring is a ring with unity, though possibly non-commutative. Examples * An injective module is its own injective hull. * The injective hull of an integral domain is its field of fractions . * The injective hull of a cyclic ''p''-group (as Z-module) is a Prüfer group . * The injective hull of ''R''/rad(''R'') is Hom''k''(''R'',''k''), where ''R'' is a finite-dimensional ''k''-algebra with Jacobson radical rad(''R'') . * A simple module is necessarily the socle of its injective hull. * The injective hull of the residue field of a discrete valuation ring (R,\mathfrak,k) where \mathfrak = x\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Category
In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of ''all'' limits (even when ''J'' is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category ''J''). Dually, a category is finitely cocomplete if all finite colimits exist. Theorems It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (category Theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite category. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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