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Almost Ring
In mathematics, almost modules and almost rings are certain objects interpolating between rings and their fields of fractions. They were introduced by in his study of ''p''-adic Hodge theory. Almost modules Let ''V'' be a local integral domain with the maximal ideal ''m'', and ''K'' a fraction field of ''V''. The category of ''K''-modules, ''K''-Mod, may be obtained as a quotient of ''V''-Mod by the Serre subcategory of torsion modules, i.e. those ''N'' such that any element ''n'' in ''N'' is annihilated by some nonzero element in the maximal ideal. If the category of torsion modules is replaced by a smaller subcategory, we obtain an intermediate step between ''V''-modules and ''K''-modules. Faltings proposed to use the subcategory of almost zero modules, i.e. ''N'' ∈ ''V''-Mod such that any element ''n'' in ''N'' is annihilated by ''all'' elements of the maximal ideal. For this idea to work, ''m'' and ''V'' must satisfy certain technical conditions. Let ''V'' be a ring (n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion (algebra)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when ''R'' has a metric given by a non-Archimedean absolute value. General construction Suppose that ''E'' is an abelian group with a descending filtration : E = F^0 E \supset F^1 E \supset F^2 E \supset \cdots \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring which is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Valuation Ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' is a local principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local Dedekind domain and not a field. # ''R'' is a Noetherian local domain whose maximal ideal is principal, and not a field.https://mathoverflow.net/a/155639/114772 # ''R'' is an integrally closed Noetherian local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element ( up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Element
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a''''j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is a root of a monic polynomial over ''A''. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. If every element of ''B'' is integral over ''A'', then we say that ''B'' is integral over ''A'', or equivalently ''B'' is an integral extension of ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every ( small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form ... [...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] |
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] |
Adjoint Functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphism s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
