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Spectrum Of A Ring
In commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ..., the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ... it is simultaneously a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... equi ...
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Commutative Algebra
Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ... that studies commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...s, their ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ..., and modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the bene ...
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B-sheaf
In mathematics, the gluing axiom is introduced to define what a sheaf (mathematics), 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 (mathematics), 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 (set theory), union X and intersection (set theory), 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 (category theory), section s of F over X is equally well given by a pair of sections :(s', s ...
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Category Of Commutative Rings
In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is Category (mathematics)#Small and large categories, large, meaning that the class (set theory), class of all rings is proper class, proper. As a concrete category The category Ring is a concrete category meaning that the objects are set (mathematics), sets with additional structure (addition and multiplication) and the morphisms are function (mathematics), functions that preserve this structure. There is a natural forgetful functor :''U'' : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :''F'' : Set → Ring which assigns to each set ''X'' ...
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Continuous Function (topology)
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a continuous function is a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... such that a continuous variation (that is a change without jump) of the argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ... induces a continuous variation of the value Value or values may refer to: * Value (ethics) In eth ...
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Ring Homomorphism
In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ..., a branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., a ring homomorphism is a structure-preserving function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... between two rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Rin ...
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Functor
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., specifically category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ..., a functor is a mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ... between categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity ...
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Category Theory
Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... and its concepts in terms of a labeled directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges often called ... called a ''category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...'', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or morphi ...
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Regular Function
In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ..., a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... f\colon Y\to k from a quasi-affine variety Y to its underlying field k, is a regular function if for an arbitrary point P in Y there exists an open neighborhood ''U'' around that point such that ''f'' can be expressed by fraction like g/h in which g, h are polynomials on the ambient affine space of Y such that ''h'' is nowhere zero on ''U''. A regular map from an arbitrary variety X to affine space \mathbb^n is a map given by n-tuple of regular functi ...
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Locally Ringed Space
In mathematics, a ringed space is a family of (Commutative ring, commutative) ring (mathematics), rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of Restriction (mathematics), restrictions. Precisely, it is a topological space equipped with a sheaf (mathematics), sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of Continuous_function#Continuous_functions_between_topological_spaces, continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germ of a function, germs of functions at a point is valid. Ringed spaces appear in mathematical analysis, analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be c ...
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Local Ring
In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., more specifically ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ..., local rings are certain rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ... that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties Variety may refer to: ...
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Quasicoherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of Sheaf (mathematics), sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernel (category theory), kernels, image (mathematics), images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X-sheaf of modules, modules which has a local presenta ...
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Localization Of A Module
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring (mathematics), ring or module (mathematics), module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of algebraic fraction, fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the ring \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf (mathematics), sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of function (mathematics), functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ...
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