Lifting Continuous Map of Spectrum Ring
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Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. [1]
159 relations: Affine, Affine space, Affine variety, Algebra homomorphism, Algebraic geometry of projective spaces, Algebraic number field, Algebraic representation, Algebraic space, Algebraic variety, Ample line bundle, Analytic space, Arakelov theory, Arithmetic and geometric Frobenius, Arithmetic surface, Arithmetic zeta function, Artin–Verdier duality, Associated prime, Associative algebra, Étale spectrum, Čech cohomology, Banach–Stone theorem, Bass conjecture, Beauville–Laszlo theorem, Borel functional calculus, Canonical map, Cartan's theorems A and B, Category of rings, Character variety, Chevalley–Iwahori–Nagata theorem, Christopher Deninger, Closed immersion, Cohen–Macaulay ring, Coherent sheaf, Coherent sheaf cohomology, Commutative algebra, Commutative ring, Compact space, Completion (algebra), Complex affine space, Conductor of an abelian variety, Cone (algebraic geometry), Connected ring, Constructible topology, Cotangent sheaf, Counterexamples in Topology, Covering space, Dedekind zeta function, Deformation ring, Discrete mathematics, Domain (ring theory), ... , Duality (mathematics), Emmy Noether, Equivalence of categories, Extensive category, Fiber product of schemes, Field of definition, Field with one element, Finite morphism, Flat topology, Formal scheme, Fred Van Oystaeyen, Galois connection, General topology, Generic point, GIT quotient, Glossary of algebraic geometry, Glossary of algebraic topology, Glossary of areas of mathematics, Glossary of arithmetic and diophantine geometry, Glossary of commutative algebra, Gordan's lemma, Grothendieck group, Group scheme, Hausdorff space, Hyperconnected space, Inclusion map, Initial and terminal objects, Injective module, Injective sheaf, Integral domain, Invertible sheaf, Irreducibility (mathematics), Irreducible ideal, Irreducible ring, J-2 ring, Kolmogorov space, Krull dimension, Lie algebra representation, Linear algebraic group, List of algebraic geometry topics, List of commutative algebra topics, List of mathematical abbreviations, Local cohomology, Localization (algebra), Localization of a module, Localization of a ring, Melvin Hochster, Metrization theorem, Model theory, Moduli space, Morphism of algebraic varieties, Morphism of schemes, Noetherian, Noetherian topological space, Noncommutative algebraic geometry, Noncommutative geometry, Normal cone, Normal space, Opposite category, Outline of category theory, Parafactorial local ring, Picard group, Prime ideal, Prime number, Primon gas, Proj construction, Projective module, Projective variety, Quasi-separated morphism, Ramanujam–Samuel theorem, Rational point, Real closed ring, Reductive group, Regular sequence, Riemann hypothesis, Ring (mathematics), Ring theory, Ringed space, Scheme (mathematics), Semistable abelian variety, Sheaf (mathematics), Sheaf of algebras, Sheaf of modules, Sierpiński space, Smooth morphism, Sober space, Spec, Specialization (pre)order, Spectral space, Spectrum (disambiguation), Spectrum of a C*-algebra, Stalk (sheaf), Stein factorization, Stein manifold, Support of a module, Syntomic topology, T1 space, Tangent cone, Tautological bundle, Tensor product of algebras, Topological space, Topologically stratified space, Torsion-free module, Type (model theory), Vector space, Von Neumann regular ring, Witt group, Zariski topology, Zero ring. Expand index (109 more) » « Shrink index
Affine
Affine may refer to.
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Affine space
In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.
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Affine variety
In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.
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Algebra homomorphism
A homomorphism between two associative algebras, A and B, over a field (or commutative ring) K, is a function F\colon A\to B such that for all k in K and x, y in A,.
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Algebraic geometry of projective spaces
Projective space plays a central role in algebraic geometry.
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Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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Algebraic representation
In mathematics, an algebraic representation of a group G on a ''k''-algebra A is a linear representation \pi: G \to GL(A) such that, for each g in G, \pi(g) is an algebra automorphism.
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Algebraic space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by for use in deformation theory.
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Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry.
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Ample line bundle
In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M into projective space.
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Analytic space
An analytic space is a generalization of an analytic manifold that allows singularities.
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Arakelov theory
In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov.
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Arithmetic and geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number.
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Arithmetic surface
In mathematics, an arithmetic surface over a Dedekind domain R with fraction field K is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes.
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Arithmetic zeta function
In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers.
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Artin–Verdier duality
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by, that generalizes Tate duality.
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Associated prime
In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually denoted by \operatorname_R(M)\,.
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Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
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Étale spectrum
In algebraic geometry, the étale spectrum of a commutative ring or an '''E'''∞-ring, denoted by Specét or Spét, is an analog of the prime spectrum Spec of a commutative ring that is obtained by replacing Zariski topology with étale topology.
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Čech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space.
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Banach–Stone theorem
In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.
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Bass conjecture
In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic ''K''-groups are supposed to be finitely generated.
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Beauville–Laszlo theorem
In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve.
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Borel functional calculus
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope.
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Canonical map
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.
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Cartan's theorems A and B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold.
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Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).
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Character variety
In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi, the G-character variety of \pi is a space of equivalence classes of group homomorphisms More precisely, G acts on \mathfrak by conjugation, and two homomorphisms are defined to be equivalent if and only if their orbit closures intersect.
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Chevalley–Iwahori–Nagata theorem
In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomials is an isomorphism if this ring is finitely generated and all orbits of G on V are closed.
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Christopher Deninger
Christopher Deninger (born 8 April 1958) is a German mathematician at the University of Münster.
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Closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies Z as a closed subset of X such that regular functions on Z can be extended locally to X. The latter condition can be formalized by saying that f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z is surjective.
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Cohen–Macaulay ring
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality.
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Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space.
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Coherent sheaf cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties.
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Commutative algebra
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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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.
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Complex affine space
Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are.
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Conductor of an abelian variety
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is.
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Cone (algebraic geometry)
In algebraic geometry, a cone is a generalization of a vector bundle.
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Connected ring
In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions.
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Constructible topology
In commutative algebra, the constructible topology on the spectrum \operatorname(A) of a commutative ring A is a topology where each closed set is the image of \operatorname (B) in \operatorname(A) for some algebra B over A. An important feature of this construction is that the map \operatorname(B) \to \operatorname(A) is a closed map with respect to the constructible topology.
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Cotangent sheaf
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of \mathcalO_X-modules that represents (or classifies) S-derivations in the sense: for any \mathcal_X-modules F, there is an isomorphism that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential d: \mathcal_X \to \Omega_ such that any S-derivation D: \mathcal_X \to F factors as D.
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Counterexamples in Topology
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.
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Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
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Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the rational numbers Q).
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Deformation ring
In mathematics, a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field.
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Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
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Domain (ring theory)
In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or.
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Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.
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Emmy Noether
Amalie Emmy NoetherEmmy is the Rufname, the second of two official given names, intended for daily use.
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Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".
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Extensive category
In mathematics, an extensive category is a category C with finite coproducts that are disjoint and well-behaved with respect to pullbacks.
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Fiber product of schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction.
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Field of definition
In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong.
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Field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist.
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Finite morphism
In algebraic geometry, a morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi.
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Flat topology
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry.
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Formal scheme
In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings.
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Fred Van Oystaeyen
Fred Van Oystaeyen (born 1947), also Freddy van Oystaeyen, is a mathematician and emeritus professor of mathematics at the University of Antwerp.
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Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).
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General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
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Generic point
In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.
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GIT quotient
In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme \operatorname A with an action by a group scheme G is the affine scheme \operatorname(A^G), the prime spectrum of the ring of invariants of A, and is denoted by X /\!/ G. A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.
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Glossary of algebraic geometry
This is a glossary of algebraic geometry.
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Glossary of algebraic topology
This is a glossary of properties and concepts in algebraic topology in mathematics.
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Glossary of areas of mathematics
This is a glossary of terms that are or have been considered areas of study in mathematics.
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Glossary of arithmetic and diophantine geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.
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Glossary of commutative algebra
This is a glossary of commutative algebra.
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Gordan's lemma
In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated.
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Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid M in the most universal way in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck–Riemann–Roch theorem.
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Group scheme
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law.
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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Hyperconnected space
In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint).
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Inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element, x, of A to x, treated as an element of B: A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (On the other hand, this notation is sometimes reserved for embeddings.) This and other analogous injective functions from substructures are sometimes called natural injections.
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Initial and terminal objects
In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.
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Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
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Injective sheaf
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext).
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Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
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Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules.
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Irreducibility (mathematics)
In mathematics, the concept of irreducibility is used in several ways.
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Irreducible ideal
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
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Irreducible ring
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways.
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J-2 ring
In commutative algebra, a J-0 ring is a ring such that the set of regular points of the spectrum contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points of the spectrum is an open subset, and a J-2 ring is a ring such that any finitely generated algebra over the ring is a J-1 ring.
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Kolmogorov space
In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other.
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Krull dimension
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals.
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Lie algebra representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator.
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Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations.
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List of algebraic geometry topics
This is a list of algebraic geometry topics, by Wikipedia page.
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List of commutative algebra topics
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings.
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List of mathematical abbreviations
This article is a listing of abbreviated names of mathematical functions, function-like operators and other mathematical terminology.
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Local cohomology
In algebraic geometry, local cohomology is an analog of relative cohomology.
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Localization (algebra)
In commutative algebra and algebraic geometry, the localization is a formal way to introduce the "denominators" to a given ring or module.
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Localization of a module
In algebraic geometry, the localization of a module is a construction to introduce denominators in a module for a ring.
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Localization of a ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.
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Melvin Hochster
Melvin Hochster (born August 2, 1943) is an eminent American mathematician, regarded as one of the leading commutative algebraists active today.
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Metrization theorem
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
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Model theory
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.
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Moduli space
In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
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Morphism of algebraic varieties
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.
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Morphism of schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety.
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Noetherian
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length.
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Noetherian topological space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition.
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Noncommutative algebraic geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
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Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).
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Normal cone
In algebraic geometry, the normal cone CXY of a subscheme X of a scheme Y is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
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Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods.
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Opposite category
In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism.
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Outline of category theory
The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions.
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Parafactorial local ring
In algebraic geometry, a Noetherian local ring R is called parafactorial if it has depth at least 2 and the Picard group Pic(Spec(R) − m) of its spectrum with the closed point m removed is trivial.
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Picard group
In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product.
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Prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Primon gas
In mathematical physics, the primon gas or free Riemann gas is a toy model illustrating in a simple way some correspondences between number theory and ideas in quantum field theory and dynamical systems.
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Proj construction
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties.
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Projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules.
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Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.
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Quasi-separated morphism
In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X×YX is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi compact).
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Ramanujam–Samuel theorem
In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal.
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Rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field.
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Real closed ring
In mathematics, a real closed ring is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the integers.
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Reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field.
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Regular sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense.
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Riemann hypothesis
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.
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Ringed space
In mathematics, a ringed space can be equivalently thought of as either Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry.
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
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Semistable abelian variety
In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Sheaf of algebras
In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of \mathcalO_X-modules.
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Sheaf of modules
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) →F(V) are compatible with the restriction maps O(U) →O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).
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Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
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Smooth morphism
In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if.
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Sober space
In mathematics, a sober space is a topological space X such that every irreducible closed subset of X is the closure of exactly one point of X: that is, this closed subset has a unique generic point.
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Spec
Spec may refer to.
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Specialization (pre)order
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.
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Spectral space
In mathematics, a spectral space (sometimes called a coherent space) is a topological space that is homeomorphic to the spectrum of a commutative ring.
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Spectrum (disambiguation)
A spectrum is a condition or value that is not limited to a specific set of values but can vary infinitely within a continuum.
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Spectrum of a C*-algebra
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces.
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Stalk (sheaf)
The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.
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Stein factorization
In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers.
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Stein manifold
In the theory of several complex variables and complex manifolds in mathematics, a Stein manifold is a complex submanifold of the vector space of n complex dimensions.
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Support of a module
In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals \mathfrak of A such that M_\mathfrak \ne 0.
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Syntomic topology
In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by.
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T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other.
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Tangent cone
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
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Tautological bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: the fiber of the bundle over a vector space V (a point in the Grassmannian) is V itself.
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Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topologically stratified space
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way.
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Torsion-free module
In algebra, a torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring.
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Type (model theory)
In model theory and related areas of mathematics, a type is an object that, loosely speaking, describes how a (real or possible) element or elements in a mathematical structure might behave.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R such that.
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Witt group
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
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Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.
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Zero ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element.
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Redirects here:
Affine scheme, Affine schemes, Category of affine schemes, Global Spec, Prime ideal spectrum, Prime ideal topology, Prime spectrum, Relative Spec, Spec of a ring, Spectrum (ring theory).
References
[1] https://en.wikipedia.org/wiki/Spectrum_of_a_ring
mcfaddenworythe1936.blogspot.com
Source: https://en.unionpedia.org/i/Spectrum_of_a_ring
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