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Artinian Subscheme
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Commutative Algebra
This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed to be commutative ring, commutative with identity 1. !$@ A B C D E F G H . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Power
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude of a -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the parallelotope defined by the constituent vectors. The alternating property that v\wedge v=0 implies a skew-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of 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. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called '' points'', which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through points in general position, a -dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Divisor
The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical example'' is often used to mean ' archetype'. Science and technology * Canonical form, a natural unique representation of an object, or a preferred notation for some object Mathematics * * Canonical coordinates, sets of coordinates that can be used to describe a physical system at any given point in time * Canonical map, a morphism that is uniquely defined by its main property * Canonical polyhedron, a polyhedron whose edges are all tangent to a common sphere, whose center is the average of its vertices * Canonical ring, a graded ring associated to an algebraic variety * Canonical injection, in set theory * Canonical representative, in set theory a standard member of each element of a set partition Differential geometry * Canonical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjunction Formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction. Adjunction for smooth varieties Formula for a smooth subvariety Let ''X'' be a smooth algebraic variety or smooth complex manifold and ''Y'' be a smooth subvariety of ''X''. Denote the inclusion map by ''i'' and the ideal sheaf of ''Y'' in ''X'' by \mathcal. The conormal exact sequence for ''i'' is :0 \to \mathcal/\mathcal^2 \to i^*\Omega_X \to \Omega_Y \to 0, where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism :\omega_Y = i^*\omega_X \otimes \operatorname(\mathcal/\mathcal^2)^\vee, where \vee denotes the dual of a line bundle. The particular case of a smooth divisor Suppose that ''D'' is a smooth div ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dualizing Sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme ''X'' of dimension ''n'' over a field ''k'' is a coherent sheaf \omega_X together with a linear functional :t_X: \operatorname^n(X, \omega_X) \to k that induces a natural isomorphism of vector spaces :\operatorname_X(F, \omega_X) \simeq \operatorname^n(X, F)^*, \, \varphi \mapsto t_X \circ \varphi for each coherent sheaf ''F'' on ''X'' (the superscript * refers to a dual vector space). The linear functional t_X is called a trace morphism. A pair (\omega_X, t_X), if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, \omega_X is an object representing the contravariant functor F \mapsto \operatorname^n(X, F)^* from the category of coherent sheaves on ''X'' to the category of ''k''-vector spaces. For a normal projective variety ''X'', the dualizing sheaf exists and it is in fact the canonical sheaf: \omega_X = \mathcal_X(K_X) where K_X is a canonical divisor. More generally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Scheme
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be holomorphically embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any Field (mathematics), field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those Complex torus, complex tori that can be holomorphic, holomorphically embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berkovich Spectrum
In mathematics, a Berkovich space, introduced by , is a version of an analytic space over a non-Archimedean field (e.g. ''p''-adic field), refining Tate's notion of a rigid analytic space. Motivation In the complex case, algebraic geometry begins by defining the complex affine space to be \Complex^n. For each U\subset\Complex^n, we define \mathcal_U, the ring of analytic functions on U to be the ring of holomorphic functions, i.e. functions on U that can be written as a convergent power series in a neighborhood of each point. We then define a local model space for f_, \ldots, f_\in\mathcal_U to be :X:=\ with \mathcal_X=\mathcal_U/(f_, \ldots,f_). A complex analytic space is a locally ringed \Complex-space (Y, \mathcal_Y) which is locally isomorphic to a local model space. When k is a complete non-Archimedean field, we have that k is totally disconnected. In such a case, if we continue with the same definition as in the complex case, we wouldn't get a good analytic theory. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Spec
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings. Zariski topology For any ideal I of R, define V_I to be the set of prime ideals containing I. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\big\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows: For f\in R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of \operatorname(R), and \big\ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: In fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, \operatorname(R) is not, in general, a T1 space. However, \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Polynomial Functions
In mathematics, the ring of polynomial functions on a vector space ''V'' over a field ''k'' gives a coordinate-free analog of a polynomial ring. It is denoted by ''k'' 'V'' If ''V'' is finite dimensional and is viewed as an algebraic variety, then ''k'' 'V''is precisely the coordinate ring of ''V''. The explicit definition of the ring can be given as follows. Given a polynomial ring k _1, \dots, t_n/math>, we can view t_i as a coordinate function on k^n; i.e., t_i(x) = x_i where x = (x_1, \dots, x_n). This suggests the following: given a vector space ''V'', let ''k'' 'V''be the commutative ''k''-algebra generated by the dual space V^*, which is a subring of the ring of all functions V \to k. If we fix a basis for ''V'' and write t_i for its dual basis, then ''k'' 'V''consists of polynomials in t_i. If ''k'' is infinite, then ''k'' 'V''is the symmetric algebra of the dual space V^*. In applications, one also defines ''k'' 'V''when ''V'' is defined over some subfield of ''k' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
