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Relative Dimension (scheme Theory)
In algebraic geometry, the dimension of a scheme is a generalization of a dimension of an algebraic variety. Scheme theory emphasizes the relative point of view and, accordingly, the relative dimension of a morphism of schemes is also important. Definition By definition, the dimension of a scheme ''X'' is the dimension of the underlying topological space: the supremum of the lengths ''â„“'' of chains of irreducible closed subsets: :\emptyset \ne V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_\ell \subset X. In particular, if X = \operatorname A is an affine scheme, then such chains correspond to chains of prime ideals (inclusion reversed) and so the dimension of ''X'' is precisely the Krull dimension of ''A''. If ''Y'' is an irreducible closed subset of a scheme ''X'', then the codimension of ''Y'' in ''X'' is the supremum of the lengths ''â„“'' of chains of irreducible closed subsets: :Y = V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_\ell \subset X. An irreducible subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Of An Algebraic Variety
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding. Dimension of an affine algebraic set Let be a field, and be an algebraically closed extension. An affine algebraic set is the set of the common zeros in of the elements of an ideal in a polynomial ring R=K _1, \ldots, x_n Let A=R/I be the algebra of the polynomial functions over . The dimension of is any of the following integers. It does not change if is enlarged, if is replaced by another algebraically closed extension of and if is replaced by another ideal having the same zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equidimensional Ring
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Definition A prime ideal ''P'' is said to be a minimal prime ideal over an ideal ''I'' if it is minimal among all prime ideals containing ''I''. (Note: if ''I'' is a prime ideal, then ''I'' is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal. A minimal prime ideal over an ideal ''I'' in a Noetherian ring ''R'' is precisely a minimal associated prime (also called isolated prime) of R/I; this follows for instance from the primary decomposition of ''I''. Examples * In a commutative artinian ring, every maximal ideal is a minimal prime ideal. * In an integral domain, the only minimal prime ideal is the zero ideal. * In the ring Z of integers, the minimal prime ideals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Scheme Theory
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] |
Kleiman's Theorem
In algebraic geometry, Kleiman's theorem, introduced by , concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection. Precisely, it states: given a connected algebraic group ''G'' acting transitively on an algebraic variety ''X'' over an algebraically closed field ''k'' and V_i \to X, i = 1, 2 morphisms of varieties, ''G'' contains a nonempty open subset such that for each ''g'' in the set, # either gV_1 \times_X V_2 is empty or has pure dimension \dim V_1 + \dim V_2 - \dim X, where g V_1 is V_1 \to X \overset\to X, # (Kleiman– Bertini theorem) If V_i are smooth varieties and if the characteristic of the base field ''k'' is zero, then gV_1 \times_X V_2 is smooth. Statement 1 establishes a version of Chow's moving lemma: after some perturbation of cycles on ''X'', their intersection has expected dimension. Sketch of proof We write f_i for V_i \to X. Let h: G \times V_1 \to X be the composition that is (1_G, f_1): ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber (mathematics)
In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context: # In naive set theory, the fiber of the element y in the set Y under a map f : X \to Y is the inverse image of the singleton \ under f. # In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed. Definitions Fiber in naive set theory Let f : X \to Y be a function between sets. The fiber of an element y \in Y (or ''fiber over'' y) under the map f is the set f^(y) = \, that is, the set of elements that get mapped to y by the function. It is the preimage of the singleton \. (One usually takes y in the image of f to avoid f^(y) being the empty set.) The collection of all fibers for the function f forms a partition of the domain X. The fiber containing an element x\in X is the set f^(f(x)). For example, the fibers of the projection map \R^2\to\R that sends (x,y) to x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A Surface (mathematics), surface, such as the Boundary (mathematics), boundary of a Cylinder (geometry), cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the Euclidean plane, plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Type Scheme
In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is integral over k\left \right/math>. This definition can be extended to the quasi-projective varieties, such that a regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties). Definition by Schemes A morphism ''f'': ''X'' → ''Y'' of schemes is a finite morphism if ''Y'' has an open cover by affine schemes :V_i = \mbox \; B_i such that for each ''i'', :f^(V_i) = U_i is an open affine subscheme Spec ''A''''i'', and the restriction of ''f'' to ''U''''i'', which induces a ring homomorphism :B_i \rightarrow A_i, makes ''A''''i'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Étale Morphism
In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word ''étale'' is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle. Definition Let \phi : R \to S be a ring homomorphism. This makes S an R-algebra. Choose a monic polynomial f in R /math> and a polynomial g in R /math> such that the derivative f' of f is a unit in (R fR _g. We say that \phi is ''standard étale'' if f and g can be chose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
