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Scheme-theoretic Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Intersection
Derive may refer to: *Derive (computer algebra system), a commercial system made by Texas Instruments * ''Dérive'' (magazine), an Austrian science magazine on urbanism *Dérive, a psychogeographical concept *Derived trait, or apomorphy See also * *Derivation (other) Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a pro ... * Derivative (other) {{disambiguation ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gysin Homomorphism
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a Fiber bundle#Sphere bundles, sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa. It was introduced by , and is generalized by the Serre spectral sequence. Definition Consider a fiber-oriented sphere bundle with total space ''E'', base space ''M'', fiber ''S''''k'' and projection map \pi: S^k \hookrightarrow E \stackrel M. Any such bundle defines a degree ''k'' + 1 cohomology class ''e'' called the Euler class of the bundle. De Rham cohomology Discussion of the sequence is clearest with de Rham cohomology. There cohomology classes are represented by differential forms, so that ''e'' can be represented by a (''k'' + 1)-form. The projection map \pi induces a map in cohomology H^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P''''n'', there should exist ''n'' − ''m'' homogeneous polynomials: :F_i(X_0,\cdots,X_n), 1\leq i\leq n - m, in the homogeneous coordinates ''X''''j'', which generate all other homogeneous polynomials that vanish on ''V''. Geometrically, each ''F''''i'' defines a hypersurface; the intersection of these hypersurfaces should be ''V''. The intersection of hypersurfaces will always have dimension at least ''m'', assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to ''m'', with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension . When then ''V'' is automatically a hypersurface and there is nothin ... [...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] |
Chow's Moving Lemma
In algebraic geometry, Chow's moving lemma, proved by , states: given algebraic cycles ''Y'', ''Z'' on a nonsingular quasi-projective variety ''X'', there is another algebraic cycle ''Z' '' which is rationally equivalent to ''Z'' on ''X,'' such that ''Y'' and ''Z' '' intersect properly. The lemma is one of the key ingredients in developing intersection theory and the Chow ring In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so ..., as it is used to show the uniqueness of the theory. Even if ''Z'' is an effective cycle, it is not, in general, possible to choose ''Z' '' to be effective. References * * * Theorems in algebraic geometry Zhou, Weiliang {{math-hist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor (algebraic Geometry)
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Cycle
In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher-dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre's Inequality On Height
In algebra, specifically in the theory of commutative rings, Serre's inequality on height states: given a (Noetherian) regular ring ''A'' and a pair of prime ideals \mathfrak, \mathfrak in it, for each prime ideal \mathfrak r that is a minimal prime ideal over the sum \mathfrak p + \mathfrak q, the following inequality on heights holds: :\operatorname(\mathfrak r) \le \operatorname(\mathfrak p) + \operatorname(\mathfrak q). Without the assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection. Sketch of Proof Serre gives the following proof of the inequality, based on the validity of Serre's multiplicity conjectures for formal power series ring over a complete discrete valuation ring. By replacing A by the localization at \mathfrak r, we assume (A, \mathfrak r) is a local ring. Then the inequality is equivalent to the following inequality: for finite A-modules M, N such that M \otimes_A N has finite length, :\dim_A M + \dim_A N \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Scheme
In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.. For an example of a regular scheme that is not smooth, see . See also *Étale morphism *Dimension of an algebraic variety *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. ... * Smooth completion References Algebraic geometry Scheme theory {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Algebraic Geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutative rings or E_-ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. Introduction Basic objects of study in the field are derived schemes and derived stacks. The oft-cited motivation is Serre's intersection formula. In the usual formulation, the formula involves the Tor functor and thus, un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
