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Purity Conjecture
In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something happens, it happens in a particular codimension". Purity of the branch locus For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski–Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a ''branch locus'', namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...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 chos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formalized by saying that f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z is surjective. An example is the inclusion map \operatorname(R/I) \to \operatorname(R) induced by the canonical map R \to R/I. Other characterizations The following are equivalent: #f: Z \to X is a closed immersion. #For every open affine U = \operatorname(R) \subset X, there exists an ideal I \subset R such that f^(U) = \operatorname(R/I) as schemes over ''U''. #There exists an open affine covering X = \bigcup U_j, U_j = \operatorname R_j and for each ''j'' there exists an ideal I_j \subset R_j such that f^(U_j) = \operatorname (R_j / I_j) as schemes over U_j. #There is a quasi-coherent sheaf of ideals \mathcal on ''X'' such that f_\ast\mathcal_Z\cong \mathcal_X/\mathcal and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Cohomological Purity Conjecture
In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states:A version of the theorem is stated at given *a regular scheme ''X'' over some base scheme, *i: Z \to X a closed immersion of a regular scheme of pure codimension ''r'', *an integer ''n'' that is invertible on the base scheme, *\mathcal a locally constant étale sheaf with finite stalks and values in \mathbb/n\mathbb, for each integer m \ge 0, the map :\operatorname^m(Z_; \mathcal) \to \operatorname^_Z(X_; \mathcal(r)) is bijective, where the map is induced by cup product with c_r(Z). The theorem was introduced in SGA 5 Exposé I, § 3.1.4. as an open problem. Later, Thomason proved it for large ''n'' and Gabber in general. See also *purity (algebraic geometry) In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SGA 4
SGA may refer to: * Old Irish language (ISO 639-3 code) * ''Schwarz-Gelbe Allianz'' (Black-Yellow Alliance), an Austrian political party * Second-generation antipsychotics * Séminaire de Géométrie Algébrique du Bois Marie, an influential mathematical seminar and the series of books it produced. * SGA Airlines (Siam General Aviation), a regional airline in Thailand * SG&A, Selling, General and Administrative expenses in income statements * Simple Genetic Algorithm * Small for gestational age, babies whose birth weight lies below the 10th percentile for that gestational age * Society of Graphic Art, a British arts organization founded in 1919 * Songwriters Guild of America, an organization to help "advance, promote, and benefit" the profession of songwriters * Southern Governors' Association * Standard Galactic Alphabet, the writing system in the ''Commander Keen'' fictional universe * ''Stargate Atlantis'', an American-Canadian science fiction television series and a spin-of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Artin
Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Faculty profile , MIT mathematics department, retrieved 2011-01-03 Life and career Michael Artin or Artinian was born in , Germany, and brought up in . His parents were Natalia Naumovna Jasny (Natascha) and[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme Theory
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 commu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). Similarly, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
