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Grothendieck's Connectedness Theorem
In mathematics, Grothendieck's connectedness theorem , states that if ''A'' is a complete Noetherian local ring whose spectrum is ''k''-connected and ''f'' is in the maximal ideal, then Spec(''A''/''fA'') is (''k'' − 1)-connected. Here a Noetherian scheme is called ''k''-connected if its dimension is greater than ''k'' and the complement of every closed subset of dimension less than ''k'' is connected. It is a local analogue of Bertini's theorem. See also * Zariski connectedness theorem *Fulton–Hansen connectedness theorem In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dim ... References Bibliography * * Theorems in algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Laskerâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠0 and the sum of any two non-units in ''R'' is a non-unit. * 1 ≠0 and if ''x'' is any element of ''R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a local ring, and the maximal right ideal is also the unique maximal le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Scheme
In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thus, a scheme is noetherian if and only if it is locally noetherian and quasi-compact. As with noetherian rings, the concept is named after Emmy Noether. It can be shown that, in a locally noetherian scheme, if \operatorname A is an open affine subset, then ''A'' is a noetherian ring. In particular, \operatorname A is a noetherian scheme if and only if ''A'' is a noetherian ring. Let ''X'' be a locally noetherian scheme. Then the local rings \mathcal_ are noetherian rings. A noetherian scheme is a noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-noetherian valuation ring. The definitions extend to formal schemes. Properties and Noetherian hypotheses Having a (locally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertini's Theorem
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0. Statement for hyperplane sections of smooth varieties Let ''X'' be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space \mathbf P^n. Let , H, denote the complete system of hyperplane divisors in \mathbf P^n. Recall that it is the dual space (\mathbf P^n)^ of \mathbf P^n and is isomorphic to \mathbf P^n. The theorem of Bertini states that the set of hyperplanes not containing ''X'' and with smooth intersection with ''X'' contains an open dense subset of the total system of divisors , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Connectedness Theorem
In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected. Statement Suppose that ''f'' is a proper surjective morphism of varieties from ''X'' to ''Y'' such that the function field of ''Y'' is separably closed In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ... in that of ''X''. Then Zariski's connectedness theorem says that the inverse image of any normal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fulton–Hansen Connectedness Theorem
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979. The formal statement is that if ''V'' and ''W'' are irreducible algebraic subvarieties of a projective space ''P'', all over an algebraically closed field, and if : \dim(V) + \dim (W) > \dim (P) in terms of the dimension of an algebraic variety, then the intersection ''U'' of ''V'' and ''W'' is connected. More generally, the theorem states that if Z is a projective variety and f\colon Z \to P^n \times P^n is any morphism such that \dim f(Z) > n, then f^\Delta is connected, where \Delta is the diagonal in P^n \times P^n. The special case of intersections is recovered by taking Z = V \times W, with f the natural inclusion. See also * Zariski ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
