|
Theorem On Formal Functions
In algebraic geometry, the theorem on formal functions states the following: :Let f: X \to S be a proper morphism of noetherian schemes with a coherent sheaf \mathcal on ''X''. Let S_0 be a closed subscheme of ''S'' defined by \mathcal and \widehat, \widehat formal completions with respect to X_0 = f^(S_0) and S_0. Then for each p \ge 0 the canonical (continuous) map: ::(R^p f_* \mathcal)^\wedge \to \varprojlim_k R^p f_* \mathcal_k :is an isomorphism of (topological) \mathcal_-modules, where :*The left term is \varprojlim R^p f_* \mathcal \otimes_ \mathcal_S/. :*\mathcal_k = \mathcal \otimes_ (\mathcal_S/^) :*The canonical map is one obtained by passage to limit. The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper morphism, proper birational morphism into a normal variety is an isomorphism. Some other corollaries (with the notations as above) are: Corollary: For any s \in S, topologically, : ... [...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] |
Proper Morphism
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ''k'' is proper over ''k''. A scheme ''X'' of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space ''X''(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. Definition A morphism ''f'': ''X'' → ''Y'' of schemes is called universally closed if for every scheme ''Z'' with a morphism ''Z'' → ''Y'', the projection from the fiber product :X \times_Y Z \to Z is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ( GAII, 5.4.. One also says that ''X'' is proper ... [...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] |
Formal Completion
In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes. A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions. Algebraic geometry based on formal schemes is called formal algebraic geometry. Definition Formal schemes are usually defined only in the Noetherian case. While t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein Factorization
In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points. Statement One version for schemes states the following: Let ''X'' be a Scheme (mathematics), scheme, ''S'' a locally noetherian scheme and f: X \to S a proper morphism. Then one can write :f = g \circ f' where g\colon S' \to S is a finite morphism and f'\colon X \to S' is a proper morphism so that f'_* \mathcal_X = \mathcal_. The existence of this decomposition itself is not difficult. See below. But, by Zariski's connectedness theorem, the last part in the above says that the fiber f'^(s) is connected for any s \in S. It follows: Corollary: For any s \in S, the set of connected components of the fiber f^(s) is in bijection with the set of points in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski's Main Theorem
In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational. Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows: *The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem. *A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset. *The total transform of a normal point under a proper birational morphism is connected. *A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's origi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Morphism
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Variety
In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and only if the ring ''O''(''X'') of regular functions on ''X'' is an integrally closed domain. A variety ''X'' over a field is normal if and only if every finite birational morphism from any variety ''Y'' to ''X'' is an isomorphism. Normal varieties were introduced by . Geometric and algebraic interpretations of normality A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve ''X'' in the affine plane ''A''2 defined by ''x''2 = ''y''3 is not normal, because there is a finite birational morphism ''A''1 → ''X'' (namely, ''t'' maps to (''t''3, ''t''2) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Existence Theorem
In mathematics, the Grothendieck existence theorem, introduced by , gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme ''S'' to schemes over ''S''. The theorem can be viewed as an instance of (Grothendieck's) formal GAGA. See also *Chow's lemma Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: :If X ... References * *. *. * *. * formal GAGA * External links * * * formal GAGA * * Theorems in algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base Change Map
In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: :g^*(R^r f_* \mathcal) \to R^r f'_*(g'^*\mathcal) where :\begin X' & \stackrel\to & X \\ f' \downarrow & & \downarrow f \\ S' & \stackrel g \to & S \end is a Cartesian square of topological spaces and \mathcal is a sheaf on ''X''. Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps ''f'', in algebraic geometry for (quasi-)coherent sheaves and ''f'' proper or ''g'' flat, similarly in analytic geometry, but also for étale sheaves for ''f'' proper or ''g'' smooth. Introduction A simple base change phenomenon arises in commutative algebra when ''A'' is a commutative ring and ''B'' and ''A' ''are two ''A''-algebras. Let B' = B \otimes_A A'. In this situation, given a ''B''-module ''M'', there is an isomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
