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Fpqc Morphism
In algebraic geometry, there are two slightly different definitions of an fpqc morphism, both variations of faithfully flat morphisms. Sometimes an fpqc morphism means one that is faithfully flat and quasicompact. This is where the abbreviation fpqc comes from: fpqc stands for the French phrase "fidèlement plat et quasi-compact", meaning "faithfully flat and quasi-compact". However it is more common to define an fpqc morphism f: X \to Y of schemes to be a faithfully flat morphism that satisfies the following equivalent conditions: # Every quasi-compact open subset of Y is the image of a quasi-compact open subset of ''X''. # There exists a covering V_i of Y by open affine subschemes such that each V_i is the image of a quasi-compact open subset of X. # Each point x \in X has a neighborhood U such that f(U) is open and f: U \to f(U) is quasi-compact. # Each point x \in X has a quasi-compact neighborhood such that f(U) is open affine. Examples: An open faithfully flat morphism is ... [...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] |
French Language
French ( or ) is a Romance language of the Indo-European family. It descended from the Vulgar Latin of the Roman Empire, as did all Romance languages. French evolved from Gallo-Romance, the Latin spoken in Gaul, and more specifically in Northern Gaul. Its closest relatives are the other langues d'oïl—languages historically spoken in northern France and in southern Belgium, which French ( Francien) largely supplanted. French was also influenced by native Celtic languages of Northern Roman Gaul like Gallia Belgica and by the ( Germanic) Frankish language of the post-Roman Frankish invaders. Today, owing to France's past overseas expansion, there are numerous French-based creole languages, most notably Haitian Creole. A French-speaking person or nation may be referred to as Francophone in both English and French. French is an official language in 29 countries across multiple continents, most of which are members of the ''Organisation internationale de la Francophonie'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faithfully Flat Morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \to \mathcal_ is a flat map for all ''P'' in ''X''. A map of rings A\to B is called flat if it is a homomorphism that makes ''B'' a flat ''A''-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: *flatness is a generic property; and *the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some finiteness conditions on ''f'', it can be shown that there is a non-empty open subscheme Y' of ''Y'', such that ''f'' restricted to ''Y''′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to ''f'' and the inclusio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-compact Morphism
In algebraic geometry, a morphism f: X \to Y between schemes is said to be quasi-compact if ''Y'' can be covered by open affine subschemes V_i such that the pre-images f^(V_i) are quasi-compact (as topological space). If ''f'' is quasi-compact, then the pre-image of a quasi-compact open subscheme (e.g., open affine subscheme) under ''f'' is quasi-compact. It is not enough that ''Y'' admits a covering by quasi-compact open subschemes whose pre-images are quasi-compact. To give an example, let ''A'' be a ring that does not satisfy the ascending chain conditions on radical ideals, and put X = \operatorname A. ''X'' contains an open subset ''U'' that is not quasi-compact. Let ''Y'' be the scheme obtained by gluing two ''Xs along ''U''. ''X'', ''Y'' are both quasi-compact. If f: X \to Y is the inclusion of one of the copies of ''X'', then the pre-image of the other ''X'', open affine in ''Y'', is ''U'', not quasi-compact. Hence, ''f'' is not quasi-compact. A morphism from a quasi-compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Topology
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term ''flat'' here comes from flat modules. There are several slightly different flat topologies, the most common of which are the fppf topology and the fpqc topology. ''fppf'' stands for ', and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. ''fpqc'' stands for ', and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined be a family which is a cover on Zariski open subsets. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent. The "pure" faithfully flat topology without any further finiteness conditions such as qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fppf Morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \to \mathcal_ is a flat map for all ''P'' in ''X''. A map of rings A\to B is called flat if it is a homomorphism that makes ''B'' a flat ''A''-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: *flatness is a generic property; and *the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some finiteness conditions on ''f'', it can be shown that there is a non-empty open subscheme Y' of ''Y'', such that ''f'' restricted to ''Y''′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to ''f'' and the inclusio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
