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Faithfully Flat Descent
Faithfully flat descent or flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open cover. In practice, from an affine point of view, this technique allows one to prove some statement about a ring or scheme after faithfully flat base change. In the language of stacks, flat descent is exactly the statement that the prestack of quasi-coherent sheaves is a stack with respect to étale (or fpqc) topology. "Vanilla" faithfully flat descent is generally false; instead, faithfully flat descent is valid under some finiteness conditions (e.g., quasi-compact or locally of finite presentation). A faithfully flat descent is a special case of Beck's monadicity theorem. Idea Given a faithfully flat ring homomorphism A \to B, the faithfully flat descent is, roughy, the statement that to give a module or an algebra ... [...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] |
Zariski Descent
Oscar Zariski (April 24, 1899 – July 4, 1986) was an American mathematician. The Russian-born scientist was one of the most influential algebraic geometers of the 20th century. Education Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kyiv. He left Kyiv in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski. Johns Hopkins University years Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Séminaire De Géométrie Algébrique Du Bois Marie
In mathematics, the (''SGA''; from French: "Seminar on Algebraic Geometry of Bois Marie") was an influential seminar run by French mathematician Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the (IHÉS) near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series. Style The material has a reputation of being hard to read for a number of reasons. More elementary or foundational parts were relegated to the EGA series of Grothendieck and Jean Dieudonné, causing long strings of logical dependencies in the statements. The style is very abstract and makes heavy use of category theory. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quot Scheme
In algebraic geometry, the Quot scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if ''X'' is a projective scheme over a Noetherian scheme ''S'' and if ''F'' is a coherent sheaf on ''X'', then there is a scheme \operatorname_F(X) whose set of ''T''-points \operatorname_F(X)(T) = \operatorname_S(T, \operatorname_F(X)) is the set of isomorphism classes of the quotients of F \times_S T that are flat over ''T''. The notion was introduced by Alexander Grothendieck. It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a Hilbert scheme. (In fact, taking ''F'' to be the structure sheaf \mathcal_X gives a Hilbert scheme.) Definition For a scheme of finite type X \to S over a Noetherian base scheme S, and a coherent sheaf \mathcal \in \text(X), there is a functor\mathcal_: (Sch/S)^ \to \textsending T \to S to\mathcal_(T) = \left\/ \simwhere X_T = X\times_ST and \mathcal_T = pr_X^*\mathcal under the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes. Hilbert scheme of projective space The Hilbert scheme \mathbf(n) of \mathbb^n classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points :\operatorname(S, \mathbf(n)) of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of \mathbb^n \times S that are flat over . The closed subschemes of \mathbb^n \times S that are flat over can informally be thought of as the families of subschemes of projective space parameterized b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fpqc Morphism
In 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; th ..., 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 quasi-compact. This is where the abbreviation fpqc comes from: fpqc stands for the French phrase "fidèlement plat et quasi-compacte", 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-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Moore Category
In category theory, a branch of mathematics, a monad is a triple (T, \eta, \mu) consisting of a functor ''T'' from a category to itself and two natural transformations \eta, \mu that satisfy the conditions like associativity. For example, if F, G are functors adjoint to each other, then T = G \circ F together with \eta, \mu determined by the adjoint relation is a monad. In concise terms, a monad is a monoid in the category of endofunctors of some fixed category (an endofunctor is a functor mapping a category to itself). According to John Baez, a monad can be considered at least in two ways: https://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html # A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category, # A monad as a tool for studying algebraic gadgets; for example, a group can be described by a certain monad. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F''). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, then \op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Field Extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ''F''. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'' with fixed field ''F'', then ''E''/''F'' is a Galois extension. The property of an extension being Galois behaves well with respect to field composition and intersection. Characterization of Galois extensions An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois: *E/F is a normal extension and a separable extension. *E is a splitting f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fpqc 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] |
Faithfully Flat Morphism
In mathematics, in particular 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 inclusion map of Y' into ''Y''. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
