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Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prestack
In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object. Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks. Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category ''C''; for example, ''C'' can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology. Informal definition Let ''F'' be a category and suppose it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Moduli Stack Of Vector Bundles
In algebraic geometry, the moduli stack of rank-''n'' vector bundles Vect''n'' is the stack parametrizing vector bundles (or locally free sheaves) of rank ''n'' over some reasonable spaces. It is a smooth algebraic stack of the negative dimension -n^2. Moreover, viewing a rank-''n'' vector bundle as a principal GL_n-bundle, Vect''n'' is isomorphic to the classifying stack BGL_n = text/GL_n Definition For the base category, let ''C'' be the category of schemes of finite type over a fixed field ''k''. Then \operatorname_n is the category where # an object is a pair (U, E) of a scheme ''U'' in ''C'' and a rank-''n'' vector bundle ''E'' over ''U'' # a morphism (U, E) \to (V, F) consists of f: U \to V in ''C'' and a bundle-isomorphism f^* F \overset\to E. Let p: \operatorname_n \to C be the forgetful functor. Via ''p'', \operatorname_n is a prestack over ''C''. That it is a stack over ''C'' is precisely the statement "vector bundles have the descent property". Note that each fib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Group-stack
In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a scheme whose sets of points have group structures in a compatible way. Examples *A group scheme is a group-stack. More generally, a group algebraic-space, an algebraic-space analog of a group scheme, is a group-stack. *Over a field ''k'', a vector bundle stack \mathcal on a Deligne–Mumford stack ''X'' is a group-stack such that there is a vector bundle ''V'' over ''k'' on ''X'' and a presentation V \to \mathcal. It has an action by the affine line \mathbb^1 corresponding to scalar multiplication. *A Picard stack is an example of a group-stack (or groupoid-stack). Actions of group-stacks The definition of a group action of a group-stack is a bit tricky. First, given an algebraic stack ''X'' and a group scheme ''G'' on a base scheme ''S'', a right action of ''G'' on ''X'' consists o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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-compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Grothendieck Construction
The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory. Definition Let F\colon \mathcal \rightarrow \mathbf be a functor from any small category to the category of small categories. The Grothendieck construction for F is the category \Gamma(F) (also written \textstyle\int_ F, \textstyle\mathcal \int F or F \rtimes \mathcal), with * objects being pairs (c,x), where c\in \operatorname(\mathcal) and x\in \operatorname(F(c)); and * morphisms in \operatorname_((c_1,x_1),(c_2,x_2)) being pairs (f, g) such that f: c_1 \to c_2 in \mathcal, and g: F(f)(x_1) \to x_2 in F(c_2). Composition of morphisms is defined by (f,g) \circ (f',g') = (f \circ f', g \circ F(f)(g')). Slogan "The Grothendieck construction takes structured, tabulated data and flattens it by throwing it all into one big space. The projection functor is then tasked with remembering which box each datum originally came from." Example If G is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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GIT Quotient
In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of invariants of ''A'', and is denoted by X /\!/ G. A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it. Taking Proj (of a graded ring) instead of \operatorname, one obtains a projective GIT quotient (which is a quotient of the set of semistable points.) A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since the categorical quotient is unique, if there is a geometric quotient, then the two notions coincide: for example, one has :G / H = G /\!/ H = \operatorname\!\big(k H\big) for an algebraic group ''G'' over a field ''k'' and closed subgroup ''H''. If ''X'' is a complex smooth projective variety and if ''G'' is a redu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Reductive Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Deligne–Mumford Stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened to " smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford. A key fact about a Deligne–Mumford stack ''F'' is that any ''X'' in F(B), where ''B'' is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme. Examples Affine Stacks Deligne–Mumford stacks are typically constructed by taking the stack quotient of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group C_n = \langle a \mid a^n =1 \rangle on \mathbb^2 given by a\cdot\colon(x,y) \mapsto (\zeta_n x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Morphism Of Algebraic Stacks
In algebraic geometry, given algebraic stacks p: X \to C, \, q: Y \to C over a base category ''C'', a morphism f: X \to Y of algebraic stacks is a functor such that q \circ f = p. More generally, one can also consider a morphism between prestacks; (a stackification would be an example.) Types One particular important example is a presentation of a stack, which is widely used in the study of stacks. An algebraic stack ''X'' is said to be smooth of dimension ''n'' - ''j'' if there is a smooth presentation U \to X of relative dimension ''j'' for some smooth scheme ''U'' of dimension ''n''. For example, if \operatorname_n denotes the moduli stack of rank-''n'' vector bundles, then there is a presentation \operatorname(k) \to \operatorname_n given by the trivial bundle \mathbb^n_k over \operatorname(k). A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact In mathematics, specifically general topology, compactness is a property that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |