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Quotient Stack
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks. Definition A quotient stack is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' an ''S''-scheme on which ''G'' acts. Let the quotient stack /G/math> be the category over the category of ''S''-schemes: *an object over ''T'' is a principal ''G''-bundle P\to T together with equivariant map P\to X; *an arrow from P\to T to P'\to T' is a bundle map (i.e., forms a commutative diagram) that is compatible with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 stack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle ''EG'' → ''BG''. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is, roughly speaking, a path-connected topological space ''X'' such that the fundam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Publications Mathématiques De L'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique. The journal was established in 1959 and was published at irregular intervals, from one to five volumes a year. It is now biannual. The editor-in-chief is Claire Voisin (Collège de France). See also *''Annals of Mathematics'' *'' Journal of the American Mathematical Society'' *''Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...'' External links * Back issues from 1959 to 2010 Mathematics journals Publications established in 1959 Springer Science+Business Media academic journals Biannual journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Of Algebraic Curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem. The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends"). Moduli stacks of stable curves The moduli stack \mathcal_ classifies families of smooth projective curves, together with their isomorphisms. When g > 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group-scheme Action
In algebraic geometry, an action of a group scheme is a generalization of a group action to a group scheme. Precisely, given a group ''S''-scheme ''G'', a left action of ''G'' on an ''S''-scheme ''X'' is an ''S''-morphism :\sigma: G \times_S X \to X such that * (associativity) \sigma \circ (1_G \times \sigma) = \sigma \circ (m \times 1_X), where m: G \times_S G \to G is the group law, * (unitality) \sigma \circ (e \times 1_X) = 1_X, where e: S \to G is the identity section of ''G''. A right action of ''G'' on ''X'' is defined analogously. A scheme equipped with a left or right action of a group scheme ''G'' is called a ''G''-scheme. An equivariant morphism between ''G''-schemes is a morphism of schemes that intertwines the respective ''G''-actions. More generally, one can also consider (at least some special case of) an action of a group functor: viewing ''G'' as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above.In details, gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Stack Of Principal Bundles
In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack given by: for any \mathbf_q-algebra ''R'', :\operatorname_G(X)(R) = the category of principal ''G''-bundles over the relative curve X \times_ \operatornameR. In particular, the category of \mathbf_q-points of \operatorname_G(X), that is, \operatorname_G(X)(\mathbf_q), is the category of ''G''-bundles over ''X''. Similarly, \operatorname_G(X) can also be defined when the curve ''X'' is over the field of complex numbers. Roughly, in the complex case, one can define \operatorname_G(X) as the quotient stack of the space of holomorphic connections on ''X'' by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of \operatorname_G(X). In the fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Bundle
In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ... , such that every bundle with the given structure group over is a pullback bundle, pullback by means of a continuous map . Existence of a universal bundle In the CW complex category When the definition of the classifying space takes place within the homotopy category (mathematics), category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem. For compact Lie groups We will first prove: :Proposition. Let be a compact Lie group. There exists a contractible space on which acts freely. The projection is a -principal fibr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Stack Of Formal Group Laws
In algebraic geometry, the moduli stack of formal group laws is a stack classifying formal group laws and isomorphisms between them. It is denoted by \mathcal_. It is a "geometric “object" that underlies the chromatic approach to the stable homotopy theory, a branch of algebraic topology. Currently, it is not known whether \mathcal_ is a derived stack or not. Hence, it is typical to work with stratifications. Let \mathcal^n_ be given so that \mathcal^n_(R) consists of formal group laws over ''R'' of height exactly ''n''. They form a stratification of the moduli stack \mathcal_. \operatorname \overline \to \mathcal^n_ is faithfully flat. In fact, \mathcal^n_ is of the form \operatorname \overline / \operatorname(\overline, f) where \operatorname(\overline, f) is a profinite group called the Morava stabilizer group. The Lubin–Tate theory describes how the strata \mathcal^n_ fit together. References * * Further reading * Topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazard Ring
In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let :F(x,y) be :x+y+\sum_ c_ x^i y^j for indeterminates c_, and we define the universal ring ''R'' to be the commutative ring generated by the elements c_, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring ''R'' has the following universal property: :For every commutative ring ''S'', one-dimensional formal group laws over ''S'' correspond to ring homomorphisms from ''R'' to ''S''. The commutative ring ''R'' constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Stack Of Principal Bundles
In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack given by: for any \mathbf_q-algebra ''R'', :\operatorname_G(X)(R) = the category of principal ''G''-bundles over the relative curve X \times_ \operatornameR. In particular, the category of \mathbf_q-points of \operatorname_G(X), that is, \operatorname_G(X)(\mathbf_q), is the category of ''G''-bundles over ''X''. Similarly, \operatorname_G(X) can also be defined when the curve ''X'' is over the field of complex numbers. Roughly, in the complex case, one can define \operatorname_G(X) as the quotient stack of the space of holomorphic connections on ''X'' by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of \operatorname_G(X). In the fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Stack
Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated mapping, the depiction of events over time on a map * Brain mapping, the techniques used to study the brain * Data mapping, data element mappings between two distinct data models * Gene mapping, the assignment of DNA fragments to chromosomes * Mind mapping, the drawing of ideas and the relations among them * Projection mapping, the projection of videos on the surface of objects with irregular shapes * Robotic mapping, creation and use of maps by robots * Satellite mapping, taking photos of Earth from space * Spiritual mapping, a practice of some religions * Texture mapping, in computer graphics * Web mapping, mapping of data delivered by Geographic Information Systems See also * * * Mapping theorem (other) * Mappings (poetry) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups ''H''''k''(''X'';''R'') on ''X'' with coefficients in a commutative ring ''R'' (typically ''R'' is Z''n'', Z, Q, R, or C) one can define the cup product, which takes the form :H^k(X;R) \times H^\ell(X;R) \to H^(X; R). The cup product gives a multiplication on the direct sum of the cohomology groups :H^\bullet(X;R) = \bigoplus_ H^k(X; R). This multiplication turns ''H''•(''X'';''R'') into a ring. In fact, it is naturally an N-graded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |