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Quotient Stack
In algebraic geometry, a quotient stack is a stack (mathematics), stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a Scheme (mathematics), scheme or a algebraic variety, variety by a Group (mathematics), 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'' group-scheme action, acts. Let the quotient stack [X/G] be the fibered category, category over the category of ''S''-schemes, where *an object over ''T'' is a torsor (algebraic geometry), principal ''G''-bundle P\to ... [...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] |
Resolution Property
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X- modules that has a local presentation, that is, every point in X has an open neighborhood U in which there is an e ... [...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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Quotient
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continuous def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Stack Of Formal Group Laws
Modulus is the diminutive from the Latin word ''modus'' meaning measure or manner. It, or its plural moduli, may refer to the following: Physics, engineering and computing * Moduli (physics), scalar fields for which the potential energy function has continuous families of global minima * The measurement of standard pitch in the teeth of a rotating gear * Bulk modulus, a measure of compression resistance * Elastic modulus, a measure of stiffness *Shear modulus, a measure of elastic stiffness * Young's modulus, a specific elastic modulus * Modulo operation (a % b, mod(a, b), etc.), in both math and programming languages; results in remainder of a division * Casting modulus used in Chvorinov's rule. Mathematics * Modulus (modular arithmetic), base of modular arithmetic * Modulus, the absolute value of a real or complex number ( ) * Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects * Conformal modulus, a measure of the size of a cur ... [...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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Stack
Mapping may refer to: * 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 * Mapping (YouTube content), a genre of audiovisual content involving countries interacting with each other. Types of mapping * Animated mapping, the depiction of events over time on a map using sequential images representing each timeframe * Brain mapping, the techniques used to study the brain * Data mapping, data element mappings between two distinct data models * Digital mapping, the use of a computer to depict spatial data on a map * 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 m ... [...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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel's Theorem
In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, .... See also * Atiyah–Bott formula Notes References * * {{topology-stub Theorems in algebraic topology Theorems in algebraic geometry ... [...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 \to 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 a path-connected topological space ''X'' such that the fundamental group of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirō Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
