Chow Group Of A Stack

	Chow Group Of A Stack In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = /G/math>, the Chow group of ''X'' is the same as the ''G''-equivariant Chow group of ''Y''. A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers). Definitions develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack. There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence. If a stack ''X'' can be written as the quotient stack X = /G/math> for some quasi-projective variety ''Y'' with a linearized action of a lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Chow Group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Derived Algebraic Geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutative rings or E_-ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. Introduction Basic objects of study in the field are derived schemes and derived stacks. The oft-cited motivation is Serre's intersection formula. In the usual formulation, the formula involves the Tor functor and thus, unless h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Gromov–Witten Invariant In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important. Definition Consider the following: ''X'': a closed symplectic manifold of dimensi ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Perfect Obstruction Theory In algebraic geometry, given a Deligne–Mumford stack ''X'', a perfect obstruction theory for ''X'' consists of: # a perfect two-term complex E = ^ \to E^0/math> in the derived category D(\text(X)_) of quasi-coherent étale sheaves on ''X'', and # a morphism \varphi\colon E \to \textbf_X, where \textbf_X is the cotangent complex of ''X'', that induces an isomorphism on h^0 and an epimorphism on h^. The notion was introduced by for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class. Examples Schemes Consider a regular embedding I \colon Y \to W fitting into a cartesian square : \begin X & \xrightarrow & V \\ g \downarrow & & \downarrow f \\ Y & \xrightarrow & W \end where V,W are smooth. Then, the complex :E^\bullet = ^N_^ \to j^\Omega_V/math> (in degrees -1, 0) forms a perfect obstruction theory for ''X''. The map comes from the composition :g^N_^\vee \to g^i^\Omega_W =j^f^\Omega_W \to j^\Omega_V This i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Białynicki-Birula Decomposition In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties). Linear action of a torus A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus ''T'' is acting on a finite-dimensional vector space ''V'', then there is a direct sum decomposition: :V = \bigoplus_ V_ where \chi: T \to \mathbb_m is a group homomorphism, a character of ''T''. V_ = \, ''T''-invariant subspace called the weight subspace of weight \chi. The decomposition exists because the linear action determines (and is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Equivariant Cohomology Ring In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^(X; \Lambda) = H^(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^(X; \Lambda) = H^(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^(X;A) ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Intersection Product In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks. Topological intersection form For a connected oriented manifold of dimension the intersection form is defined on the -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class in . Stated precisely, there is a bilinear form :\lambda_M \colon H^n(M,\partial M) \times H^n(M,\partial M)\to \mathbf given by :\lambda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Graded Ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Chow Ring In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Equivariant Cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^(X; \Lambda) = H^(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^(X; \Lambda) = H^(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^(X;A) of X ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Sumihiro's Theorem In algebraic geometry, Sumihiro's theorem, introduced by , states that a normal algebraic variety with an action of a torus can be covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover Book design Back cover copy, part of co ... by torus-invariant affine open subsets. The "normality" in the hypothesis cannot be relaxed. The hypothesis that the group acting on the variety is a torus can also not be relaxed. Notes References . External links Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ind-scheme In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. Examples \mathbbP^ = \varinjlim \mathbbP^N is an ind-scheme. Perhaps the most famous example of an ind-scheme is an infinite grassmannian (which is a quotient of the loop group of an algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ... ''G''.) See also * formal scheme References A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary versioV.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conferenceExpanded versionhttp://ncatlab.org/nlab/show/ ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]