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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 fiber ... [...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] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Free Sheaf
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 which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Stack
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 open immersion followed by an affine morphism.§ 8.6 of F. MeyerNotes on algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves \mathcal_ and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin. Definition Motivation One of the motivating examples of an algebraic stack is to consider a groupoid scheme (R,U,s,t,m) over a fixed scheme S. For example, if R = \mu_n\times_S\mathbb^n_S (where \mu_n is the group scheme of roots of unity), U = \mathbb^n_S, s = \text_U is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying 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] |
Descent (mathematics)
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vector bundles The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose ''X'' is a topological space covered by open sets ''Xi''. Let ''Y'' be the disjoint union of the ''Xi'', so that there is a natural mapping :p: Y \rightarrow X. We think of ''Y'' as 'above' ''X'', with the ''Xi'' projection 'down' onto ''X''. With this language, ''descent'' implies a vector bundle on ''Y ''(so, a bundle given on each ''Xi''), and our concern is to 'glue' those bundles ''Vi'', to make a single bundle ''V'' on X. What we mean is that ''V'' should, when restricted to ''Xi'', give back ''Vi'', up to a bundle isomorphism. The data needed is then this: on each overlap :X_, ... [...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] |
