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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, \zeta_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Morphism
In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) means that each geometric fiber of ''f'' is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If ''S'' is the spectrum of an algebraically closed field and ''f'' is of finite type, then one recovers the definition of a nonsingular variety. Equivalent definitions There are many equivalent definitions of a smooth morphism. Let f: X \to S be locally of finite presentation. Then the following are equivalent. # ''f'' is smooth. # ''f'' is formally smooth (see below). # ''f'' is flat and the sheaf of relative differentials \Omega_ is locally free of rank equal to the relative dimension of X/S. # For any x \in X, there exists a neighborhood \operatornameB of x and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack Quotient
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] |
Groupoid Scheme
In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Definition A groupoid object in a category ''C'' admitting finite fiber products consists of a pair of objects R, U together with five morphisms :s, t: R \to U, \ e: U \to R, \ m: R \times_ R \to R, \ i: R \to R satisfying the following groupoid axioms # s \circ e = t \circ e = 1_U, \, s \circ m = s \circ p_1, t \circ m = t \circ p_2 where the p_i: R \times_ R \to R are the two projections, # (associativity) m \circ (1_R \times m) = m \circ (m \times 1_R), # (unit) m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R, # (inverse) i \circ i = 1_R, s \circ i = t, \, t \circ i = s, m \circ (1_R, i) = e \circ s, \, m \circ (i, 1_R) = e \circ t. Examples Group objects A group object is a special case of a groupoid object, where R = U and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial function replacing the binary operation; *''Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g:A \rightarrow B, h:B \rightarrow C, say. Composition is then a total function: \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C , so that h \circ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent 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 ex ... [...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 which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem). Definition There are two common ways to define algebraic spaces: they can be defined as either quotients of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Artin
Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Faculty profile , MIT mathematics department, retrieved 2011-01-03 Life and career Michael Artin or Artinian was born in , Germany, and brought up in . His parents were Natalia Naumovna Jasny (Natascha) and[...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] |
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] |
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] |
