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Artin's Criterion
In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces or as Algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of pointed curves. Notation and technical notes Throughout this article, let S be a scheme of finite-type over a field k or an excellent DVR. p:F \to (Sch/S) will be a category fibered in groupoids, F(X) will be the groupoid lying over X \to S. A stack F is called limit preserving if it is compatible with filtered direct limits in Sch/S, meaning given a filtered system \_ there is an equivalence of categories\lim_F(X_i) \to F(\lim_X_i)An element of x \in F(X) is called an algebraic element if it is the henselization of an \mathcal_S-algebra of finite type. A limit preserving stack F over Sch/S is called an algebraic sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient Condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations betw ... [...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] |
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] |
Moduli Stack Of Elliptic Curves
In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme S to it correspond to elliptic curves over S. The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in \mathcal_. Properties Smooth Deligne-Mumford stack The moduli stack of elliptic curves is a smooth separated Deligne–Mumford stack of finite type over \text(\mathbb), but is not a scheme as elliptic curves have non-trivial automorphisms. j-invariant There is a proper morphism of \mathcal_ to the affine line, the coarse moduli space of elliptic curves, given by the ''j''-inv ... [...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] |
Finite Type Scheme
In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is integral over k\left \right/math>. This definition can be extended to the quasi-projective varieties, such that a regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties). Definition by Schemes A morphism ''f'': ''X'' → ''Y'' of schemes is a finite morphism if ''Y'' has an open cover by affine schemes :V_i = \mbox \; B_i such that for each ''i'', :f^(V_i) = U_i is an open affine subscheme Spec ''A''''i'', and the restriction of ''f'' to ''U''''i'', which induces a ring homomorphism :B_i \rightarrow A_i, makes ''A''''i'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic (algebra), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Fibered In Groupoids
Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people with the name) * Parnell "Stacks" Edwards, a key associate in the Lufthansa heist * Robert Stack Pierce (1933–2016), an American actor and baseball player * Brian "Stack" Stevens (1941–2017), a Cornish rugby player Arts, entertainment, and media * ''Stack magazine'', a bimonthly publication about high school sports * ''Stacks'' (album), a 2005 album by Bernie Marsden * Stacks, trailer parks that were made vertical, in the film ''Ready Player One'' Computing * Stack (abstract data type), abstract data type and data structure based on the principle of last in first out * Stack (Haskell), a tool to build Haskell projects and manage their dependencies * Stack in Macintosh, one of a collection of documents created with HyperCard (as in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Approximation Theorem
In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field ''k'' are well-approximated by the algebraic functions on ''k''. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case k = \Complex); and an algebraic version of this theorem in 1969. Statement of the theorem Let \mathbf = x_1, \dots, x_n denote a collection of ''n'' indeterminates, k \mathbf the ring of formal power series with indeterminates \mathbf over a field ''k'', and \mathbf = y_1, \dots, y_n a different set of indeterminates. Let :f(\mathbf, \mathbf) = 0 be a system of polynomial equations in k mathbf, \mathbf/math>, and ''c'' a positive integer. Then given a formal power series solution \hat(\mathbf) \in k \mathbf, there is an algebraic solution \mathbf(\mathbf) consisting of algebraic functions (more precisely, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlessinger's Theorem
In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck. Definitions Λ is a complete Noetherian local ring with residue field ''k'', and ''C'' is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field ''k''. A small extension in ''C'' is a morphism ''Y''→''Z'' in ''C'' that is surjective with kernel a 1-dimensional vector space over ''k''. A functor is called representable if it is of the form ''h''''X'' where ''h''''X''(''Y'')=hom(''X'',''Y'') for some ''X'', and is called pro-representable if it is of the form ''Y''→lim hom(''X''''i'',''Y'') for a filtered direct limit over ''i'' in some filtered ordered set. A morphism of functors ''F''→''G'' from ''C'' to sets is called smooth if whenever ''Y''→''Z'' is an epimorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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