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Log Structure
In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others. Motivation The idea is to study some algebraic variety (or scheme) ''U'' which is smooth but not necessarily proper by embedding it into ''X'', which is proper, and then looking at certain sheaves on ''X''. The problem is that the subsheaf of \mathcal_X consisting of functions whose restriction to ''U'' is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of \mathcal_X , multiplicatively. Remembering this additional structure on ''X'' corresponds to remembering the inclusion j \colon U \to X , which likens ''X'' with this extra structure to a variety wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Normal Crossings
In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings being integrally closed). Normal crossing divisors Normal crossing divisors are a class of divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way. Let ''A'' be an algebraic variety, and Z= \bigcup_i Z_i a reduced Cartier divisor, with Z_i its irreducible components. Then ''Z'' is called a smooth normal crossing divisor if either :(i) ''A'' is a curve, or :(ii) all Z_i are smooth, and for each component Z_k, (Z-Z_k), _ is a smooth normal crossing divisor. Equivalently, one says that a reduced divisor has normal crossings if each point étale locally looks like the intersection of coordinate hyperplanes. Normal crossing singularity A normal crossings singularity is a point in an algebraic variety ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Log Geometry
Log most often refers to: * Trunk (botany), the stem and main wooden axis of a tree, called logs when cut ** Logging, cutting down trees for logs ** Firewood, logs used for fuel ** Lumber or timber, converted from wood logs * Logarithm, in mathematics Log, LOG or LoG may also refer to: Arts, entertainment and media * ''Log'' (magazine), an architectural magazine * ''The Log'', a boating and fishing newspaper published by the Duncan McIntosh Company * Lamb of God (band) or LoG, an American metal band * The Log, an electric guitar by Les Paul * Log, a fictional product in ''The Ren & Stimpy Show'' * The League of Gentlemen or LoG, a British comedy show. Places * Log, Russia, the name of several places * Log, Slovenia, the name of several places Science and mathematics *Logarithm, a mathematical function * Log file, a computer file in which events are recorded * Laplacian of Gaussian or LoG, an algorithm used in digital image processing Other uses * Logbook, or log, a record ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Galois Representation
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Crystalline Cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H''''n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . Crystalline cohomology is partly inspired by the ''p''-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety ''X'' in characteristic ''p'' is the de Rham cohomology of a smooth lift of ''X'' to characteristic 0, while de Rham cohomology of ''X'' is the crystalline cohomology reduced mod ''p'' (after taking into account higher ''Tor''s). The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by tak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus (topology), genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mixed Hodge Structure
In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. In mixed Hodge theory, where the decomposition of a cohomology group H^k(X) may have subspaces of different weights, i.e. as a direct sum of Hodge structures :H^k(X) = \bigoplus_i (H_i, F_i^\bullet) where each of the Hodge structures have weight k_i. One of the early hints that such structures should exist comes from the long exact sequence \dots \to H^(Y) \to H^i_c(U) \to H^i(X) \to \dotsassociated to a pair of smooth projective varieties Y \subset X . This sequence suggests that the cohomology groups H^i_c(U) (for U = X - Y ) should have differing weights coming from both H^(Y) and H^i(X) . Motivation Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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étale Morphism
In algebraic geometry, an étale morphism () is a morphism of Scheme (mathematics), schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word ''étale'' is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle. Definition Let \phi : R \to S be a ring homomorphism. This makes S an R-algebra. Choose a monic polynomial f in R[x] and a polynomial g in R[x] such that the Formal derivative, derivative f' of f is a unit in (R[x]/fR[x])_g. We say that \phi is ''stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Discrete Valuation Ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' that satisfies any and all of the following equivalent conditions: # ''R'' is a local ring, a principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local ring, a Dedekind domain, and not a field. # ''R'' is Noetherian and a local domain whose unique maximal ideal is principal, and not a field. # ''R'' is integrally closed, Noetherian, and a local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element (up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |