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Hodge–de Rham Spectral Sequence
In mathematics, the Hodge–de Rham spectral sequence (named in honor of W. V. D. Hodge and Georges de Rham) is an alternative term sometimes used to describe the Frölicher spectral sequence (named after Alfred Frölicher, who actually discovered it). This spectral sequence describes the precise relationship between the Dolbeault cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler manifold, the sequence degenerates, thereby leading to the Hodge decomposition of the de Rham cohomology. Description of the spectral sequence The spectral sequence is as follows: :H^q(X, \Omega^p) \Rightarrow H^(X, \mathbf C) where ''X'' is a complex manifold, H^(X, \mathbf C) is its cohomology with complex coefficients and the left hand term, which is the E_1-page of the spectral sequence, is the cohomology with values in the sheaf of holomorphic differential forms. The existence of the spectral sequence as stated above follows from the Poincaré lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges De Rham
Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. Biography Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in Switzerland. He was the fifth born of the six children in the family of Léon de Rham, a constructions engineer. Georges de Rham grew up in Roche but went to school in nearby Aigle, the main town of the district, travelling daily by train. By his own account, he was not an extraordinary student in school, where he mainly enjoyed painting and dreamed of becoming a painter. In 1919 he moved with his family to Lausanne in a rented apartment in Beaulieu Castle, where he would live for the rest of his life. Georges de Rham started the Gymnasium in Lausanne with a focus on humanities, following his passion for literature and philosophy but learning little mathematics. On graduating from the Gymnasium in 1921 however, he decided not to continue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Complex
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Theories
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2021 impact factor of 12.779. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the mass media, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Ideal
In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let \mathcal(x_1,\ldots,x_n) denote the ring of smooth functions in n variables and f a function in the ring. The Jacobian ideal of f is : J_f := \left\langle \frac, \ldots, \frac \right\rangle. Relation to deformation theory In deformation theory, the deformations of a hypersurface given by a polynomial f is classified by the ring \frac This is shown using the Kodaira–Spencer map. Relation to Hodge theory In Hodge theory, there are objects called real Hodge structures which are the data of a real vector space H_\mathbb and an increasing filtration F^\bullet of H_\mathbb = H_\mathbb\otimes_\mathbb satisfying a list of compatibility structures. For a smooth projective variety X there is a canonical Hodge structure. Statement for degree d hypersurfaces In the special case X is defined by a homogeneous degree d polynomial f \in \Gamma(\mathbb^,\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Hochschild Homology
In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic p. For instance, if we consider the \mathbb-algebra \mathbb_p then HH_k(\mathbb_p/\mathbb) \cong \begin \mathbb_p & k \text \\ 0 & k \text \end but if we consider the ring structure on \begin HH_*(\mathbb_p/\mathbb) &= \mathbb_p\langle u \rangle \\ &= \mathbb_p ,u^2/2!, u^3/3!,\ldots \end (as a divided power algebra structure) then there is a significant technical issue: if we set u \in HH_2(\mathbb_p/\mathbb), so u^2 \in HH_4(\mathbb_p/\mathbb), and so on, we have u^p = 0 from the resolution of \mathbb_p as an algebra over \mathbb_p\otimes^\mathbf\mathbb_p, i.e. HH_k(\mathbb_p/\mathbb) = H_k(\mathbb_p\otimes_\mathbb_p) This calculation is further elaborated on the Hochschild homology page, but the key point is the pathological behavior of the ring structure on the Hochschild homology of \mathbb_p. In contrast, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Complex
In algebra, a perfect complex of modules over a commutative ring ''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a bounded complex of finite projective ''A''-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if ''A'' is Noetherian, a module over ''A'' is perfect if and only if it is finitely generated and of finite projective dimension. Other characterizations Perfect complexes are precisely the compact objects in the unbounded derived category D(A) of ''A''-modules. They are also precisely the dualizable objects in this category. A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect;http://www.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf see also module spectrum. Pseudo-coherent sheaf When the structure sheaf \mathcal_X is not coherent, working with coherent sheaves has awkwardness (namely the k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Cyclic Homology
Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values that repeat periodically * Periodic mapping Physical sciences * Periodic table of chemical elements * Periodic trends, relative characteristics of chemical elements observed * Redshift periodicity, astronomical term for redshift quantization Other uses * Fokker periodicity blocks, which mathematically relate musical intervals * Periodic acid, a compound of iodine * Principle of periodicity, a concept in generally accepted accounting principles * Quasiperiodicity, property of a system that displays irregular periodicity See also * Aperiodic (other) * Cycle (other) * Frequency (other) * Period (other) Period may refer to: Common uses * Era, a length or span of time * Full stop (or period ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hochschild Homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by . Definition of Hochschild homology of algebras Let ''k'' be a field, ''A'' an associative ''k''-algebra, and ''M'' an ''A''-bimodule. The enveloping algebra of ''A'' is the tensor product A^e=A\otimes A^o of ''A'' with its opposite algebra. Bimodules over ''A'' are essentially the same as modules over the enveloping algebra of ''A'', so in particular ''A'' and ''M'' can be considered as ''Ae''-modules. defined the Hochschild homology and cohomology group of ''A'' with coefficients in ''M'' in terms of the Tor functor and Ext functor by : HH_n(A,M) = \operatorname_n^(A, M) : HH^n(A,M) = \operatorname^n_(A, M) Hochschild complex Let ''k'' be a ring, ''A'' an associative ''k''-algebra th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dg Category
In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded \Z-module. In detail, this means that \operatorname(A,B), the morphisms from any object ''A'' to another object ''B'' of the category is a direct sum :\bigoplus_\operatorname_n(A,B) and there is a differential ''d'' on this graded group, i.e., for each ''n'' there is a linear map :d\colon \operatorname_n(A,B) \rightarrow \operatorname_(A,B), which has to satisfy d \circ d = 0. This is equivalent to saying that \operatorname(A,B) is a cochain complex. Furthermore, the composition of morphisms \operatorname(A,B) \otimes \operatorname(B,C) \rightarrow \operatorname(A,C) is required to be a map of complexes, and for all objects ''A'' of the category, one requires d(\operatorname_A) = 0. Examples * Any additive category may be considered to be a DG-category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartier Isomorphism
In algebraic geometry, the Cartier isomorphism is a certain isomorphism between the cohomology sheaves of the de Rham complex of a smooth algebraic variety over a field of positive characteristic, and the sheaves of differential forms on the Frobenius twist of the variety. It is named after Pierre Cartier. Intuitively, it shows that de Rham cohomology in positive characteristic is a much larger object than one might expect. It plays an important role in the approach of Deligne and Illusie to the degeneration of the Hodge–de Rham spectral sequence. Statement Let ''k'' be a field of characteristic ''p'' > 0, and let f: X \to S be a morphism of ''k''-schemes. Let X^ = X \times_ S denote the Frobenius twist and let F: X \to X^ be the relative Frobenius. The Cartier map is defined to be the unique morphismC^: \bigoplus_ \Omega^i_ \to \bigoplus_ \mathcal^i(F_* \Omega^_)of graded \mathcal_-algebras such that C^(d(x \otimes 1)) = ^ dx/math> for any local section ''x'' of \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of order p is the ring of p-adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers. The main idea behind Witt vectors is instead of using the standard p-adic expansiona = a_0+a_1p+a_2p^2 + \cdotsto represent an element in \mathbb_p, we can instead consider an expansion using the Teichmüller character\omega: \mathbb_p^* \to \mathbb_p^*which sends each element in the solution set of x^-1 in \mathbb_p to an element in the solution set of x^-1 in \mathbb_p. That is, we expand out elements in \mathbb_p in terms of roots of unity instead of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
