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Zeeman's Comparison Theorem
In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman (), gives conditions for a morphism of spectral sequences to be an isomorphism. Statement Illustrative example As an illustration, we sketch the proof of Borel's theorem In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. See also *Atiyah–Bott formula In algebraic geometry, the Atiyah–Bott formula says ..., which says the cohomology ring of a classifying space is a polynomial ring. First of all, with ''G'' as a Lie group and with \mathbb as coefficient ring, we have the Serre spectral sequence E_2^ for the fibration G \to EG \to BG. We have: E_ \simeq \mathbb since ''EG'' is contractible. We also have a theorem of Hopf stating that H^*(G; \mathbb) \simeq \Lambda(u_1, \dots, u_n), an exterior algebra generated by finitely many homogeneous elements. Next, we let E(i) be the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has played a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Zeeman
Sir Erik Christopher Zeeman FRS (4 February 1925 – 13 February 2016), was a British mathematician, known for his work in geometric topology and singularity theory. Overview Zeeman's main contributions to mathematics were in topology, particularly in knot theory, the piecewise linear category, and dynamical systems. His 1955 thesis at the University of Cambridge described a new theory termed "dihomology", an algebraic structure associated to a topological space, containing both homology and cohomology, introducing what is now known as the Zeeman spectral sequence. This was studied by Clint McCrory in his 1972 Brandeis thesis following a suggestion of Dennis Sullivan that one make "a general study of the Zeeman spectral sequence to see how singularities in a space perturb Poincaré duality". This in turn led to the discovery of intersection homology by Robert MacPherson and Mark Goresky at Brown University where McCrory was appointed in 1974. From 1976 to 1977 he was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as wel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. This was still not the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Module
In algebra, a flat module over a ring ''R'' is an ''R''- module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper ''Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel's Theorem
In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. See also *Atiyah–Bott formula In algebraic geometry, the Atiyah–Bott formula says the cohomology ring :\operatorname^*(\operatorname_G(X), \mathbb_l) of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The origi ... Notes References * * {{topology-stub Theorems in algebraic topology Theorems in algebraic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Theorem Of Hopf
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey ɑ. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, " a", and its variant " an", are indefinite articles. History The earliest certain ancestor of "A" is aleph (also written 'aleph), the first letter of the Phoenician alphabet, which consisted entirely of consonants (for that reason, it is also called an abjad to distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meanin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserving: ::f(a+b)=f(a)+f(b) for all ''a'' and ''b'' in ''R'', :multiplication preserving: ::f(ab)=f(a)f(b) for all ''a'' and ''b'' in ''R'', :and unit (multiplicative identity) preserving: ::f(1_R)=1_S. Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. If in addition ''f'' is a bijection, then its inverse ''f''−1 is also a ring homomorphism. In this case, ''f'' is called a ring isomorphism, and the rings ''R'' and ''S'' are called ''isomorphic''. From the standpoint of ring theory, isomorphic rings cannot be distinguished. If ''R'' and ''S'' are rngs, then the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Sequences
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Discovery and motivation Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology. To compute sheaf cohomology, Leray introduced a computational technique now known as the Leray spectral sequence. This gave a relation between cohomology groups of a sheaf and cohomology groups of the pushforward of the sheaf. The relation involved an infinite process. Leray found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. This was still not the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
