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Bockstein Spectral Sequence
In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''. It is named after Meyer Bockstein. Definition Let ''C'' be a chain complex of torsion-free abelian groups and ''p'' a prime number. Then we have the exact sequence: :0 \longrightarrow C \overset\longrightarrow C \overset \longrightarrow C \otimes \Z/p \longrightarrow 0. Taking integral homology ''H'', we get the exact couple of "doubly graded" abelian groups: :H_*(C) \overset \longrightarrow H_*(C) \overset \longrightarrow H_*(C \otimes \Z/p) \overset \longrightarrow. where the grading goes: H_*(C)_ = H_(C) and the same for H_*(C \otimes \Z/p),\deg i = (1, -1), \deg j = (0, 0), \deg k = (-1, 0). This gives the first page of the spectral sequence: we take E_^1 = H_(C \otimes \Z/p) with the differential ^1 d = j \circ k. The derived couple of the above exact couple then gives the second page and so forth. Expli ... [...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] |
Meyer Bockstein
Meyer Bockstein (also Меер Феликсович Бокштейн or Meer Feliksovich Bokshtein or Bokstein) (4 October 1913 to 2 May 1990) was a topologist from Moscow who introduced the Bockstein homomorphism. The Bockstein spectral sequence In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''. It is named after Meyer Bockstein. Definition Let ''C'' be a chain complex o ... is named after him. References * * Russian Jewish Encyclopedia(Entry 791) {{DEFAULTSORT:Bockstein, Meyer F. Topologists 1913 births 1990 deaths Mathematicians from Moscow Soviet mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion-free Abelian Group
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case. Definitions An abelian group \langle G, + ,0\rangle is said to be torsion-free if no element other than the identity e is of finite order. Explicitly, for any n > 0, the only element x \in G for which nx = 0 is x = 0. A natural example of a torsion-free group is \langle \mathbb Z,+,0\rangle , as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the free abelian group \mathbb Z^r is torsion-free for any r \in \mathbb N. An important step in the proof of the classification of finitely generated abel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which alw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Couple
In mathematics, an exact couple, due to , is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple. For the definition of an exact couple and the construction of a spectral sequence from it (which is immediate), see . For a basic example, see Bockstein spectral sequence. The present article covers additional materials. Exact couple of a filtered complex Let ''R'' be a ring, which is fixed throughout the discussion. Note if ''R'' is \Z, then modules over ''R'' are the same thing as abelian groups. Each filtered chain complex of modules determines an exact couple, which in turn determines a spectral sequence, as follows. Let ''C'' be a chain complex graded by integers and suppose it is given an increasing filtration: for each integer ''p'', there is an inclusion of complexes: :F_ C \subset F_p C. From the filtration one can form the associated graded comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Couple
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] |
Universal Coefficient Theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely determine its ''homology groups with coefficients in'' , for any abelian group : : Here might be the simplicial homology, or more generally the singular homology: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients may be used, at the cost of using a Tor functor. For example it is common to take to be , so that coefficients are modulo 2. This becomes straightforward in the absence of 2- torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers of and the Betti numbers with coefficients in a field . These can differ, but only when the characteristic of is a prime numb ... [...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] |
