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Quillen–Lichtenbaum Conjecture
In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by , who was inspired by earlier conjectures of . and proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of Markus Rost, proved the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes. Statement The conjecture in Quillen's original form states that if ''A'' is a finitely-generated algebra over the integers and ''l'' is prime, then there is a spectral sequence analogous to the Atiyah–Hirzebruch spectral sequence, starting at :E_2^ = H^p_(\textA ell^ Z_\ell(-q/2)), (which is understood to be 0 if ''q'' is odd) and abutting to :K_A\otimes Z_\ell for −''p'' − ''q'' > 1 + dim ''A''. ''K''-theory of the integers Assuming the Quillen–Lichtenbaum conjecture and the Vandiver conjecture, the ''K''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Hirzebruch Spectral Sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, it relates the generalized cohomology groups : E^i(X) with 'ordinary' cohomology groups H^j with coefficients in the generalized cohomology of a point. More precisely, the E_2 term of the spectral sequence is H^p(X;E^q(pt)), and the spectral sequence converges conditionally to E^(X). Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where E=H_. It can be derived from an exact couple that gives the E_1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E. In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
