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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 Jean Leray , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.

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rdf:type	software yago:WikicatSpectralSequences yago:Abstraction100002137 yago:Arrangement107938773 yago:Group100031264 yago:Ordering108456993 yago:Sequence108459252 yago:Series108457976
rdfs:label	Spektralsequenz (de) Suite spectrale (fr) Successione spettrale (it) スペクトル系列 (ja) 스펙트럼 열 (ko) Spectral sequence (en) Спектральная последовательность (ru) 譜序列 (zh)
rdfs:comment	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 Jean Leray , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. (en) In algebra omologica, topologia algebrica e geometria algebrica, una successione spettrale è un modo di calcolare i gruppi di omologia considerandone approssimazioni successive. Le successioni spettrali sono una generalizzazione delle successioni esatte. A partire dalla loro introduzione da parte di Jean Leray nel 1946 sono diventate degli importanti strumenti computazionali. (it) スペクトル系列（スペクトルけいれつ、英: Spectral sequence）とは、ホモロジー代数学や代数的位相幾何学で用いられる、ホモロジー群を逐次近似により計算する方法のことである。スペクトル系列は完全系列の一般化であり、ジャン・ルレイによって初めて用いられたときから、特に代数的位相幾何学、代数幾何学、ホモロジー代数学といった分野において重要な計算ツールとなっている。 (ja) 호몰로지 대수학에서 스펙트럼 열(spectrum列, 영어: spectral sequence)은 어떤 호몰로지 또는 코호몰로지에 대한 일련의 근사들을 나타내는 수학적 대상이다. (ko) В гомологической алгебре и алгебраической топологии спектральная последовательность — это средство вычисления групп гомологий путём последовательных приближений. С момента их введения Жаном Лере они стали важным вычислительным средством, особенно в алгебраической топологии, алгебраической геометрии и гомологической алгебре. (ru) 在同調代數中，譜序列是一種藉著逐步逼近以計算同調或上同調群的技術，由讓·勒雷在1946年首創。其應用見諸代數拓撲、群上同調與同倫理論。 (zh) Eine Spektralsequenz oder Spektralfolge ist ein Grenzwertprozess zur Berechnung von Homologiegruppen im mathematischen Teilgebiet der homologischen Algebra.Nach J. F. Adams sind Spektralsequenzen wie exakte Sequenzen, nur komplizierter. Wie für exakte Sequenzen gelte auch für Spektralsequenzen: sie bieten keine Erfolgsgarantie, sind aber trotzdem in den Händen der Fachleute häufig ein effektives Werkzeug. (de) En algèbre homologique et en topologie algébrique, une suite spectrale est une suite de modules différentiels (En,dn) tels que En+1 = H(En) = Ker dn / Im dn est l'homologie de En. Elles permettent donc de calculer des groupes d'homologie par approximations successives. Elles ont été introduites par Jean Leray en 1946. (fr)
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dct:subject	Spectral sequences
Wikipage page ID	803894 (xsd:integer)
Wikipage revision ID	1099971977 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Module (mathematics) Algebraic topology Annals of Mathematics Cup product Universal coefficient theorem Derived category Derived functor EHP spectral sequence Notices of the American Mathematical Society Pullback Quillen spectral sequence Arnold's spectral sequence Long exact sequence Macaulay2 Bockstein spectral sequence Singularity theory Snake lemma Stable homotopy groups of spheres Comptes rendus de l'Académie des Sciences Frölicher spectral sequence Kernel (algebra) Koszul cohomology Leray spectral sequence MathOverflow May spectral sequence Adams spectral sequence Spectral sequences Dolbeault cohomology K-theory Additive relation Algebraic geometry Extraordinary cohomology theory Paper Chromatic spectral sequence Gysin sequence Ring (mathematics) Group cohomology Atiyah–Hirzebruch spectral sequence Čech-to-derived functor spectral sequence Stable homotopy theory Abelian category Abelian group Cohomology ring Eilenberg–Moore spectral sequence Hodge–de Rham spectral sequence Homological algebra Homology (mathematics) Tor functor Transgression map Differential graded algebra Spectral sequence Fibration Grothendieck spectral sequence Chain complex Sheaf (mathematics) Čech cohomology Image (mathematics) Lyndon–Hochschild–Serre spectral sequence Exact sequence Five-term exact sequence Sheaf cohomology Serre spectral sequence Künneth spectral sequence Les Comptes rendus de l'Académie des sciences Filtration (abstract algebra) Direct image of a sheaf Hyperhomology spectral sequence William Schumacher Massey Adams–Novikov spectral sequence

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