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In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp.

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rdf:type	yago:WikicatCohomologyTheories yago:WikicatTopologicalMethodsOfAlgebraicGeometry yago:Ability105616246 yago:Abstraction100002137 yago:Cognition100023271 yago:Explanation105793000 yago:HigherCognitiveProcess105770664 yago:Know-how105616786 yago:Method105660268 yago:Process105701363 yago:PsychologicalFeature100023100 saint yago:Theory105989479 yago:Thinking105770926
rdfs:label	Garbenkohomologie (de) Cohomologie des faisceaux (fr) 층 코호몰로지 (ko) 層係数コホモロジー (ja) Sheaf cohomology (en) Когомологии пучков (ru) Когомологія пучків (uk)
rdfs:comment	Garbenkohomologie ist in der Mathematik, hauptsächlich in der algebraischen Geometrie und in der komplexen Analysis, eine Technik, mit der man globale Eigenschaften topologischer Räume und auf ihnen definierter Garben studieren kann. Im einfachsten Fall beschreibt die erste Kohomologiegruppe die Schwierigkeiten, um aus lokalen Lösungen eine globale Lösung zu erhalten. (de) Les groupes de cohomologie d'un faisceau de groupes abéliens sont les groupes de cohomologie du complexe de cochaines. (fr) 수학에서 층 코호몰로지(層 cohomology, 영어: sheaf cohomology)는 아벨 군 값을 가진 층에 정의되는 호몰로지 이론이다. 대역 단면(global section) 함자의 유도 함자이다. 체흐 코호몰로지보다 더 추상적이지만, 대수기하학에서 다루기 더 편하다. (ko) 数学において、層コホモロジー (sheaf cohomology) は、アーベル群の層に関連する層の理論の一面であり、ホモロジー代数を用いて、層 F の大域切断の具体的な計算を可能とする。数値的な領域での幾何学的な問題の記述として、層コホモロジーの理論は、重要な幾何学的な不変量の次元を計算することへ有用なツールとして使うことができる。 1950年以後の数年間で急速に発展した層コホモロジーは、リーマン・ロッホの定理のより古典的な方法や代数幾何学の(linear system of divisors)の解析や多変数複素函数論やホッジ理論へ結びついた。層コホモロジー群のランク、もしくは次元は、幾何学的なデータの新しい情報源になったり以前の研究の新しい解釈を与えたりする。 (ja) In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. (en) Когомологии пучков — это результат использования гомологической алгебры для исследования глобальных сечений пучков. Грубо говоря, когомологии пучков описывают препятствия к глобальному решению геометрической проблемы, когда она может быть решена локально. (ru) Когомологія пучків — результат використання гомологічної алгебри для дослідження глобальних перетинів пучків. Грубо кажучи, когомології пучків описують перешкоди до глобального вирішення геометричної проблеми, коли вона може бути вирішена локально. (uk)
dcterms:subject	Cohomology theories Topological methods of algebraic geometry Homological algebra Sheaf theory
Wikipage page ID	1055357 (xsd:integer)
Wikipage revision ID	1096656657 (xsd:integer)
Link from a Wikipage to another Wikipage	Principal ideal domain Quasi-coherent Roger Godement Epimorphism Monomorphism Non-abelian cohomology Algebraic topology Holomorphic vector bundle Cup product Unit (ring theory) Vector space De Rham's theorem Derived category Derived functor Injective sheaf John Wiley & Sons Quasi-isomorphism Ringed space Perfect pairing Compact space Complex analytic space Connected component (topology) Continuous function Continuous map Mathematics Mayer–Vietoris sequence Alexander duality Cohomology theories Function (mathematics) Graded-commutative Connected space Constant sheaf Constructible sheaf Crystalline cohomology Homotopic Homotopy equivalent Orientation sheaf Topological methods of algebraic geometry Left exact functor Locally compact Locally contractible Long exact sequence Short exact sequence Smooth function Stack Exchange Closed manifold Étale cohomology Functor Leray spectral sequence Picard group Stalk (sheaf) MathOverflow Topologically stratified space Weil conjectures Coherent sheaves Countable Hausdorff space Local cohomology Sheaf of modules Nisnevich topology Poincaré duality Alexander Grothendieck Algebraic Geometry (book) Algebraic geometry Algebraic variety Dual space

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