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In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold.

Attributes	Values
rdf:type	organisation
rdfs:label	Local system (en) 局部系統 (zh)
rdfs:comment	In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. (en) 在數學中，局部系統或稱局部係數是源於代數拓撲的一種觀念，它是常係數的同調或上同調理論的推廣。這個觀念也能應用於代數幾何。用層論的語言來講，局部系統是局部上同構於的阿貝爾群層。若此層整體來看也同構於常數層，則就回到了傳統的常係數理論。例子包括了帶有的向量叢，基本群的線性表示則給出了局部同構於向量空間常數層的局部系統。 (zh)
dcterms:subject	Algebraic topology Sheaf theory
Wikipage page ID	3461103 (xsd:integer)
Wikipage revision ID	1112260682 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic topology Homology theory D-module Intersection homology Mathematics Gauss–Manin connection Fundamental groups Module over a ring Monodromy Connected space Constant sheaf Constructible sheaf Locally simply connected Stack Exchange Leray spectral sequence Covariant functor Locally constant sheaf Perverse sheaf Sheaf of modules Poincaré duality Algebraic topology Curvature Norman Steenrod Abelian group Sheaf theory Hodge structure Twisted Poincaré duality Group representation D-modules Sheaf cohomology Topological space Universal cover Orientation covering
Link from a Wikipage to an external page	https://webusers.imj-prg.fr/~fouad.elzein/elzein-snoussif.pdf%7Ctitle= http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf%7Ctitle= https://www.math.stonybrook.edu/~cschnell/pdf/notes/locsys.pdf%7Ctitle= http://faculty.tcu.edu/gfriedman/notes/ih.pdf https://math.stackexchange.com/q/13332%7Ctitle=What
sameAs	Local system Local system Local system Local system
dbp:wikiPageUsesTemplate	dbt:Cite_web dbt:Clarify dbt:More_citations_needed dbt:Reflist dbt:Short_description dbt:Math_proof
proof	Take local system and a loop at x. It's easy to show that any local system on is constant. For instance, is constant. This gives an isomorphism , i.e. between L and itself. Conversely, given a homomorphism , consider the constant sheaf on the universal cover of X. The deck-transform-invariant sections of gives a local system on X. Similarly, the deck-transform-ρ-equivariant sections give another local system on X: for a small enough open set U, it is defined as : where is the universal covering. (en)
title	Proof of equivalence (en)
has abstract	In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. (en) 在數學中，局部系統或稱局部係數是源於代數拓撲的一種觀念，它是常係數的同調或上同調理論的推廣。這個觀念也能應用於代數幾何。用層論的語言來講，局部系統是局部上同構於的阿貝爾群層。若此層整體來看也同構於常數層，則就回到了傳統的常係數理論。例子包括了帶有的向量叢，基本群的線性表示則給出了局部同構於向量空間常數層的局部系統。 (zh)
gold:hypernym	Idea
prov:wasDerivedFrom	wikipedia-en:Local_system?oldid=1112260682&ns=0
page length (characters) of wiki page	11736 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Local_system
is Link from a Wikipage to another Wikipage of	Decomposition theorem of Beilinson, Bernstein and Deligne Orientation character Fundamental groupoid Glossary of algebraic topology Constructible sheaf Deligne–Lusztig theory Fundamental group Perverse sheaf Riemann–Hilbert correspondence Base change theorems Group cohomology Twisted Poincaré duality Flat vector bundle ∞-groupoid Homology with local coefficients Local coefficient system Local coefficients
is Wikipage redirect of	Homology with local coefficients Local coefficient system Local coefficients
is foaf:primaryTopic of	wikipedia-en:Local_system

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