About: Direct image functor

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In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F. This assignment gives rise to a functor f∗ from the category of sheaves on X to the category of sheaves on Y, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme.

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rdf:type	Thing yago:WikicatContinuousMappings yago:Abstraction100002137 yago:Function113783816 yago:MathematicalRelation113783581 yago:Relation100031921
rdfs:label	Direct image functor (en) 順像関手 (ja) Функтор прямого образа (ru) Прямий образ пучка (uk)
rdfs:comment	In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F. This assignment gives rise to a functor f∗ from the category of sheaves on X to the category of sheaves on Y, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme. (en) 数学の層論や代数幾何学の分野に現れる順像関手（じゅんぞうかんしゅ、英: direct image functor）とは、層の切断の概念を相対的な場合へ一般化するものである。 (ja) Функтор прямого образа — это обобщение понятия сечения пучка на относительный случай. (ru) Прямий образ пучка — узагальнення поняття перетину пучка на відносний випадок. (uk)
rdfs:seeAlso	Coherent duality
dcterms:subject	Continuous mappings Sheaf theory
Wikipage page ID	3022744 (xsd:integer)
Wikipage revision ID	1111343223 (xsd:integer)
Link from a Wikipage to another Wikipage	Morphism of sheaves Restriction of scalars Derived functor Inverse image functor Ringed space Continuous map Mathematics Quasi-separated morphism Equivalence of categories Functor Topology Adjoint functor Algebraic geometry Quasi-compact morphism Proper morphism Abelian group Continuous mappings Sheaf theory Topos Category (mathematics) Scheme (mathematics) Étale topology Exact functor Exceptional inverse image functor Sheaf cohomology Proper base change theorem Topological space Springer-Verlag Fiber product Quasi-coherent sheaves Derived categories Global sections functor Section of a sheaf Sheaf theory Étale sheaf
sameAs	Direct image functor Direct image functor Direct image functor Direct image functor Direct image functor Direct image functor Direct image functor
dbp:wikiPageUsesTemplate	dbt:Citation dbt:See_also dbt:Images_of_sheaves
has abstract	In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F. This assignment gives rise to a functor f∗ from the category of sheaves on X to the category of sheaves on Y, which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme. (en) 数学の層論や代数幾何学の分野に現れる順像関手（じゅんぞうかんしゅ、英: direct image functor）とは、層の切断の概念を相対的な場合へ一般化するものである。 (ja) Функтор прямого образа — это обобщение понятия сечения пучка на относительный случай. (ru) Прямий образ пучка — узагальнення поняття перетину пучка на відносний випадок. (uk)
prov:wasDerivedFrom	wikipedia-en:Direct_image_functor?oldid=1111343223&ns=0
page length (characters) of wiki page	6573 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Direct_image_functor
is Link from a Wikipage to another Wikipage of	Derived category Dévissage Inverse image functor Ringed space Transfer operator Adjoint functors Base change theorems Direct image with compact support Fourier–Mukai transform Grothendieck–Riemann–Roch theorem Sheaf (mathematics) Image functors for sheaves Six operations Direct image Direct image (functor) Direct image of a sheaf Direct image sheaf Higher direct image Higher direct image functor
is Wikipage redirect of	Direct image Direct image (functor) Direct image of a sheaf Direct image sheaf Higher direct image Higher direct image functor
is foaf:primaryTopic of	wikipedia-en:Direct_image_functor

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