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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| - Direct image functor (en)
- 順像関手 (ja)
- Функтор прямого образа (ru)
- Прямий образ пучка (uk)
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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. (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)
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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. (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)
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