Continuous mappings | Sheaf theory

Direct image functor

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. (Wikipedia).

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Related pages

Topological space | Sheaf cohomology | Étale topology | Topology | Inverse image functor | Topos | Quasi-separated morphism | Ringed space | Mathematics | Proper morphism | Equivalence of categories | Algebraic geometry | Category (mathematics) | Derived functor | Functor | Quasi-compact morphism | Scheme (mathematics) | Exceptional inverse image functor | Abelian group | Exact functor