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Exceptional Inverse Image Functor
In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form. Definition Let ''f'': ''X'' → ''Y'' be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor :R''f''!: D(''Y'') → D(''X'') where D(–) denotes the derived category of sheaves of abelian groups or modules over a fixed ring. It is defined to be the right adjoint of the total derived functor R''f''! of the direct image with compact support. Its existence follows from certain properties of R''f''! and general theorems about existence of adjoint functors, as does the unicity. The notation R''f''! is an abuse of notation insofar as there is in general no functor ''f''! whose derived functor would be R''f''!. Examples and properties *If ''f'': ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
