Sheaf (mathematics)
Extension by zero

For the inclusion j: U → X {\displaystyle j:U\to X} of an open subset, the extension by zero j ! F {\displaystyle j_{!}{\mathcal {F}}} (pronounced "j lower shriek of F") of a sheaf F {\displaystyle {\mathcal {F}}} of abelian groups on U {\displaystyle U} is the sheafification of the presheaf defined by

For a sheaf G {\displaystyle {\mathcal {G}}} on X {\displaystyle X}, this construction is in a sense complementary to i ∗ {\displaystyle i_{*}}, where i: X ∖ U → X {\displaystyle i:X\setminus U\to X} is the inclusion of the complement of U {\displaystyle U}:

More generally, if A ⊂ X {\displaystyle A\subset X} is a locally closed subset, then there exists an open U {\displaystyle U} of X {\displaystyle X} containing A {\displaystyle A} such that A {\displaystyle A} is closed in U {\displaystyle U}. Let f: A → U {\displaystyle f:A\to U} and j: U → X {\displaystyle j:U\to X} be the natural inclusions. Then the extension by zero of a sheaf F {\displaystyle {\mathcal {F}}} on A {\displaystyle A} is defined by j ! f ∗ F {\displaystyle j_{!}f_{*}F}.

Due to its nice behavior on stalks, the extension by zero functor is useful for reducing sheaf-theoretic questions on X {\displaystyle X} to ones on the strata of a stratification, i.e., a decomposition of X {\displaystyle X} into smaller, locally closed subsets.
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