Image functors for sheaves
In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses.
| Image functors for sheaves |
|---|
| direct image f∗ |
| inverse image f∗ |
| direct image with compact support f! |
| exceptional inverse image Rf! |
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| Base change theorems |
Given a continuous mapping f: X → Y of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topological space. The functors in question are
- direct image f∗ : Sh(X) → Sh(Y)
- inverse image f∗ : Sh(Y) → Sh(X)
- direct image with compact support f! : Sh(X) → Sh(Y)
- exceptional inverse image Rf! : D(Sh(Y)) → D(Sh(X)).
The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "f shriek" or "f lower shriek" and "f upper shriek"—see also shriek map.
The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes.
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