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^{!} 

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.
Adjointness
The functors are adjoint to each other as depicted at the right, where, as usual, means that F is left adjoint to G (equivalently G right adjoint to F), i.e.
 Hom(F(A), B) ≅ Hom(A, G(B))
for any two objects A, B in the two categories being adjoint by F and G.
For example, f^{∗} is the left adjoint of f_{*}. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms and for on Y and on X, respectively. However, these are almost never isomorphisms—see the localization example below.
Verdier duality
Verdier duality gives another link between them: morally speaking, it exchanges "∗" and "!", i.e. in the synopsis above it exchanges functors along the diagonals. For example the direct image is dual to the direct image with compact support. This phenomenon is studied and used in the theory of perverse sheaves.
Base Change
Another useful property of the image functors is base change. Given continuous maps and , which induce morphisms and , there exists a canonical isomorphism .
Localization
In the particular situation of a closed subspace i: Z ⊂ X and the complementary open subset j: U ⊂ X, the situation simplifies insofar that for j^{∗}=j^{!} and i_{!}=i_{∗} and for any sheaf F on X, one gets exact sequences
 0 → j_{!}j^{∗} F → F → i_{∗}i^{∗} F → 0
Its Verdier dual reads
 i_{∗}Ri^{!} F → F → Rj_{∗}j^{∗} F → i_{∗}Ri^{!} F[1],
a distinguished triangle in the derived category of sheaves on X.
The adjointness relations read in this case
and
 .
References
 Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: SpringerVerlag, ISBN 9783540163893, MR 0842190 treats the topological setting
 Artin, Michael (1972). Alexandre Grothendieck; JeanLouis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie  196364  Théorie des topos et cohomologie étale des schémas  (SGA 4)  vol. 3. Lecture notes in mathematics (in French). 305. Berlin; New York: SpringerVerlag. pp. vi+640. doi:10.1007/BFb0070714. ISBN 9783540061182. treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.
 Milne, James S. (1980), Étale cohomology, Princeton University Press, ISBN 9780691082387 is another reference for the étale case.