Let f : X → Y be a morphism of schemes. The morphism f induces several functors. Specifically, it gives adjoint functors f* and f* between the categories of sheaves on X and Y, and it gives the functor f! of direct image with proper support. In the derived category, Rf! admits a right adjoint f!. Finally, when working with abelian sheaves, there is a tensor product functor ⊗ and an internal Hom functor, and these are adjoint. The six operations are the corresponding functors on the derived category: Lf*, Rf*, Rf!, f!, ⊗L, and RHom.
Suppose that we restrict ourselves to a category of -adic torsion sheaves, where is coprime to the characteristic of X and of Y. In SGA 4 III, Grothendieck and Artin proved that if f is smooth of relative dimension d, then Lf* is isomorphic to f!(−d)[−2d], where (−d) denote the dth inverse Tate twist and [−2d] denotes a shift in degree by −2d. Furthermore, suppose that f is separated and of finite type. If g : Y′ → Y is another morphism of schemes, if X′ denotes the base change of X by g, and if f′ and g′ denote the base changes of f and g by g and f, respectively, then there exist natural isomorphisms:
Again assuming that f is separated and of finite type, for any objects M in the derived category of X and N in the derived category of Y, there exist natural isomorphisms:
If i is a closed immersion of Z into S with complementary open immersion j, then there is a distinguished triangle in the derived category:
where the first two maps are the counit and unit, respectively of the adjunctions. If Z and S are regular, then there is an isomorphism:
where 1Z and 1S are the units of the tensor product operations (which vary depending on which category of -adic torsion sheaves is under consideration).
If S is regular and g : X → S, and if K is an invertible object in the derived category on S with respect to ⊗L, then define DX to be the functor RHom(—, g!K). Then, for objects M and M′ in the derived category on X, the canonical maps:
are isomorphisms. Finally, if f : X → Y is a morphism of S-schemes, and if M and N are objects in the derived categories of X and Y, then there are natural isomorphisms: