In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism.
[1] It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f : X → Y.
The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms.
These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold.
The six operations formalism has since been shown to apply to contexts such as D-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives.
form an adjoint functor pair, as do
Finally, when working with abelian sheaves, there is a tensor product functor ⊗ and an internal Hom functor, and these are adjoint.
In SGA 4 III, Grothendieck and Artin proved that if f is smooth of relative dimension d, then
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
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: