In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra. 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.
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The operations
The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors.
The functors
Six operations in étale cohomology
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
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: