Six operations

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 . 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.
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 formula_2 and formula_1 form an adjoint functor pair, as do formula_3 and formula_4. Similarly, internal tensor product is left adjoint to internal Hom.
Six operations in étale cohomology.
Let 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: , , , , , and .
Suppose that we restrict ourselves to a category of formula_9-adic torsion sheaves, where formula_9 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 , where denote the "d"th inverse Tate twist and denotes a shift in degree by . Furthermore, suppose that "f" is separated and of finite type. If is another morphism of schemes, if 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 and are the units of the tensor product operations (which vary depending on which category of formula_9-adic torsion sheaves is under consideration).
If "S" is regular and , and if "K" is an invertible object in the derived category on "S" with respect to , then define "D""X" to be the functor . Then, for objects "M" and "M"′ in the derived category on "X", the canonical maps:
are isomorphisms. Finally, if 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: