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Concept
Six operations
Summary
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. 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.
The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors.
the
the
the
the
internal tensor product
internal Hom
The functors and form an adjoint functor pair, as do and . Similarly, internal tensor product is left adjoint to internal Hom.
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 , 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.
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