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 The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors.
the direct image f ∗ {\displaystyle f_{*}}
the inverse image f ∗ {\displaystyle f^{*}}
the proper (or extraordinary) direct image f ! {\displaystyle f_{!}}
the proper (or extraordinary) inverse image f ! {\displaystyle f^{!}}
internal tensor product internal Hom The functors f ∗ {\displaystyle f^{*}} and f ∗ {\displaystyle f_{*}} form an adjoint functor pair, as do f ! {\displaystyle f_{!}} and f ! {\displaystyle f^{!}} . Similarly, internal tensor product is left adjoint to internal Hom.
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 ∗ {\displaystyle f^{*}} and f ∗ {\displaystyle f_{*}} between the categories of sheaves on X and Y, and it gives the functor f ! {\displaystyle f_{!}} of direct image with proper support. In the derived category, Rf! admits a right adjoint f ! {\displaystyle 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 ℓ {\displaystyle \ell } -adic torsion sheaves, where ℓ {\displaystyle \ell } 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 L f ∗ {\displaystyle Lf^{*}} is isomorphic to f!(−d)[−2d], where (−d) denotes 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:
L g ∗ ∘ R f ! → R f ! ′ ∘ L g ′ ∗ , {\displaystyle Lg^{*}\circ Rf_{!}\to Rf'_{!}\circ Lg'^{*},}
R g ∗ ′ ∘ f ′ ! → f ! ∘ R g ∗ . {\displaystyle Rg'_{*}\circ f'^{!}\to f^{!}\circ Rg_{*}.}
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:
( R f ! M ) ⊗ Y N → R f ! ( M ⊗ X L f ∗ N ) , {\displaystyle (Rf_{!}M)\otimes _{Y}N\to Rf_{!}(M\otimes _{X}Lf^{*}N),}
RHom Y ( R f ! M , N ) → R f ∗ RHom X ( M , f ! N ) , {\displaystyle \operatorname {RHom} _{Y}(Rf_{!}M,N)\to Rf_{*}\operatorname {RHom} _{X}(M,f^{!}N),}
… excerpt ends here. Continue reading the full article.
