Six 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 ${\displaystyle f_{*}}$
• the inverse image ${\displaystyle f^{*}}$
• the proper (or extraordinary) direct image ${\displaystyle f_{!}}$
• the proper (or extraordinary) inverse image ${\displaystyle f^{!}}$
• internal tensor product
• internal Hom

The functors ${\displaystyle f^{*}}$ and ${\displaystyle f_{*}}$ form an adjoint functor pair, as do ${\displaystyle f_{!}}$ and ${\displaystyle f^{!}}$.[2] 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 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 ${\displaystyle \ell }$-adic torsion sheaves, where ${\displaystyle \ell }$ 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 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. 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:

${\displaystyle Lg^{*}\circ Rf_{!}\to Rf'_{!}\circ Lg'^{*},}$

${\displaystyle Rg'_{*}\circ f'^{!}\to f^{!}\circ Rg_{*}.}$

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:

${\displaystyle (Rf_{!}M)\otimes _{Y}N\to Rf_{!}(M\otimes _{X}Lf^{*}N),}$

${\displaystyle \operatorname {RHom} _{Y}(Rf_{!}M,N)\to Rf_{*}\operatorname {RHom} _{X}(M,f^{!}N),}$

${\displaystyle f^{!}\operatorname {RHom} _{Y}(M,N)\to \operatorname {RHom} _{X}(Lf^{*}M,f^{!}N).}$

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:

${\displaystyle Rj_{!}j^{!}\to 1\to Ri_{*}i^{*}\to Rj_{!}j^{!}[1],}$

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:

${\displaystyle 1_{Z}(-c)[-2c]\to i^{!}1_{S},}$

where 1_Z and 1_S are the units of the tensor product operations (which vary depending on which category of ${\displaystyle \ell }$-adic torsion sheaves is under consideration).

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 D_X to be the functor RHom(—, g^!K). Then, for objects M and M′ in the derived category on X, the canonical maps:

${\displaystyle M\to D_{X}(D_{X}(M)),}$

${\displaystyle D_{X}(M\otimes D_{X}(M'))\to \operatorname {RHom} (M,M'),}$

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:

${\displaystyle D_{X}(f^{*}N)\cong f^{!}(D_{Y}(N)),}$

${\displaystyle D_{X}(f^{!}N)\cong f^{*}(D_{Y}(N)),}$

${\displaystyle D_{Y}(f_{!}M)\cong f_{*}(D_{X}(M)),}$

${\displaystyle D_{Y}(f_{*}M)\cong f_{!}(D_{X}(M)).}$

• Coherent duality
• Grothendieck local duality
• Image functors for sheaves
• Verdier duality
• Change of rings

• Laszlo, Yves; Olsson, Martin (2005). "The six operations for sheaves on Artin stacks I: Finite coefficients". arXiv:math/0512097.
• Ayoub, Joseph. Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (PDF) (Thesis).
• Cisinski, Denis-Charles; Déglise, Frédéric (2019). Triangulated categories of mixed motives. Springer Monographs in Mathematics. arXiv:0912.2110. doi:10.1007/978-3-030-33242-6. ISBN 978-3-030-33241-9. S2CID 115163824.
• Mebkhout, Zoghman (1989). Le formalisme des six opérations de Grothendieck pour les D_X-modules cohérents. Travaux en Cours. Vol. 35. Paris: Hermann. ISBN 2-7056-6049-6.

• six operations at the nLab
• What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?