nLab
six operations

Idea

For $f:X\to Y$ a morphism of sites such that the corresponding derived direct image $f_{*}$ and inverse image functor $f^{*}$ on the derived category of abelian sheaves are related to a further adjoint functor pair $(f_{!}⊣f^{!})$.

These together with the internal Ext and Tor functors are often called Grothendieck’s six operations. There is a rich calculus induced by this structure.

Related concepts

• Grothendieck duality

References

A general abstract discussion is in

• H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints (pdf)

The traditional applications are discussed in

• Yves Laszlo, Martin Olsson, The six operations for sheaves on Artin stacks I: Finite Coefficients (arXiv:math/0512097)

• Yoseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescants dans le monde motivique PhD thesis, Paris (pdf)

A quick list of the axioms with a Grothendieck’s six operations with an eye towards the definition of motives is in section A.5 of

• Denis-Charles Cisinski, Frédéric Déglise, Triangulated categories of mixed motives (arXiv:0912.2110)