# Contents

## 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 $\left({f}_{!}⊣{f}^{!}\right)$.

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.

## 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

Revised on December 20, 2012 22:29:02 by Urs Schreiber (131.174.40.34)