cohomology

# Contents

## Idea

### General

What is known as (Grothendieck’s) six operations is a formalization of structure that

These are four operations and together with

form six operations.

(All this is usually interpreted as derived functors/(infinity,1)-functors so that for instance in the usual application to derived categories of abelian sheaves the last two operations are really Tor and Ext.)

With a list of compatibility conditions between these (for instance (Cisinski-Déglise 09, p. x) this is a structure of Grothendieck’s six operations.

These consistency conditions include the following:

1. The adjunctions are functorial, hence form 2-functors $f \mapsto f_*$, $f \mapsto f_!$:

2. There is a natural transformation $f_! \to f_*$ which is a natural equivalence when $f$ is a proper map.

3. Beck-Chevalley condition: given a (homotopy) pullback diagram

$\array{ && Q_1 \underset{X}{\times} Q_2 \\ & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ Q_1 && && Q_2 \\ & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{g}} \\ && X }$

such that $f$ is a separated morphism, then there are natural equivalences

$g^\ast \circ f_! \simeq (p_2)_! \circ (p_1)^\ast$
$f^! \circ g_\ast \simeq (p_1)_\ast\circ (p_2)^! \,.$

Morover one imposes a formalization of Verdier duality with dualizing object

### Specializations

Often specializations of the general concept play a role:

• Wirthmüller context: $f^! \simeq f^\ast$ and $f^\ast$ is a strong closed monoidal functor

• Verdier-Grothendieck context: the “projection formula” $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$

• Grothendieck context: $f_! \simeq f_\ast$ and the projection formula holds $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$.

## Properties

### Relation to motivic homotopy theory

The initial object in the (infinity,2)-category of functors to stable (infinity,1)-categories which satisfy the six operations formalism (and a bit more, such that A1-homotopy invariance) is stable motivic homotopy theory. See there for more.

## References

General abstract discussion is in

• H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (TAC, pdf)

• Roy Joshua, Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories (pdf)

The traditional applications are discussed in

• Joseph Lipman, Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Math., no. 1960, Springer-Verlag, New York, 2009, 1–259. (pdf)

• 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 9, 2013 01:13:39 by Urs Schreiber (82.113.121.146)