# Contents

## Idea

The Nisnevich site is a certain Grothendieck topology on a category of schemes.

The (∞,1)-topos of ∞-stacks on $\mathrm{Nis}$ (specifically: its homotopy stabilization: A^1 homotopy theory) is the $\left(\infty ,1\right)$-topos whose cohomology theory is motivic cohomology.

## Definition

The Nisnevich site relative to a scheme $S$ is the site given by

• the category of smooth schemes of finite type over $S$;

• equipped with the Grothendieck topology whose covering families for an $S$-scheme $U$ are finite families of etale maps ${V}_{i}\to U$ that form a cover in the etale site and such that for all fields $k$ every morphism $\mathrm{Spec}k\to U$ lifts to one of the ${V}_{i}$.

fpqc-site $\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site

## References

A quick overview is at the beginning of the talk slides

• Jardine, Motivic spaces and the motivic stable category (pdf) .

A detailed discussion is in section 3.1.1 of

or in the lecture notes

Revised on September 5, 2011 09:40:38 by Urs Schreiber (89.204.153.80)