nLab
Nisnevich site

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 $(\mathrm{\infty },1)$-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 $V_i$.

Related concepts

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

• Fabien Morel, Vladimir Voevodsky, ${𝔸}^1$-homotopy theory of schemes , K-theory, 0305 (web pdf)

or in the lecture notes

• Eric Friedlander, Algebraic Cycles and algebraic K-theory, II (lecture 6 (pdf))