Nisnevich site

The *Nisnevich topology* is a certain Grothendieck topology on the category of schemes which is finer than the Zariski topology but coarser than the étale topology. It retains many desirable properties from both topologies:

- The Nisnevich cohomological dimension (and even the homotopy dimension) of a scheme is bounded by its Krull dimension? (like Zariski)
- Fields have trivial shape for the Nisnevich topology (like Zariski)
- Algebraic K-theory satisfies Nisnevich descent (like Zariski)
- Any smooth closed pair $(Z,X)$ (i.e., $Z$ is a closed subscheme of $X$ and $X$ and $Z$ are both smooth over a common base scheme) is Nisnevich-locally isomorphic to $(\mathbb{A}^n,\mathbb{A}^{n+k})$ (like étale)
- Pushforward along a finite morphism? is an exact functor on Nisnevich sheaves of abelian groups (like étale)
- Nisnevich cohomology can be computed using Čech cohomology (like étale)

The Nisnevich topology plays a central rôle in motivic homotopy theory.

An family of morphisms of Noetherian schemes $\{p_i:V_i\to U\}$ is a **Nisnevich cover** if each $p_i$ is an étale map and if every field-valued point $Spec k\to U$ lifts to one of the $V_i$. This is a pretopology on the category of Noetherian schemes, and the associated topology is the **Nisnevich topology**.

The **Nisnevich site** over a Noetherian scheme $S$ usually refers to the site given by the category of smooth schemes of finite type over $S$ equipped with the Nisnevich topology. The **small Nisnevich site** of $S$ is the subsite consisting of étale $S$-schemes.

For a general affine scheme $X$, one defines a sieve $S$ on $X$ to be a covering sieve for the Nisnevich topology if there exist a Noetherian affine scheme $Y$, a morphism $f: X\to Y$, and a Nisnevich covering sieve $T$ on $Y$ such that $f^\ast(T)\subset S$. On an arbitrary scheme $X$, a sieve $S$ is a Nisnevich covering sieve if there exists an open cover $\{U_i\to X\}$ by affine schemes such that $S_{/U_i}$ is a Nisnevich covering sieve on $U_i$ for all $i$.

Let $Et/S$ be the category of étale schemes of finite type over a Noetherian scheme $S$. An (∞,1)-presheaf $F$ on $Et/S$ is said to satisfy *Nisnevich excision* if the following conditions hold:

- $F(\emptyset)$ is contractible.
- If $Z$ is a closed subscheme of $X\in Et/S$ and if $X'\to X$ is a morphism in $Et/S$ which is an isomorphism over $Z$, then the square

$\array{
F(X) &\to& F(X-Z)
\\
\downarrow && \downarrow
\\
F(X') &\to& F(X'-Z)
}$

is an (∞,1)-pullback square. Intuitively, this says that the space of sections of $F$ over $X$ with support in $Z$ (i.e., the homotopy fiber of $F(X) \to F(X-Z)$) does not depend on $X$. This is Definition 2.5 in DAG XI.

An (∞,1)-presheaf on $Et/S$ is an (∞,1)-sheaf for the Nisnevich topology if and only if it satisifes Nisnevich excision.

This is Morel-Voevosky, Prop. 1.16 or DAG XI, Thm. 2.9.

If $S$ is a Noetherian scheme of finite Krull dimension?, then the (∞,1)-topos of (∞,1)-sheaves on the small Nisnevich site of $S$ has homotopy dimension $\leq\dim(S)$.

This is DAG XI, Theorem 2.24. As a consequence, Postnikov towers are convergent in the (∞,1)-topos of (∞,1)-sheaves on the Nisnevich site over $S$, and in particular that (∞,1)-topos is hypercomplete.

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

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,
*$\mathbb{A}^1$-homotopy theory of schemes*, K-theory, 0305 (web pdf)

or in the lecture notes

A self-contained account of the Nisnevich $(\infty,1)$-topos including the non-Noetherian case is in

- Jacob Lurie,
*Derived Algebraic Geometry XI: Descent Theorems*(pdf)

Revised on July 21, 2013 18:06:22
by Marc Hoyois
(92.104.194.50)