nLab Nisnevich site

Idea

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 descent over the Nisnevich site – as is true for the Zariski site but not in full generality for the etale site, see at algebraic K-theory – Descent for more;

• For $Z\subset X$ a closed immersion between affine schemes that are smooth over a base $S$, the Nisnevich sheaf $X/(X-Z)$ is isomorphic to $N_{X,Z}/(N_{X,Z}-Z)$, where $N_{X,Z}$ is the normal bundle of $Z$ in $X$ (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 ?ech cohomology (like étale)

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

Definition

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 non-Noetherian 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 $i$.

As an excision property

Let $Et/S$ be the category of étale schemes of finite presentation over a quasi-compact quasi-separated 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.

Proposition

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.

Properties

Homotopy dimension

Proposition

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.

Proposition

More generally, if $S$ is a pro-algebraic space limit of a cofiltered diagram of qcqs algebraic spaces of Krull dimension $\leq d$, then the (∞,1)-topos of (∞,1)-sheaves on the small Nisnevich site of $S$ has homotopy dimension $\leq d$.

This is Clausen and A. Mathew, Hyperdescent and etale K-theory, 2019, arXiv:1905.06611, Cor.3.11, Thm.3.12, and Thm.3.17.

Related concepts

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

• cd-structure

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, $\mathbb{A}^1$-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))

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)