1 The Nisnevich topology

Nisnevich localization is relatively standard. This is obtained by defining the Nisnevich topology on $\mathrm{Sm}_S$ , and then imposing the usual sheaf condition. Consequently, the Nisnevich localization functor is a standard sheafification functor, and automatically enjoys the nice formal properties of sheafification. For example, it is an exact functor.

Definition 1.1

Let $X \in \mathrm{Sm}_S$ . A Nisnevich cover of $X$ is a finite family of étale morphisms $\{ p_i: U_i \to X\} _{i \in I}$ such that there is a filtration

\emptyset \subseteq Z_n \subseteq Z_{n - 1} \subseteq \cdots \subseteq Z_1 \subseteq Z_0 = X

of $X$ by finitely presented closed subschemes such that for each strata $Z_m \setminus Z_{m + 1}$ , there is some $p_i$ such that

p_i^{-1}(Z_m \setminus Z_{m + 1}) \to Z_m \setminus Z_{m + 1}

admits a section.

Example 1.2

Any Zariski cover is a Nisnevich cover. Any Nisnevich cover is an étale cover.

Example 1.3

Let $k$ be a field of characteristic not $2$ , $S = \operatorname{Spec}k$ and $a \in k^\times$ . Consider the covering

$\begin{useimager} \[ \begin{tikzcd}[column sep=small] & \A^1 \setminus \{0\} \ar[d, "x^2"]\\ \A^1 \setminus \{a\} \ar[r, hook] & \A^1 \end{tikzcd}. \] \end{useimager}$

This forms a Nisnevich cover with the filtration $\emptyset \subseteq \{ a\} \subseteq \mathbb {A}^1$ iff $\sqrt{a} \in k$ .

It turns out to check that something is a Nisnevich sheaf, it suffices to check it for very particular covers with two opens.

Definition 1.4

An elementary distinguished square is a pullback diagram

$\begin{useimager} \[ \begin{tikzcd} U \times_X V \ar[r] \ar[d] & V \ar[d, "p"]\\ U \ar[r, "i"] & X \end{tikzcd} \] \end{useimager}$

of $S$ -schemes in $\mathrm{Sm}_S$ such that $i$ is a Zariski open immersion, $p$ is étale, and $p^{-1}(X \setminus U) \to X \setminus U$ is an isomorphism of schemes, where $X \setminus U$ is equipped with the reduced induced scheme structure.

\{ U, V\}

forms a Nisnevich cover of

X

with filtration

\emptyset \subseteq X \setminus U \subseteq X

Definition 1.5

We define $\mathrm{Spc}_S = L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_S)$ to be the full subcategory of $\mathcal{P}(\mathrm{Sm}_S)$ consisting of presheaves that satisfy descent with respect to Nisnevich covers. Such presheaves are also said to be Nisnevich local. This is an accessible subcategory of $\mathcal{P}(\mathrm{Sm}_S)$ and admits a localization functor $L_{\mathrm{Nis}} : \mathcal{P}(\mathrm{Sm}_S) \to \mathrm{Spc}_S$ .

Example 1.6

Every representable functor satisfies Nisnevich descent, since they in fact satisfy étale descent. Note that these functors are valued in discrete spaces.

Lemma 1.7

Then $F \in \mathcal{P}(\mathrm{Sm}_S)$ is Nisnevich local iff $F(\emptyset ) \simeq *$ and for every elementary distinguished square

$\begin{useimager} \[ \begin{tikzcd} U \times_X V \ar[r] \ar[d] & V \ar[d, "p"]\\ U \ar[r, "i"] & X \end{tikzcd} \] \end{useimager}$

the induced diagram

$\begin{useimager} \[ \begin{tikzcd} F(X) \ar[r] \ar[d] & F(V) \ar[d]\\ F(U) \ar[r] & F(U \times_X V) \end{tikzcd} \] \end{useimager}$

is a pullback diagram.

Note that the “only if” direction is immediate from definition, and doesn't require the assumption on

S

Corollary 1.8

$\begin{useimager} \[ \begin{tikzcd} U \times_X V \ar[r] \ar[d] & V \ar[d, "p"]\\ U \ar[r, "i"] & X \end{tikzcd} \] \end{useimager}$

is an elementary distinguished square, then, when considered a square $\mathrm{Spc}_S$ , this is a pushout diagram.

In particular, this holds when $\{ U, V\}$ is a Zariski cover.