Motivic Homotopy Theory — The Nisnevich topology

# 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

$p_ i^{-1}(Z_ m \setminus Z_{m + 1}) \to Z_ m \setminus Z_{m + 1}$

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

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

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

the induced diagram

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

If

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.