Motivic Homotopy Theory — The Nisnevich topology

1 The Nisnevich topology

Nisnevich localization is relatively standard. This is obtained by defining the Nisnevich topology on SmS\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 XSmSX \in \mathrm{Sm}_ S. A Nisnevich cover of XX is a finite family of étale morphisms {pi:UiX}iI\{ p_ i: U_ i \to X\} _{i \in I} such that there is a filtration

ZnZn1Z1Z0=X \emptyset \subseteq Z_ n \subseteq Z_{n - 1} \subseteq \cdots \subseteq Z_1 \subseteq Z_0 = X

of XX by finitely presented closed subschemes such that for each strata ZmZm+1Z_ m \setminus Z_{m + 1}, there is some pip_ i such that

pi1(ZmZm+1)ZmZm+1 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 kk be a field of characteristic not 22, S=SpeckS = \operatorname{Spec}k and ak×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 {a}A1\emptyset \subseteq \{ a\} \subseteq \mathbb {A}^1 iff ak\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 SS-schemes in SmS\mathrm{Sm}_ S such that ii is a Zariski open immersion, pp is étale, and p1(XU)XUp^{-1}(X \setminus U) \to X \setminus U is an isomorphism of schemes, where XUX \setminus U is equipped with the reduced induced scheme structure.

{U,V}\{ U, V\} forms a Nisnevich cover of XX with filtration XUX\emptyset \subseteq X \setminus U \subseteq X.

Definition 1.5

We define SpcS=LNisP(SmS)\mathrm{Spc}_ S = L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_ S) to be the full subcategory of P(SmS)\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 P(SmS)\mathcal{P}(\mathrm{Sm}_ S) and admits a localization functor LNis:P(SmS)SpcSL_{\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 FP(SmS)F \in \mathcal{P}(\mathrm{Sm}_ S) is Nisnevich local iff F()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 SS.

Corollary 1.8

If

\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 SpcS\mathrm{Spc}_ S, this is a pushout diagram.

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