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\section*{Nisnevich site}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_nonnoetherian_schemes}{For non-Noetherian schemes}\dotfill \pageref*{for_nonnoetherian_schemes} \linebreak
\noindent\hyperlink{as_an_excision_property}{As an excision property}\dotfill \pageref*{as_an_excision_property} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_dimension}{Homotopy dimension}\dotfill \pageref*{homotopy_dimension} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Nisnevich topology} is a certain [[Grothendieck topology]] on the [[category]] of [[scheme]]s which is finer than the [[Zariski topology]] but coarser than the [[étale topology]]. It retains many desirable properties from both topologies:

\begin{itemize}%
\item The Nisnevich [[cohomological dimension]] (and even the [[homotopy dimension]]) of a scheme is bounded by its [[Krull dimension]] (like Zariski)


\item [[field|Fields]] have [[shape of an (∞,1)-topos|trivial shape]] for the Nisnevich topology (like Zariski)


\item [[algebraic K-theory|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 \emph{\href{algebraic%20K-theory#Descent}{algebraic K-theory -- Descent}} for more;


\item For $Z\subset X$ a [[closed subscheme|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 \'e{}tale)


\item Pushforward along a [[finite morphism]] is an [[exact functor]] on Nisnevich sheaves of abelian groups (like \'e{}tale)


\item Nisnevich cohomology can be computed using [[?ech cohomology]] (like \'e{}tale)



\end{itemize}
The Nisnevich topology plays a central r\^o{}le in [[motivic homotopy theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An family of morphisms of [[Noetherian scheme]]s $\{p_i:V_i\to U\}$ is a \textbf{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 \textbf{Nisnevich topology}.

The \textbf{Nisnevich site} over a [[Noetherian scheme]] $S$ usually refers to the [[site]] given by the [[category]] of [[smooth scheme]]s of finite type over $S$ equipped with the Nisnevich topology. The \textbf{small Nisnevich site} of $S$ is the subsite consisting of [[étale map|étale]] $S$-schemes.

\hypertarget{for_nonnoetherian_schemes}{}\subsubsection*{{For non-Noetherian schemes}}\label{for_nonnoetherian_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 $i$.

\hypertarget{as_an_excision_property}{}\subsubsection*{{As an excision property}}\label{as_an_excision_property}

Let $Et/S$ be the category of [[étale scheme]]s of [[morphism of finite presentation|finite presentation]] over a [[quasi-compact morphism|quasi-compact]] [[quasi-separated morphism|quasi-separated]] scheme $S$. An [[(∞,1)-presheaf]] $F$ on $Et/S$ is said to satisfy \emph{Nisnevich excision} if the following conditions hold:

\begin{itemize}%
\item $F(\emptyset)$ is [[contractible]].
\item 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

\end{itemize}
\begin{displaymath}
\itexarray{
     F(X) &\to& F(X-Z)
     \\
     \downarrow && \downarrow
     \\
     F(X') &\to& F(X'-Z)
  }
\end{displaymath}
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 \hyperlink{DAGXI}{DAG XI}.

\begin{prop}
\label{}\hypertarget{}{}
An [[(∞,1)-presheaf]] on $Et/S$ is an [[(∞,1)-sheaf]] for the Nisnevich topology if and only if it satisifes Nisnevich excision.

\end{prop}
This is \hyperlink{MV99}{Morel-Voevosky, Prop. 1.16} or \hyperlink{DAGXI}{DAG XI, Thm. 2.9}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{homotopy_dimension}{}\subsubsection*{{Homotopy dimension}}\label{homotopy_dimension}

\begin{prop}
\label{}\hypertarget{}{}
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]] $\leq\dim(S)$.

\end{prop}
This is \hyperlink{DAGXI}{DAG XI, Theorem 2.24}. As a consequence, [[Postnikov tower]]s are convergent in the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] on the Nisnevich site over $S$, and in particular that (∞,1)-topos is [[hypercomplete]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ \textbf{Nisnevich site} $\to$ [[Zariski site]]


\item [[cd-structure]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A quick overview is at the beginning of the talk slides

\begin{itemize}%
\item Jardine, \emph{Motivic spaces and the motivic stable category} (\href{http://www.aimath.org/WWN/motivesdessins/motivic.pdf}{pdf}) .

\end{itemize}
A detailed discussion is in \href{http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf#page=61}{section 3.1.1} of

\begin{itemize}%
\item [[Fabien Morel]], [[Vladimir Voevodsky]], \emph{$\mathbb{A}^1$-homotopy theory of schemes} , K-theory, 0305 (\href{http://www.math.uiuc.edu/K-theory/0305/}{web} \href{http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf}{pdf})

\end{itemize}
or in the lecture notes

\begin{itemize}%
\item [[Eric Friedlander]], \emph{\href{http://www.math.northwestern.edu/~eric/lectures/}{Algebraic Cycles and algebraic K-theory, II}} (\href{http://www.math.northwestern.edu/~eric/lectures/ihp/ihplec6.pdf}{lecture 6 (pdf)})

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

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Derived Algebraic Geometry XI: Descent Theorems} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf}{pdf})

\end{itemize}
[[!redirects Nisnevich topology]] [[!redirects Nisnevich topos]] [[!redirects Nisnevich (∞,1)-topos]]

[[!redirects Nisnevich sheaf]] [[!redirects Nisnevich sheaves]]

[[!redirects Nisnevich descent]]



\end{document}