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\begin{document}

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\section*{Zariski 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{points}{Points}\dotfill \pageref*{points} \linebreak
\noindent\hyperlink{as_a_site}{As a site}\dotfill \pageref*{as_a_site} \linebreak
\noindent\hyperlink{sheafification}{Sheafification}\dotfill \pageref*{sheafification} \linebreak
\noindent\hyperlink{KripkeJoyal}{Kripke--Joyal semantics}\dotfill \pageref*{KripkeJoyal} \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}

There is a [[little site]] notion of the [[Zariski topology]], and a [[big site]] notion. As for the little site notion: the Zariski topology on the set of prime ideals of a commutative ring $R$ is the smallest topology that contains, as open sets, sets of the form $\{p\; \text{prime}: a \notin p\}$ where $a$ ranges over elements of $R$.

As for the big site notion, the \emph{Zariski topology} is a [[coverage]] on the [[opposite category]] [[CRing]]${}^{op}$ of commutative rings. This article is mainly about the big site notion.

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

For $R$ a commutative [[ring]], write $Spec R \in CRing^{op}$ for its [[spectrum of a commutative ring]], hence equivalently for its incarnation in the [[opposite category]].

\begin{defn}
\label{StandardOpenImmersion}\hypertarget{StandardOpenImmersion}{}
For $S \subset R$ a [[multiplicative subset]], write $R[S^{-1}]$ for the corresponding [[localization of a ring|localization]] and

\begin{displaymath}
Spec(R[S^{-1}]) \longrightarrow Spec(R)
\end{displaymath}
for the dual of the canonical ring homomorphism $R \to R[S^{-1}]$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The maps as in def. \ref{StandardOpenImmersion} are not [[open immersion of schemes|open immersions]] for arbitrary multiplicative subsets $S$ (see \href{http://mathoverflow.net/questions/20782/ring-theoretic-characterization-of-open-affines}{a MathOverflow discussion}). They are for subsets of the form $S = \{ f^0, f^1, f^2, \ldots \}$, in which case they are called the \emph{standard opens} of $Spec(R)$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A family of [[morphisms]] $\{Spec A_i \to Spec R\}$ in $CRing^{op}$ is a Zariski-[[covering]] precisely if

\begin{itemize}%
\item each ring $A_i$ is the [[localization]]

\begin{displaymath}
A_i = R[r_i^{-1}]
\end{displaymath}
of $R$ at a single element $r_i \in R$


\item $Spec A_i \to Spec R$ is the canonical inclusion, dual to the canonical ring homomorphism $R \to R[r_i^{-1}]$;


\item There exists $\{f_i \in R\}$ such that

\begin{displaymath}
\sum_i f_i r_i = 1
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Geometrically, one may think of

\begin{itemize}%
\item $r_i$ as a function on the [[space]] $Spec R$;


\item $R[r_i^{-1}]$ as the [[open subset]] of points in this space on which the function is not 0;


\item the covering condition as saying that the functions form a [[partition of unity]] on $Spec R$.



\end{itemize}
\end{remark}
\begin{defn}
\label{ZariskiTopos}\hypertarget{ZariskiTopos}{}
Let $CRing_{fp} \hookrightarrow CRing$ be the [[full subcategory]] on [[finitely presented object]]s. This inherits the Zariski [[coverage]].

The [[sheaf topos]] over this [[site]] is the [[big topos]] version of the \textbf{Zariski topos}.

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

\hypertarget{points}{}\subsubsection*{{Points}}\label{points}

The [[maximal ideal]] in $R$ correspond precisely to the [[closed subset|closed]] points of the [[prime spectrum]] $Spec(R)$ in the Zariski topology.

\hypertarget{as_a_site}{}\subsubsection*{{As a site}}\label{as_a_site}

\begin{prop}
\label{}\hypertarget{}{}
The Zariski coverage is [[subcanonical coverage|subcanonical]].

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
\begin{itemize}%
\item $CRing_{fp}^{op}$ is the [[syntactic category]] of the [[cartesian theory]] of commutative rings;


\item $CRing_{fp}^{op}$ equipped with the Zariski topology is the [[syntactic site]] of the [[geometric theory]] of [[local ring]]s.



\end{itemize}
Hence

\begin{itemize}%
\item the big Zariski topos, def. \ref{ZariskiTopos}, is the [[classifying topos]] for [[local ring]]s.


\item a [[locally ringed topos]] is equivalently a topos $\mathcal{E}$ equipped with a [[geometric morphism]] into the big Zariski topos.



\end{itemize}
\end{prop}
See [[classifying topos]] and [[locally ringed topos]] for more details on this.

\hypertarget{sheafification}{}\subsubsection*{{Sheafification}}\label{sheafification}

If $F$ is a presheaf on $CRing^{op}$ and $F^{++}$ denotes its [[sheafification]], then the canonical morphism $F(R) \to F^{++}(R)$ is an isomorphism for all [[local ring|local rings]] $R$. This follows from the explicit description of the [[plus construction]] and the fact that a local ring admits only the trivial covering.

\hypertarget{KripkeJoyal}{}\subsection*{{Kripke--Joyal semantics}}\label{KripkeJoyal}

Writing $R \models \varphi$ for the interpretation of a formula $\varphi$ of the [[internal language]] of the big Zariski topos over $Spec(R)$ with the [[Kripke–Joyal semantics]], the forcing relation can be expressed as follows.

\begin{displaymath}
\begin{array}{lcl}
  R \models x = y : F &\Longleftrightarrow&
    x = y \in F(R). \\
  R \models \top &\Longleftrightarrow&
    1 = 1 \in R. \\
  R \models \bot &\Longleftrightarrow&
    1 = 0 \in R. \\
  R \models \phi \wedge \psi &\Longleftrightarrow&
    R \models \phi \,\text{and}\, R \models \psi. \\
  R \models \phi \vee \psi &\Longleftrightarrow&
    \text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i,
      R[f_i^{-1}] \models \phi \,or\, R[f_i^{-1}] \models \psi. \\
  R \models \phi \Rightarrow \psi &\Longleftrightarrow&
    \text{for any}\, R\text{-algebra}\, S \,\text{it holds that}\, (S \models \phi) \,implies\, (S \models \psi). \\
  R \models \forall x:F. \phi &\Longleftrightarrow&
    \text{for any}\, R\text{-algebra}\, S \,\text{and any element}\, x \in F(S) \,\text{it holds that}\, S \models \phi[x]. \\
  R \models \exists x.F. \phi &\Longleftrightarrow&
    \text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i,
    \text{there exists an element}\, x_i \in F(R[f_i^{-1}]) \,\text{such that}\,
    R[f_i^{-1}] \models \phi[x_i].
\end{array}
\end{displaymath}
The only difference to the Kripke--Joyal semantics of the \emph{little} Zariski topos is that in the clauses for $\Rightarrow$ and $\forall$, one has to restrict to $R$-algebras $S$ of the form $S = R[f^{-1}]$.

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

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

\begin{itemize}%
\item [[spectrum of a commutative ring]]

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

Examples A2.1.11(f) and D3.1.11 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Section VIII.6 of

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}

\end{itemize}
\begin{itemize}%
\item [[The Stacks Project]], chapter 33 \emph{Topologies on Schemes}


\item Nick Duncan, \emph{Gros and Petit Toposes}, talk notes, \href{http://cheng.staff.shef.ac.uk/pssl88/}{88th Peripatetic Seminar on Sheaves and Logic}, \href{http://cheng.staff.shef.ac.uk/pssl88/pssl88-duncan.pdf}{pdf}.


\item Daniel Murfet, \href{http://therisingsea.org/notes/ZariskiTopology.pdf}{The Zariski Site}



\end{itemize}
category: algebraic geometry

[[!redirects Zariski topos]] [[!redirects big Zariski topos]]

[[!redirects Zariski sheaf]] [[!redirects Zariski sheaves]]

[[!redirects Zariski descent]]

[[!redirects theory of local rings]] [[!redirects geometric theory of local rings]]



\end{document}