## Tag `020M`

## 33.2. The general procedure

In this section we explain a general procedure for producing the sites we will be working with. Suppose we want to study sheaves over schemes with respect to some topology $\tau$. In order to get a site, as in Sites, Definition 7.6.2, of schemes with that topology we have to do some work. Namely, we cannot simply say ''consider all schemes with the Zariski topology'' since that would give a ''big'' category. Instead, in each section of this chapter we will proceed as follows:

- We define a class $\text{Cov}_\tau$ of coverings of schemes satisfying the axioms of Sites, Definition 7.6.2. It will always be the case that a Zariski open covering of a scheme is a covering for $\tau$.
- We single out a notion of standard $\tau$-covering within the category of affine schemes.
- We define what is an ''absolute'' big $\tau$-site $\textit{Sch}_\tau$. These are the sites one gets by appropriately choosing a set of schemes and a set of coverings.
- For any object $S$ of $\textit{Sch}_\tau$ we define the big $\tau$-site $(\textit{Sch}/S)_\tau$ and for suitable $\tau$ the small
^{1}$\tau$-site $S_\tau$.- In addition there is a site $(\textit{Aff}/S)_\tau$ using the notion of standard $\tau$-covering of affines whose category of sheaves is equivalent to the category of sheaves on $(\textit{Sch}/S)_\tau$.
The above is a little clumsy in that we do not end up with a canonical choice for the big $\tau$-site of a scheme, or even the small $\tau$-site of a scheme. If you are willing to ignore set theoretic difficulties, then you can work with classes and end up with canonical big and small sites...

- The words big and small here do not relate to bigness/smallness of the corresponding categories. ↑

The code snippet corresponding to this tag is a part of the file `topologies.tex` and is located in lines 31–73 (see updates for more information).

```
\section{The general procedure}
\label{section-procedure}
\noindent
In this section we explain a general procedure for producing the
sites we will be working with. Suppose we want to study sheaves
over schemes with respect to some topology $\tau$. In order to
get a site, as in Sites, Definition \ref{sites-definition-site},
of schemes with that topology we have to do some work. Namely,
we cannot simply say ``consider all schemes with the Zariski topology''
since that would give a ``big'' category. Instead, in each section of
this chapter we will proceed as follows:
\begin{enumerate}
\item We define a class $\text{Cov}_\tau$ of coverings of schemes
satisfying the axioms of Sites, Definition \ref{sites-definition-site}.
It will always be the case that a Zariski open covering of
a scheme is a covering for $\tau$.
\item We single out a notion of standard
$\tau$-covering within the category of affine schemes.
\item We define what is an ``absolute'' big $\tau$-site $\Sch_\tau$.
These are the sites one gets by appropriately choosing a set of schemes
and a set of coverings.
\item For any object $S$ of $\Sch_\tau$
we define the big $\tau$-site $(\Sch/S)_\tau$ and for suitable
$\tau$ the small\footnote{The words big and
small here do not relate to bigness/smallness of the corresponding
categories.} $\tau$-site $S_\tau$.
\item In addition there is a site $(\textit{Aff}/S)_\tau$ using the
notion of standard $\tau$-covering of affines whose category of sheaves
is equivalent to the category of sheaves on $(\Sch/S)_\tau$.
\end{enumerate}
The above is a little clumsy in that we do not end up with a canonical
choice for the big $\tau$-site of a scheme, or even the small
$\tau$-site of a scheme. If you are willing to ignore set theoretic
difficulties, then you can work with classes and end up with
canonical big and small sites...
```

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