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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:

  1. 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 $\tau$.
  2. We single out a notion of standard $\tau$-covering within the category of affine schemes.
  3. 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.
  4. For any object $S$ of $\textit{Sch}_\tau$ we define the big $\tau$-site $(\textit{Sch}/S)_\tau$ and for suitable $\tau$ the small1 $\tau$-site $S_\tau$.
  5. 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...

  1. 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...

Comments (1)

Comment #2588 by Ingo Blechschmidt on June 1, 2017 a 5:58 pm UTC

I have seen people define the big toposes not using the site $\mathrm{Aff}/S$ (consisting of $S$-schemes which are affine as absolute schemes) but by using the site of affine $S$-schemes ($S$-schemes whose structural morphism to $S$ is affine). The resulting sheaf toposes are equivalent in case that the diagonal morphism $S \to S \times S$ is affine, since in this case morphisms of the form $\operatorname{Spec} A \to S$ are affine. If the diagonal is not affine, then the two toposes probably differ.

Should we add a remark along these lines?

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