The Stacks project

72.2 The general procedure

In this section we explain a general procedure for producing the sites we will be working with. This discussion will make little or no sense unless the reader has read Topologies, Section 34.2.

Let $S$ be a base scheme. Take any category $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with $S$ and any set of schemes over $S$ you want to be included. Choose any set of coverings $\text{Cov}_{fppf}$ on $\mathit{Sch}_\alpha $ as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of fppf coverings. Let $\mathit{Sch}_{fppf}$ denote the big fppf site so obtained, and let $(\mathit{Sch}/S)_{fppf}$ denote the corresponding big fppf site of $S$. (The above is entirely as prescribed in Topologies, Section 34.7.)

Given choices as above the category of algebraic spaces over $S$ has a set of isomorphism classes. One way to see this is to use the fact that any algebraic space over $S$ is of the form $U/R$ for some étale equivalence relation $j : R \to U \times _ S U$ with $U, R \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, see Spaces, Lemma 64.9.1. Hence we can find a full subcategory $\textit{Spaces}/S$ of the category of algebraic spaces over $S$ which has a set of objects such that each algebraic space is isomorphic to an object of $\textit{Spaces}/S$. We fix a choice of such a category.

In the sections below, given a topology $\tau $, the big site $(\textit{Spaces}/S)_\tau $ (resp. the big site $(\textit{Spaces}/X)_\tau $ of an algebraic space $X$ over $S$) has as underlying category the category $\textit{Spaces}/S$ (resp. the subcategory $\textit{Spaces}/X$ of $\textit{Spaces}/S$, see Categories, Example 4.2.13). The procedure for turning this into a site is as usual by defining a class of $\tau $-coverings and using Sets, Lemma 3.11.1 to choose a sufficiently large set of coverings which defines the topology.

We point out that the small étale site $X_{\acute{e}tale}$ of an algebraic space $X$ has already been defined in Properties of Spaces, Definition 65.18.1. Its objects are schemes étale over $X$, of which there are plenty by definition of an algebraic spaces. However, a more natural site, from the perspective of this chapter (compare Topologies, Definition 34.4.8) is the site $X_{spaces, {\acute{e}tale}}$ of Properties of Spaces, Definition 65.18.2. These two sites define the same topos, see Properties of Spaces, Lemma 65.18.3. We will not redefine these in this chapter; instead we will simply use them.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03Y6. Beware of the difference between the letter 'O' and the digit '0'.