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.