The Stacks project

97.3 What we already know

The analogue of this chapter for algebraic spaces is the chapter entitled “Bootstrap”, see Bootstrap, Section 80.1. That chapter already contains some representability results. Moreover, some of the preliminary material treated there we already have worked out in the chapter on algebraic stacks. Here is a list:

1. We discuss morphisms of presheaves representable by algebraic spaces in Bootstrap, Section 80.3. In Algebraic Stacks, Section 94.9 we discuss the notion of a $1$-morphism of categories fibred in groupoids being representable by algebraic spaces.

2. We discuss properties of morphisms of presheaves representable by algebraic spaces in Bootstrap, Section 80.4. In Algebraic Stacks, Section 94.10 we discuss properties of $1$-morphisms of categories fibred in groupoids representable by algebraic spaces.

3. We proved that if $F$ is a sheaf whose diagonal is representable by algebraic spaces and which has an étale covering by an algebraic space, then $F$ is an algebraic space, see Bootstrap, Theorem 80.6.1. (This is a weak version of the result in the next item on the list.)

4. We proved that if $F$ is a sheaf and if there exists an algebraic space $U$ and a morphism $U \to F$ which is representable by algebraic spaces, surjective, flat, and locally of finite presentation, then $F$ is an algebraic space, see Bootstrap, Theorem 80.10.1.

5. We have also proved the “smooth” analogue of (4) for algebraic stacks: If $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ and if there exists a stack in groupoids $\mathcal{U}$ over $(\mathit{Sch}/S)_{fppf}$ which is representable by an algebraic space and a $1$-morphism $u : \mathcal{U} \to \mathcal{X}$ which is representable by algebraic spaces, surjective, and smooth then $\mathcal{X}$ is an algebraic stack, see Algebraic Stacks, Lemma 94.15.3.

Our first task now is to prove the analogue of (4) for algebraic stacks in general; it is Theorem 97.16.1.