The Stacks project

65.1 Introduction

Algebraic spaces were first introduced by Michael Artin, see [ArtinI], [ArtinII], [Artin-Theorem-Representability], [Artin-Construction-Techniques], [Artin-Algebraic-Spaces], [Artin-Algebraic-Approximation], [Artin-Implicit-Function], and [ArtinVersal]. Some of the foundational material was developed jointly with Knutson, who produced the book [Kn]. Artin defined (see [Definition 1.3, Artin-Implicit-Function]) an algebraic space as a sheaf for the étale topology which is locally in the étale topology representable. In most of Artin's work the categories of schemes considered are schemes locally of finite type over a fixed excellent Noetherian base.

Our definition is slightly different from Artin's original definition. Namely, our algebraic spaces are sheaves for the fppf topology whose diagonal is representable and which have an étale “cover” by a scheme. Working with the fppf topology instead of the étale topology is just a technical point and scarcely makes any difference; we will show in Bootstrap, Section 80.12 that we would have gotten the same category of algebraic spaces if we had worked with the étale topology. In that same chapter we will prove that the condition on the diagonal can in some sense be removed, see Bootstrap, Section 80.6.

After defining algebraic spaces we make some foundational observations. The main result in this chapter is that with our definitions an algebraic space is the same thing as an étale equivalence relation, see the discussion in Section 65.9 and Theorem 65.10.5. The analogue of this theorem in Artin's setting is [Theorem 1.5, Artin-Implicit-Function], or [Proposition II.1.7, Kn]. In other words, the sheaf defined by an étale equivalence relation has a representable diagonal. It follows that our definition agrees with Artin's original definition in a broad sense. It also means that one can give examples of algebraic spaces by simply writing down an étale equivalence relation.

In Section 65.13 we introduce various separation axioms on algebraic spaces that we have found in the literature. Finally in Section 65.14 we give some weird and not so weird examples of algebraic spaces.