In this chapter we use the material from the preceding sections to give criteria under which a presheaf of sets on the category of schemes is an algebraic space. Some of this material comes from the work of Artin, see [ArtinI], [ArtinII], [Artin-Theorem-Representability], [Artin-Construction-Techniques], [Artin-Algebraic-Spaces], [Artin-Algebraic-Approximation], [Artin-Implicit-Function], and [ArtinVersal]. However, our method will be to use as much as possible arguments similar to those of the paper by Keel and Mori, see [K-M].
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