The purpose of this chapter is to find criteria guaranteeing that a stack in groupoids over the category of schemes with the fppf topology is an algebraic stack. Historically, this often involved proving that certain functors were representable, see Grothendieck's lectures [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI]. This explains the title of this chapter. Another important source 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].
Some of the notation, conventions and terminology in this chapter is awkward and may seem backwards to the more experienced reader. This is intentional. Please see Quot, Section 98.2 for an explanation.
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