## 108.1 Introduction

In this chapter we verify basic properties of moduli spaces and moduli stacks such as $\mathit{Hom}$, $\mathit{Isom}$, $\mathcal{C}\! \mathit{oh}_{X/B}$, $\mathrm{Quot}_{\mathcal{F}/X/B}$, $\mathrm{Hilb}_{X/B}$, $\mathcal{P}\! \mathit{ic}_{X/B}$, $\mathrm{Pic}_{X/B}$, $\mathit{Mor}_ B(Z, X)$, $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$, $\mathcal{P}\! \mathit{olarized}$, and $\mathcal{C}\! \mathit{omplexes}_{X/B}$. We have already shown these algebraic spaces or algebraic stacks under suitable hypotheses, see Quot, Sections 99.3, 99.4, 99.5, 99.6, 99.8, 99.9, 99.10, 99.11, 99.12, 99.13, 99.14, and 99.16. The stack of curves, denoted $\textit{Curves}$ and introduced in Quot, Section 99.15, is discussed in the chapter on moduli of curves, see Moduli of Curves, Section 109.3.

In some sense this chapter is following the footsteps of Grothendieck's lectures [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI].

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