The Stacks project

99.1 Introduction

As initially conceived, the purpose of this chapter was to write about Quot and Hilbert functors and to prove that these are algebraic spaces provided certain technical conditions are satisfied. This material, in the setting of schemes, is covered in Grothendieck's lectures in the séminair Bourbaki, see [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI]. For projective schemes the Quot and Hilbert schemes live inside Grassmannians of spaces of sections of suitable very ample invertible sheaves, and this provides a method of construction for these schemes. Our approach is different: we use Artin's axioms to prove Quot and Hilb are algebraic spaces.

Upon further consideration, it turned out to be more convenient for the development of theory in the Stacks project, to start the discussion with the stack $\mathcal{C}\! \mathit{oh}_{X/B}$ of coherent sheaves (with proper support over the base) as introduced in [lieblich_remarks]. For us $f : X \to B$ is a morphism of algebraic spaces satisfying suitable technical conditions, although this can be generalized (see below). Given modules $\mathcal{F}$ and $\mathcal{G}$ on $X$, under suitably hypotheses, the functor $T/B \mapsto \mathop{\mathrm{Hom}}\nolimits _{X_ T}(\mathcal{F}_ T, \mathcal{G}_ T)$ is an algebraic space $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ over $B$. See Section 99.3. The subfunctor $\mathit{Isom}(\mathcal{F}, \mathcal{G})$ of isomorphisms is shown to be an algebraic space in Section 99.4. This is used in the next sections to show the diagonal of the stack $\mathcal{C}\! \mathit{oh}_{X/B}$ is representable. We prove $\mathcal{C}\! \mathit{oh}_{X/B}$ is an algebraic stack in Section 99.5 when $X \to B$ is flat and in Section 99.6 in general. Please see the introduction of this section for pointers to the literature.

Having proved this, it is rather straightforward to prove that $\mathrm{Quot}_{\mathcal{F}/X/B}$, $\mathrm{Hilb}_{X/B}$, and $\mathrm{Pic}_{X/B}$ are algebraic spaces and that $\mathcal{P}\! \mathit{ic}_{X/B}$ is an algebraic stack. See Sections 99.8, 99.9, 99.11, and 99.10.

In the usual manner we deduce that the functor $\mathit{Mor}_ B(Z, X)$ of relative morphisms is an algebraic space (under suitable hypotheses) in Section 99.12.

In Section 99.13 we prove that the stack in groupoids

parametrizing flat families of proper algebraic spaces satisfies all of Artin's axioms (including openness of versality) except for formal effectiveness. We've chosen the very awkward notation for this stack intentionally, because the reader should be careful in using its properties.

In Section 99.14 we prove that the stack $\mathcal{P}\! \mathit{olarized}$ parametrizing flat families of polarized proper algebraic spaces is an algebraic stack. Because of our work on flat families of proper algebraic spaces, this comes down to proving formal effectiveness for polarized schemes which is often known as Grothendieck's algebraization theorem.

In Section 99.15 we prove that the stack $\mathcal{C}\! \mathit{urves}$ parametrizing families of curves is algebraic.

In Section 99.16 we study moduli of complexes on a proper morphism and we obtain an algebraic stack $\mathcal{C}\! \mathit{omplexes}_{X/B}$. The idea of the statement and the proof are taken from [lieblich-complexes].

What is not in this chapter? There is almost no discussion of the properties the resulting moduli spaces and moduli stacks possess (beyond their algebraicity); to read about this we refer to Moduli Stacks, Section 108.1. In most of the results discussed, we can generalize the constructions by considering a morphism $\mathcal{X} \to \mathcal{B}$ of algebraic stacks instead of a morphism $X \to B$ of algebraic space. We will discuss this (insert future reference here). In the case of Hilbert spaces there is a more general notion of “Hilbert stacks” which we will discuss in a separate chapter, see (insert future reference here).