I’ve been trying to learn a bit about stacks lately. This is actually going to tie in quite nicely into the homotopy-theoretic story I was talking about earlier: the moduli stack of formal groups turns out to be a close approximation to stable homotopy theory. But before that, there are other more fundamental examples of stacks.

Here is a toy example of an Artin stack I would like to understand. Let ${X}$ be a noetherian scheme; we’d like a stack ${\mathrm{Coh}(X)}$ which parametrizes coherent sheaves on ${X}$. This is definitely a “stack” rather than a scheme, because coherent sheaves are likely to have plenty of automorphisms.

Let’s be a little more precise. Given a test scheme (say, affine) ${T}$, a map

$\displaystyle T \rightarrow \mathrm{Coh}(X)$

should be a “family of coherent sheaves on ${X}$” parametrized by ${T}$. One way of saying this is that we just have a quasi-coherent sheaf ${\mathcal{F}}$ of finite presentation on ${X \times_{\mathbb{Z}} T}$. In order to impose a “continuity” condition on this family, we require that ${\mathcal{F}}$ be flat over ${T}$.

To make this more precise, we define it relative to a base:

Definition 1 Let ${S}$ be a base noetherian scheme, and let ${X \rightarrow S}$ be an ${S}$-scheme of finite type. We define a functor

$\displaystyle \mathrm{Coh}_{X/S}: \mathrm{Aff}_{/S}^{op} \rightarrow \mathrm{Grpd}$

sending an affine ${T \rightarrow S}$ to the groupoid of finitely presented quasi-coherent sheaves on ${T \times_S X}$, flat over ${T}$. The pull-back morphisms are given by pull-backs of sheaves.

In this post, we will see that this is an Artin stack under certain conditions.