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.