Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:09:16Z http://mathoverflow.net/feeds/question/41428 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41428/can-deligne-mumford-stacks-be-characterized-by-their-restriction-to-a-small-subca Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory? Jon Skowera 2010-10-07T16:15:07Z 2010-10-09T08:32:29Z <p>If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For example, let's say we assume the fppf topology (to be concrete), then does the restriction to the fully faithful subcategory $T$ with objects</p> <p>$\mathrm{Ob}(T) :=$ { $I_n := \mathrm{Spec~} k[\epsilon]/(\epsilon^n) \mid n \in \mathbf{N}$ }</p> <p>completely determine the stack? I think I've read that the restriction to $(\mathrm{Aff}/k)$ is enough, but since the fibers over the $I_n$ determine the $n^{th}$-order formal neighborhoods ($\mathrm{HOM}(I_n, X) \cong X(I_n)$), I wonder if the restriction to $T$ is enough?</p> http://mathoverflow.net/questions/41428/can-deligne-mumford-stacks-be-characterized-by-their-restriction-to-a-small-subca/41443#41443 Answer by Anatoly Preygel for Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory? Anatoly Preygel 2010-10-07T18:59:31Z 2010-10-07T18:59:31Z <p>Restricting to Aff is certainly enough, but Aff isn't small (there are e.g., polynomial algebras on arbitrary sets). If your DM stack is <em>finitely presented</em> over $k$ (which is probably good to include in the definition, to avoid these issues), then it is determined by it's restriction to finitely-presented affines (which is essentially small).</p> <p>Without some finiteness hypothesis, no set of finitely-presented algebras can suffice (even for affine schemes, nevermind DM stacks). (And I suppose no small category of test objects can suffice: Take Spec of a field generated by a set of cardinality larger than that of global sections of any of your test schemes.)</p> <p>The set you give is insufficient even for smooth varieties over an alg. closed field: you will have a morphism whenever you have an (arbitrary) map on $k$-points.</p>