Ok, so I’ve finally found (what I think will be) a “classical” solution to getting a deformation theory for the stack of coherent sheaves in the non-flat setting. I quickly recall the setting.

The problem: Suppose you have a finite type morphism X —> S of Noetherian algebraic spaces. Let A be a finite type S-algebra. Let F be a coherent sheaf on the base change X_{A} which is flat over A and has proper support over A. We want to write down some pseudo-coherent complex L on X_{A} such that for every surjection of S-algebras A’ —> A with square zero kernel I the ext groups

Ext

^{i}_{XA}(L, F ⊗_{A}I), i = 0, 1, 2

give infinitesimal automorphisms, infinitesimal defos, and obstructions.

Derived solution: If you know derived algebraic geometry, then you know how to solve this problem. I tried to sketch the approach in this remark and now I can answer the question formulated at the end of that remark as follows.

Namely, the question is to construct a complex L such that H^{0}(L) = F and H^{-2}(L) = Tor_{1}^{OS}(O_{X}, A) ⊗ F. The ingredient I was missing is a canonical map

c : L

_{XA/A}—> Tor_{1}^{OS}(O_{X}, A)[2]

You get this map quite easily from the Lichtenbaum-Schlessinger description of the cotangent complex (again, in terms of derived schemes, this follows as X_{A} is cut out in the derived base change by an ideal which starts with the Tor_{1} sheaf sitting in cohomological degree -1, but remember that the point here is to NOT use derived methods). OK, now use the Atiyah class

F —> L

_{XA/A}⊗ F[1]

and compose it with the map above to get F —> Tor_{1}^{OS}(O_{X}, A) ⊗ F[3]. The cone on this map is the desired complex L.

Yay!

PS: Of course, to actually prove that L “works” may be somewhat painful.

Cool. Is it any easier to prove this works when X is a closed (locally finitely presented) subscheme of an fppf S-scheme? I vaguely remember from the introduction of “Complexes Cotangents” that one of the motivations for Illusie was to prove a similar result relating the Atiyah class and the obstruction to deforming a coherent sheaf (of course in the case that X is flat over S).

I think it shouldn’t matter.

The Ext^0 and Ext^1 properties are already clear from standard stuff because the cohomology sheaf in degree -2 does not interfere. Thus the only thing to do is to construct the obstruction class in Ext^2. Once that is done a local calculation should show that vanishing of this Ext^2 class means that the map I ⊗ F —> Ker(O_{X_{A’}} —> O_{X_A}) ⊗ F is an isomorphism, and then the usual arguments will suffice.

Now that I think about it this is totally doable with what is already available in the stacks project.