Finally!

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 XA which is flat over A and has proper support over A. We want to write down some pseudo-coherent complex L on XA such that for every surjection of S-algebras A’ —> A with square zero kernel I the ext groups

ExtiXA(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 H0(L) = F and H-2(L) = Tor1OS(OX, A) ⊗ F. The ingredient I was missing is a canonical map

c : LXA/A —> Tor1OS(OX, 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 XA is cut out in the derived base change by an ideal which starts with the Tor1 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 —> LXA/A ⊗ F[1]

and compose it with the map above to get F —> Tor1OS(OX, 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.

2 thoughts on “Finally!”

1. 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.