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.<\/p>\n
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<\/sub> which is flat over A and has proper support over A. We want to write down some pseudo-coherent complex L on XA<\/sub> such that for every surjection of S-algebras A’ —> A with square zero kernel I the ext groups<\/p>\n Exti<\/sup>XA<\/sub><\/sub>(L, F ⊗A<\/sub> I), i = 0, 1, 2<\/p><\/blockquote>\n give infinitesimal automorphisms, infinitesimal defos, and obstructions.<\/p>\n 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<\/a> and now I can answer the question formulated at the end of that remark as follows.<\/p>\n Namely, the question is to construct a complex L such that H0<\/sup>(L) = F and H-2<\/sup>(L) = Tor1<\/sub>OS<\/sub><\/sup>(OX<\/sub>, A) ⊗ F. The ingredient I was missing is a canonical map<\/p>\n c : LXA<\/sub>\/A<\/sub> —> Tor1<\/sub>OS<\/sub><\/sup>(OX<\/sub>, A)[2]<\/p><\/blockquote>\n 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<\/sub> is cut out in the derived base change by an ideal which starts with the Tor1<\/sub> sheaf sitting in cohomological degree -1, but remember that the point here is to NOT use derived methods). OK, now use the Atiyah class<\/p>\n F —> LXA<\/sub>\/A<\/sub> ⊗ F[1]<\/p><\/blockquote>\n and compose it with the map above to get F —> Tor1<\/sub>OS<\/sub><\/sup>(OX<\/sub>, A) ⊗ F[3]. The cone on this map is the desired complex L.<\/p>\n Yay!<\/p>\n PS: Of course, to actually prove that L “works” may be somewhat painful.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading