Derived things are flat

In this post I try to come to grips with what Jason was trying to say to me in the comment on this post and in an email he sent me.

So we have currently enough background material built up in the Stacks project to start discussing an application of Artin’s axioms to a particular moduli problem. I plan to use the Quot functor as the first example. We discussed this in the blog post above, but not in complete generality. Jason’s suggestion addresses the issue of how to deal with the most general case (in some sense).

Setup: S is a Noetherian scheme, f : X —> S is a morphism of an algebraic space to S which is locally of finite type, and F is a finite type O_X-module. The Quot functor associates to a scheme T over S the set of quotients

F_T —> Q

which are of finite presentation as O_{X_T}-modules, have support proper over T, and are flat over T. Here X_T is the base change of X to T and F_T is the pullback of F to X_T.

I first discuss two cases where I believe I know exactly what to do. But… I haven’t worked out all the details so I could still be wrong.

Case I: F is flat over S. In this case everything works exactly as explained in the blog post above. If T = Spec(A) and 0 —> E —> F_A —> Q —> 0 is a quotient, then the obstruction group for a square zero thickening 0 —> I —> A’ —> A —> 0 of A is Ext^1_{O_{X_A}}(E, Q ⊗ I).

Case II: X is flat over S. In this case the suggestion in the fifth remark of the post works. In other words, given a quotient F_A —> Q we choose a distinguished triangle

E —> F ⊗^L A —> G —> E[1]

where F ⊗^L A means the derived pullback of F to X_A. In this case the obstruction group is still Ext^1_{O_{X_A}}(E, Q ⊗ I) except that the extension group is computed in the derived category of O_{X_A}-modules.

Case III: general. The trick used in Case II does not work anymore. A naive explanation of the problem is that to get the obstruction class one uses a distinguished triangle

F ⊗^L I —> F ⊗^L A’ —> F ⊗^L A —> F ⊗^L I[1]

But if X isn’t flat over S we cannot define the derived tensor products F ⊗^L A’ and F ⊗^L A as the derived pullback of F to X_{A’} and X_A. Jason’s suggestion (as I understand it) is that instead of using the usual base change X_{A’} we should work with the derived base change X ⊗^L A’ and similarly for A and then form the derived pullback of F to that. The idea is that X, viewed as a derived algebraic space, is flat over S. This strategy requires us to leave the world of algebraic spaces and enter the world of derived algebraic spaces.

Of course, we can try to approximate derived algebraic spaces. One remark is that if there exists a closed immersion i : X —> Y with Y an algebraic space of finite type and flat over S, then we can simply work with i_*F on Y. In general I do not know how to construct such an embedding. Locally we can, and we can try to use a Cech cover approach, but I don’t know how to make this work. Another idea (closer to Jason’s suggestion) is to choose a sheaf of flat f^{-1}O_S-algebras P on X and a surjection P —> O_X. Then we can try to work in the category of P-modules, etc. Finally, we can go whole hog and find a simplicial resolution of O_X by flat f^{-1}O_S-algebras and use that.

However, in each of these approaches the end product is an extension group on a derived object, let’s call it Y_A. Now, we are going to enter the realm of pure speculation (if we haven’t done so already — maybe all of this is a fictional reality — I can say that here since nobody has read this far into the post — have you?). Say we are looking at Ext^1_{Y_A}(E, Q ⊗ I). It seems quite likely to me that RHom_{O_{Y_A}}(E, Q ⊗ I) is a well defined object of D(O_{X_A}). Thus there still is a chance that we can define this gadget using non-derived methods and make things work in that way…

What do you think?

[Update 01/08/2013: It occurs to me now that, in the hypothetical situation above, we have

Ext^1_{Y_A}(E, Q ⊗ I) = Ext^1_{X_A}(Li_A^*E, Q ⊗ I)

where i_A : X_A —> Y_A is the embedding of the usual base change of X into the base change of any of our approximations Y of X as above. So actually the thing we should use on X isn’t E but Li_A^*E! And again there is a chance we can construct Li_A^*E using non-derived methods. This could be quite useful to the overall strategy. I’ll think more.]