# Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack which sends a ring $R$ to the category of formal group laws over it where the morphisms are the isomorphisms between FGLs (given by invertible power series). According to Lurie, a quasi-coherent sheaf on this stack should be the assignment of an $R$-module $M_R$ to each morphism $\mathrm{Spec} R \to M_{FG}$ (i.e. to each "formal group"* over $R$ an $R$-module) together with compatibility isomorphisms $M_R \otimes_R R' \simeq M_{R'}$ for every commutative diagram involving $\mathrm{Spec} R'$ (i.e. for every formal group over $R$ and morphism $R \to R'$ carrying that to a formal group over $R'$). This makes sense, but what confuses me is that it's not obvious when presented in this way why quasi-coherent sheaves should form an abelian category. The cokernel is easy enough (i.e. do it pointwise), but taking the kernel pointwise doesn't seem to work because of the need for these compatibility isomorphisms of modules. How should I then think of the kernel?

(I don't know anything about stacks, but I thought that usually one restricted to the case of étale morphisms of rings, in which case this problem does not occur.)

Also, more specifically to these notes: is there any particular reason that a distinction needs to be made between what Lurie calls $M^s_{FG}$ (i.e. the stack of FGLs and isomorphisms which are the identity to first order) and $M_{FG}$ defined above? For instance, the Landweber exact functor theorem is formulated as a criterion for a sheaf associated to an $L$-module to be flat over $M_{FG}$; does it work any differently for $M_{FG}^s$? (My impression was that the exact functor theorem was classically stated for modules over $L$ which are comodules over $MU_*(MU)$, i.e. quasi-coherent sheaves on $M_{FG}^s$, if I understand correctly. What's the distinction?)

*A formal group here is something obtained by gluing formal group laws along isomorphisms Zariski locally.

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I am not sure if the answer to your question will be in the following, but I always like having more sources than fewer, have you looked at any of the stuff Paul Goerss has written on the subject? He even gave a mini course on it at the fields institute (you can listen to the streaming audio, but his voice is a little quiet). –  Sean Tilson Jan 7 '12 at 3:55
No, I haven't. I should probably have a look at his paper "Quasi-coherent sheaves on the moduli stack of formal groups." –  Akhil Mathew Jan 7 '12 at 4:08
Since $M_{FG}$ is a stack in the étale topology, it is enough to consider étale $R \to R'$ in the definition of a quasi-coherent sheaf. –  Martin Brandenburg Jan 7 '12 at 10:08
@Martin: Are you saying that it gives the same category of quasi-coherent sheaves? –  Akhil Mathew Jan 7 '12 at 14:08
I don't think it is enough to impose compatibility under etale maps. For example, setting M_R = Omega^1_{R/Z} with the evident pullback maps satisfies M_R (x)_R R' =~ M_{R'} for any R ---> R' etale, but does not come from a quasi-coherent sheaf. However, if X is Deligne-Mumford, then giving a quasi-coherent sheaf on X is equivalent to specifying an R-module M_R for each etale map Spec(R) ----> X, and the compatibility isomorphism for each etale map Spec(R') ----> Spec(R) over X (i.e., work on the etale site of X). For X an Artin stack, replace 'etale' with 'smooth' in the previous sentence. –  Bhargav Jan 7 '12 at 21:32