Fppf Extensions

Let X = Spec(A) be an affine scheme. Let M, N be A-modules. Let F, G be the sheaves of O_{big}-modules on the big fppf site of X associated to M and N, e.g., F(Spec(B)) = B ⊗_A M and similarly for G. As a by-product of the material on adequate modules I proved the following formula

Ext^i_{O_{big}}(F, G) = Pext^i_A(M, N)

The ext group on the left is the ext group in the category of all O_{big}-modules. The ext group on the right is the ith pure extension group of M by N over A. This group is computed by taking a universally exact resolution 0 -> N -> I^0 -> I^1 -> … with each I^j pure injective and taking the ith cohomology group of Hom_A(M, I^*). An A-module I is pure injective if for any universally injective map M_0 -> M_1 the map Hom_A(M_1, I) -> Hom_A(M_0, I) is surjective.

There seems to be a lot of papers on pure modules, pure injectivity, etc. Gruson and Jensen characterized pure injective modules as those modules such that the functor – ⊗_A M : mod-A —> Ab is injective in the functor category (mod-A, Ab). Here mod-A is the category of finitely presented A-modules. It follows that

Ext^i_{(mod-A, Ab)}(- ⊗_A M, – ⊗_A N) = Pext^i_A(M, N)

Our formula above is about O_{big}-modules, which in terms of functors means functors F : Alg_A —> Ab such that F(B) has the structure of a B-module for every A-algebra B and such that B —> B’ gives a B-linear map F(B) —> F(B’). These are called module-valued functors (terminology due to Jaffe). Then we can rewrite the first equality above as

Ext^i_{module-valued functors}(F, G) = Pext^i_A(M, N)

where F(B) = B ⊗_A M and G(B) = B ⊗_A N. In this formula you can let Alg_A be any sufficiently large category of A-algebras, e.g., the category of finitely presented A-algebras.

The two results seem related. But there is a big difference between the functor categories (mod-A, Ab) and (Alg_A, Ab). Namely, if we look at Ext^i_{(Alg_A, Ab)}(F, G) then we get a completely different animal. For example suppose that G_a(B) = B for all A-algebras B and suppose that A is an F_p algebra. Then we see that Hom_{(Alg_A, Ab)}(G_a, G_a) contains the frobenius map frob : G_a —> G_a which on values over B raises every element to the pth power. In fact, the work of Breen on ext groups of abelian sheaves on the fppf-site (warning: this is not exactly what he studies there) implies some of the higher ext groups Ext^i_{(Alg_A, Ab)}(G_a, G_a) are nonzero also (lowest case seems to be i = 2p)!

Conclusion: module-valued functors over Alg_A and abelian group valued functors on mod-A somehow ends up giving the same ext groups for the functors associated to A-modules described above.