This is just a quick note so if people google this issue they see that one needs to be careful (I think there are some positive things in the literature in special situations).

Let k be a field. Let R be an algebra over k. Let M be an R-module.

For an Artinian local k-algebra (A, m) with residue field k denote F(A) the set of isomorphism classes of pairs (M_A, phi) where M_A is an R \otimes_k A module and phi is an isomorphism of M_A/m M_A with M.

The functor F satisfies condition H1 of Schlessinger’s paper as one can see by using 07RU. Details omitted.

However, the functor F doesn’t satisfy condition H2 in general. As an example, with notation as in Schlessinger‘s Theorem 2.11, take R = k, M = k, A = k, A’ = k[epsilon_1], and A” = k[epsilon_2] where the epsilon have square zero. Then A’ ×_A A” = k[epsilon_1, epsilon_2]/(epsilon_i epsilon_j) and we see that the modules M(c) = A’ ×_A A’/(epsilon_1 – c epsilon_2] for any nonzero c in k map to the same elements of F(A’) and F(A”) for all c. But these modules are all pairwise nonisomorphic.

So of course(!) the functor F in the special case doesn’t have a hull.