We continue the discussion. Let R be a regular complete local ring with residue field k. Let S = R[[x_1, …, x_n]]. Let I ⊂ R and J ⊂ S be ideals such that IS &sub J and such that A = R/I —> B = S/J is a flat ring homomorphism. Consider the map
(**) I/m_RI —> J/m_SJ
In the previous post we claimed that the cokernel of this map is (J + m_RS)/(m_SJ + m_RS). To see this choose f’_1, …, f’_r ∈ J whose images f_1, …, f_r in S/m_RS form a minimal system of generators for the ideal (J + m_RS)/m_RS which is the ideal cutting out B/m_AB in S/m_RS. Think of B as a flat deformation of B/m_AB over A. Then by the discussion in this post, the flatness assumption implies that any relation between f_1, …, f_r in S/m_RS lifts to a relation in S/IS. Hence any element h of J ∩ m_RS is an element of IS + m_RJ as desired.
But more is true. Namely, recall from this post that with these choices we obtain an obstruction map
Ob : Rel/TrivRel —> S/(JS + m_RS) \otimes_k I/m_RI
(unfortunately the notation between these two posts isn’t compatible) where Rel is the module of relations between the f_1, …, f_r in S/m_RS. Now because J ⊂ m_S we can compose Ob with the canonical map S/(JS + m_R S) —> k to get a reduced map
Ob_reduced : Rel/TrivRel —> I/m_RI.
At this point an argument along the lines of the argument in the first paragraph shows that (**) is injective if this reduced obstruction map is zero (in fact I think it is equivalent, but I didn’t check this).
Having arrived at this point we see that it suffices to prove the following (changing back to the notation in the post on deformation theory):
(***) Given a proper ideal I in R = k[[x_1, …, x_n]], a minimal set of generators f_1, …, f_r for I with module of relations Rel and submodule of trivial relations TrivRel ⊂ Rel. Then any R-module homomorphism Rel/TrivRel —> R/I has image contained in m_R/I.
It turns out that this result is contained in a paper by Vasconcelos and with a little bit more detail on the proof it is Lemma 2 in this paper by Rodicio. The key appears to be the technique of Tate to find divided power dga resolutions of R/I combined with a technique for constructing derivations on dgas which is due to Gulliksen. We will return to this in a future post.