The key to the solution to (***) which we formulated here is the construction of a derivation on the Tate resolution of R/I. I am not going to explain this in detail, partly because I do not have a good intuition for why this derivation should exist. So I am only going to give you the flavor of the thing: although the following is correct in spirit it is likely not completely correct in all details (in particular, some of the arguments below should not be done on the level of co/homology but rather on the level of complexes).

Let R = k[[x_1, …, x_n]] and I = (f_1, …, f_r) ⊂ m_R with r minimal for I. Consider as before the sequence

0 — > Rel —> R^{⊕ r} —> I —> 0

Note that if (***) is false, then we obtain a surjective map ξ : Rel —> R/I which annihilates the submodule TrivRel of trivial relations. This in particular implies that Rel/TrivRel = R/I ⊕ (Other part). Because R^{⊕ r} —> R —> R/I —> 0 is the beginning of a minimal resolution we see that we obtain an element e(ξ) ∈ Ext_R^2(R/I, R/I). Cupping with e(ξ) determines an operation

e(ξ) : Tor_n(R/I, M) —> Tor_{n – 2}(R/I, M)

for any module M (in particular M = k or M = R/I). Use Tate’s method to find a free dga with divided powers A which is a resolution of R/I as in the post here. It turns out that the fact that ξ is zero on trivial relations implies that we can represent e(ξ) by a derivation j : A —> A compatible with divided power structure which is homogeneous of degree -2 (this is where Gulliksen whose name I mentioned before comes in). On the other hand, the decomposition Rel/TrivRel = R/I ⊕ (Other part) implies that there exists a nonzero element x in Tor_2(R/I, k) such that δ(x) is nonzero (as an element of k). But this is a contradiction because some divided power of x is zero (as R is regular so finite projective dimension) and on the other hand δ^n(γ_n(x)) = (δ(x))^n is nonzero.

It is really a beautiful trick (apparently due to Gulliksen) to play off against each other the derivation pushing things down in degree and the divided power structure to go back up into the area where the Tors are zero.