Suppose $R$ is a local Noetherian complete intersection ring that is a finite $A$-algebra, where $A$ is a DVR.
If the module of differentials of $R$ is free as an $R/\mathfrak a$-module for some ideal $\mathfrak a$ of $R$, and if i take $a_1,...,a_n \in m_R$ (where $m_R$ is the maximal ideal of $R$) such that $da_1,...,da_n$ is a basis, is it true that $A[[a_1,...,a_n]]=R$? And suppose this holds: is it true that the map $x_i \to a_i$ induces an isomorphism $A[[a_1,...,a_n]]\simeq A[[x_1,...,x_n]]/(p_1,...,p_n)$ for some $p_1,...,p_n \in A[[x_1,....,x_n]]$?