# Local complete intersection ring

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 $p_1,...,p_n \in A[[x_1,....,x_n]]$?

• What is "the" maximal ideal of $R$? Is $R$ local or graded? – Jesko Hüttenhain Sep 9 '14 at 17:45
• Yes sorry i forgot to say R local – user174627 Sep 9 '14 at 19:07
• What exactly do you mean by $A[[a_1, \ldots, a_n]]$? Also what does the ideal $\mathfrak{a}$ have to do with $a_1, \ldots, a_n \in m_R$? – zcn Sep 9 '14 at 20:05
• I mean the ring of power series in $a_1,...,a_n$. It is well defined since they live in the maximal ideal and you have that the ring is complete(so all the series are converging and to something in the ring). The connection between the variable and the ideal is that $da_1,...,da_n$ is a $A/a$ basis for the differentials – user174627 Sep 9 '14 at 20:35
• @user174627: There seems to be some duplication of this question - I assume you are the same asker of this question, which I have answered. You may ask the moderators to merge your accounts. Also, you can use the @ symbol to reply directly to a comment – zcn Sep 9 '14 at 20:57