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

user174627
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  • What is "the" maximal ideal of $R$? Is $R$ local or graded? – Jesko Hüttenhain Sep 09 '14 at 17:45
  • Yes sorry i forgot to say R local – user174627 Sep 09 '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 09 '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 09 '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 09 '14 at 20:57
  • @zcn yes thanks a lot. I tought that no one would have answered that one because it was too messy so i forgot to check again, and was waiting for an answer here. Anyway yes, your answer there gives me exactly what i needed(i was not aware of that theorem)! – user174627 Sep 09 '14 at 21:13
  • @user174627: In that case, I recommend you delete some of the duplicate questions, as well as merge your accounts (to do so you can flag a moderator with a custom reason). Also, please consider accepting the answer given, to remove it from the unanswered queue. – zcn Sep 09 '14 at 21:27
  • @user174627: If you worry that a question is messy, simply edit it. It is far messier for the site to have multiple copies of the same question floating about. It is also easier to have a single account. Then you don't need to change accounts to handle different questions or answers. – robjohn Sep 10 '14 at 17:00
  • @robjohn:Right i tought of editing but there was a problem, that here question fastly disappear downstairs with more question posted; and it happened to me in the past(with yet another account :-)) that also when the question was polished nobody answered because it was too far from the top question. Then i re-wrote the question in another form in a new post and tadaaà immediate answer. That should be enough for you to understand my motivation in re posting – user174627 Sep 10 '14 at 19:19
  • @user174627: if you had only one account, it would be easy to find all of the questions you've written. When you edit it to be neater, the question will rise to the top of the list of questions again. There is never any good motivation to repost. Please, just improve the original question. – robjohn Sep 10 '14 at 20:04

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