# When do we know a quotient of the polynomial ring over a local ring is torsion free

Let $R$ be a local ring (e.g. the discrete valuation ring $\mathbb C[[T]]$) and $\mathfrak{m}$ its maximal ideal. Consider the polynomial ring $R[X_1,\dots, X_n]$ in $n$ variables and a finitely generated ideal $I=(f_1,\dots,f_l)$.

Question:

1. What conditions can guarantee $A=R[X_1,\dots,X_n]/I$ is torsion free (i.e. if $\mathfrak{m}^N x=0$ for some positive integer $N$ and $x\in A$ then $x=0$)? (it would be better if there is no Noetherian condition)
2. How about the ring of formal power series, $R[[X_1,\dots,X_n]]/I$ ?
• Have you worked through a simple case? Say one variable and the localization of $\mathbb Z$ at a prime? – hardmath Jul 15 '18 at 14:24
• I considered the case $R=\mathbb C[[T]]$ and $\mathfrak{m}=(T)$, which I prefer. – Hang Jul 15 '18 at 14:37
• If you're looking at valuation rings, then torsion free is equivalent to flat. And in that case, I think this paper of Jack Ohm will be your friend ams.org/journals/tran/1972-171-00/S0002-9947-1972-0306176-6/… – Badam Baplan Jul 15 '18 at 16:26
• @BadamBaplan Thanks. It seems that our notions of torsion-free are slightly different. Here we'd better to say adic torsion free. – Hang Jul 15 '18 at 17:08