# Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where $c_{n-1},\ldots,c_1,c_0 \in \mathbb{C}[x,y]$ and $c_0 \neq 0$.

What additional conditions are needed in order that $R/I$ will be a regular ring (=a noetherian ring for which every localization at a maximal ideal is a regular local ring)?

Is it true that $f \notin m^2$ for every maximal ideal of $R$ is a sufficient condition for $R/I$ to be regular?, see https://mathoverflow.net/questions/125469/when-r-f-is-regular

If so, since a maximal ideal $m$ of $R$ is of the following form: $m=(x-\alpha,y-\beta,z-\gamma)$, where $\alpha,\beta,\gamma \in \mathbb{C}$, see https://mathoverflow.net/questions/85915/maximal-ideals-in-a-polynomial-ring-over-the-real-numbers, how one shows that the above $f$ is not in $m^2$ ($m$ maximal)? Should we add additional conditions on $f$ in order to guarantee the regularity of $R/I$?

In "A generalized Jacobian criterion for regularity", Peter Seibt shows that a noetherian $k$-algebra $B$ ($k$ is a field of characteristic zero) is regular iff the $B$-module of $k$-differentials is $B$-flat. Is this theorem may help in my "specific" case, or no? I suspect no (since probably there is not enough information in order to show that the module of differentials is flat. Here $B=R/I$)-- Am I right?

• For a concrete answer see here: math.stackexchange.com/a/1121034/121097. (It took me a while to find it!) May 17, 2015 at 23:21
• Thanks! I really appreciate your help. I will read carefully the above concrete answer. May 17, 2015 at 23:38

Yes, $f\notin m^2$ for any maximal ideal $m$ is sufficient for proving the regularity.
For showing $f\notin m^2$ you need to check that not all the partial derivatives of $f$ at $(\alpha,\beta,\gamma)$ are $0$. (If $f(\alpha,\beta,\gamma)\ne 0$ there is no need to go further.)
• Thank you very much! Please, is the partial derivatives criterion appears in Eisenbud's book "Commutative algebra with a view toward algebraic geometry"? (Theorem 16.19 and Corollary 16.20). I guess the above $f$ (irreducible, etc.) is too general in order to check this sufficient criterion? May 17, 2015 at 23:24
• @user237522 Yes, you can use the Jacobian criterion. (I also think that your $f$ is a little too general.) May 17, 2015 at 23:28