# Elementary questions about regular rings and Zariski tangent spaces

So I've got 3 rather related questions, which all seem to be true, except maybe the third. I'm asking because I remember thinking about this in the past and encountering a difficulty with all 3.

First question:

Let $S$ be an integral domain, and $\mathfrak{m}$ a maximal ideal, and $\mathfrak{p}$ a prime ideal contained in $\mathfrak{m}$. Is the Zariski tangent space of $S/\mathfrak{p}$ at $\mathfrak{m}$ just $\mathfrak{m}/(\mathfrak{p}+\mathfrak{m}^2)$ ?

Second question: To what extent are these following (increasingly general) statements true?

Let $R$ be a DVR. Let $f\in R[x_1,\ldots,x_n]$ be such that $(f)$ is prime, then $R[x_1,\ldots,x_n]/(f)$ is regular if and only if $f\notin\mathfrak{m}^2$ for any maximal ideal $\mathfrak{m}$ of $R[x_1,\ldots,x_n]$ containing $f$.

Let $S$ be an integral domain of dimension $d$, and $\mathfrak{p}$ a prime ideal such that $S/\mathfrak{p}$ has dimension $d-k$, then $S/\mathfrak{p}$ is regular if and only if for every maximal ideal $\mathfrak{m}$ of $S$ containing $\mathfrak{p}$, the image of $\mathfrak{p}$ in $\mathfrak{m}/\mathfrak{m}^2$ generates a sub-(vector)-space of dimension $k$.

Thorough proofs or references to thorough proofs would be appreciated.

thanks,

• will
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For the second question you don't need $(f)$ to be prime. In my answer here I gave a proof for the local case that extends easily to the global case. – user26857 Dec 6 '12 at 9:22
As far as I can see you have also a third question. This seems to be false for $\mathfrak p=0$. – user26857 Dec 6 '12 at 9:37

(1) the cotangent space is $(\mathfrak{m}/\mathfrak{p})/(\mathfrak{m}/\mathfrak{p})^2 = \mathfrak{m}/(\mathfrak{m}^2 + \mathfrak{p})$.