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