Let $(R,\mathfrak{m},k)$ be a discrete valuation ring, (of characteristic $p$ if you need).

Let $n\geq 1$ be an integer.

Is the ring $\frac{R}{\mathfrak{m}^n}[x]$ regular?

Note that: Regularity can be checked at localisation of maximal ideals.

  • $\begingroup$ This is part of a general phenomenon: if $A$ is regular and $a\in A$ belongs to the square of a maximal ideal $\mathfrak m$, then $A/(a)$ is not regular. $\endgroup$ – user26857 Feb 17 at 15:50
  • $\begingroup$ In your case $A=R[x]$, and $a=t^n$. Then $a\in M^2$, where $M=(\mathfrak m,x)$. In fact, the localization of $A$ at $M$ is not regular. $\endgroup$ – user26857 Feb 17 at 15:55
  • 1
    $\begingroup$ @user26857 Thank you very much, it's very helpful. $\endgroup$ – Lance Wu Feb 17 at 15:58

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