Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bX-a)$. Is $S$ regular? If this is not the case are there some conditions on $a$ and $b$ (or on $R$) that imply regularity? For example $a=1$ is enough.
1 Answer
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If $R = k[x,y]$ and we throw in $\frac {x^2}y$, we get $k[x,y,z]/(zy-x^2)$, not regular. This shows that it's not enough to assume $a,b$ form a regular sequence. The only sufficient condition I can come up with is that $a$ be outside the square of (every/the) maximal ideal
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$\begingroup$ Thanks! Actually I'm interested in the case where $R= \mathbb Z_p[Y]$, with $a=Y$ and $b=p^n$ for some integer $n$. I think that in this case $S$ is always regular, do you agree with me? $\endgroup$– RickyMay 12, 2011 at 9:01
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$\begingroup$ Yes, $\mathbb{Z}_p[Y,Z]/(Zp^n-Y)$ is regular. $\endgroup$ May 12, 2011 at 10:44