Is a localization at a maximal ideal of a polynomial ring a perfect ring?

Question

There are several equivalent definitions for a perfect ring $R$ (not necessarily a commutative ring), for example: Every flat left $R$-module is projective; see wikipedia.

Also, there is the notion of semiperfect rings, which include, for example, local rings.

Questions: Is a regular local ring perfect? (probably no). At least, is a localization of a polynomial ring at a maximal ideal perfect?

For example, is $R=\mathbb{C}[x]_{(x)}$ perfect? Is $\mathbb{C}[x,y]_{(x,y)}$ perfect?

Recall that finitely generated flat modules over an integral doamin are projective, though I am not sure if this result helps here.

Also see this relevant question.

Any hints and comments are welcome! Thank you.

Answer 1

The answer to both question is "no" in a kind of extreme way.

A corollary of Bass's Theorem $P$ says that a commutative ring is perfect iff it satisfies the DCC on principal ideals.

A domain satisfying the DCC on principal ideals is already a field.

So it is not very fruitful idea to look for perfect rings among domains (in particular regular local rings or any polynomial ring over a field.)

Answer 2

The answer is 'no'. $\mathbb{C}[x]_{(x)}$ is a principal ideal domain (PID), even a discrete valuation ring (DVR). Over PIDs, we have

• flat $\iff$ torsionfree
• projective $\iff$ free

So it suffices to give an example of a torsionfree module which is not free. Just take the field of fractions (in case your PID is not a field itself).