Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am reading this paper

and I was trying to construct examples of excellent Henselian regular local rings containing a field that are not complete, but could not come up with any. Can someone enlighten with an example?

share|cite|improve this question

1 Answer 1

up vote 4 down vote accepted

Inside the ring $k[[ x]]$, consider the subring $R$ of elements which are algebraic over $k(x)$. The ring $R$ is Henselian because the Henselian condition is all about equations being solvable, so you can always satisfy the Henselian condition with algebraic elements. I leave it to you to see that $R$ is a regular local ring (of dimension $1$) and is not complete.

share|cite|improve this answer
Thank you, David. I don't think I would have come up with something like this on my own. – Sasukaro T Oct 5 '12 at 3:58
Just noted the adjective "excellent" above. Assuming $char(k)=0$, this is a characteristic zero Dedekind domain, so we're OK. I assume we are also OK in finite characteristic, but I was never good at working with the excellent condition. – David Speyer Oct 10 '12 at 19:40

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.