2
$\begingroup$

I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity.

I'd want to call a DVR $(R,\mathfrak{m})$ pseudo-Henselian if for every [insert appropriate adjective(s)?] variety $V$ defined over $R$, whose generic fiber is absolutely irreducible, say, then if $V$ has a smooth point over $R/\mathfrak{m}$, then that point lifts to a point of $V$ defined over $R$.

This would be in analogy with pseudo-algebraically closed (PAC) fields: a field $K$ is PAC if every geometrically irreducible variety $V/K$ has a $K$-rational point.

----------------EDIT---------------

I guess I should have made it more clear what I am asking.

a) Does this definition seem appropriate/does it exist already by another name?

b) Is it actually equivalent to Henselian?

c) If not, can you give an example of a local ring that satisfies this property but is not Henselian?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ I'm not sure that I understand the distinction between your notion of "pseudo-Henselian" and the usual definition of Henselian. Did you want to impose that the geometric generic fiber of $V$ is irreducible? $\endgroup$
    – Jason Starr
    Commented Sep 26, 2013 at 11:32
  • $\begingroup$ @Jason I want to say something like that in the spot where I said [insert appropriate adjectives]. Certainly I want this definition to look weaker than Henselian. If it is equivalent to Henselian, that would be nice to know. I'll edit it but I'm still not sure I have the ``right'' definition. $\endgroup$
    – Bobby Grizzard
    Commented Sep 26, 2013 at 12:15

0

Reset to default

You must log in to answer this question.