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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 17th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexunmotivated stub for [[Henselian ring]]

   unmotivated stub for Henselian ring

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeDec 17th 2010
   • (edited Dec 17th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexHenselian rings are local rings closed under pointed etale extensions. They are the local rings classified by the large nisnevich topos. Here, by a pointed etale extension of a local ring $R$, we mean a local etale $R$-algebra (by local, we mean that it's a local homomorphism of local rings) that induces an isomorphism on residue fields. The Henselization of a local ring is the maximal pointed etale extension. Strict henselization and strictly Henselian rings are what you get when you look at not-necessarily-pointed etale extensions.

   Henselian rings are local rings closed under pointed etale extensions. They are the local rings classified by the large nisnevich topos.

   Here, by a pointed etale extension of a local ring $R$, we mean a local etale $R$-algebra (by local, we mean that it’s a local homomorphism of local rings) that induces an isomorphism on residue fields.

   The Henselization of a local ring is the maximal pointed etale extension.

   Strict henselization and strictly Henselian rings are what you get when you look at not-necessarily-pointed etale extensions.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 17th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexGiven your MO success with Henselian rings, you should have enough confidence to put that stuff into the nLab entry. I won't do it for you. I will go to bed now! :-)

   Given your MO success with Henselian rings, you should have enough confidence to put that stuff into the nLab entry. I won’t do it for you. I will go to bed now! :-)

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 2nd 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI have been expanding a bit at [[Henselian ring]], but there's loads more one could put down. While poking around for this, I ran into Mike's [MO question](http://mathoverflow.net/questions/48690/what-does-an-etale-topos-classify) and Harry's answer, but Mike's last question in the comments lies unanswered, and thus the answer seemed ever so slightly inconclusive. There's the big etale topos of a (commutative) ring $R$ which I guess classifies strict Henselian local algebras over $R$ (hm, is that right?), and then there's the little etale topos of $R$ which classifies... strict henselizations of... what? Localizations at primes of $R$? There are some old topos-theory oriented books from the 70's which go into such matters, I believe, but I no longer recall which ones. I seem to associate the name Gavin Wraith with one article I had found sort of helpful at the time I was reading such things.

   I have been expanding a bit at Henselian ring, but there’s loads more one could put down.

   While poking around for this, I ran into Mike’s MO question and Harry’s answer, but Mike’s last question in the comments lies unanswered, and thus the answer seemed ever so slightly inconclusive. There’s the big etale topos of a (commutative) ring $R$ which I guess classifies strict Henselian local algebras over $R$ (hm, is that right?), and then there’s the little etale topos of $R$ which classifies… strict henselizations of… what? Localizations at primes of $R$?

   There are some old topos-theory oriented books from the 70’s which go into such matters, I believe, but I no longer recall which ones. I seem to associate the name Gavin Wraith with one article I had found sort of helpful at the time I was reading such things.