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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 3rd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI finally created [[Hensel's lemma]], using the formulation in Bourbaki's Commutative Algebra III.4.3. I also want to put in the formulation as alluded to in [this M.SE answer](https://math.stackexchange.com/a/1347065/3835) to an old question of mine, which is more geometric. I also want to point out that at [[Henselian ring]] it might be worth expanding (and I can do this in a few weeks) to consider Henselian couples, where one no longer considers just the maximal ideal in a local ring, but any ideal contained in the radical. (This is a generalisation of the usual result in a different direction from Bourbaki's, and no doubt one can form the pushout of these lemmas.) This point of view is very geometric. There is no doubt an interesting treatment using the internal language of an appropriate topos of this circle of ideas.

   I finally created Hensel’s lemma, using the formulation in Bourbaki’s Commutative Algebra III.4.3. I also want to put in the formulation as alluded to in this M.SE answer to an old question of mine, which is more geometric.

   I also want to point out that at Henselian ring it might be worth expanding (and I can do this in a few weeks) to consider Henselian couples, where one no longer considers just the maximal ideal in a local ring, but any ideal contained in the radical. (This is a generalisation of the usual result in a different direction from Bourbaki’s, and no doubt one can form the pushout of these lemmas.) This point of view is very geometric.

   There is no doubt an interesting treatment using the internal language of an appropriate topos of this circle of ideas.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 3rd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI also needed to create [[restricted formal power series]], to describe one of the hypothesis of Bourbaki's version of Hensel's lemma.

   I also needed to create restricted formal power series, to describe one of the hypothesis of Bourbaki’s version of Hensel’s lemma.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 3rd 2017
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI will actually type up the proof before too long. There are too many proofs on the internet of Hensel's _original_ lemma only. Ditto for Raynaud's proof for characterisations of Henselian couples.

   I will actually type up the proof before too long. There are too many proofs on the internet of Hensel’s original lemma only.

   Ditto for Raynaud’s proof for characterisations of Henselian couples.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 11th 2017
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   Author: DavidRoberts
   Format: MarkdownItexAdded references at [[Hensel's lemma]].

   Added references at Hensel’s lemma.