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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2012
   • (edited Mar 19th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSince its creation, the entry _[[étale map]]_ started out as wanting to be about the abstraction of the notion of _[[local homeomorphism]]_ and _[[étale morphism]]_ to more general contexts. I have added references to such axiomatizations now. But I am not sure that it is good to have "étale map" be interpreted so much differently from "étale morphism". I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is _[[étale morphism]]_ should be renamed to "étale morphism of schemes". Finally, I changed the wording of the Idea-section at the beginning of [[étale map]]: it used to say that an étale map is like a "local isomorphism". As we have recently seen in discussion, this is a very misleading thing to say, since it is _not_ like a _[[local isomorphism]]_ but like a _[[local homeomorphism]]_. These two concepts, maybe unfortunately termed, are not about the same idea.

   Since its creation, the entry étale map started out as wanting to be about the abstraction of the notion of local homeomorphism and étale morphism to more general contexts.

   I have added references to such axiomatizations now. But I am not sure that it is good to have “étale map” be interpreted so much differently from “étale morphism”. I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is étale morphism should be renamed to “étale morphism of schemes”.

   Finally, I changed the wording of the Idea-section at the beginning of étale map: it used to say that an étale map is like a “local isomorphism”. As we have recently seen in discussion, this is a very misleading thing to say, since it is not like a local isomorphism but like a local homeomorphism. These two concepts, maybe unfortunately termed, are not about the same idea.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2012
   • (edited Mar 19th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI wrote: > But I am not sure that it is good to have "étale map" be interpreted so much differently from "étale morphism". I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is étale morphism should be renamed to "étale morphism of schemes". I went ahead and implemented this. Give me a minute or two now to clear the cache to avoid the cache bug kicking in... So there is now 1. [[etale map]] = [[etale morphism]] pointing to the general notion 1. [[etale morphism of schemes]].

   I wrote:

   But I am not sure that it is good to have “étale map” be interpreted so much differently from “étale morphism”. I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is étale morphism should be renamed to “étale morphism of schemes”.

   I went ahead and implemented this. Give me a minute or two now to clear the cache to avoid the cache bug kicking in…

   So there is now

   1. etale map = etale morphism pointing to the general notion

   2. etale morphism of schemes.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt _[[etale map]]_ I removed the following stubby lines, as they are redundant by the discussion at _[[etale morphism of schemes]]_: *** An étale map between [[commutative rings]] is usually called a _[[étale morphism]]_ of rings: a ring homomorphism with the property that, when viewed as a morphism between [[affine schemes]], it is étale. See [this comment by Harry Gindi](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=716&page=1#Item_25) for a purely ring-theoretic characterisation. +-- {: .query} Zoran: that is the infinitesimal lifting property for smooth morphisms, need an additional condition in general. =-- *** By the way, Zoran has still another old query box sitting at _[[etale map]]_. would be nice if somebody would take care of editing this.

   At etale map I removed the following stubby lines, as they are redundant by the discussion at etale morphism of schemes:

   ---

   An étale map between commutative rings is usually called a étale morphism of rings: a ring homomorphism with the property that, when viewed as a morphism between affine schemes, it is étale. See this comment by Harry Gindi for a purely ring-theoretic characterisation.

   +– {: .query}

   Zoran: that is the infinitesimal lifting property for smooth morphisms, need an additional condition in general.

   =–

   ---

   By the way, Zoran has still another old query box sitting at etale map. would be nice if somebody would take care of editing this.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJan 7th 2020
   • (edited Jan 7th 2020)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI\'ve corrected an imprecision that Zoran objected to (the query box referred to in Urs\'s previous comment from over 6 years ago). Hopefully my correction is correct. <a href="https://ncatlab.org/nlab/revision/diff/%C3%A9tale+map/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/%C3%A9tale+map/19">v19</a>, <a href="https://ncatlab.org/nlab/show/%C3%A9tale+map">current</a>

   I've corrected an imprecision that Zoran objected to (the query box referred to in Urs's previous comment from over 6 years ago). Hopefully my correction is correct.

   diff, v19, current