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created geometric fibre. Can someone lease check these algebraic geometry entries as that area is quite far from my safety zone! so I will get some things wrong.
I have
expanded the text
added formatting
added hyperlinks
added redirects
cross-linked with fiber and geometric point
harmonized the notation to be the same in the Idea-section and the Definition-section
A question: why do you require in the end the bundle to be finite étale? I would rather state the definition generally and then add a Properties-section that discusses what happens in this special case.
I side very much mathematically with your sentiment about removing finite etale, but with algebraic geometry having quite well thought and detailed terminology lets not change this unless we find some authority who used it more generally.
unless we find some authority who used it more generally.
Which “authority” says, conversely, that the term must be restricted to finite etale maps?
Sorry. I hurriedly ’finished’ that entry. The finite étale idea was initially gioing to be an example rather than integrated in the text, but something came up here and I stopped editing it. When I restarted that idea had got lost somewhere! I will look at the present version and adapt it so as to include the idea I had had.
I am trying to find a reasonable reference for the definition, but to no avail as yet. I note that there is more work to do on EGA, but do not see quite what is needed.
There is still work to do on this. I found a reference in EGA p.112 (I.3.4.5), but would prefer to include a more recent one, as the terminology is still not quite correct I think.
Thanks.
Notice that things like finite set, local ring etc. all have Lab entries. And notably also EGA does! I added these as hyperlinks.
And I turned the Remark-section into an actual Remark environment to go right after the definition. I think that’s where it belongs, not after the example.
I have generally kept removing “Remark”-sections. We used to have them a lot in the beginning of the Lab. But I find it a bit awkward. No mathematical text has a section of remarks. If remarks are actually really remarks they should then go right after the definition or theorem that they are remarking on, or if they are in fact propositions themselves, then they should go in the Properties section.
I think. Of course you may disagree. But have a look at the entry and give the idea a chance.
My one query is whether EGA should point to the n-Lab entry or to the pdf file. Otherwise it is beginning to shape up nicely.
I was ’spurred’ on to look at these pages partially by your question as to the ’interest’ of the abc conjecture. It seemed to me that your question could only be answered by attempting to fill in some of the background from the algebraic and arithmetic geometry. I am also involved in writing some stuff on profinite homotopy theory and wanted to refresh my memory about the algebraic geometry that is related to it and the anabelian stuff.
My one query is whether EGA should point to the n-Lab entry or to the pdf file.
If we have nLab pages on references entries should always point to thes pages. That page should then provide all the further relevant links. This way we can provide much more useful information than if we directly linked to the pdf-s.
That is fine by me, but more work needs doing on EGA. I will see what I can added there.
Ah, I see. Yes the entry EGA should start with a one-sentence explanation “EGA is…” then right afterwards there should be the link to the pdf/numbdam prominently. And then after that can come more background discussion as is currently in the entry.
But I see that you are editing it. Thanks for taking care of this.
EGA is unfinished. The listed volumes I-IV are just a part of the original plan. AG outlines what should be in ch. V,VI,VII, at least and some handwritten notes existed for small part of those. See e.g. these pre-notes for some parts of EGA V.
Wikipedia lists titles of planned chapters I-XIII.
I noted this in the entry.
For chapter VI, wikipedia says
Did not appear. Descent theory and related construction techniques treated by Grothendieck in FGA.
We should complement this at wikipedia noting that the descent is treated also by Gabriel and Grothendieck in SGA I. FGA has pseudofunctors and SGA I.6 introduces the fibered category point of view.
I was trying to find a usable list of all chapters plus missing bits, but did not find one that I liked. My idea is that FGA and possibly SGA should be left at EGA as summaries as they are now, and new fuller pages on FGA (at least) be created, but then that page should have the list of sections in the FGA and I could only find a scanned version, so left it for the moment. Do add what you suggested? Perhaps we need to expand SGA1 a bit as well although some of what I would have put there already exists at Galois theory. What do people think?
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