Timeline for Inaccessible cardinals and Andrew Wiles's proof
Current License: CC BY-SA 4.0
32 events
| when toggle format | what | by | license | comment | |
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| May 22 at 16:52 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https
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| Jun 18, 2021 at 13:40 | comment | added | David Roberts♦ | @BCnrd for what it's worth check the edit at the end of my answer for more recent developments on proving universes and indeed full ZFC unnecessary for all the relevant "large structure" technology inside SGA. | |
| Jun 18, 2021 at 13:38 | comment | added | David Roberts♦ | @PeteL.Clark for what it's worth Colin McLarty has results that show not even ZFC is needed for derived functor cohomology (and he claims "all of the relevant EGA/SGA"), see the edit to my answer to this question. It's not just arguments like in the paper you link, or historical citation tracing by involved parties and special case analysis. | |
| Jun 18, 2021 at 13:22 | comment | added | David Roberts♦ | @PeterLeFanuLumsdaine I have put in a proper reference and link, though some kind soul had added a Wayback machine link in the meantime. | |
| Jun 18, 2021 at 13:22 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Added paper details and doi link
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| S Aug 31, 2020 at 15:14 | history | suggested | CommunityBot | CC BY-SA 4.0 |
changing the link to an archived version — current one is dead.
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| Aug 31, 2020 at 14:37 | review | Suggested edits | |||
| S Aug 31, 2020 at 15:14 | |||||
| May 13, 2019 at 15:33 | comment | added | Peter LeFanu Lumsdaine | Unfortunately the link to the article is no longer active, and it’s no longer clear what article was linked. | |
| Aug 17, 2010 at 8:50 | vote | accept | Cosmonut | ||
| Aug 17, 2010 at 1:27 | comment | added | Harry Gindi | Pete and I have both removed our comments. | |
| Aug 16, 2010 at 23:01 | comment | added | Yemon Choi | While the last two comments (by HG and PLC) hold some amusement/interest, might I respectfully suggest they don't really belong as MO comments? | |
| Aug 16, 2010 at 15:10 | history | edited | BCnrd | CC BY-SA 2.5 |
made it gender neutral
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| Aug 16, 2010 at 14:34 | comment | added | Torsten Ekedahl | Dear BCnrd, I kinda figured it was something like that. I think I'll pass on that issue until I find myself needing it. In practice crystalline cohomology seem to me to be mostly a formalism for constructing appropriate complexes of sheaves in the Zariski topology. | |
| Aug 16, 2010 at 14:23 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 281 characters in body
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| Aug 16, 2010 at 14:16 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
added 533 characters in body
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| Aug 16, 2010 at 13:57 | comment | added | BCnrd | Dear Torsten: For the crystalline theory, the issue is to control the "size" of the nil-thickenings one allows to consider in the sheaf theory. In the etale case, by restricting to small etale site on a fixed scheme (sheaves determined by affines etale over the base) we see that Hom between sheaves is a set (so Tohoku proof of enough inj's via transfinite induction on Hom sets can be applied). For crystalline case, seek analogous "small" site for same needs (e.g., want Ext's) while also retaining a form of the crucial ring fiber product trick early in the theory. Perhaps another 550 pts... | |
| Aug 16, 2010 at 13:46 | comment | added | BCnrd | Dear Torsten: Thanks for clarification on Deligne's comment. Milne indeed refers to SGA4 in some places. When something I wanted to understand wasn't proved in Milne's book or F-K then either relevant parts from sga4 were covered nicely in sga 4.5 or could be understood directly from SGA4 without going back into the frightening volume 1....or in some cases (like aspects of torsion Kunneth in derived categories, which is really fearsome in sga4 and whose proof in F-K is too slick because the crucial relation with cup product is omitted) it was easier to work out the proofs by oneself. | |
| Aug 16, 2010 at 13:30 | comment | added | BCnrd | Dear Junkie: There are 2 issues. One is sheaf-theoretic and cohomological stuff on general topoi (sheafification, construction of enough injectives, making Hom between two sheaves into a set, etc.). For real examples like small etale site, Berkovich etale cohom, etc., no need for metathm or "elimination recipe" if one pays attention to it when reading all proofs. The 2nd issue is univ. mapping properties on "entire category" (your comment about "whole categories of sheaves"). That's an issue for category theory in general, not just SGA; handled as for flatness against "all" mods over a ring. | |
| Aug 16, 2010 at 13:25 | comment | added | Torsten Ekedahl | @BCnrd: As I recall it Milne's book refers to SGA 4 at a few places. I have not read Freitag-Kiehl however. What Deligne agreed with was that the answer to Manin's problem was that inaccessible cardinals are not needed (the comment about recursivity was just to show that the concern about the use of inaccessible cardinals seems to focus on the wrong level). I am not sure which detail of the crystalline theory you are referring to (in practice it seems that at least the original crystalline theory is dealing even less with general sheaves than the étale one does). | |
| Aug 16, 2010 at 12:49 | comment | added | Junkie | Link has W2/SGA4.5 in #7: While Deligne often uses Unis he stresses in conversation that they are a convenience technically eliminable in favor of ZFC. The theorems used in practice can always be given in terms of individual sheaves on small sites, without ever looking at whole categories of sheaves let alone categories of categories of them..This is a recipe for eliminating Unis from any use of Gr’s cohomology in NT..Though its obvious in practice that it could always be done, its not done in publications, and has never been made a precise metatheorem. Anyone interested should give it a try. | |
| Aug 16, 2010 at 12:31 | comment | added | BCnrd | Dear Pete: There remains a certain aspect of the "sheafification" process on the crystalline site that I haven't figured out how to carry out on general schemes (in a manner consistent with how the proofs of the theory work, especially a certain fiber-product construction with subrings) without the universe stuff. Some day I'll figure it out. (Torsten, do you know how to do it?) But this is exactly why I have always been careful to reject using universes in proofs of things I care about: so I'd never get dragged into concerns of the sort in the article you linked. | |
| Aug 16, 2010 at 12:26 | comment | added | BCnrd | Dear Torsten: I learned etale cohom. mainly from books of Milne and Freitag-Kiehl and later parts of SGA 4 (and some SGA 4.5 and Verdier's duality paper) which don't depend on the theory of cohomology on an arbitrary topos once one has the cohomology theory up and running from somewhere (such as the other sources). I never saw anything which required universes in that, and it gave all I needed to read Weil II. So I wonder: what specifically was Deligne agreeing with? Is there a specific example? (Etale cohom. may not be "recursively computable", but that seems to be a separate matter.) | |
| Aug 16, 2010 at 12:25 | comment | added | Torsten Ekedahl | Just a clarification: My last comment was aimed at pointing out that there may be questions remaining about the (prooftheoretic?) complexity of étale cohomology but they lie in a completely different part of the complexity hierarchy than does dependence on inaccessible cardinals. | |
| Aug 16, 2010 at 12:12 | comment | added | BCnrd | Section 1 of link shows the author "reasoning" about math he does not understand, and rests on the bogus argument that since there is one general reference on cohomology by Gr. written with universes, appeal to any cohomological results of Gr. has the same dependence. The result of Tate which concerns the author was inspired by Gr. duality but is just a piece of pure comm. algebra. Deligne-Rap. use Gr. duality, but Gr. duality requires no universes. (Improvements in the method removed Gr. duality, by the way.) Proofs exist to impart understanding; the author should have read some of them. | |
| Aug 16, 2010 at 12:09 | comment | added | Torsten Ekedahl | (cont'd) Having seen that I asked Deligne about what he thought and he agreed with me. (He also told me that he had spent some time trying to see if étale cohomology was recursively computable and had been able to conclude that with torsion coefficients it was but had been unable to get recursively computable bounds on the order of torsion in $\ell$-adic cohomology.) | |
| Aug 16, 2010 at 12:08 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
small change in language to make a more nuanced statement
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| Aug 16, 2010 at 12:06 | comment | added | Torsten Ekedahl | I completely agree with BCnrd opinions here but as far as I know there is no published complete proof of the relevant results that bypasses SGA IV so in that sense it is not an established fact. It is perhaps also interesting to note that (as far as I can remember) Manin poses the question on whether Deligne's proof of the Riemann hypotheses can be made independent of inaccessible cardinals in the problem list in Mathematical Developments Arising from Hilbert Problems: Proceedings (Proceedings of Symposia in Pure Mathematics, V. 28 parts 1 & 2). | |
| Aug 16, 2010 at 12:01 | comment | added | Pete L. Clark | @B: Obviously when I refer to workers in the field taking things on faith, I didn't mean you! On the other hand, I seem to recall you telling me a while ago (i.e., about ten years) that there were some issues as above with the crystalline site. Do you still feel that way? | |
| Aug 16, 2010 at 11:56 | vote | accept | Cosmonut | ||
| Aug 17, 2010 at 8:45 | |||||
| Aug 16, 2010 at 11:38 | comment | added | BCnrd | For etale cohomology all of that universe stuff is entirely irrelevant. I say this not as an "article of faith", but because I've read all of the proofs of the theorems of etale cohomology. Perhaps if one wants to make a super-general theory of cohomology for "all" topoi there are these problems, but if one only cares about more "real" examples such as etale cohomology then there are no issues. | |
| Aug 16, 2010 at 11:29 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Aug 16, 2010 at 11:23 | history | answered | Pete L. Clark | CC BY-SA 2.5 |