Timeline for History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?
Current License: CC BY-SA 3.0
9 events
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| Dec 30, 2016 at 3:22 | history | edited | Joël | CC BY-SA 3.0 |
some spelling mistakes corrected
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| Jan 20, 2014 at 8:02 | comment | added | ACL | For a categorical point of view in an essentially classical framework, an interesting reference is the paper « Gèbres » by J-P. Serre, Ens. Math 39 (1993) p. 33-85. | |
| Jan 20, 2014 at 7:08 | history | edited | Joël | CC BY-SA 3.0 |
added 200 characters in body
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| Jan 20, 2014 at 6:01 | history | edited | Joël | CC BY-SA 3.0 |
deleted 84 characters in body
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| Jan 20, 2014 at 5:51 | comment | added | Pedro Lauridsen Ribeiro | @AlexanderChervov: Indeed, neither Tannaka nor Krein used categories in their contributions. Again, mostly because their viewpoint was that of "classical" abstract harmonic analysis. As mentioned in my answer, it seems that Hochschild was the first to realize that the algebraic structure Krein identified for the dual of a compact group (a "Krein algebra") was actually a Hopf algebra structure. Tannaka-Krein duality can then be rephrased as an equivalence between the (opposite) category of compact groups and a subcategory of the category of Hopf algebras. I don't know who did this last step... | |
| Jan 19, 2014 at 19:14 | comment | added | Joël | Dear Alexander and Anon, thanks for your corrections. I'll try to do some research and come up with an improved version of my post, if possible free of mistakes. | |
| Jan 19, 2014 at 18:33 | comment | added | abz | Some corrections: the error in Saavedra's thesis is where he "proves" that fibre functors are locally isomorphic (not the existence). In fact, with his definition of Tannakian category they aren't (so the fibre functors don't form a gerbe). For this it is necessary to require that End(1)=k. The error was discovered by Deligne, pointed out in Deligne and Milne, and corrected later (with some difficulty) by Deligne. | |
| Jan 19, 2014 at 13:57 | comment | added | Alexander Chervov | Thank you very much for your very interesting answer ! If it would be possible to add more comments, why topic is important in algebraic geometry that would be very kind of you. As a small remark on "...and the language of category, which was becoming popular at that time...", it seems neither Tanaka (1938), nor Krein (1949) used the language of categories. Krein seems used something "block-algebras" also called "Krein-algebras". I wonder when the language of categories stared to be used for this topic ? | |
| Jan 18, 2014 at 17:23 | history | answered | Joël | CC BY-SA 3.0 |