Timeline for Motivic Galois theory and Betti realizations?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| S Aug 24, 2017 at 16:55 | history | bounty ended | CommunityBot | ||
| S Aug 24, 2017 at 16:55 | history | notice removed | CommunityBot | ||
| Aug 22, 2017 at 20:46 | comment | added | user40276 | Just recalling that usually the motivic t-strucure is defined in terms of the Betti realization when it exists. | |
| Aug 16, 2017 at 17:37 | comment | added | Will Sawin | @AlexSaad There are actually two different conjectural faithful functors related to the Hodge-de Rham isomorphism. The first is based on the $\mathbb C$-de Rham cohomology w/ its Hodge filtration and the comparison isomorphism, and this faithfulness is the Hodge conjecture. The second is based on the $\mathbb Q$-de Rham cohomology and the comparison isomorphism after tensoring with $\mathbb C$, and this faithfulness is the period conjecture. | |
| Aug 16, 2017 at 16:44 | comment | added | Alex Saad | I'm quite sketchy on this so please correct me if I'm wrong. There are many realisations of a motive, including Betti and de Rham realisations, which together with the Betti-de Rham comparison isomorphism form the "Hodge realisation". There are also $l$-adic realisations for prime numbers $l$. Conjecturally, the Hodge and $l$-adic realisations should both be fully faithful functors. Thus the Betti fibre functor, and the Tannaka group it defines, is a good approximation to the full "motivic" Galois group, and the extra info provided by the Hodge realisation should capture everything motivic. | |
| Aug 16, 2017 at 16:28 | comment | added | Eoin | Grothendieck's original motivation was to find a Tannakian category of motives. If you have a Tannakian category, and a fiber functor on that category, then you can construct an algebraic group from its automorphisms (Deligne's reconstruction theorem). Probably the reason these are related are the many incarnations of the Galois group (or fundamental group, or something else) and their actions on appropriate cohomology groups. It's only a step from seeing these groups act on the cohomology to trying to functorialize and find the root of where the two come from. | |
| S Aug 16, 2017 at 15:41 | history | bounty started | tttbase | ||
| S Aug 16, 2017 at 15:41 | history | notice added | tttbase | Draw attention | |
| Feb 1, 2017 at 3:03 | review | Close votes | |||
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| Jan 26, 2017 at 18:04 | review | Close votes | |||
| Jan 26, 2017 at 18:58 | |||||
| Jan 26, 2017 at 17:31 | review | First posts | |||
| Jan 26, 2017 at 17:47 | |||||
| Jan 26, 2017 at 17:30 | history | asked | peter | CC BY-SA 3.0 |