Timeline for Clarification on relationship between Grothendieck-Messing and Honda systems
Current License: CC BY-SA 3.0
9 events
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| Oct 9, 2017 at 2:43 | history | edited | SomeGuy | CC BY-SA 3.0 |
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| Oct 8, 2017 at 23:47 | comment | added | nfdc23 | I suggest that you email Shin Hattori to ask about this. | |
| Oct 8, 2017 at 20:46 | comment | added | SomeGuy | that I didn't actually know they coincided. Maybe this is not a good enough reason to attack it, but seems to be an obvious question (possibly worthy of some devoted time). Anyways, I'm just rambling now. Thanks again! Best, Alex. | |
| Oct 8, 2017 at 20:46 | comment | added | SomeGuy | @nfdc23 Dear nfcd23, I see. That makes a lot of sense. I guess I, somewhat naively, had the thought process "if the two classifications look the same, they're the same classification" which, of course, doesn't hold any real water. Do you think it's worth thinking about comparing the two? To be honest, my immediate motivation was that I was giving a talk on Breuil and Kisin's work, and wanted to explain what happens in the unramified case. I kind of wanted to just unify the classification in the literature (i.e. Fontaine's work for urnamified and Conrad's for low ramification) but realized | |
| Oct 8, 2017 at 20:37 | comment | added | nfdc23 | Anyway, one has shown that G-M theory meshes well with classical Dieudonne theory (up to a Frobenius twist), which is the main conclusion of the Mazur-Messing book, and one knows how classical Dieudonne theory encodes the tangent space (all done in Fontaine's book), one does recover as you say a classification looking just like Honda systems. But does it really match Fontaine's log-style construction of $L$ on the nose (in terms of the submodule of $D(G_k)$ associated to $G$)? Perhaps for all purposes it doesn't matter, so nobody has the incentive to dig far down to check. | |
| Oct 8, 2017 at 20:30 | comment | added | nfdc23 | With these matters, it can be such a pain to match up two very distant-looking constructions that if one doesn't truly need the consistency (i.e., maybe one can get by purely from the G-M side abstractly, or from the Honda system side concretely, depending on the intended purpose) then it can be rather unappetizing to try to cook up a proof of agreement (maybe up to some Frobenius twist or duality or whatever). So unless one really needs to know the agreement, what is the incentive to sweat out making a proof? (Sure, it is nice to know...and even nicer if someone else grinds out that task.) | |
| Oct 8, 2017 at 20:20 | comment | added | SomeGuy | @nfdc23 Dear nfdc23, that makes a lot of sense. Can you clarify what you mean by 'needs agreement'? Also, I hadn't realized that this was written by B. Conrad as a graduate student--the fact that B. Conrad didn't know Grothendieck-Messing at the time didn't even occur to me--I assumed that he was born knowing everything about $p$-divisible groups. :P I was more confused because not only was it not in that article, but I couldn't find a single reference that says you can deduce Honda theory from GM--I thought I might be missing some major subtlety. Thanks again! Best, Alex. | |
| Oct 8, 2017 at 20:09 | comment | added | nfdc23 | The Honda system formalism for (commutative, $p$-power order) finite flat group schemes $G$ over $W(k)$ and Dieudonne theory for $G_k$ set up from scratch by Fontaine in his Asterisque book can sometimes (as in the link you gave) be a simpler self-contained way to proceed than via G-M theory for $p$-divisible groups (+ Mazur-Messing, combined with Raynaud's theorem in [BBM, 3.3.1]). And perhaps B. Conrad (a graduate student when that link was written) didn't know G-M theory at that time. So doesn't seem strange that G-M isn't mentioned in that link. It depends on whether one needs agreement. | |
| Oct 8, 2017 at 6:10 | history | asked | SomeGuy | CC BY-SA 3.0 |