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7:51 AM
@PiotrPstrągowski I would like to know this as well. I once had the impression that a level structure on a p-divisible group is the same thing as a monomorphism from a constant elementary p-power torsion group.
 
8:02 AM
I'm not so sure about that anymore.
 
8:12 AM
"constant elenentary..." i just mean something like $(\mathbb{Z}/p^l \mathbb{Z})^k$.
 
 
1 hour later…
9:13 AM
@PiotrPstrągowski the universal deformation of a formal group of height n lies over the Lubin-Tate ring so formal groups of height n at the special fiber over Z_p are classified by algebra maps LT_n-->Z_p. There are no such maps (since there are not even Z_p-algebra maps W(F_p^n)-->Z_p) so the thing you want to consider just dont exist. You should really take W(F_p^n) to obtain such a lift to char 0, and then you get Lubin-Tate extensions over W(F_p^n).
Ohh oops ignore me it was nonsense of course!
I guess you should get the unramified extension again but with a different filtration on the it.
 
 
9 hours later…
6:45 PM
Does anyone have any sense of whether or not the category of simplicial sets satisfies this condition that Kelly uses in his enriched category theory book: every object has but a small set of extremal epimorphic quotients.
 
7:15 PM
Here, someone put me out of my misery: mathoverflow.net/questions/350752/…
 
 
2 hours later…
8:52 PM
@SaalHardali in certain contexts, this is true: definitely when the p-series of the formal group completely splits and has no repeated roots, and maybe also without the splitting. i think that this is true over the lubin-tate ring, but certainly it’s not true over the residue field, where the p-series might look like x^(p^k). over the intermediate object W(k), i do not have a guess
definitely level structures are an attempt to capture monomorphisms w/ an admission that those have bad functoriality w/r/t base change. the divisibility condition is an attempt to keep the points in the selected subgroup from clustering / collapsing, provided the group itself has space for them to not cluster / collapse
again, the p-typical formal group with p-series x^(p^k) is an example of a group with no room available for monomorphisms to exist, but the monomorphism coming from the fattening over the (separable closure of the) lubin tate ring has to get imaged somewhere 🤷
 

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