@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.
@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.
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.
@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 🤷
