Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-m}\mathbb Z/\mathbb Z)^h\to G[p^m](S)$ on a 1-dimensional $p$-divisible group $G$ over a scheme $S$ satisfying certain conditions, then we can form "partial images" in a sense, i.e. direct summands $M$ of $(p^{-m}\mathbb Z/\mathbb Z)^h$ "give us" certain unique finite flat subgroup schemes $H_M$ of $G[p^m]$, so that $\alpha|_M$ is a level structure for $H_M$.
The conditions on $S$ are: locally noetherian and having a dense set of points with residue field algebraic over $\mathbb F_p$.
The second condition "bothers" me, particularly the algebraic part.
My main question is: is this condition necessary (and why)? In particular, are there counterexamples? Since 1-dimensionality guarantees (local) embeddability in a curve, it would somewhat bother me if there were counterexamples for schemes such as $S=\mathbb F_p[T]$ and its fraction field $S=\mathbb F_p(T)$. On the other hand, for example elliptic curves over such base schemes are less known to me, so maybe funny stuff happens.
Thoughts:
I've spent quite a bit of time on the proof but still have some questions. As far as I understand, 1-dimensionality gives a closed subscheme of $S$ for which the result of the lemma holds. Requiring it to be a finite flat subgroup cuts out further closed conditions yielding a scheme $S'$, so we have to check that $S'=S$. The proof then involves several reductions:
- One reduces (using the density condition assumption) to local Artin rings $A=\mathcal O_{S,s}/\mathfrak m_s^i$ for various points $s$ and for $i\ge0$.
I am ok with this step. Seems just usual AG.
- Tensoring with $W(\overline{\mathbb F_p})$ we can assume $A$ has residue field $\overline{\mathbb F_p}$.
I am ok with this step, seems to rely on flatness of $W(\overline{\mathbb F_p})$ over $\mathbb Z_p$ (?). But then we could just use $W(\overline k)$ instead for an arbitrary point $s$? $W(R)$ is flat over $\mathbb Z_p$ for any perfect ring $R$, in particularly for any alg. closed field of char. $p$.
- We replace $G/\text{Spec} A$ by Drinfeld's universal deformation of $G\times\overline{\mathbb F_p}$ and its universal level structure, then pass to its generic point for which $G$ becomes etale and the result holds (i.e. images exist and $S'=S$).
This is a bit of a mouthful and is where I am not sure we can just work with any residue field $k$ or just ones contained in $\overline{\mathbb F_p}$. While previous lemmas (e.g. II.2.2 and II.2.3) deal with (moduli problems) on local Artin algebras with any residue field (not just algebraic over $\mathbb F_p$) it seems that some of Drinfeld's (pro-)representability results only hold (or have only been written) for functors on local Artin algebras with residue field $\overline{\mathbb F_p}$.
Any answers/comments/clarifcations or references will be greatly appreciated.
