# lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)

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:

1. 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.

1. 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. $$p$$.

1. 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 $$\overline{\mathbb F_p}$$.