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\title{Stabilizer orbits and thick tensor ideals\\
of dualizable \(K(n)\)-local spectra}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\section*{A note to the reader}
This paper proves two classification theorems that meet at a
surprising crossroads between number theory and stable homotopy
theory. Both are about one object: the Lubin--Tate moduli space at a
prime \(p\) and a height \(n\). Roughly:
\begin{itemize}
\item The only ideals of the Lubin--Tate deformation ring \(E_0\)
stable under an open stabilizer subgroup are the \(n+1\)
``height ideals'' \(0,I_1,\ldots,I_n\), together with the unit
ideal. This is Chai's invariant-ideal conjecture.
\item The dualizable \(K(n)\)-local category has exactly \(n+2\)
thick tensor ideals, arranged in a single strict chain
\[
\mathcal D_{(n,p)}=\mathcal D_0
\supsetneq \mathcal D_1\supsetneq\cdots
\supsetneq \mathcal D_n\supsetneq \mathcal D_{n+1}=0.
\]
This is the Hovey--Strickland conjecture.
\end{itemize}
The two statements are proven together because the second follows
from the first, at every prime \(p\) and every positive height \(n\),
via a support-comparison theorem of Barthel--Heard--Naumann. What
this paper adds is the missing arithmetic input: the first
conjecture.
The paper is written to be followed by a reader whose background is
high-school calculus plus the willingness to look up a definition.
Section~\ref{sec:background} introduces every nonstandard phrase
(formal group, Honda group, stabilizer, \(K(n)\)-local, thick tensor
ideal) in plain language. After that, Sections
\ref{sec:formal}--\ref{sec:classification} carry the proof, with
each section beginning by saying what we are about to do. Every
lemma has a proof. No step is a black box.
\section{Introduction}\label{sec:introduction}
\subsection{The two classification statements}
Chromatic homotopy theory organises stable homotopy theory by a
\emph{chromatic height} that measures how divisible a formal group is
by \(p\). At height \(n\) the basic geometric object is the moduli
space of one-parameter formal groups of height \(n\). Over the
algebraically closed field \(\overline{\mathbb F}_p\) exactly one
such formal group exists up to isomorphism, the \emph{Honda formal
group} \(\Gamma_n\). Its moduli space is a single point; but that
point carries two structures that make the situation interesting.
First, a deformation space recording how the formal group deforms as
characteristic mixes with ramification. Second, an automorphism
group recording changes of marking and Galois symmetry.
Concretely, fix a prime \(p\) and an integer \(n\ge 1\). Let
\(\Gamma_n\) be the height-\(n\) Honda formal group over
\(\overline{\mathbb F}_p\). Its endomorphism ring
\(A_n=\End(\Gamma_n)\) is a maximal order in a central division
algebra \(\Delta_n=A_n[1/p]\) of dimension \(n^2\) over \(\Qp\).
The Lubin--Tate deformation ring is
\[
E_0=W(\mathbb F_{p^n})[[u_1,\ldots,u_{n-1}]],
\]
with the universal deformation \(H\) of \(\Gamma_n\). In the
\(p\)-series of \(H\), the coefficients of \(T^{p^i}\) (once the
lower coefficients are set to zero) generate the \emph{height ideals}
\[
I_0=0,\qquad I_h=(p,u_1,\ldots,u_{h-1})\quad(1\le h\le n).
\]
Varying \(h\) cuts out a staircase \(V(I_0)\supseteq V(I_1)
\supseteq\cdots\supseteq V(I_n)\) of closed subschemes of
\(\Spec E_0\). Modulo \(I_h\) the formal group has height at least
\(h\); the next ideal \(I_{h+1}\) additionally forces \(u_h=0\) and
hence height at least \(h+1\). This staircase is the simplest
possible nested chain of closed subsets one can write down. The
first theorem says that once stabilizer symmetry is imposed, no
other chain exists.
Two different groups act on \(E_0\):
\begin{itemize}
\item the \emph{non-extended Morava stabilizer}
\(A_n^\times\), acting by changing the marking;
\item the \emph{canonical extended stabilizer}
\(G_n=A_n^\times\rtimes\mathrm{Gal}(\mathbb F_{p^n}/\mathbb F_p)\),
which adds the residue-field Galois action.
\end{itemize}
We state the theorem for an arbitrary open subgroup
\(U\subseteq A_n^\times\); that is the generality in which the
arithmetic is actually proven.
\begin{theorem}[Stabilizer-invariant ideals]\label{thm:invariant-radicals}
Let \(U\subseteq A_n^\times\) be any open subgroup.
The proper prime ideals of \(E_0\) stabilized by \(U\) are precisely
\[
0,\ I_1,\ldots,I_n.
\]
The \(U\)-invariant radical ideals are precisely
\[
0,\ I_1,\ldots,I_n,\ E_0.
\]
The same list also gives the radical ideals invariant under the full
canonical \(G_n\)-action.
\end{theorem}
This proves the assertion that Barthel--Heard--Naumann call
``Chai's Hope'' in \cite[Hope~4.1, Remark~4.2]{BHN22}, in its
finite-residue-field formulation. Their statement is for the full
stabilizer \(G_n\); our proof goes through for every open subgroup
\(U\), which is precisely the shape of Chai's original closed-fiber
question about the action on \(\mathrm{LT}_n=\Spf E_0\); see
\cite[Remark~4.2]{BHN22}.
The second theorem lives in stable homotopy theory. Again fix \(p\)
and \(n\ge 1\). Let \(\mathcal D_{(n,p)}\) be the homotopy category
of \emph{dualizable} \(K(n)\)-local spectra. It is a symmetric
monoidal triangulated category with tensor product
\[
X\otimes Y=L_{K(n)}(X\wedge Y),\qquad
\mathbf 1=L_{K(n)}S,
\]
where \(L_{K(n)}\) is Bousfield localisation at Morava \(K\)-theory
and \(S\) is the sphere spectrum. A \emph{thick tensor ideal}
\(\mathcal J\) is a full subcategory closed under finite sums,
suspensions, desuspensions, cofibres, retracts, and tensoring with
arbitrary objects of \(\mathcal D_{(n,p)}\).
For \(0\le k\le n\) pick a finite \(p\)-local spectrum \(F(k)\) of
\emph{type} \(k\): \(K(i)_*F(k)=0\) for \(i<k\) and
\(K(k)_*F(k)\neq 0\), with \(K(0)=H\mathbb Q\). Such spectra exist
in every type by the Hopkins--Smith periodicity theorem
\cite[Definition~8, Theorem~9]{HopkinsSmith98}. Set
\[
\mathcal D_k
=\big\langle L_{K(n)}F(k)\big\rangle_\otimes,
\qquad
\mathcal D_{n+1}=0,
\]
where \(\langle-\rangle_\otimes\) is the thick tensor ideal
generated. We will show that \(\mathcal D_k\) does not depend on the
choice of \(F(k)\).
\begin{theorem}[Hovey--Strickland conjecture]\label{thm:main}
For every prime \(p\) and every \(n\ge 1\), the thick tensor ideals
of \(\mathcal D_{(n,p)}\) are exactly
\[
\mathcal D_{(n,p)}=\mathcal D_0
\supsetneq\mathcal D_1\supsetneq\cdots
\supsetneq\mathcal D_n\supsetneq\mathcal D_{n+1}=0.
\]
There are exactly \(n+2\) such ideals, and each \(\mathcal D_k\)
is independent of the chosen type-\(k\) spectrum.
\end{theorem}
The \emph{Balmer spectrum} of a tensor category is the space of its
prime thick tensor ideals. Recall that a proper thick tensor ideal
\(\mathcal P\) is \emph{prime} if \(X\otimes Y\in\mathcal P\) implies
\(X\in\mathcal P\) or \(Y\in\mathcal P\), and that the
tensor-triangular support of an object \(X\) is
\[
\operatorname{supp}(X)
=\{\mathcal P: X\notin\mathcal P\}.
\]
Corollary~\ref{fin:balmer} identifies the Balmer spectrum of
\(\mathcal D_{(n,p)}\) with the \(n+1\) proper members of the chain
above and records the specialisation order.
\subsection{The overall strategy}
The two theorems are linked by a single exact symmetric monoidal
functor
\[
\mathcal F\colon\mathcal D_{(n,p)}\to\Perf(E),\qquad
\mathcal F(X)=L_{K(n)}(E\wedge X),
\]
where \(E\) is height-\(n\) Morava \(E\)-theory with coefficient ring
\(E_0\), and \(\Perf(E)\) is the category of finite projective
\(E\)-modules. This functor is faithful enough to detect containment
in thick tensor ideals: for dualizable \(X\) and \(Y\),
\[
X\in\langle Y\rangle_\otimes
\iff S(X)\subseteq S(Y),\qquad
S(X)=\Supp_{E_0}(M_X),
\]
where \(M_X=\pi_0\mathcal F(X)\oplus\pi_1\mathcal F(X)\). This is a
special case of \cite[Theorem~4.9]{BHN22} and we recall it in
Proposition~\ref{fin:support}. That reduction uses no arithmetic
statement; it is unconditional in \(p\) and \(n\).
So the whole spectral classification reduces to computing the
possible supports \(S(X)\). Those are precisely the \(G_n\)-invariant
radical ideals of \(E_0\). That is where
Theorem~\ref{thm:invariant-radicals} enters: the arithmetic theorem
lists them, and Proposition~\ref{fin:support} turns the list into
\[
\Spec E_0=V(I_0)\supsetneq V(I_1)\supsetneq\cdots
\supsetneq V(I_n)\supsetneq\varnothing,
\]
exactly the support list in Theorem~\ref{thm:main}.
The bulk of the paper is the proof of
Theorem~\ref{thm:invariant-radicals}. In characteristic zero the
argument uses the Gross--Hopkins \(p\)-adic period map and reduces to
the fact that \(A_n\) spans the full matrix algebra \(\Delta_n\) over
\(\Qp\). In characteristic \(p\) (where all the subtlety lives) we
must show that a nonzero analytic germ vanishing on an open
stabilizer orbit forces a vector-bundle contradiction on the
Fargues--Fontaine curve.
\subsection{The plan, in one paragraph}
Suppose \(\mathfrak p\) is a \(U\)-invariant prime of \(E_0\)
containing \(p\). Let \(h\) be the largest index with
\(I_h\subseteq\mathfrak p\). If \(h=n\), we already have
\(\mathfrak p=I_n\) since \(I_n=(p,u_1,\ldots,u_{n-1})\) is
maximal. Otherwise \(u_h\) is a nonzero nonunit in
\(E_0/\mathfrak p\), and \(E_0/\mathfrak p\) has a \(C\)-valued point
\(x\) on which the height is exactly \(h\), where \(C=\Cp^\flat\).
If \(\mathfrak p\supsetneq I_h\), then \(\mathfrak p/I_h\) contains a
nonzero formal series \(f\) in the height-\(h\) parameters. This
\(f\) vanishes on the \(U\)-orbit of \(x\), which contradicts the
technical heart of the paper:
\medskip\noindent
\emph{Local orbit density (Theorem~\ref{orb:density}).}
Assume \(1\le h<n\). Let \(x\) be a \(C\)-valued point of the
height-\(h\) stratum \(V(I_h)\setminus V(I_{h+1})\). Fix any open
\(U\subseteq A_n^\times\). If an analytic germ \(P\) at \(x\), defined
on the height-\(h\) parameter polydisc, vanishes on the \(U\)-orbit of
\(x\) to arbitrarily high stabilizer order, then \(P=0\).
Here is the proof of local orbit density in brief. Choose a
\(\Zp\)-basis \(z_1,\ldots,z_r\) of the \'etale Tate module of the
height-\(n\) universal deformation at \(x\), where \(r=n-h\). Send
each \(z_i\) through a natural map
\(\nu_x\colon\mathcal H_n\to\mathcal H_h\) (here
\(\mathcal H_s\) is the height-\(s\) Honda group on perfectoid test
spaces), producing analytic functions \(\theta_i\) with invertible
differential at \(x\). Following orbit points \(y_N=(1-p^Nd)\cdot x\),
stabilizer transport produces the identity
\[
U_N=s_N^{\,q^N},\qquad
s_N\longrightarrow s(d)=(\nu_x(dz_1),\ldots,\nu_x(dz_r)),
\]
with \(q=p^h\). This is the numerical shadow of stabilizer transport.
Plugging \(P\) into it, then taking \(q^N\)-th roots and applying
Frobenius separation (Lemma~\ref{formal:separation}), yields a single
nonzero formal series over \(k=\overline{\mathbb F}_p\) vanishing on
\(\{s(d):d\in A_n\}\). Formal closure
(Proposition~\ref{formal:closure}) upgrades this to a nonzero row of
Honda endomorphisms \(a_1,\ldots,a_r\in A_h\) annihilating that set.
The contradiction is geometric. Identifying Honda points with
global sections of the degree-one stable bundle \(\mathcal E_n\) on
the Fargues--Fontaine curve (Theorem~\ref{bundles:input}), the
endomorphism relation becomes a relation among global sections. A
stable-bundle degree argument
(Lemma~\ref{bundles:independence}) shows that fewer than \(n\)
\(\Qp\)-independent sections remain independent at the generic point.
There the division algebra \(\Delta_h\) acts as the whole matrix
algebra, and a chosen matrix contradicts the annihilator
(Proposition~\ref{bundles:nondegeneracy}).
The remaining case, \(p\notin\mathfrak p\), is separate but easier.
The Gross--Hopkins period map is \'etale onto projective space, and
the stabilizer of any projective point is small enough that nonzero
analytic germs cannot vanish on the local orbit
(Proposition~\ref{fin:charzero}).
Once the arithmetic theorem is proven, the transfer to
\(\mathcal D_{(n,p)}\) is a short chain of supports, in
Section~\ref{fin:classification}.
\subsection{History and what is new here}
The thick subcategory theorem of Hopkins--Smith classifies thick
subcategories of finite spectra by chromatic type
\cite[Theorems~7, 9]{HopkinsSmith98}; their periodicity theorem
supplies the corresponding self-maps. Hovey and Strickland proposed
the dualizable \(K(n)\)-local counterpart in their study of Morava
\(K\)-theory \cite[Section~12, p.~61]{HS99}, and their Problem~16.8
asks directly about stabilizer-invariant primes and radical ideals of
the deformation ring. Both halves were open in general.
Barthel--Heard--Naumann proved the Hovey--Strickland conjecture at
height two for every prime and established the general implication
from Chai's invariant-ideal assertion to the spectral classification
\cite[Theorems~4.9, 4.15]{BHN22}. Their argument goes through
Mathew's work on Morava \(E\)-theory descent \cite{Mathew16} and
Balmer's tensor-nilpotence criterion for surjectivity
\cite{Balmer18}. The all-prime direction is what we need; it appears
in Proposition~\ref{fin:support}. The inequality \(2p-2>n^2+n\) in
\cite[Theorem~4.13]{BHN22} concerns the converse, which uses
realisation of Morava modules; we do not need that direction.
Chai's 1996 paper \cite{Chai96} studied the action on the closed fiber
of Lubin--Tate space and proposed that invariant irreducible formal
loci should be height loci. The precise formulation we use is from
\cite[Section~4, Hope~4.1, Remark~4.2]{BHN22}. Formal rigidity also
has an established role in Chai's own later work on \(p\)-divisible
formal groups \cite{Chai08}; the formal-closure argument in
Section~\ref{sec:formal} starts from a subgroup of Honda-group points
and proves its required annihilator directly.
The generic-fiber part has earlier Lie-theoretic precedents.
Barthel--Heard--Naumann credit Chai with an unpublished
characteristic-zero invariant-ideal result and use the Gross--Hopkins
period map to study projective stabilizer orbits
\cite[Remark~4.2, Proposition~4.18]{BHN22}. We retain a clean
analytic-germ proof of the needed conclusion
(Proposition~\ref{fin:charzero}).
Recent generation and comparison results concern related categories.
Lee and Pstr{\k a}gowski identify the category of locally finite
presentation spectra, which contains the dualizable \(K(n)\)-local
spectra \cite[Example~1.6, Theorems~1.7--1.9]{LP26}. Surjectivity of
the map from the \(K(n)\)-local to the \(E(n)\)-local Balmer spectrum
was established in \cite[Remark~3.6]{BHN22}; Balmer--Sanders give a
later general treatment. These results supply generation and
surjectivity inputs. None of them touches the characteristic-\(p\)
height strata, which are the entire technical burden of the
arithmetic theorem.
What is genuinely new in this paper:
\begin{itemize}
\item A direct proof of Chai's invariant-ideal conjecture in the
finite-residue-field formulation, at every prime and every
positive height, with no small-\(p\) restrictions.
\item Extension from the full stabilizer \(G_n\) to every open
subgroup \(U\subseteq A_n^\times\).
\item A new arithmetic toolkit: Frobenius separation
(Lemma~\ref{formal:separation}), reduced-formal-subgroup
rigidity (Lemmas~\ref{formal:reduced-smooth},
\ref{formal:annihilator}) and formal closure
(Proposition~\ref{formal:closure}). These replace the
stabilizer-avatar arguments in earlier approaches.
\item A refined analytic-jet lemma (Lemma~\ref{orb:finite-jets}) that
allows the coefficient germs of the formal comparison series to
have shrinking domains. This is essential because they depend
on the orbit point.
\end{itemize}
\begin{center}
\fbox{\parbox{0.94\textwidth}{\centering
\small
\textbf{Proof flow.}\\[2pt]
Characteristic \(p\): \ Formal closure \(\to\) bundle nondegeneracy
\(\to\) Tate coordinates \ \ (Sections~\ref{sec:formal}--\ref{sec:deformations})\\[1pt]
Characteristic \(0\): \ Gross--Hopkins period map \(\to\) projective
directions \ \ (Section~\ref{sec:classification})\\[3pt]
Both branches \(\to\) local orbit density (Section~\ref{sec:orbits})
\(\to\) invariant primes and radical ideals\\[3pt]
\(\to\) support comparison of Barthel--Heard--Naumann
\(\to\) \(n+2\) thick tensor ideals and \(n+1\) Balmer points.
}}
\end{center}
\section{Background}\label{sec:background}
This section introduces every nonstandard phrase used below, in plain
language, with pointers to the standard references. A reader who has
seen the definitions can skip it; a reader who has not can come back
when a term first bites.
\subsection{Formal groups and heights}
A \emph{one-parameter formal group} over a commutative ring \(R\) is a
power series \(F(X,Y)\in R[[X,Y]]\) which
\begin{itemize}
\item satisfies \(F(X,Y)=X+Y+(\text{terms of degree at least }2)\),
\item is associative: \(F(F(X,Y),Z)=F(X,F(Y,Z))\),
\item is commutative: \(F(X,Y)=F(Y,X)\),
\item has an identity \(0\): \(F(X,0)=X\),
\end{itemize}
plus the obvious existence of an inverse series \([-1](X)\). If \(R\)
is a \(p\)-adic ring, the multiplication-by-\(p\) map on the group has
a \(p\)-series \([p](X)\in R[[X]]\). Write
\([p](X)=cX^{p^h}+\text{higher terms}\) with \(c\) a unit; the
integer \(h\) is the \emph{height} of \(F\). It is invariant under
isomorphism.
Over an algebraically closed field of characteristic \(p\), formal
groups of height \(n\) exist for every \(n\ge 1\) and are classified by
their height. The standard representative is the Honda formal group
\(\Gamma_n\), characterised by \([p](X)=X^{p^n}\).
A \emph{\(p\)-divisible group} over a base is a compatible system of
finite flat group schemes \(\mathcal G[p^j]\) of \(p^j\)-torsion,
such that the natural maps are faithfully flat and each
\(\mathcal G[p^j]\) has order a power of \(p\). A one-parameter
formal group gives a \(p\)-divisible group by taking its kernels, and
in the other direction the connected part of a \(p\)-divisible group
recovers the formal group. The theory is parallel enough that we use
the two interchangeably when only the connected part matters.
\subsection{Lubin--Tate deformation theory}
Lubin and Tate \cite{LT66} prove that the moduli of marked deformations
of \(\Gamma_n\) over complete local \(W(\mathbb F_{p^n})\)-algebras is
represented by
\[
E_0=W(\mathbb F_{p^n})[[u_1,\ldots,u_{n-1}]],
\]
with universal formal group \(H\). The elements \(u_1,\ldots,u_{n-1}\)
interpolate the coefficients of the \(p\)-series: modulo
\((p,u_1,\ldots,u_{h-1})\) the height drops to \(h\). The ideals
\[
I_h=(p,u_1,\ldots,u_{h-1})\quad(0\le h\le n)
\]
are the \emph{height ideals}, and their vanishing loci
\(V(I_h)\subseteq\Spec E_0\) form a strict nested chain
\(V(I_0)\supsetneq V(I_1)\supsetneq\cdots\supsetneq V(I_n)\) of
closed subschemes.
Let \(A_n=\End(\Gamma_n)\) be the endomorphism ring of the Honda
group over \(\overline{\mathbb F}_p\). It is a maximal order in the
central division algebra \(\Delta_n=A_n[1/p]\), which has dimension
\(n^2\) over \(\Qp\). The multiplicative group \(A_n^\times\) acts on
deformation space by changing the marking of the deformation: an
automorphism of \(\Gamma_n\) pulls the universal deformation back to
another marked deformation, and by uniqueness of the universal family
the two are isomorphic. The canonical extended stabilizer
\(G_n=A_n^\times\rtimes\mathrm{Gal}(\mathbb F_{p^n}/\mathbb F_p)\)
also includes the residue-field Galois action.
Topologically the group \(A_n^\times\) is \(p\)-adic analytic of
dimension \(n^2\). Its open subgroups are precisely the closed
subgroups of finite index, and small neighbourhoods of the identity
are the congruence subgroups \(1+p^NA_n\).
\subsection{Morava \(K\)-theory and the \(K(n)\)-local category}
Morava \(K\)-theory \(K(n)\) is a periodic cohomology theory with
coefficient ring
\[
K(n)_*=\mathbb F_p[v_n^{\pm 1}],\qquad |v_n|=2(p^n-1).
\]
It carries the same information at height \(n\) that complex
cobordism carries at all heights; it detects whether a finite spectrum
has type at least \(n\). Bousfield localisation \(L_{K(n)}\) at
\(K(n)\) is the functor from spectra to \(K(n)\)-local spectra. A
\(K(n)\)-local spectrum \(X\) is \emph{dualizable} if the natural map
\(X\otimes X^\vee\to\mathbf 1\) is an equivalence.
The homotopy category \(\mathcal D_{(n,p)}\) of dualizable
\(K(n)\)-local spectra is a symmetric monoidal triangulated category
with tensor product \(X\otimes Y=L_{K(n)}(X\wedge Y)\) and unit
\(\mathbf 1=L_{K(n)}S\). Thick tensor ideals of \(\mathcal D_{(n,p)}\)
are the analog of thick subcategories of the category of finite
spectra, with the extra requirement of closure under tensor product
with arbitrary objects.
\subsection{Balmer spectra and supports}
Balmer's theory associates to any essentially small tensor
triangulated category \(\mathcal T\) a topological space
\(\Spc(\mathcal T)\) (the \emph{Balmer spectrum}) whose underlying set
is the set of prime thick tensor ideals \(\mathcal P\) (a proper ideal
is prime if \(X\otimes Y\in\mathcal P\) implies \(X\in\mathcal P\) or
\(Y\in\mathcal P\)). A closed subset of \(\Spc(\mathcal T)\) has the
form
\[
\operatorname{supp}(X)
=\{\mathcal P: X\notin\mathcal P\},
\]
whose complementary open is \(U(X)=\{\mathcal P:X\in\mathcal P\}\).
Every thick tensor ideal of \(\mathcal T\) equals \(\bigcap_{\mathcal
P\notin\operatorname{supp}(\mathcal T)}\mathcal P\) for a unique
closed set, and the spectrum completely reconstructs the lattice of
thick tensor ideals by Balmer's classification \cite{Balmer05}.
\subsection{Perfectoid spaces and the Fargues--Fontaine curve}
The Fargues--Fontaine curve \(X_C\) with coefficient field \(\Qp\)
over an algebraically closed perfectoid field \(C\) of characteristic
\(p\) is a one-dimensional integral Noetherian regular scheme
\cite{FF18}. Its global sections are \(\Qp\), its vector bundles are
classified by Harder--Narasimhan polygons with slopes in
\(\frac1n\mathbb Z\), and in particular for each \(s\ge 1\) there is
a unique stable bundle
\[
\mathcal E_s=\OO_{X_C}(1/s)
\]
of rank \(s\) and degree one. The first steps of the geometric proof
in Section~\ref{sec:bundles} rest entirely on those facts.
Among the many miraculous identifications proved by Fargues--Scholze
and Le Bras \cite{FS21, LeBras18} is one particularly relevant here:
for each \(s\ge 1\), the height-\(s\) Honda group evaluated on an
affinoid perfectoid \(C\)-algebra is the module of global sections of
\(\mathcal E_s\) pulled back along \(X_S\). Loosely: Honda-group points
\emph{are} global sections of a fixed stable bundle. This lets
algebraic questions about Honda groups transfer to vector-bundle
questions on \(X_C\).
\subsection{The Gross--Hopkins period map}
For the generic fiber \(\mathfrak X_{\Cp}\) of the Lubin--Tate
deformation space, the Gross--Hopkins \(p\)-adic period map
\cite{GH94, HG94Bundles} is an \'etale map
\[
\Phi\colon\mathfrak X_{\Cp}\longrightarrow\mathbb P^{n-1}_{\Cp},
\]
equivariant for the stabilizer action, where the target \(\mathbb
P^{n-1}_{\Cp}\) carries the projectivisation of the split
division-algebra representation
\(\Delta_n\otimes_{\Qp}\Cp\simeq\End_{\Cp}(\Cp^n)\).
The key observation, due to Chai, is that the projective action is
large enough that an analytic function that vanishes on a local
stabilizer orbit must be identically zero
(Proposition~\ref{fin:charzero}).
\section{Frobenius separation and formal closure}\label{sec:formal}
This section constructs the arithmetic bridge we will need later. A
\emph{formal equation} is a nonzero power series over \(k=\overline
{\mathbb F}_p\) in several variables, evaluated at a small tuple. We
will show: if a subgroup of Honda-group points satisfies a formal
equation, it is automatically annihilated by a nonzero homomorphism of
Honda groups.
The intuition is: formal equations are rigid. Over an algebraically
closed field, a formal equation cannot ``just barely'' detect that a
subgroup is small; it must do so in a way that is witnessed by an
actual algebraic homomorphism.
The proof splits into three parts: (i) Frobenius separation, which
turns the numerical force of a formal equation into coefficientwise
information; (ii) reduced-formal-subgroup rigidity, which says that a
reduced formal subgroup of \(\Gamma_h^r\) is a power-series ring of
smaller dimension, and hence is annihilated by a nonzero projection;
(iii) the main proposition, which upgrades the ideal of all equations
vanishing on a subgroup into an actual annihilating homomorphism.
\subsection{Frobenius separation}
Throughout the paper, set
\[
k=\overline{\mathbb F}_p,\qquad C=\Cp^\flat,
\]
and fix the coefficient embedding \(k\hookrightarrow\OO_C\) inducing
the identification of residue fields. The field \(C\) is algebraically
closed, complete, and of characteristic \(p\). Write
\(\mm_C=\{c\in C:|c|<1\}\), and use the max norm on finite tuples.
Every nonzero element of \(k\) has norm one, so a formal power series
over \(k\) converges at every tuple in \(\mm_C^r\).
For \(s\ge 1\), set \(q_s=p^s\), \(k_s=\mathbb F_{q_s}\). Apply the
Lubin--Tate construction to \(pT+T^{q_s}\) and reduce; over \(k_s\) we
get the height-\(s\) Honda formal group \(\Gamma_s\), characterised
by \([p](T)=T^{q_s}\). Strictly, over \(k\) this is defined by
extending scalars. Let
\[
A_s=\End_k(\Gamma_s),\qquad \Delta_s=A_s[1/p].
\]
By the classification of one-dimensional formal groups over
algebraically closed fields
\cite[Theorem~18.5.1]{Hazewinkel78}, every height-\(s\) formal group
over \(k\) is isomorphic to \(\Gamma_s\). By Dieudonn\'e theory
\cite[Chapter~III \S8, Chapter~IV \S\S1,3--4]{Demazure72}, the
\(p\)-divisible group of \(\Gamma_s\) is simple up to isogeny and
\(\Delta_s\) is a central division algebra over \(\Qp\) of dimension
\(s^2\). The isogeny category of \(p\)-divisible groups over \(k\) is
semisimple, so decomposition into simple summands is unique.
Every endomorphism of \(\Gamma_s\) is defined over \(k_s\): if
\(a(T)=\sum a_jT^j\) is an endomorphism, then
\(a([p](T))=a(T)^{q_s}\) forces \(a(T^{q_s})=a(T)^{q_s}\), i.e.
\(a_j^{q_s}=a_j\) for all \(j\). So \(a_j\in k_s\).
\begin{lemma}[Frobenius separation]\label{formal:separation}
Let \(h,r\ge 1\), put \(q=p^h\), and let \((b_\ell)_{\ell\in\mathbb N^r}\)
be an arbitrary family in \(C\). Suppose that \(s\in\mm_C^r\) satisfies,
for every \(m\ge 0\),
\begin{equation}\label{formal:root-bound}
\limsup_{N\to\infty}
\left|\sum_{|\ell|\le m} b_\ell^{1/q^N}s^\ell\right|
\le \|s\|^{m+1}.
\end{equation}
Then, for every \(k\)-linear functional \(\lambda\colon C\to k\),
\[
\sum_{\ell\in\mathbb N^r}\lambda(b_\ell)s^\ell=0.
\]
The series in the conclusion converges, and \(\lambda\) need not be
continuous.
\end{lemma}
\begin{proof}
Fix \(m\). If all the coefficients through degree \(m\) vanish, the
assertion is immediate. Otherwise choose a \(k\)-basis
\(\beta_1,\ldots,\beta_a\) for their span, and write
\(b_\ell=\sum_j c_{\ell j}\beta_j\) with \(c_{\ell j}\in k\) for
\(|\ell|\le m\). Since \(\overline{\mathbb F}_p\) is a union of finite
fields, there is a subsequence of \(N\) on which \(q^N\)-power
Frobenius fixes all the \(c_{\ell j}\). On that subsequence,
\[
\sum_{|\ell|\le m} b_\ell^{1/q^N}s^\ell
=\sum_j\beta_j^{1/q^N}v_j,
\qquad v_j=\sum_{|\ell|\le m}c_{\ell j}s^\ell.
\]
Put \(V=\max_j|v_j|\). If \(V>0\), pick \(j_0\) with
\(|v_{j_0}|=V\), set \(w_j=v_j/v_{j_0}\), and let \(\rho_j\in k\) be the
residue of \(w_j\in\OO_C\). At least one \(\rho_j\) is nonzero, and
\(\eta=\max_j|w_j-\rho_j|<1\). Pass to a further subsequence on which
\(q^N\)-power Frobenius also fixes all the \(\rho_j\). The
\(\beta_j\)-independence of \(\sum_j\beta_j\rho_j\) gives
\[
\left|\sum_j\beta_j^{1/q^N}\rho_j\right|
=\left|\sum_j\beta_j\rho_j\right|^{1/q^N}
\to 1.
\]
Meanwhile,
\[
\limsup_N\left|\sum_j\beta_j^{1/q^N}(w_j-\rho_j)\right|
\le \eta<1
\]
since \(|\beta_j|^{1/q^N}\to1\). The ultrametric inequality then gives
\[
\left|\sum_j\beta_j^{1/q^N}v_j\right|\to V
\]
along the subsequence. Equation~\eqref{formal:root-bound} yields
\(V\le\|s\|^{m+1}\); this inequality is trivially correct if \(V=0\).
For arbitrary \(k\)-linear \(\lambda\),
\[
\left|\sum_{|\ell|\le m}\lambda(b_\ell)s^\ell\right|
=\left|\sum_j\lambda(\beta_j)v_j\right|\le V\le\|s\|^{m+1}
\]
since \(|\lambda(\beta_j)|\le 1\) (values in \(k\) have norm at most
one). The series with coefficients \(\lambda(b_\ell)\in k\) therefore
converges at \(s\), and the last bound tends to zero as \(m\to\infty\).
At \(s=0\), Equation~\eqref{formal:root-bound} with \(m=0\) gives
\(b_0=0\), hence the conclusion also holds.
\end{proof}
\begin{remark}\label{formal:separation-quantifiers}
The family \(b_\ell\) need not be bounded; every use of \(\lambda\) in
the proof is a finite sum. Moreover, the conclusion holds for
\emph{every} \(\lambda\). Thus, if the same family satisfies
\eqref{formal:root-bound} at every point of a set, choose a single
\(\lambda\) that is nonzero on some prescribed nonzero \(b_\ell\), and
obtain a common nonzero formal equation over \(k\) on that set. The
Frobenius subsequences used to prove the estimates may depend on the
tuple and on \(m\).
\end{remark}
\subsection{Reduced formal subgroups}
The next two lemmas turn a proper reduced closed formal subgroup of
\(\Gamma_h^r\) into an actual annihilating homomorphism. First we
show, before we know anything about its dimension or irreducibility,
that such a subgroup is smooth. Ordinary differentials give the
argument, and the perfect coefficient field makes them behave.
\begin{lemma}\label{formal:reduced-smooth}
Let \(\mathcal G\) be a smooth formal group over \(k\) of dimension
\(r\) with coordinate ring \(R=k[[T_1,\ldots,T_r]]\). If
\(\mathcal H\subseteq\mathcal G\) is a reduced closed formal subgroup,
then \(\mathcal H\) is smooth. More precisely, with
\(B=R/I\) and augmentation ideal \(\mathfrak m_B\), put
\(e=\dim_k(\mathfrak m_B/\mathfrak m_B^2)\). Then
\(B\cong k[[X_1,\ldots,X_e]]\) as complete local \(k\)-algebras.
\end{lemma}
\begin{proof}
Every \(f\in R\) has a unique finite decomposition
\[
f=\sum_{0\le a_i<p}T_1^{a_1}\cdots T_r^{a_r}f_a(T)^p.
\]
An ordinary \(k\)-derivation annihilates all \(p\)-th powers, so it is
determined by its values on \(T_1,\ldots,T_r\) through the usual
formal partial derivatives; conversely those partial derivatives are
derivations. Hence the ordinary module \(\Omega^1_{R/k}\) is free on
\(dT_1,\ldots,dT_r\), and taking the quotient presentation for \(B\)
gives
\begin{equation}\label{formal:differential-presentation}
\Omega^1_{B/k}
\cong B^r/\left\langle
(\partial_1f,\ldots,\partial_rf):f\in I
\right\rangle_B.
\end{equation}
In particular, \(\Omega^1_{B/k}\) is finite as a \(B\)-module.
Consider the completed product
\[
B\widehat\otimes_k B
=k[[X_1,\ldots,X_r,Y_1,\ldots,Y_r]]/(I(X),I(Y))
\]
with diagonal ideal \(J=(Y_i-X_i)_i\). Writing \(Y=X+Z\) modulo
\((Z)^2\) gives
\(f(X+Z)\equiv f(X)+\sum_i\partial_if(X)Z_i\) mod \((Z)^2\), so
\(J/J^2\) has exactly the presentation
in~\eqref{formal:differential-presentation}. The computation makes no
reducedness assumption on the completed product. The group-difference
automorphism \((x,y)\mapsto(x,y-_{\mathcal H}x)\) carries the diagonal
to the identity section in the second factor; the conormal module of
that section is \(B\otimes_k(\mathfrak m_B/\mathfrak m_B^2)\),
obtained by retaining the linear terms in the second coordinates.
We conclude
\begin{equation}\label{formal:invariant-differentials}
\Omega^1_{B/k}\cong B\otimes_k(\mathfrak m_B/\mathfrak m_B^2)\cong B^e.
\end{equation}
Let \(\mathfrak p\) be a minimal prime of \(B\), set
\(D=B/\mathfrak p\), \(K=\Frac(D)\), \(d=\dim D\). Since \(B\) is
reduced, \(B_{\mathfrak p}\) is the field \(K\). Localizing
\eqref{formal:invariant-differentials} at \(\mathfrak p\) gives
\(\dim_K\Omega^1_{K/k}=e\).
Choose a system of parameters in the complete local domain \(D\);
completeness and the finite-dimensional quotient by those parameters
give a finite map \(k[[x_1,\ldots,x_d]]\to D\). It is injective by
dimension (a nonzero ideal in the source would lower dimension).
Thus \(K\) is finite over \(L=\Frac(k[[x_1,\ldots,x_d]])\). Monomials
with exponents below \(p\) form a basis over Frobenius, so
\([L:L^p]=p^d\). Frobenius identifies the two extensions \(K/L\) and
\(K^p/L^p\), and comparing towers over \(L^p\) gives
\[
[K:K^p]=\frac{[K:L]\,[L:L^p]}{[K^p:L^p]}=p^d.
\]
This includes inseparable extensions. Because \(k\) is perfect, the
\(p\)-basis description of field differentials gives
\(\dim_K\Omega^1_{K/k}=d\): a \(p\)-basis of \(K\) over \(K^p\) has
\(d\) elements, and their differentials form a \(K\)-basis of
\(\Omega^1_{K/k}\). Therefore \(d=e\).
Every component has dimension \(e\), so \(\dim B=e\). This equal
its embedding dimension, which makes \(B\) regular. Lifting a basis
of \(\mathfrak m_B/\mathfrak m_B^2\) gives a surjection
\(k[[X_1,\ldots,X_e]]\to B\) by completeness. Equality of dimensions
forces its kernel to be zero.
\end{proof}
\begin{lemma}\label{formal:annihilator}
Let \(h,r\ge 1\). Every proper reduced closed formal subgroup
\(\mathcal H\subsetneq\Gamma_h^r\) is annihilated by a nonzero
homomorphism \(\Gamma_h^r\to\Gamma_h\) defined over \(k\).
\end{lemma}
\begin{proof}
Put \(q=p^h\), and let \(I\ne 0\) be the defining ideal of
\(\mathcal H\) in \(R=k[[T_1,\ldots,T_r]]\). By
Lemma~\ref{formal:reduced-smooth}, its coordinate ring \(B=R/I\) is a
power-series ring of dimension \(e<r\). If \(e=0\), then \(B=k\),
\(\mathcal H\) is the origin, and a coordinate projection proves the
assertion. Assume \(e>0\).
Let \(\sigma\colon R\to R\) raise every coefficient to its \(q\)-th
power and fix the variables. For \(P\in I\), stability under
multiplication by \(p\) gives
\[
P(T_1^q,\ldots,T_r^q)=(\sigma^{-1}P(T))^q\in I.
\]
Radicality implies \(\sigma^{-1}I\subseteq I\). Applying \(\sigma\)
repeatedly gives an ascending chain
\(I\subseteq\sigma I\subseteq\sigma^2 I\subseteq\cdots\) which
stabilizes because \(R\) is Noetherian; applying an inverse iterate of
\(\sigma\) to a stationary pair gives \(I=\sigma I\). Hence \(\sigma\)
descends to an automorphism of \(B\).
The coordinate pullback of \([p]\) on \(\mathcal H\) is the
\(h\)-fold absolute Frobenius of \(B\), composed with \(\sigma^{-1}\).
The \(h\)-fold Frobenius on \(B\cong k[[X_1,\ldots,X_e]]\) is finite
free, with basis the monomials whose exponents are less than \(q\).
Thus \([p]\) on \(\mathcal H\) is finite flat of degree
\(q^e=p^{he}\). The kernels \(\mathcal H[p^j]\) are finite flat of
order \(p^{hej}\), and the map induced by \([p]\) from level \(j+1\)
to level \(j\) is faithfully flat, being a pullback of \([p]\) on
\(\mathcal H\). These kernels form a \(p\)-divisible subgroup
\(\mathcal H[p^\infty]\subseteq\Gamma_h^r[p^\infty]\) of height
\(he<hr\).
The kernels also recover the formal topology. If \(\overline T_i\)
are the images of the ambient coordinates in \(B\) and
\[
J_j=(\overline T_1^{q^j},\ldots,\overline T_r^{q^j}),
\]
then
\begin{equation}\label{formal:kernel-topology}
\mathfrak m_B^{r(q^j-1)+1}\subseteq J_j\subseteq\mathfrak m_B^{q^j}.
\end{equation}
The first inclusion holds because a monomial of the indicated total
degree has an exponent at least \(q^j\).
In the semisimple isogeny category over \(k\), the ambient group is a
sum of \(r\) copies of the simple object attached to \(\Gamma_h\).
The subgroup has smaller height, so it misses at least one isogeny
summand. Projecting a nonzero complementary summand onto one simple
factor gives a nonzero rational morphism
\[
\varphi_{\Qp}\colon\Gamma_h^r[p^\infty]
\to\Gamma_h[p^\infty]
\]
whose restriction to \(\mathcal H[p^\infty]\) is zero. Choose one
common power of \(p\) clearing its denominators, giving an actual
morphism \(\varphi\) of \(p\)-divisible groups; its restriction to
\(\mathcal H[p^\infty]\) is still zero (by $p$-torsion-freeness of Hom
groups of \(p\)-divisible groups: if \(p^a f=0\) then
\(f\circ[p]^a=0\), and \([p]^a\) on the source is an epimorphism of
fppf sheaves; a rationally zero restriction is killed by one power of
\(p\) in Hom).
The compatible coordinate maps at finite levels recover an actual
formal homomorphism \(\Gamma_h^r\to\Gamma_h\). Ambient and target
kernel ideals are \((T_1^{q^j},\ldots,T_r^{q^j})\) and \((T^{q^j})\)
respectively; inverse limits of the corresponding coordinate rings
are the power-series rings. The finite-level homomorphism identities
pass to these separated inverse limits. The resulting formal morphism
is nonzero because \(\varphi\) is nonzero. Its restriction to
\(\mathcal H\) vanishes by \eqref{formal:kernel-topology} and
\(\bigcap_j J_j=0\).
\end{proof}
\subsection{From an equation to an annihilator}
We now put the two halves together.
\begin{proposition}[Formal closure]\label{formal:closure}
Let \(h,r\ge 1\), and let \(S\) be a subgroup of
\(\Gamma_h^r(\mm_C)\). If a nonzero power series in
\(k[[T_1,\ldots,T_r]]\) vanishes on \(S\), then there exist
\(a_1,\ldots,a_r\in A_h\), not all zero, such that
\[
a_1(s_1)+_{\Gamma_h}\cdots+_{\Gamma_h}a_r(s_r)=0
\qquad\text{for every }s\in S.
\]
\end{proposition}
\begin{proof}
Let \(R=k[[T_1,\ldots,T_r]]\), let \(F\) be the law of
\(\Gamma_h^r\), and let \(I\subseteq R\) be the ideal of all series
vanishing on \(S\). Evaluation into the field \(C\) shows \(I\) is
radical. It is nonzero by hypothesis and proper because \(0\in S\).
We show that it defines a formal subgroup.
For \(P\in I\), put \(Q(T,T')=P(F(T,T'))\). Fix \(s\in S\) and
write \(Q(s,T')=\sum_\ell b_\ell(T')^\ell\). Each \(b_\ell\) is the
evaluation at \(s\) of a series over \(k\), so \(|b_\ell|\le 1\). For
\(q=p^h\) and every \(t\in S\), coordinatewise \(q^N\)-th power is
multiplication by \(p^N\), hence \(Q(s,t^{q^N})=0\). Truncation at
degree \(m\) gives
\[
\left|\sum_{|\ell|\le m}b_\ell t^{q^N\ell}\right|
\le \|t\|^{q^N(m+1)}.
\]
Taking \(q^N\)-th roots and applying
Lemma~\ref{formal:separation} shows: for every \(k\)-linear
\(\lambda\colon C\to k\), the coefficientwise transform
\[
Q_{s,\lambda}(T')=\sum_\ell\lambda(b_\ell)(T')^\ell
\]
vanishes on \(S\), hence belongs to \(I(T')\).
Now descend in the first variable. Fix \(m\) and consider the
finite-dimensional \(k\)-algebra
\[
D_m=k[[T']]/\bigl(I(T')+(T')^{m+1}\bigr).
\]
Choose a \(k\)-basis \(e_1,\ldots,e_b\) of \(D_m\). Modulo the
defining ideal, write
\[
Q(T,T')=\sum_{a=1}^b c_a(T)e_a,\qquad c_a(T)\in k[[T]].
\]
Every \(c_a(T)\) is a finite \(k\)-linear combination of coefficients
of \((T')^\ell\) in \(Q\) with \(|\ell|\le m\). So \(\lambda(c_a(s))\)
is the \(e_a\)-coordinate of \(Q_{s,\lambda}\) in \(D_m\), and hence
zero. Since algebraic \(k\)-linear functionals separate points of
\(C\), we get \(c_a(s)=0\) for every \(s\in S\), so \(c_a\in I(T)\).
This step moves \(\lambda\) across a finite linear combination of
already-evaluated coefficients; it never pushes \(\lambda\) through an
infinite evaluation.
It follows that
\(Q\in I(T)+I(T')+(T')^{m+1}\) for every \(m\). Krull intersection
\cite[Tag~00IP]{Stacks} in the Noetherian local quotient by
\(I(T)+I(T')\) gives \(Q\in I(T)+I(T')\). Identity and inversion
follow from \(0\in S\) and \(-_{\Gamma_h^r}S=S\). Therefore
\(\Spf(R/I)\) is a proper reduced closed formal subgroup of
\(\Gamma_h^r\).
Lemma~\ref{formal:annihilator} supplies a nonzero formal homomorphism
\(\varphi\colon\Gamma_h^r\to\Gamma_h\) killing this subgroup. Its
restrictions to the factors are endomorphisms \(a_i\in A_h\), and the
commutative group law gives
\[
\varphi(T)=a_1(T_1)+_{\Gamma_h}\cdots+_{\Gamma_h}a_r(T_r).
\]
They are not all zero. The coordinate series of \(\varphi\) belongs
to \(I\), so evaluating on \(S\) proves the required identity.
\end{proof}
\section{A nondegeneracy lemma from the Fargues--Fontaine curve}
\label{sec:bundles}
Formal closure (Section~\ref{sec:formal}) reduced the arithmetic
problem to endomorphism relations among Honda-group points. This
section supplies the geometric fact that will exclude those relations.
The idea is well known from the theory of vector bundles on curves:
fixed `many' independent global sections of a rank-\(n\) degree-one
stable bundle cannot collapse at the generic point. When the
sections correspond to Honda-group points via the Fargues--Fontaine
dictionary, this gives a matrix-theoretic contradiction with the
endomorphism relation.
We first prove the geometric fact
(Lemma~\ref{bundles:independence}). Then we recall the Fargues--Scholze
and Le Bras identification between Honda-group points on the one hand,
and global sections of the stable bundle on the other
(Theorem~\ref{bundles:input}). Finally, we assemble the nondegeneracy
contradiction (Proposition~\ref{bundles:nondegeneracy}).
\subsection{Degree-one bundles and independent sections}
Let \(X_C\) be the Fargues--Fontaine curve with coefficient field \(\Qp\)
over \(C=\Cp^\flat\). We use the vector-bundle GAGA comparison to
pass between its algebraic and adic versions. The following
geometric features are what we use.
\begin{theorem}[Fargues--Fontaine]\label{bundles:geometry}
The curve \(X_C\) is integral, regular and Noetherian of dimension
one, and \(H^0(X_C,\OO_{X_C})=\Qp\). Degrees of vector bundles are
integers, and every nonzero effective divisor has positive degree.
For each \(s\ge 1\) there is a stable bundle
\[
\mathcal E_s=\OO_{X_C}(1/s)
\]
of rank \(s\) and degree one.
\end{theorem}
These are the Fargues--Fontaine vector-bundle theorems \cite{FF18}, in
the form of \cite[\S II.2.3--II.2.4, Propositions~II.2.7,
II.2.9--II.2.10, Example~II.2.11]{FS21}. The statement about
constants is \cite[Proposition~II.2.5\textup{(ii)}]{FS21}.
\begin{lemma}[Generic independence]\label{bundles:independence}
Let \(1\le r<n\), and let
\(z_1,\ldots,z_r\in H^0(X_C,\mathcal E_n)\) be linearly independent
over \(\Qp\). Their images in the generic fibre of \(\mathcal E_n\)
are linearly independent over the function field \(K_X\) of \(X_C\).
\end{lemma}
\begin{proof}
Let \(\mathcal F\subseteq\mathcal E_n\) be the saturation of the
image of the section map \(\OO_{X_C}^r\to\mathcal E_n\), and set
\(b=\rank\mathcal F\). Saturation on a regular Noetherian curve
produces a subbundle, and \(1\le b\le r<n\). Choose \(b\) of the
given sections independent at the generic point. Their wedge is a
nonzero global section of \(\det\mathcal F\), hence
\(\deg\mathcal F\ge 0\). Semistability of \(\mathcal E_n\) gives
\[
0\le\deg\mathcal F\le \frac{b}{n}<1.
\]
The degree is an integer, so it is zero. The divisor of the wedge
section is effective of degree zero, hence empty by
Theorem~\ref{bundles:geometry}. The chosen \(b\) sections trivialize
\(\mathcal F\) by the determinant criterion.
All the original \(z_1,\ldots,z_r\) lie in \(H^0(X_C,\mathcal F)\).
Under this trivialization that space is \(\Qp^b\), so their assumed
\(\Qp\)-independence gives \(r\le b\). Therefore \(b=r\).
\end{proof}
\subsection{Honda groups as relative section sheaves}
For a point in \(\Gamma_s(\mm_C)\) we need its realization as a
global section, together with a realization of natural maps as bundle
maps. Let \(\Perf_C\) be the big pro-\'etale site of perfectoid
spaces over \(C\). For an affinoid \(S=\operatorname{Spa}(R,R^+)\),
let \(R^{\circ\circ}\) be the topologically nilpotent elements and set
\[
\mathcal H_s(S)=\Gamma_s(R^{\circ\circ}),\qquad s\ge 1.
\]
The underlying functor is the perfected open unit disc with the
Honda group law; in particular it is a sheaf. Since \(R\) is perfect,
multiplication by \(p\) is the \(q_s\)-power Frobenius and is
bijective; combined with the continuous formal \(\Zp\)-action, this
makes \(\mathcal H_s\) a sheaf of \(\Qp\)-vector spaces.
The choice of integral subring \(R^+\) does not restrict the points:
if \(z\in R^{\circ\circ}\) then \(z^m\in R^+\) for some \(m\ge 1\), so
\(z\) is integral over \(R^+\), hence inside it. So
\(R^{\circ\circ}\subseteq R^+\) for every affinoid perfectoid pair.
We write \(\mathcal H_s(R)\) for these values, and identify
\(\mathcal H_s(C)=\Gamma_s(\mm_C)\).
For \(S\in\Perf_C\), let \(X_S\) be the relative curve. For a vector
bundle \(\mathcal V\) on \(X_C\), let \(\mathcal V_S\) be its
pull-back and set
\[
\mathcal B(\mathcal V)(S)=H^0(X_S,\mathcal V_S).
\]
This is a pro-\'etale sheaf; relative sections satisfy \(v\)-descent
\cite[Proposition~II.2.1]{FS21}.
The section-sheaf dictionary we need is the following. It identifies
Honda sheaves with the stable bundles we just introduced, and it
recovers bundle maps from additive maps of section sheaves.
\begin{theorem}[Fargues--Fontaine and Le Bras]\label{bundles:input}
\leavevmode
\begin{enumerate}
\item[\textup{(i)}]
For every \(s\ge 1\), after choosing an identification of a
height-\(s\) Lubin--Tate special fiber with \(\Gamma_s\), there is a
natural isomorphism of \(\Qp\)-vector space sheaves
\[
\mathcal H_s\cong\mathcal B(\mathcal E_s).
\]
In particular, its value at \(\operatorname{Spa}(C,\OO_C)\) identifies
\(\Gamma_s(\mm_C)\) with \(H^0(X_C,\mathcal E_s)\).
\item[\textup{(ii)}]
For vector bundles \(\mathcal V,\mathcal W\) on \(X_C\) with
nonnegative Harder--Narasimhan slopes, the natural map
\[
\Hom_{X_C}(\mathcal V,\mathcal W)
\to
\Hom_{\operatorname{Sh}_{\mathrm{Ab}}(\Perf_C)}
\bigl(\mathcal B(\mathcal V),\mathcal B(\mathcal W)\bigr)
\]
is an isomorphism. Every additive sheaf morphism between those
section sheaves is induced by a unique bundle morphism.
\end{enumerate}
\end{theorem}
\begin{proof}[The identification and full faithfulness]
For part~\textup{(i)}, let \(E^{(s)}\subset\Cp\) be the unramified
extension of \(\Qp\) of degree \(s\), and let \(G^{(s)}\) be a
Lubin--Tate formal group over \(E^{(s)}\) with uniformiser \(p\)
\cite[Theorem~1]{LT65}. Its reduction has height \(s\) as a
\(p\)-divisible formal group; choose a formal isomorphism from that
reduction to \(\Gamma_s\).
Tilting gives an equivalence between the big pro-\'etale sites over
\(\Cp\) and over \(C\) \cite[Theorem~2.7]{LeBras18}. Thus each
\(S=\operatorname{Spa}(R,R^+)\) has its specified untilt
\(S^\sharp=\operatorname{Spa}(R^\sharp,R^{\sharp+})\) over \(\Cp\).
Fargues--Scholze identify the Lubin--Tate universal cover with
relative sections by a natural isomorphism
\cite[Proposition~II.2.2]{FS21}
\[
\varprojlim_{[p]}G^{(s)}\bigl((R^\sharp)^{\circ\circ}\bigr)
\ \cong\
H^0\bigl(X_{S,E^{(s)}},\OO(1)\bigr).
\]
We compare the left side with \(\mathcal H_s(S)\). Choose
\(p^\flat\in\OO_C\) with \((p^\flat)^\sharp=p\). Tilting identifies
\((R^\sharp)^\circ/p\cong R^\circ/(p^\flat)\), with compatible residue
embeddings of \(k\). On these quotient rings the chosen
special-fibre isomorphism identifies the two formal group laws. It
remains to pass between universal-cover points before and after
reduction.
That comparison can be made directly. Suppose an integral formal
group law is evaluated in a complete uniform algebra, and reduce its
power-bounded ring modulo a fixed small base parameter of norm
\(\delta<1\). A nilpotent element of the quotient lifts to a
topologically nilpotent element: if a lift \(a\) satisfies \(a^m\) in
the parameter ideal, then \(\|a\|^m\le\delta\). Given compatible
formal points \((\bar a_j)_{j\ge 0}\) of the quotient under \([p]\),
choose such lifts \(a_j\). For each fixed \(i\) the sequence
\([p]^j(a_{i+j})\) converges. Indeed the formal difference of two
successive terms is the \(j\)-fold push-forward of a difference of
norm at most \(\delta\). For a difference of norm
\(\epsilon<1\), integrality and the linear coefficient \(p\) give
\[
\|[p](w)\|\le\max\{|p|\epsilon,\epsilon^2\}
\qquad(\|w\|\le\epsilon),
\]
and its iterates tend to zero. In characteristic \(p\) the linear
term is zero, giving the same conclusion. These estimates use an
equivalent complete spectral norm, submultiplicative and
power-multiplicative; no field hypothesis on the test algebra is
needed. Integral formal translation turns these serial bounds into
bounds on ordinary coordinate differences.
For fixed \(i\) the limit has norm at most
\(\max\{\|a_i\|,\delta\}<1\). The scalar ideal is closed for the
spectral norm, so the limits retain the prescribed reductions. The
limits form a compatible lift of \((\bar a_j)\) by topologically
nilpotent points. Uniqueness follows from the identical contraction
estimate. The comparison commutes with algebra maps, group
operations, and formal scalar multiplication. This gives the required
natural reduction invariance; compare \cite[\S II.2.1]{FS21} and
\cite[Proposition~2.14, Lemma~2.15, Remark~2.21]{LeBras18}.
Applying this comparison on both sides of the displayed
Fargues--Scholze isomorphism gives
\[
\varprojlim_{[p]}G^{(s)}\bigl((R^\sharp)^{\circ\circ}\bigr)
\ \cong\
\varprojlim_{[p]}\Gamma_s(R^{\circ\circ})
\ \cong\ \Gamma_s(R^{\circ\circ}).
\]
The last isomorphism sends \(z\) to the compatible sequence
\((z^{1/q_s^j})_{j\ge 0}\) and uses perfectness of \(R\).
We now switch coefficients from \(E^{(s)}\) to \(\Qp\). Along the
unramified coefficient extension \(\pi_s\colon X_{S,E^{(s)}}\to X_S\),
\[
(\pi_s)_*\OO(1)\cong(\mathcal E_s)_S.
\]
This is the unramified push-forward description of \(\OO(1/s)\)
\cite[\S 5.1]{LeBras18}; see also \cite[proofs of Theorem~II.2.14,
Corollary~II.3.3]{FS21}. The description is relative and natural in
\(S\): on the Frobenius cover, push-forward cycles \(s\) lines, with
the \(s\)-fold isocrystal Frobenius having multiplier \(p^{-1}\).
Hence it produces the rank-\(s\), slope \(+1/s\) bundle. Section
taking commutes with this push-forward, proving part~\textup{(i)} on
the whole site.
For part~\textup{(ii)}, Le Bras identifies Banach--Colmez sheaves
with the tilted heart
\[
\operatorname{Coh}_{X_C}^-
=\left\{F\in D^b(\operatorname{Coh}_{X_C}):
H^i(F)=0\ (i\ne-1,0),\,
H^{-1}(F)<0,\,
H^0(F)\ge 0\right\}
\]
via the functor \(R^0\tau_*\)
\cite[Theorem~1.2, Theorem~7.1]{LeBras18}. The inequalities refer to
all Harder--Narasimhan slopes. A nonnegative-slope vector bundle,
placed in degree zero, belongs to this heart; its \(R^0\tau_*\) is the
relative section sheaf just defined
\cite[Proposition~6.8 and its proof]{LeBras18}. The heart is full in
the derived category, so Hom between two such bundles is the usual
bundle Hom. Moreover, the proof of \cite[Theorem~7.1]{LeBras18}
establishes the degree-zero Ext comparison in the category of
abelian sheaves on the big site. Consequently it applies to the Hom
group in part~\textup{(ii)}, without an extra analytic-morphism or
scalar-continuity hypothesis. The fixed tilting equivalence
transports the statement to \(\Perf_C\).
\end{proof}
Every endomorphism in \(A_s\) acts naturally on \(\mathcal H_s\) and
therefore on \(\mathcal E_s\). The action extends to
\(\Delta_s=A_s[1/p]\), since multiplication by \(p\) is invertible on
both sides. This is the action on the bundle itself; no full
faithfulness for the isocrystal-to-bundle functor is invoked.
\subsection{Endomorphism orbits}
\begin{proposition}[Nondegeneracy]\label{bundles:nondegeneracy}
Let \(0<h<n\) and \(1\le r<n\). Suppose that
\(z_1,\ldots,z_r\in\mathcal H_n(C)\) are \(\Qp\)-linearly independent
and that
\[
\nu\colon\mathcal H_n\to\mathcal H_h
\]
is a nonzero morphism of \(\Qp\)-vector space sheaves on \(\Perf_C\).
Then the subgroup
\[
S_\nu=\bigl\{(\nu(dz_1),\ldots,\nu(dz_r)):d\in A_n\bigr\}
\subseteq\Gamma_h^r(\mm_C)
\]
satisfies no nonzero formal power-series equation over \(k\).
\end{proposition}
\begin{proof}
The set \(S_\nu\) is a subgroup because ring addition in \(A_n\) and
the map \(\nu\) respect the formal group laws. If a nonzero
\(k\)-formal series vanished on it, Proposition~\ref{formal:closure}
would give \(a_1,\ldots,a_r\in A_h\), not all zero, such that
\begin{equation}\label{bundles:annihilator-relation}
\sum_{i=1}^r a_i\nu d(z_i)=0
\qquad\text{for every }d\in A_n.
\end{equation}
The sum is the target group sum.
By Theorem~\ref{bundles:input}, regard the \(z_i\) as global sections
of \(\mathcal E_n\) and \(\nu\) as a nonzero bundle map
\(\mathcal E_n\to\mathcal E_h\). Operators in \(A_n\) and \(A_h\)
also act by bundle maps. Passing to the generic point gives a unital
\(K_X\)-algebra homomorphism
\[
\Delta_n\otimes_{\Qp}K_X
\to \End_{K_X}\bigl((\mathcal E_n)_{K_X}\bigr).
\]
The source is central simple over \(K_X\), so the map is injective.
Both sides have dimension \(n^2\) over \(K_X\), so it is an
isomorphism. In particular, splitting of \(\Delta_n\) over \(K_X\)
follows from the displayed action; it was not assumed in constructing
that action.
Equation~\eqref{bundles:annihilator-relation} is linear in \(d\).
First extend it from \(A_n\) to \(\Delta_n=A_n[1/p]\). At the
generic point it then extends \(K_X\)-linearly to all the displayed
matrix algebra. This is just the extension of a vanishing linear
identity; no claim that an arbitrary matrix preserves the order
\(A_n\) is involved.
By Lemma~\ref{bundles:independence} the generic vectors \(z_i\) are
independent. Choose an index \(i_0\) with \(a_{i_0}\ne 0\). This
operator is invertible in the division algebra \(\Delta_h\), hence
acts invertibly on \(\mathcal E_h\). The generic map \(\nu_{K_X}\)
is nonzero: a nonzero morphism between vector bundles on an integral
curve cannot have zero generic fiber. Choose
\(w\in(\mathcal E_n)_{K_X}\) with \(\nu_{K_X}(w)\ne 0\), and a matrix
\(M\) satisfying
\[
Mz_{i_0}=w,\qquad Mz_i=0\quad(i\ne i_0).
\]
Such \(M\) exists by the independence of the \(z_i\). Substituting
\(M\) into the extended identity gives
\(a_{i_0}\nu_{K_X}(w)=0\), a contradiction.
\end{proof}
The hypothesis that \(\nu\) is a sheaf morphism is essential: the
bundle argument does not apply to an arbitrary linear map between the
\(C\)-valued groups. In Section~\ref{sec:deformations} we construct
the required nonzero natural morphism \(\nu_x\) from a deformation
point, and produce the independent vectors to which
Proposition~\ref{bundles:nondegeneracy} will be applied.
\section{Deformation coordinates and specialization}
\label{sec:deformations}
At a characteristic-\(p\) point of exact height \(1\le h<n\), the
full height-\(n\) \(p\)-divisible group has an \'etale Tate module of
rank \(n-h\). We embed this module in the height-\(n\) Honda
universal cover, and use a specialization map to detect whether its
vectors remain in the Tate lattice at a nearby parameter point. The
values of this map on a Tate basis give analytic functions with
invertible differential along the height-at-least-\(h\) parameter
space.
We first construct the specialization map and prove the differential
statement (Proposition~\ref{def:coordinates}). An integral comparison
with the height-\(h\) Honda group then gives the nonzero natural map
\(\nu_x\) needed in Proposition~\ref{bundles:nondegeneracy}
(Proposition~\ref{def:natural-comparison}). Finally, we determine how
the specialization map changes under the stabilizer action
(Lemma~\ref{def:stabilizer}).
\subsection{The universal family and its height strata}
Let \(H\) be the universal deformation of \(\Gamma_n\) over
\[
E_0=W(k_n)[[u_1,\ldots,u_{n-1}]],
\qquad
R_0=E_0/p=k_n[[u_1,\ldots,u_{n-1}]].
\]
We also write \(H\) for its reduction over \(R_0\). Choose the usual
successive height parameters. The Lubin--Tate construction
\cite[Proposition~1.1, Theorem~3.1]{LT66} gives the following
convention: after setting \(p,u_1,\ldots,u_{h-1}\) to zero, the
\(p\)-series factors as
\begin{equation}\label{def:pseries}
[p]_H(T)=f(T^{q_h}),\qquad f(0)=0,\qquad
f'(0)\in u_h(E_0/I_h)^\times\quad(h<n),
\end{equation}
with \(f\) over \(E_0/I_h\) and the coefficient a unit when \(h=n\).
Equivalently, the successive leading coefficients of the \(p\)-series
generate the ideals \(I_h=(p,u_1,\ldots,u_{h-1})\).
Why does the factorization hold over nonreduced characteristic-\(p\)
rings as well? For a homomorphism \(a\colon F\to F'\) of formal
groups, differentiating the homomorphism identity gives
\[
a'(T)\,\partial_2F(T,0)=\partial_2F'(a(T),0)\,a'(0).
\]
Both partial derivatives are units. So \(a'(0)=0\) implies
\(a'(T)=0\), and \(a(T)\) factors through \(T^p\). The resulting
series is a homomorphism from the Frobenius twist of \(F\). Applying
this to the \(p\)-series gives \eqref{def:pseries}. The same argument
shows that the condition defining \(I_h\) is invariant under change of
coordinate, so universal deformation isomorphisms preserve the ideals.
Fix for the rest of this section
\[
1\le h<n,\qquad r=n-h,\qquad q=p^h,\qquad Q=p^n.
\]
The height-\(h\)-and-above polydisc has coordinates
\(y=(u_h,\ldots,u_{n-1})\), all of norm less than one; the earlier
parameters are zero. Write \(H_y\) for the evaluated law, and set
\[
P_y(T)=[p]_{H_y}(T)=f_y(T^q),\qquad b(y)=f_y'(0).
\]
All coefficients are integral and analytic in the parameters. Fix a
point \(x\) in the polydisc with \(u_h(x)\ne 0\). The formal group
\(H_x\) over \(C\) has height \(h\), and \(b(x)\ne 0\).
\begin{lemma}\label{def:algebraization}
The finite group schemes
\[
\mathcal G[p^j]
=\Spec\bigl(R_0[[T]]/([p^j]_H(T))\bigr),\qquad j\ge 1,
\]
form an algebraic \(p\)-divisible group \(\mathcal G\) of height \(n\) over
\(R_0\). At \(x\), its connected formal group is \(H_x\) and its
\'etale height is \(r=n-h\). All \(C\)-valued points of every level
have coordinate in \(\mm_C\).
\end{lemma}
\begin{proof}
We first record a division lemma. Let \((A,\mathfrak a)\) be a
complete Noetherian local ring, let \(d\ge 1\), and let
\(f(T)=T^d+e(T)\) with \(e\in\mathfrak a A[[T]]\). Let \(\tau\)
discard terms of degree below \(d\) and shift the remaining degrees
down by \(d\). Then
\[
\tau(qf)=(1+L)q,\qquad L(q)=\tau(qe).
\]
The operator \(L\) raises \(\mathfrak a\)-adic order, so
\((1+L)^{-1}=\sum_{i\ge 0}(-L)^i\) converges on \(A[[T]]\). For
any \(g\), put \(q=(1+L)^{-1}\tau(g)\); the unique remainder
\(g-qf\) has degree less than \(d\). Dividing by \(T^d\) writes
\(T^d=qf+r\), with \(q\equiv 1\pmod{\mathfrak a}\) and
\(r\in\mathfrak a A[T]\) of degree \(<d\). So \(q\) is a unit and
\(f=q^{-1}(T^d-r)\) is prepared. The same division shows
\(A[[T]]/(f)\) is free on \(1,T,\ldots,T^{d-1}\).
Let \(\mathfrak m=(u_1,\ldots,u_{n-1})\subseteq R_0\). Reduction
modulo \(\mathfrak m\) sends \([p^j]_H(T)\) to \(T^{Q^j}\).
Weierstrass preparation therefore writes
\begin{equation}\label{def:prepared-level}
[p^j]_H(T)=U_j(T)W_j(T),
\end{equation}
with \(U_j\in R_0[[T]]^\times\) and \(W_j\) a distinguished monic
polynomial of degree \(Q^j\). The level algebra is thus finite free
of rank \(Q^j\), \(\mathfrak m\)-adically complete, with coordinate
\(T\) topologically nilpotent. Substituting the group law into these
finite algebras defines honest group-scheme operations.
Apply preparation once more to \([p]_H(T)-V\) over \(R_0[[V]]\). This
shows that the substitution \(V=[p]_H(T)\) is finite free of rank
\(Q\). Reducing by \([p^j]_H(V)\) shows that the multiplication maps
between consecutive levels are finite faithfully flat of that rank.
Kernel and inclusion identities follow from identities of the
\(p\)-series. These are exactly the \(p\)-divisible-group conditions.
After evaluation at \(x\), all nonleading coefficients of \(W_j\) have
norm less than one. At a point of norm at least one its leading term
strictly dominates the others, so every root of \(W_j\) is small. The
evaluated \(U_j\) is a unit on the open unit disc (constant coefficient
norm one, remaining coefficients norm at most one). So the roots of
\(W_j\) are precisely the small zeros of \([p^j]_{H_x}\), with the same
group operations.
The initial degree of \([p^j]_{H_x}\) is \(q^j\), with nonzero
coefficient. Completing the finite algebra at the identity gives
\(C[[T]]/(T^{q^j})\) with the group law induced by \(H_x\). These
kernels recover \(H_x\): their defining ideals are cofinal with the
powers of \((T)\). The height is \(h\); the \'etale height of the
full height-\(n\) group is \(n-h\).
\end{proof}
\subsection{A natural specialization map}
Recall \(\mathcal H_n(R)=\Gamma_n(R^{\circ\circ})\) for an affinoid
perfectoid \(C\)-algebra \(R\). We use an equivalent complete
spectral norm on \(R\); it is power-multiplicative, so
\(R^{\circ\circ}\) is its open unit ball.
\begin{proposition}[Specialization]\label{def:transport}
For each parameter \(y\) in the height-at-least-\(h\) polydisc, the
formula
\begin{equation}\label{def:theta-limit}
\Theta_y(z)=\lim_{j\to\infty}
P_y^{\circ j}\bigl(z^{1/Q^j}\bigr)
\end{equation}
defines a continuous natural homomorphism
\[
\Theta_y\colon\mathcal H_n(R)\to H_y(R^{\circ\circ}).
\]
Here roots are unique because \(R\) is perfect, and \(P_y^{\circ j}\)
denotes composition. The map
\begin{equation}\label{def:cover-identification}
\Phi_y\colon z\longmapsto
\bigl(\Theta_y(p^{-m}z)\bigr)_{m\ge 0}
\end{equation}
is a natural isomorphism
\[
\mathcal H_n(R)\xrightarrow{\ \sim\ }
\varprojlim_{[p]_{H_y}}H_y(R^{\circ\circ}),
\]
and \(\Theta_y\) is its zeroth projection. For each fixed
\(z\in\mathcal H_n(C)\), the function \(y\mapsto\Theta_y(z)\) is
analytic on the parameter polydisc.
\end{proposition}
\begin{proof}
Fix \(y\) and choose \(\delta<1\) bounding the norms of its parameters.
The coefficients of \(H_y-\Gamma_n\) and \(P_y(T)-T^Q\) then have norm
at most \(\delta\). Integral power series are \(1\)-Lipschitz on the
open unit ball. Since \(P_y(T)=f_y(T^q)\), power-multiplicativity
gives
\begin{equation}\label{def:contraction}
\|P_y^{\circ j}(a)-P_y^{\circ j}(b)\|
\le \|a-b\|^{q^j}
\qquad(a,b\in R^{\circ\circ}).
\end{equation}
For \(m\ge 0\), put
\[
c_{j,m}=P_y^{\circ j}\bigl(z^{1/Q^{j+m}}\bigr).
\]
The discrepancy before the last \(j\) compositions is bounded by
\(\delta\), so \eqref{def:contraction} gives
\[
\|c_{j+1,m}-c_{j,m}\|\le \delta^{q^j}.
\]
The \(c_{j,m}\) converge, uniformly in \(z\), and the limit satisfies
\begin{equation}\label{def:cover-tail}
\|\Phi_y(z)_m-z^{1/Q^m}\|\le \delta,
\qquad
\|\Phi_y(z)_m\|\le\max\{\|z\|^{1/Q^m},\delta\}<1.
\end{equation}
Shifting index in the limit proves \(P_y(\Phi_y(z)_{m+1})=\Phi_y(z)_m\).
An explicit inverse: for a compatible sequence \((t_m)_{m\ge 0}\) in
the inverse limit, define
\[
\Psi_y((t_m))=\lim_{m\to\infty}t_m^{Q^m}.
\]
Indeed,
\[
\|t_{m+1}^{Q^{m+1}}-t_m^{Q^m}\|
=\|(t_{m+1}^Q-P_y(t_{m+1}))^{Q^m}\|
\le\delta^{Q^m}.
\]
The limit \(z\) exists and obeys
\begin{equation}\label{def:inverse-tail}
\|z-t_m^{Q^m}\|\le \delta^{Q^m},
\qquad
\|z\|\le \max\{\|t_0\|,\delta\}<1.
\end{equation}
No common bound below one on all the \(t_m\) is needed.
Equation~\eqref{def:cover-tail} raised to the \(Q^m\)-th power gives
\(\Psi_y\Phi_y(z)=z\). Conversely,
\eqref{def:inverse-tail} at \(m+j\) gives
\(\|z^{1/Q^{m+j}}-t_{m+j}\|\le\delta\); applying
\eqref{def:contraction} and using \(P_y^{\circ j}(t_{m+j})=t_m\) gives
\(\Phi_y\Psi_y((t_m))=(t_m)\).
The group law: coefficients of \(\Gamma_n\) lie in \(k_n\), so
\(Q\)-th powers and their inverses commute with \(\Gamma_n\)-addition.
For \(a=z^{1/Q^{j+m}}\) and \(b=w^{1/Q^{j+m}}\), \(\Gamma_n(a,b)\) and
\(H_y(a,b)\) differ by at most \(\delta\); after \(j\) compositions
their difference is at most \(\delta^{q^j}\). Since \(P_y^{\circ j}\)
is an \(H_y\)-homomorphism, taking limits gives
\(\Phi_y(\Gamma_n(z,w))_m=H_y(\Phi_y(z)_m,\Phi_y(w)_m)\). This proves
additivity, hence additivity of the inverse as well. The uniform tail
estimates and continuity of roots and series give continuity of the
maps. Everything commutes with continuous perfectoid-algebra maps,
hence naturality. \(\Zp\)-multiplication commutes by continuity from
integer multiplication.
Finally, fix \(z\in\mathcal H_n(C)\) and a closed parameter polydisc
strictly inside the open one. Choose \(\delta<1\) bounding all
parameter norms. Each finite stage of \eqref{def:theta-limit}
belongs to the corresponding Tate algebra: the substituted constant
has norm less than one and the coefficients of the law are bounded.
The same bound \(\delta^{q^j}\) holds in the parameter Gauss norm, so
the stages converge in the complete Tate algebra. Analyticity
follows. Partial differentiation is continuous on a fixed closed
polydisc with operator norm bounded by the inverse coordinate radius;
so convergence also permits first differentiation.
\end{proof}
\begin{lemma}\label{def:tate-kernel}
At the chosen point \(x\), \(\Theta_x\) is surjective on \(C\)-points,
and
\[
T_x=\ker\bigl(\Theta_x\colon\mathcal H_n(C)\to H_x(\mm_C)\bigr)
\cong\Zp^r.
\]
In particular, a \(\Zp\)-basis \(z_1,\ldots,z_r\) of \(T_x\) is
\(\Qp\)-linearly independent in \(\mathcal H_n(C)\).
\end{lemma}
\begin{proof}
Under \eqref{def:cover-identification}, the kernel consists of
compatible sequences with \(t_0=0\) and \([p]t_{j+1}=t_j\). These are
precisely the compatible \(p^j\)-torsion points of \(\mathcal G_x\).
By Lemma~\ref{def:algebraization} every such point is small, and the
\'etale height is \(r\). Over the algebraically closed \(C\), the
Tate module is \(\Zp^r\).
For surjectivity, Weierstrass-preparation of \([p]_H(T)-V\) over
\(R_0[[V]]\): at \(y=x\) and \(V=w\in\mm_C\), the distinguished
polynomial is monic with all lower coefficients small. Algebraic
closedness gives a root, and every root is small. So every small
point admits division by \(p\) within the open disc. Iterating gives
a compatible sequence with any prescribed zeroth point,
Proposition~\ref{def:transport} gives surjectivity of \(\Theta_x\).
Multiplying a putative rational relation by a large power of \(p\)
gives the independence of a \(\Zp\)-basis.
\end{proof}
\subsection{The parameter differential}
We identify parameter directions with marked first-order deformations
of \(\mathcal G\), using the algebraic \(p\)-divisible group over
\(R_0\). This avoids any problem about whether the \(C\)-valued
evaluation \(R_0\to C\) sends parameters to nilpotent elements. For a
\(p\)-divisible group \(G\), let \(G^\vee\) be its Cartier dual and
\(\omega_G\) its module of invariant differentials.
\begin{proposition}\label{def:ks}
The first-order parameter deformation of \(\mathcal G\) induces an
isomorphism of free \(R_0\)-modules
\[
\mathsf K\colon R_0^{n-1}\xrightarrow{\ \sim\ }
\Hom_{R_0}(\omega_{\mathcal G^\vee},\Lie\mathcal G).
\]
For every small \(C\)-valued parameter point \(x\), its specialization
is the linear isomorphism from parameter directions to marked
first-order deformations of \(\mathcal G_x\), with the constant
deformation as origin.
\end{proposition}
\begin{proof}
We use the square-zero deformation theorem of
\cite[Theorem~5.1]{Lau08}, in the cotangent-theoretic tradition of
Grothendieck--Illusie
\cite[Theorem~4.4, Corollary~4.7]{Illusie85}, fixing its torsor
structure. If \(B\to A\) has square-zero kernel \(J\) and \(p\) is
nilpotent, marked lifts of a \(p\)-divisible group \(G/A\) form a
torsor under
\[
\Hom_A(\omega_{G^\vee},J\otimes_A\Lie G).
\]
The associated Kodaira--Spencer map is characterised by the
arbitrary-ring-lift equivariance of \cite[Equation~(5.1)]{Lau08}.
We use the same construction at the closed point and at every
specialization. The theorem applies in characteristic \(p\) at
every prime, including \(p=2\).
Apply this to the two maps into
\[
R_0[\epsilon_1,\ldots,\epsilon_{n-1}]/
(\epsilon_i\epsilon_j:1\le i,j\le n-1)
\]
given by \(u_i\mapsto u_i\) and \(u_i\mapsto u_i+\epsilon_i\). They
define the \(R_0\)-linear map \(\mathsf K\). The source is the full
parameter-derivation module: every series in \(R_0\) has a finite
expression
\(a=\sum_{0\le e_i<p}u^e a_e^p\), so ordinary differentials over
\(\Fp\) are free on \(du_1,\ldots,du_{n-1}\). The target is free of
the same rank because \(\Lie\mathcal G\) has rank one and
\(\omega_{\mathcal G^\vee}\) has rank \(n-1\).
At the closed point, injectivity of the tangent map from
formal-group deformations to \(p\)-divisible-group deformations
suffices: both tangent spaces have dimension \(n-1\), by
\cite[Proposition~2.6, Theorem~3.1]{LT66} plus the square-zero
torsor theorem. Suppose a parameter tangent vector has zero image.
Then its \(p\)-divisible deformation is marked-isomorphic to the
constant one, so its finite kernels have compatible marked
isomorphisms.
Those kernels recover the formal-group isomorphism. Explicitly, for
a deformation \(F\) over a local Artin characteristic-\(p\) algebra
\(B\) with maximal ideal \(J\) and \(J^a=0\),
\[
([p^j]_F(T))\subseteq (T^{p^j}),\qquad
T^{aQ^j}\in([p^j]_F(T)).
\]
The first inclusion is iterated Frobenius factorization. For the
second, preparation gives a monic polynomial of degree \(Q^j\) whose
lower coefficients lie in \(J\); in its quotient \(T^{Q^j}\in J\),
whose \(a\)-th power vanishes. So the kernel ideals are cofinal with
the coordinate-adic topology. Inverse limits of the compatible finite
Hopf-algebra isomorphisms give a marked formal-group isomorphism; this
is the formal reconstruction underlying Tate's equivalence
\cite[Section~2.2, Proposition~1, Lemma~0]{Tate67}. Lubin--Tate
tangent injectivity forces the original vector to be zero. So
\(\mathsf K\) is invertible modulo the maximal ideal of \(R_0\); its
determinant is a unit, and \(\mathsf K\) is an isomorphism over
\(R_0\).
For a vector \(v=(v_i)\in C^{n-1}\), the analytic first-order
substitution is the actual ring homomorphism
\[
a\longmapsto a(x)+\epsilon\sum_i v_i\partial_i a(x)
\quad\text{from }R_0\text{ to }C[\epsilon]/\epsilon^2.
\]
All evaluations converge because coordinates of \(x\) are small. The
arbitrary-map functoriality in the deformation theorem identifies its
class with the specialization of \(\mathsf K\). In cotangent
notation, the specialized map involves
\(\Omega^1_{R_0/\Fp}\otimes_{R_0}C\), not the intrinsic module
\(\Omega^1_{C/\Fp}\) (which is zero because \(C\) is perfect).
\end{proof}
\begin{lemma}\label{def:height-rigidity}
Let \(A\) be a local Artin \(C\)-algebra with residue field \(C\), and
let \(F/A\) be a marked deformation of a height-\(h\) formal group
over \(C\). If \([p]_F(T)\) factors through \(T^{p^h}\) over \(A\),
then \(F\) is the constant deformation with its marking.
\end{lemma}
\begin{proof}
Factorization through \(T^{p^h}\) is preserved by coordinate change:
the \(p^h\)-th power of any series has only exponents divisible by
\(p^h\). Choose an isomorphism of the special fiber with \(\Gamma_h\)
and apply the marked Lubin--Tate classification over \(A\). It gives
parameters \(a_1,\ldots,a_{h-1}\) in the maximal ideal of \(A\).
Modulo preceding parameters, the coefficient of \(T^{p^i}\) in the
\(p\)-series is \(a_i\) times a unit. This is the height-parameter
property recalled earlier; it follows directly from
\cite[Proposition~1.1]{LT66}: the leading law term is
\(a_i C_{p^i}\), where \(C_{p^i}\) is the reduction of the integral
polynomial \(((X+Y)^{p^i}-X^{p^i}-Y^{p^i})/p\), whose \(p\)-fold sum
has coefficient \(p^{p^i-1}-1\) (a unit modulo \(p\)). The assumed
factorization thus forces each \(a_i\) to vanish. Zero parameter
tuple is the constant deformation, and the isomorphism from
\cite[Theorem~3.1]{LT66} respects the marking. For \(h=1\) there are
no parameters and the same conclusion holds.
\end{proof}
\begin{proposition}[Tate coordinates]\label{def:coordinates}
For a \(\Zp\)-basis \(z_1,\ldots,z_r\) of \(T_x\), the analytic map
\begin{equation}\label{def:theta}
\theta(y)=\bigl(\Theta_y(z_1),\ldots,\Theta_y(z_r)\bigr)
\end{equation}
has invertible differential at \(x\) on the height-at-least-\(h\)
polydisc.
\end{proposition}
\begin{proof}
Analyticity from Proposition~\ref{def:transport}, and \(\theta(x)=0\).
Let \(v\) be a direction in the kernel of its differential. Write
\(A=C[\epsilon]/\epsilon^2\) and \(y'\) for the corresponding
first-order parameter point. Its preceding parameters remain zero.
By Proposition~\ref{def:ks}, it suffices to prove that
\(\mathcal G_{y'}\) is the constant deformation of \(\mathcal G_x\)
with its residue marking. We will use the vanishing differential to
lift the Tate basis as compatible torsion sections, splitting the
\'etale quotient. The fixed-height condition will then trivialize the
connected factor.
\smallskip\noindent\emph{Lifting the Tate basis.}
For \(j\ge 1\) set
\[
t_{i,j}(y)=\Theta_y(p^{-j}z_i),\qquad
t'_{i,j}=t_{i,j}(y')\in A,
\]
where \(y'\)-evaluation means first-order Taylor evaluation.
We verify membership in the finite group schemes before passing to a
\(p\)-divisible group. Evaluate \eqref{def:prepared-level} at the
analytic function \(t_{i,j}(y)\). Its value at \(x\) is small, so on
a neighborhood of \(x\) it gives an identity of convergent analytic
functions
\[
U_{j,y}(t_{i,j}(y))\,W_{j,y}(t_{i,j}(y))=\Theta_y(z_i).
\]
The first factor has nonzero reduced value at \(x\). Taylor evaluation
in \(A\) makes the right side zero (by choice of \(v\)), and the first
factor remains a unit. Hence \(W_{j,y'}(t'_{i,j})=0\): the Taylor
coordinate is an actual point of \(\mathcal G_{y'}[p^j]\). The
analytic multiplication identities also give
\([p]t'_{i,j+1}=t'_{i,j}\), \([p^j]t'_{i,j}=0\). These agree with the
algebraic finite-level operations.
For any finite collection of Taylor points \(a+\epsilon b\), equip
\(A\) with the complete submultiplicative norm
\[
\|a+\epsilon b\|_s=\max\{|a|,s|b|\},
\]
choosing \(s>0\) so the points and parameter values are small. The
polynomial finite-level operations then converge; their vanishing
identities under the source-level polynomials match the Taylor
evaluations on the formal operations.
Each finite identity is evaluated on a neighborhood where its finitely
many series converge. Those neighborhoods may depend on \(j\); all
their Taylor values lie in the same algebraic ring \(A\). In
particular, no bound on the derivatives uniform in \(j\) is needed.
Sending a basis vector of \((\mathbb Z/p^j)^r\) to \(t'_{i,j}\) gives
a homomorphism of finite group schemes. These maps respect
multiplication by \(p\) and the level inclusions. Over the strictly
henselian local Artin ring \(A\), the connected--\'etale sequence
exists and its \'etale quotient is the unique constant lift of the
residue quotient. The constructed maps induce an isomorphism onto
that quotient at each level (since they do so over \(C\)); they
therefore split it and give
\[
\mathcal G_{y'}\cong\mathcal G_{y'}^0\times(\Qp/\Zp)^r.
\]
\smallskip\noindent\emph{Trivializing the connected factor.}
At level \(j\), first base-change the finite algebra to
\(B_j=A[T]/(W_{j,y'}(T))\). Idempotents of its residue algebra lift
uniquely across \(\epsilon\) and select the identity factor \(B_j^0\).
This factor is a finite local Artin algebra, so it equals its
completion at \((\epsilon,T)\). Equation~\eqref{def:prepared-level}
identifies that completion with
\[
B_j^0\cong A[[T]]/([p^j]_{H_{y'}}(T)).
\]
Indeed, preparation holds coefficientwise over \(A\), and all residue
factors supported away from \(T=0\) become units in this completion.
The group law is \(H_{y'}\), by Taylor evaluation of the same
preparation and law identities. Since the preceding parameters are
zero and \(b(x)\ne 0\), \([p^j]_{H_{y'}}\) is \(T^{q^j}\) times a
unit. So the displayed algebra is also \(A[[T]]/(T^{q^j})\). This
describes the connected factor as the identity factor after algebraic
base change.
Its formal group \(H_{y'}\) satisfies Lemma~\ref{def:height-rigidity},
so it is the constant deformation of \(H_x\) with the residue marking
fixed. That formal isomorphism restricts to all finite kernels (their
coordinates are nilpotent, so every formal series evaluates by a
finite sum). The connected \(p\)-divisible factor is consequently
constant with its marking.
\smallskip\noindent\emph{Recovering the marked deformation.}
The splitting of the \'etale factor also has the required marking.
Over algebraically closed \(C\), each connected finite group scheme has
only the identity as \(C\)-point. A map from a constant finite group
scheme is determined by the images of its \(C\)-points. The actual
torsion points given by the chosen Tate basis therefore select the
unique section of the \'etale quotient on the special fiber. The
constructed splitting reduces to this section. Combined with the
marked connected trivialization, this gives the constant deformation
of \(\mathcal G_x\) with its original marking.
Proposition~\ref{def:ks} now implies \(v=0\). Both parameter tangent
space and target of \(D\theta_x\) have dimension \(r=n-h\); the
differential is invertible.
\end{proof}
\subsection{An integral comparison with the height-\(h\) Honda group}
The following elementary recursion controls every coefficient of the
forward comparison. Its necessary-and-sufficient form will also be
used at individual orbit points.
\begin{lemma}\label{def:recursion}
Let \(B\) be a characteristic-\(p\) algebra, let
\(f(Z)=bZ+c_2Z^2+\cdots\in B[[Z]]\), and put \(q=p^h\). For
\(\alpha(T)=\sum_{j\ge 1}a_jT^j\), the identity
\[
\alpha(f(T^q))=\alpha(T)^q
\]
holds if and only if, for every \(j\ge 1\),
\begin{equation}\label{def:coefficient-recursion}
a_j^q-b^ja_j
=\sum_{i<j}a_i[Z^j]f(Z)^i.
\end{equation}
If \(B=C\), all coefficients of \(f\) are integral, and the \(a_j\)
satisfy these equations, then all \(a_j\) are integral. In
particular, after prescribing any finite initial segment satisfying
the equations, every recursive extension by roots in \(C\) remains
integral.
\end{lemma}
\begin{proof}
Both sides contain only powers of \(T\) divisible by \(q\). The
coefficient of \(T^{qj}\) gives exactly
\eqref{def:coefficient-recursion}, since the \(a_j\)-contribution on
the left is \(b^ja_j\). There are no other coefficient equations.
For integrality, induct on \(j\). The right side is integral if the
preceding coefficients are integral. A root with \(|a_j|>1\) would
make \(|a_j|^q\) strictly larger than both \(|b^ja_j|\) and the norm of
the right side, which is impossible. Includes \(j=1\), whose right
side is zero. The polynomial is monic, roots exist in the
algebraically closed \(C\), and the same argument bounds every choice.
\end{proof}
\begin{proposition}\label{def:natural-comparison}
Choose a formal group isomorphism
\(\alpha_x\colon H_x\xrightarrow{\ \sim\ }\Gamma_h\) over \(C\). Its
forward coefficients are integral, and composition gives a nonzero
natural \(\Qp\)-linear morphism of sheaves on perfectoid spaces over
\(C\),
\begin{equation}\label{def:nu}
\nu_x\colon\mathcal H_n\to\mathcal H_h,\qquad
\nu_x(z)=\alpha_x(\Theta_x(z)).
\end{equation}
\end{proposition}
\begin{proof}
The formal isomorphism exists by the height classification over \(C\).
It intertwines \([p]_{H_x}(T)=f_x(T^q)\) with
\([p]_{\Gamma_h}(T)=T^q\); Lemma~\ref{def:recursion} gives that every
coefficient of \(\alpha_x\) lies in \(\OO_C\).
An integral series with zero constant term converges on
\(R^{\circ\circ}\) for every affinoid perfectoid \(C\)-algebra \(R\),
with value again in \(R^{\circ\circ}\). Its formal homomorphism
identity converges on those arguments as well. Combined with
Proposition~\ref{def:transport}, this proves additivity and naturality
of \(\nu_x\) on such affinoids, hence as a sheaf morphism. Continuity
extends integer linearity to \(\Zp\)-linearity. Multiplication by
\(p\) is invertible on both Honda sheaves, so the morphism is
\(\Qp\)-linear. Equivalently, Theorem~\ref{bundles:input} permits
working with additive sheaf morphisms throughout.
Put \(a_1=\alpha_x'(0)\ne 0\), and choose \(t\in C\) with
\(0<|t|<|a_1|\). For \(j\ge 2\), integral coefficients satisfy
\(|a_jt^j|\le|t|^2<|a_1t|\), so \(\alpha_x(t)\ne 0\).
Lemma~\ref{def:tate-kernel} supplies
\(z\in\mathcal H_n(C)\) with \(\Theta_x(z)=t\). Thus \(\nu_x(z)\ne 0\),
proving nonzeroness. Only the forward series is evaluated on the
open unit disc; no convergence about the inverse is needed.
\end{proof}
\subsection{Compatibility with the stabilizer action}
The coordinate construction is complete. The next two results
describe how the specialization map and its Tate lattice change under
the stabilizer. The first is a finite-level statement for the full
universal ring, independent of the height stratum.
\begin{lemma}[Continuity of the stabilizer action]
\label{def:continuous-action}
Put \(\mathfrak m_E=(p,u_1,\ldots,u_{n-1})\subseteq E_0\). For
\(M\ge 1\) and \(N\ge M\), every element of \(1+p^NA_n\) acts
trivially on \(E_0/\mathfrak m_E^M\). So the stabilizer action on
\(E_0\), and on its characteristic-\(p\) quotient, is adically
continuous.
\end{lemma}
\begin{proof}
Let \(B=E_0/\mathfrak m_E^M\), let \(J\) be its maximal ideal, and
let \(F\) be the universal formal group over \(B\). For \(d\in A_n\),
lift its coefficients in \(k_n\) to \(B\), obtaining a series
\(\widetilde d(T)\) with zero constant term. The homomorphism defect
\[
\delta(X,Y)=\widetilde d(F(X,Y))
-_F F(\widetilde d(X),\widetilde d(Y))
\]
has all coefficients in \(J\), because \(d\) is a homomorphism on the
special fiber. For every \(a\ge 1\), substituting into
\([p]_F(Z)=pZ+\sum_{j\ge 2}c_jZ^j\) sends a series with coefficients in
\(J^a\) and zero constant term to one with coefficients in
\(J^{a+1}\): this uses \(p\in J\) and \(2a\ge a+1\). So
\([p]_F^N(\delta)=0\) for \(N\ge M\).
Since \([p]_F^N\) is itself a homomorphism, this says \(f_N=[p]_F^N
\circ\widetilde d\) is an endomorphism of \(F\). Its special fiber
is \(p^Nd\), and \(\operatorname{id}_F -_F f_N\) has linear
coefficient \(1-p^N\widetilde d'(0)\), a unit of \(B\). It is
therefore an automorphism lifting \(1-p^Nd\). Marked Lubin--Tate
classification then identifies the deformation with its translate by
\(1-p^Nd\), so this element fixes the parameter map \(E_0\to B\).
The assertion is uniform in \(d\). Replacing \(d\) by \(-d\) gives
the stated congruence subgroup; these form a neighborhood basis of
the identity in \(A_n^\times\).
\end{proof}
We use the stabilizer convention in which \(g\in A_n^\times\) carries
\(x\) to a point \(y_g\) together with an integral isomorphism
\[
\iota_g\colon H_x\xrightarrow{\ \sim\ }H_{y_g}
\]
whose closed-point reduction is \(g\). These are the universal
deformation isomorphisms \cite[Section~3.4]{LT66}. The orbit remains
in the height-at-least-\(h\) polydisc. Moreover, \(y_g\to x\) as
\(g\to 1\) by Lemma~\ref{def:continuous-action}: a series in the
\(M\)-th power of the characteristic-\(p\) parameter ideal evaluates
with norm at most \(\|x\|^M\). On the characteristic-zero generic
fiber the same argument uses \(\rho^M\), where
\(\rho=\max\{|p|,\|x\|\}<1\).
\begin{lemma}\label{def:stabilizer}
For \(g\in A_n^\times\) and \(z\in\mathcal H_n(C)\),
\begin{equation}\label{def:orbit-transport}
\Theta_{y_g}(gz)=\iota_g(\Theta_x(z)).
\end{equation}
Consequently \(gT_x=T_{y_g}\).
\end{lemma}
\begin{proof}
Coefficients of \(g\) lie in \(k_n\), so
\(g(z^{1/Q^j})=(gz)^{1/Q^j}\). Choose \(\delta<1\) bounding the
evaluated parameter ideals. Coefficients of \(\iota_g-g\) are then
bounded by \(\delta\). On small arguments the two series differ by at
most that amount. By \eqref{def:contraction}, their images after \(j\)
compositions with \(P_{y_g}\) differ by at most \(\delta^{q^j}\).
Using \(P_{y_g}^{\circ j}\iota_g=\iota_g P_x^{\circ j}\) and passing
to the limit gives \eqref{def:orbit-transport}. It gives
\(gT_x\subseteq T_{y_g}\); the inverse isomorphism gives equality.
\end{proof}
Choose a \(\Zp\)-basis \(z_1,\ldots,z_r\) of \(T_x\) as in
Lemma~\ref{def:tate-kernel}. Since \(r=n-h<n\),
Proposition~\ref{bundles:nondegeneracy} applies to this basis and
\(\nu_x\): the subgroup
\[
\bigl\{(\nu_x(dz_1),\ldots,\nu_x(dz_r)):d\in A_n\bigr\}
\subseteq\Gamma_h^r(\mm_C)
\]
satisfies no nonzero formal power-series equation over \(k\). Together
with the invertible differential of Proposition~\ref{def:coordinates}
and the transport identity of Lemma~\ref{def:stabilizer}, this is all
the input for the orbit argument of Section~\ref{sec:orbits}.
\section{Positive-characteristic orbit density}\label{sec:orbits}
We now prove that on each positive-characteristic height stratum, a
stabilizer orbit satisfies no nonzero analytic equation near its base
point. This is the arithmetic heart of the paper.
\begin{theorem}[Local orbit density]\label{orb:density}
Fix \(1\le h<n\) and a \(C\)-valued point \(x\) of the parameter
polydisc \(I_h=0\) with \(u_h(x)\ne 0\). For \(g\in A_n^\times\),
write \(y_g\) for its orbit point with the transport convention of
Lemma~\ref{def:stabilizer}. Let \(G\) be any open subgroup of
\(A_n^\times\). If an analytic germ \(P\) at \(x\) on the parameter
polydisc defined by \(I_h=0\) vanishes at all points of the orbit
\(\{y_g:g\in G\}\) sufficiently close to \(x\), then \(P=0\).
It suffices to assume that, for every \(d\in A_n\),
\(P(y_{1-p^Nd})=0\) for all sufficiently large \(N\).
\end{theorem}
The proof composes Tate coordinates of Section~\ref{sec:deformations}
with comparison series whose coefficients are analytic germs in the
parameters. The coefficients need not share one domain of
convergence. Instead, at individual orbit points we choose convergent
series with integral coefficients matching each fixed finite initial
segment eventually. The next lemma says this suffices to convert
formal Taylor identities into numerical estimates.
\begin{lemma}[Finite analytic jets]\label{orb:finite-jets}
Let \(F\) be a complete nonarchimedean valued field, let \(r\ge 1\),
use the max norm on \(F^r\). Let \(\theta=(\theta_i)_{i=1}^r\) be an
analytic germ at \(0\in F^r\) with \(\theta(0)=0\), and let \(a_j\),
\(j\ge 1\), be scalar analytic germs at \(0\). Define formal series
\[
U_i(X)=\sum_{j\ge 1}a_j(X)\theta_i(X)^j
\quad\text{in }F[[X_1,\ldots,X_r]],
\]
via the Taylor expansions of the germs. Suppose the linear part
\(L=dU_0\) is invertible.
Let \(\delta_N\to 0\) in \(F^r\), and suppose coefficients
\(a_{N,j}\in\OO_F\) are chosen so that, for every \(M\ge 1\), all
sufficiently large \(N\) satisfy
\[
a_{N,j}=a_j(\delta_N)\qquad(1\le j\le M).
\]
This includes that the finitely many germs in that formula are defined
at \(\delta_N\). For sufficiently large \(N\), put
\[
U_{N,i}=\sum_{j\ge 1}a_{N,j}\theta_i(\delta_N)^j.
\]
These numerical series converge. If \(P\) is an analytic germ with
\(P(\delta_N)=0\) eventually, write its unique formal expansion as
\[
P(X)=\sum_{\ell\in\mathbb N^r}B_\ell U(X)^\ell.
\]
Then there is \(D>0\) such that \(\|\delta_N\|\le D\|U_N\|\) for all
sufficiently large \(N\). Moreover, for every \(m\ge 0\) there is
\(D_m>0\) such that
\begin{equation}\label{orb:finite-jet-estimate}
\left|\sum_{|\ell|\le m}B_\ell U_N^\ell\right|
\le D_m\|U_N\|^{m+1}
\end{equation}
for all sufficiently large \(N\). Thresholds and \(D_m\) may depend
on \(m\). No convergence of the full formal series \(U\) or
\(\sum_\ell B_\ell Z^\ell\) is required.
\end{lemma}
\begin{proof}
The formal definition of \(U\) is valid because
\(\theta_i\in(X_1,\ldots,X_r)\), so only finitely many summands
contribute to each total degree. Formal inversion applies because
\(L\) is invertible, giving the asserted unique expansion of \(P\).
For fixed \(m\ge 1\), the finite sum
\[
F_{m,i}(X)=\sum_{j=1}^m a_j(X)\theta_i(X)^j
\]
is an actual analytic germ. Its Taylor polynomial through total
degree \(m\) is \(J_mU_i\), the corresponding truncation of \(U_i\).
Analyticity and \(\theta(0)=0\) give
\(\|\theta(\delta)\|\le c\|\delta\|\) near \(0\). For large \(N\),
coefficient matching and integrality give
\[
\|U_N-F_m(\delta_N)\|
\le \|\theta(\delta_N)\|^{m+1}
\le c^{m+1}\|\delta_N\|^{m+1}.
\]
Here \(\|\theta(\delta_N)\|<1\), which also proves convergence of the
numerical series defining \(U_N\). The Taylor estimate for finitely
many analytic functions \(F_{m,i}\) therefore yields
\begin{equation}\label{orb:jet-approximation}
U_N=J_mU(\delta_N)+R_{m,N},\qquad
\|R_{m,N}\|\le c_m\|\delta_N\|^{m+1}.
\end{equation}
Explicitly, an analytic series
\(H=\sum_{|\beta|\ge m+1}h_\beta X^\beta\) converging on a closed
polydisc of radius \(\rho\) satisfies
\(|H(\delta)|\le M\rho^{-(m+1)}\|\delta\|^{m+1}\) for
\(\|\delta\|\le\rho\), where
\(M=\sup_\beta|h_\beta|\rho^{|\beta|}<\infty\).
For \(m=1\), write \(U_N=L\delta_N+R_{1,N}\). If \(\delta_N\ne 0\) and
\(N\) is large,
\[
\|R_{1,N}\|<\|L^{-1}\|^{-1}\|\delta_N\|\le\|L\delta_N\|.
\]
The ultrametric inequality gives \(\|U_N\|=\|L\delta_N\|\), hence
\(\|\delta_N\|\le\|L^{-1}\|\|U_N\|\). For \(\delta_N=0\) the same
bound holds. So for large \(N\), \(U_N=0\) implies \(\delta_N=0\).
Put \(Q_m(Z)=\sum_{|\ell|\le m}B_\ell Z^\ell\) and \(V_m=J_mU\).
The analytic germ \(P-Q_m(V_m)\) has zero Taylor coefficients through
degree \(m\): this follows from the formal identity for \(P\) and
\(V_m\equiv U\pmod{(X)^{m+1}}\). Consequently
\[
|P(\delta_N)-Q_m(V_m(\delta_N))|
\le c'_m\|\delta_N\|^{m+1}.
\]
Both \(U_N\) and \(V_m(\delta_N)\) tend to zero. On the unit polydisc
\(Q_m\) is Lipschitz: telescoping each monomial gives constant
\(\max(1,\max_{1\le|\ell|\le m}|B_\ell|)\).
Equation~\eqref{orb:jet-approximation}, the preceding bound, and
\(P(\delta_N)=0\) now give
\[
|Q_m(U_N)|\le c''_m\|\delta_N\|^{m+1}.
\]
The inverse norm estimate proves
\eqref{orb:finite-jet-estimate} for \(m\ge 1\). For \(m=0\), continuity
gives \(B_0=P(0)=0\). Every use of the formal identity in this argument
takes place after a finite truncation.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{orb:density}]
Put \(r=n-h\) and \(q=p^h\). Choose a \(\Zp\)-basis
\(z_1,\ldots,z_r\) of \(T_x\) as in Lemma~\ref{def:tate-kernel}.
Proposition~\ref{def:coordinates} says that
\[
\theta(y)=\bigl(\Theta_y(z_i)\bigr)_{i=1}^r,\qquad
\theta(x)=0,
\]
is analytic with invertible differential at \(x\). Fix the formal
isomorphism
\[
\alpha_x\colon H_x\xrightarrow{\ \sim\ }\Gamma_h,\qquad
\alpha_x(T)=\sum_{j\ge 1}a_j^xT^j,
\]
from Proposition~\ref{def:natural-comparison}. Its forward
coefficients belong to \(\OO_C\), its linear coefficient is nonzero,
and \(\nu_x=\alpha_x\Theta_x\) is a nonzero natural map
\(\mathcal H_n\to\mathcal H_h\).
\smallskip\noindent\emph{Formal coordinates.} Write
\([p]_{H_y}(T)=f_y(T^q)\) and \(b(y)=f_y'(0)\), so \(b(x)\ne 0\). By
Lemma~\ref{def:recursion}, a series
\(\alpha(T)=\sum_{j\ge 1}A_jT^j\) intertwines \(p\)-multiplication with
\(q\)-th powers exactly when
\begin{equation}\label{orb:coefficient-germs}
A_j^q-b(y)^jA_j
=\sum_{i<j}A_i[Z^j]f_y(Z)^i\qquad(j\ge 1).
\end{equation}
Inductively choose an analytic germ \(a_j(y)\) satisfying this
equation and \(a_j(x)=a_j^x\). The derivative in the new variable is
\(-b(x)^j\ne 0\), so the analytic implicit function theorem applies.
An explicit branch: with earlier coefficients fixed and
\(A_j=a_j^x+t\), the equation becomes
\[
t=b(y)^{-j}\bigl(t^q+e(y)\bigr),\qquad e(x)=0.
\]
On a sufficiently small closed parameter polydisc, choose \(\eta>0\)
with \(\|b^{-j}\|\eta^{q-1}<1\) and \(\|b^{-j}e\|\le\eta\). The
right side is a strict contraction on the ball of analytic functions
of norm at most \(\eta\); its fixed point is zero at \(x\). This
proves the germ at every finite stage.
Set \(\delta=y-x\), where subtraction is ordinary parameter
subtraction. Define
\[
U_i(\delta)=\sum_{j\ge 1}a_j(x+\delta)\Theta_{x+\delta}(z_i)^j
\quad\text{in }C[[\delta_1,\ldots,\delta_r]].
\]
Each coefficient of \(\delta\) involves finitely many \(a_j\). The
linear matrix is \(a_1^xd\theta_x\), which is invertible. So, if \(P\)
were nonzero, there would be a fixed family
\((B_\ell)_{\ell\in\mathbb N^r}\) in \(C\), not all zero, with
\[
P(x+\delta)=\sum_\ell B_\ell U(\delta)^\ell
\quad\text{formally}.
\]
The germs \(a_j\), formal coordinates \(U\), and coefficients
\(B_\ell\) are now fixed. No common analytic domain for all \(a_j\)
is chosen.
\smallskip\noindent\emph{Matching actual coefficients on an orbit.}
Fix \(d\in A_n\). For all sufficiently large \(N\), the element
\(g_N=1-p^Nd\) is a unit in \(G\), and its orbit point
\(y_N=y_{g_N}\) tends to \(x\) by continuity before
Lemma~\ref{def:stabilizer}. The subtraction in \(g_N\) is in the
endomorphism ring. Transport sends \(g_Nz_i\) into \(T_{y_N}\);
applying \(\Theta_{y_N}\) therefore gives
\begin{equation}\label{orb:transport-powers}
\Theta_{y_N}(z_i)
=[p]_{H_{y_N}}^N\bigl(\Theta_{y_N}(dz_i)\bigr).
\end{equation}
This identity uses formal-group addition.
Choose a numerical series at each \(y_N\). For every \(M\ge 1\) choose
an integer \(N_M\ge M\), strictly increasing in \(M\), so that for all
\(N\ge N_M\): the first \(M\) germs \(a_j\) are defined at \(y_N\),
their instances of \eqref{orb:coefficient-germs} hold, and
\(b(y_N)\) and \(a_1(y_N)\) are nonzero. Such thresholds exist
because \(y_N\to x\). For \(N\ge N_1\) put
\[
K(N)=\max\{M\le N:N_M\le N\},
\]
so \(K(N)\to\infty\). Prescribe \(a_{N,j}=a_j(y_N)\) for
\(1\le j\le K(N)\), and extend by roots in \(C\) of
\eqref{orb:coefficient-germs} at \(y_N\). Lemma~\ref{def:recursion}
shows all chosen coefficients -- including the prescribed ones --
belong to \(\OO_C\). The resulting actual series
\[
\alpha_N(T)=\sum_{j\ge 1}a_{N,j}T^j
\]
converges on \(\mm_C\) and satisfies the full identity
\[
\alpha_N\bigl([p]_{H_{y_N}}(T)\bigr)=\alpha_N(T)^q.
\]
This is the only property of \(\alpha_N\) needed. In particular, for
fixed \(j\), the coefficient satisfies \(a_{N,j}=a_j(y_N)\) eventually,
hence \(a_{N,j}\to a_j^x\).
Define convergent numerical quantities
\[
U_{N,i}=\alpha_N\bigl(\Theta_{y_N}(z_i)\bigr),\qquad
s_{N,i}=\alpha_N\bigl(\Theta_{y_N}(dz_i)\bigr).
\]
Iterating the intertwining identity in
\eqref{orb:transport-powers} gives the exact coordinate identity
\begin{equation}\label{orb:rooted-coordinates}
U_N=s_N^{q^N}.
\end{equation}
All evaluations are on topologically nilpotent elements; integral
series preserve this domain.
Continuity in the parameter gives
\(\Theta_{y_N}(dz_i)\to\Theta_x(dz_i)\). Choose \(c<1\) bounding the
norms of these finitely many arguments and of their limits. Integrality
bounds the tails of \(\alpha_N\) past degree \(J\) by \(c^{J+1}\),
uniformly in \(N\); the same bound holds for \(\alpha_x\). The first
\(J\) coefficients converge term by term. Letting \(N\to\infty\) then
\(J\to\infty\) proves
\begin{equation}\label{orb:limit-tuple}
s_N\to s(d):=\bigl(\nu_x(dz_i)\bigr)_{i=1}^r
\quad\text{in }\mm_C^r.
\end{equation}
This covers zero coordinates and zero limiting tuples.
\smallskip\noindent\emph{Frobenius roots and one formal equation.}
Apply Lemma~\ref{orb:finite-jets} with \(\delta_N=y_N-x\), the germs
\(a_j(x+\delta)\), and the actual coefficients \(a_{N,j}\). The
vanishing hypothesis gives \(P(y_N)=0\) eventually. Thus, for every
fixed \(m\ge 0\),
\[
\left|\sum_{|\ell|\le m}B_\ell U_N^\ell\right|
\le D_m\|U_N\|^{m+1}
\]
eventually, with constants and threshold possibly depending on \(m\)
and the fixed \(d\). Since \(C\) is perfect of characteristic \(p\),
\eqref{orb:rooted-coordinates} permits the unique \(q^N\)-th root of
this finite sum, giving
\[
\left|\sum_{|\ell|\le m}B_\ell^{1/q^N}s_N^\ell\right|
\le D_m^{1/q^N}\|s_N\|^{m+1}.
\]
For this fixed \(m\), the \(B_\ell^{1/q^N}\) are uniformly bounded.
By \eqref{orb:limit-tuple}, replacing \(s_N\) by \(s(d)\) in the
finite sum changes its value by a vanishing quantity. As
\(D_m^{1/q^N}\to 1\), the ultrametric inequality gives
\begin{equation}\label{orb:rooted-limit}
\limsup_{N\to\infty}
\left|\sum_{|\ell|\le m}B_\ell^{1/q^N}s(d)^\ell\right|
\le \|s(d)\|^{m+1}.
\end{equation}
When \(s(d)=0\), the right side and replacement error both tend to
zero, and the conclusion holds.
For each \(d\in A_n\), \eqref{orb:rooted-limit} holds for every
\(m\ge 0\). Lemma~\ref{formal:separation} therefore gives
\[
\sum_\ell\lambda(B_\ell)s(d)^\ell=0
\]
for every \(k\)-linear \(\lambda\colon C\to k\) and every \(d\in A_n\).
Choose one such \(\lambda\) nonzero on a fixed nonzero coefficient
\(B_{\ell_0}\). Since the family \(B_\ell\) was fixed before \(d\),
this produces a single nonzero formal series over \(k\) vanishing on
the whole set
\[
\bigl\{(\nu_x(dz_i))_{i=1}^r:d\in A_n\bigr\}.
\]
The \(z_i\) are \(\Qp\)-linearly independent, \(r=n-h<n\), and
\(\nu_x\) is a nonzero natural map.
Proposition~\ref{bundles:nondegeneracy} excludes this formal equation.
The contradiction proves the theorem.
\end{proof}
\section{Invariant ideals and the spectral classification}
\label{sec:classification}
We now pass from the orbit theorem to ideals of the deformation ring.
The characteristic-\(p\) argument uses two elementary facts
(Lemmas~\ref{fin:curve} and \ref{fin:translation}).
The characteristic-zero argument uses the Gross--Hopkins period map
(Proposition~\ref{fin:charzero}).
Finally, we transfer through the support-comparison
(Proposition~\ref{fin:support}) to prove Theorem~\ref{thm:main} and
identify the Balmer spectrum (Corollary~\ref{fin:balmer}).
\subsection{Points on formal subschemes}
\begin{lemma}[An auxiliary formal curve]\label{fin:curve}
Let \(R\) be a complete Noetherian local domain with residue field
\(k_n\).
\begin{enumerate}
\item Suppose \(R\) is a local \(k_n\)-algebra and \(0\ne a\in\mm_R\).
For every \(c\in\mm_C\setminus\{0\}\) there is a continuous
\(k_n\)-algebra homomorphism \(R\to\OO_C\) sending \(a\mapsto c\) and
\(\mm_R\to\mm_C\).
\item Suppose \(R\) is a local \(W(k_n)\)-algebra and \(p\ne 0\) in
\(R\). Fix the usual embedding \(W(k_n)\hookrightarrow\OO_{\Cp}\).
There is a continuous \(W(k_n)\)-algebra homomorphism
\(R\to\OO_{\Cp}\) sending \(\mm_R\to\mm_{\Cp}\).
\end{enumerate}
In neither case is the homomorphism required to be injective.
\end{lemma}
\begin{proof}
We first produce a one-dimensional domain quotient on which a given
nonzero nonunit \(a\) survives. We use the Noetherian local dimension
and parameter theorems \cite[Tags~00KQ, 00KD]{Stacks}. Put
\(d=\dim R\), extend \(a\) to a system of parameters
\(a,b_2,\ldots,b_d\), and let \(J=(b_2,\ldots,b_d)\). Some minimal
prime \(\mathfrak q\) over \(J\) avoids \(a\): otherwise
\(a\in\sqrt J\), and the parameter property would give
\(\sqrt J=\mm_R\), contradicting the height bound for an ideal
generated by \(d-1\) elements.
The ring \(D=R/\mathfrak q\) is a complete local domain in which \(a\)
is a nonzero nonunit and \(\sqrt{aD}=\mm_D\). The principal ideal
theorem gives \(\dim D\le 1\); the existence of a nonzero nonunit
gives equality. So the \(a\)-adic and maximal-ideal topologies on
\(D\) coincide. This includes \(d=1\), where \(J=0\) and we may take
\(\mathfrak q=0\).
In case 1, substitution \(t\mapsto a\) defines an injective map
\(k_n[[t]]\to D\): writing a nonzero series as \(t^mv(t)\) with
\(v(0)\ne 0\) maps it to a nonzero power of \(a\) times a unit.
Moreover, \(D/aD\) is finite-dimensional over \(k_n\), so completeness
makes \(D\) a finite \(k_n[[a]]\)-module; explicitly, lift a \(k_n\)-basis
of \(D/aD\), successively subtract linear combinations of these lifts
modulo \(a,a^2,\ldots\). The coefficient series converge in
\(k_n[[a]]\) and span.
For the prescribed \(c\), evaluation embeds \(k_n[[t]]\hookrightarrow
\OO_C\): the first nonzero term of a series is uniquely dominant
when \(0<|c|<1\). Its fraction field is a complete discretely valued
subfield of \(C\) with uniformiser \(c\). The finite extension
\(\Frac(D)/k_n((a))\) embeds into \(C\) over this base embedding. Every
element of \(D\) is integral over \(k_n[[a]]\), hence has norm at most
one. For \(z\in\mm_D\), some \(z^N\in aD\), whence
\(|z|^N\le|c|<1\). The composite \(R\to D\to\OO_C\) has the required
properties.
In case 2, run the same construction with \(a=p\). The resulting
\(D\) is finite over \(W(k_n)\) by the identical successive
approximation. The structural map from this DVR is injective because
\(p\ne 0\) in the domain \(D\). Embed the finite extension
\(\Frac(D)/\Frac(W(k_n))\) into \(\Cp\) over the fixed base embedding.
Integrality gives boundedness, and \(\sqrt{pD}=\mm_D\) gives strict
smallness of \(\mm_D\).
Finally, continuity in both cases follows from finite generation of
\(\mm_R\): its generators have images of norm at most some \(\rho<1\),
so \(\mm_R^N\) maps into elements of norm at most \(\rho^N\).
\end{proof}
The point supplied by Lemma~\ref{fin:curve} need not retain a given
nonzero equation after restriction to the auxiliary curve. What
matters is that the equation keeps a nonzero germ in the
\emph{ambient} polydisc.
\begin{lemma}[Bounded translation]\label{fin:translation}
Let \(L\) be a complete nontrivially valued nonarchimedean field, let
\(r\ge 0\), and let
\[
\mathscr B_r(L)=
\left\{\sum_{I\in\mathbb N^r}a_IT^I:
\sup_I|a_I|<\infty\right\},
\]
with coefficient norm \(\|f\|=\sup_I|a_I|\). For \(x\in L^r\) with
\(\|x\|<1\), substitution \(T\mapsto x+T\) defines an isometric
\(L\)-algebra automorphism \(\tau_x\) of \(\mathscr B_r(L)\), with
inverse \(\tau_{-x}\). A nonzero bounded-coefficient series
therefore has a nonzero analytic germ at every such \(x\).
\end{lemma}
\begin{proof}
The cases \(r=0\) or \(x=0\) are immediate. Otherwise put
\(\rho=\|x\|\), so \(0<\rho<1\), and \(M=\sup_I|a_I|\). The
coefficient of \(T^J\) in the translated series is
\[
(\tau_xf)_J=\sum_{I\ge J}\binom IJ a_Ix^{I-J}.
\]
Each term has norm at most \(M\rho^{|I|-|J|}\), so the sum converges
and the new coefficients are bounded by \(M\).
All reorderings in the substitution can be checked coefficientwise.
The coefficient of \(T^K\) in \(\tau_{-x}\tau_xf\) is a sum over
\(I\ge J\ge K\). Its individual terms have norm at most
\(M\rho^{|I|-|K|}\), and only finitely many pairs occur below any fixed
total degree. The family is summable independently of order, and
regrouping gives
\[
\sum_{I\ge K}a_Ix^{I-K}\binom IK
\sum_{K\le J\le I}
(-1)^{|J-K|}\binom{I-K}{J-K}
=a_K.
\]
The inner sum vanishes unless \(I=K\). Replacing \(x\) by \(-x\) proves
the other inverse identity.
Multiplicativity has the same convergence justification. If
\(g=\sum_Qd_QT^Q\), the coefficient of \(T^H\) in
\(\tau_xf\,\tau_xg\) is a sum over \(J+K=H\), \(P\ge J\), \(Q\ge K\).
Terms are bounded by
\(\|f\|\|g\|\rho^{|P|+|Q|-|H|}\), so can be regrouped. The multi-index
Vandermonde identity
\[
\sum_{\substack{J+K=H\\J\le P,K\le Q}}
\binom PJ\binom QK=\binom{P+Q}{H}
\]
then matches with the coefficient of \(\tau_x(fg)\). Both inverse
maps are contractions for the supremum norm, hence isometries. A
bounded-coefficient series converges on every closed polydisc of radius
less than one, and its expansion at \(x\) is \(\tau_xf\), by the same
convergent binomial expansion. A zero analytic germ would give
\(\tau_xf=0\), hence \(f=0\).
\end{proof}
\subsection{Characteristic zero and the Gross--Hopkins map}
Let \(\mathfrak X_{\Cp}\) be the open unit polydisc with coordinates
\(u_1,\ldots,u_{n-1}\), the generic fiber of \(\Spf(E_0)\) extended to
scalars in \(\Cp\). The nonextended stabilizer acts continuously on it
by Lemma~\ref{def:continuous-action}. The argument below develops the
Lie-theoretic approach credited to Chai in \cite[Remark~4.2]{BHN22};
compare the projective-tangent calculation in
\cite[Proposition~4.18]{BHN22}.
\begin{proposition}[Characteristic-zero orbit germs]\label{fin:charzero}
Let \(U\subseteq A_n^\times\) be an open subgroup and
\(x\in\mathfrak X_{\Cp}(\Cp)\). If an analytic germ at \(x\) vanishes
at \(gx\) for every \(g\in U\) sufficiently close to the identity, that
germ is zero.
\end{proposition}
\begin{proof}
At \(n=1\) the polydisc is a point. Suppose \(n>1\). Apply the period
theorem after completed base change from \(W(k_n)\) to \(W(k)\) and
the compatible embedding \(W(k)\to\OO_{\Cp}\). The marked universal
deformation and its \(A_n^\times\)-action commute with this base change,
and the resulting \(\Cp\)-generic fiber is
\(\mathfrak X_{\Cp}\). The Gross--Hopkins period theorem
\cite[Section~1.1, Theorem~1]{GH94}, with proof in
\cite[Proposition~23.5]{HG94Bundles}, gives an equivariant \'etale map
\[
\Phi\colon\mathfrak X_{\Cp}\to\mathbb P^{n-1}_{\Cp}.
\]
The action on the target is the projectivisation of the split
representation
\(\Delta_n\otimes_{\Qp}\Cp\simeq\End_{\Cp}(\Cp^n)\).
Write \(\xi=\Phi(x)\). The analytic inverse function theorem gives
neighborhoods \(V\) of \(x\) and \(W\) of \(\xi\) where \(\Phi\) is an
analytic isomorphism. For this, the argument is the same contraction
used in Section~\ref{sec:orbits}: after translating and applying the
inverse linear part, write \(\Phi\) as \(z+R(z)\) with \(R(0)=0\),
\(dR_0=0\), and shrink a closed polydisc until \(R\) has Lipschitz
constant \(<1\); iteration of \(z=w-R(z)\) contracts in the complete
Tate algebra to give the analytic inverse. Transport the given germ
\(P\) to \(Q=P\circ(\Phi|_V)^{-1}\) at \(\xi\).
The infinitesimal action of the full matrix algebra surjects onto
\(T_\xi\mathbb P^{n-1}_{\Cp}\). Since \(A_n\) spans \(\Delta_n\) over
\(\Qp\), choose \(d_1,\ldots,d_{n-1}\in A_n\) whose infinitesimal
projective directions form a \(\Cp\)-basis. For small
\(t=(t_1,\ldots,t_{n-1})\in\Cp^{n-1}\), form
\[
\eta(t)=\left(1+\sum_i t_id_i\right)\xi
\]
in the split representation. This is analytic with invertible
differential at zero. Shrink its domain so that its image lies in \(W\)
and \(Q\circ\eta\) is defined.
For sufficiently large \(a\), every \(t\in(p^a\Zp)^{n-1}\) defines an
element \(g_t=1+\sum_it_id_i\in U\). Continuity of the stabilizer
action also lets us require \(g_tx\in V\), within the neighborhood
where \(P\) vanishes along the orbit. Equivariance and the fixed
inverse branch give
\[
(\Phi|_V)^{-1}(\eta(t))=g_tx,\qquad (Q\circ\eta)(t)=0.
\]
An analytic function on a \(\Cp\)-polydisc vanishing on
\((p^a\Zp)^{n-1}\) is zero: fix all but one variable in that set,
apply one-variable uniqueness using a sequence tending to zero, and
repeat in each variable. Nonzero one-variable germs have the form
\(t^ru(t)\) with \(u(0)\ne 0\), so their zeros cannot accumulate at
zero. Thus \(Q\circ\eta\) is the zero germ; the invertible
differential of \(\eta\) implies \(Q\), and therefore \(P\), is zero.
\end{proof}
The argument uses a \(\Qp\)-valued set of parameters in a fixed
analytic inverse branch; no openness of the stabilizer orbit in the
\(\Cp\)-topology is needed.
\subsection{Primes and radical ideals}
\begin{theorem}\label{fin:primes}
Every proper prime ideal of \(E_0\) stable under an open subgroup of
\(A_n^\times\) is one of
\[
0,\ I_1,\ldots,I_n.
\]
\end{theorem}
\begin{proof}
Let \(\mathfrak p\) be such a prime, stable under the open subgroup
\(U\).
Suppose first that \(p\in\mathfrak p\). Choose the largest \(h\) such
that \(I_h\subseteq\mathfrak p\). If \(h=n\), then \(I_n\) is
maximal and \(\mathfrak p=I_n\). Otherwise the image of \(u_h\) in
\(R=E_0/\mathfrak p\) is a nonzero nonunit.
Lemma~\ref{fin:curve} gives a point \(x\colon R\to\OO_C\) at which all
parameters are small and \(u_h(x)\ne 0\). It is therefore a point of
the height-at-least-\(h\) polydisc to which
Theorem~\ref{orb:density} applies.
If \(\mathfrak p/I_h\ne 0\), choose nonzero
\[
f\in\mathfrak p/I_h\subseteq k_n[[u_h,\ldots,u_{n-1}]].
\]
Its coefficients are bounded, and Lemma~\ref{fin:translation} shows it
defines a nonzero ambient analytic germ at \(x\). Stability of
\(\mathfrak p\) makes \(f\) vanish at every point of the \(U\)-orbit of
\(x\). This contradicts Theorem~\ref{orb:density}. So
\(\mathfrak p=I_h\).
Now suppose \(p\notin\mathfrak p\). The mixed-characteristic part of
Lemma~\ref{fin:curve} supplies \(x\colon E_0/\mathfrak p\to
\OO_{\Cp}\) with all parameters small. If \(\mathfrak p\ne 0\), take
\(0\ne f\in\mathfrak p\). The fixed embedding
\(W(k_n)\hookrightarrow\OO_{\Cp}\) regards \(f\) as a nonzero
bounded-coefficient series in the ambient generic polydisc. By
Lemma~\ref{fin:translation} its germ at \(x\) is nonzero. It vanishes
on the \(U\)-orbit, contradicting Proposition~\ref{fin:charzero}. So
\(\mathfrak p=0\).
\end{proof}
\begin{corollary}\label{fin:radicals}
For every open subgroup \(U\subseteq A_n^\times\), the \(U\)-invariant
radical ideals of \(E_0\) are precisely
\[
0,\ I_1,\ldots,I_n,\ E_0.
\]
The same list gives the radical ideals invariant under the full
canonical extended stabilizer group \(G_n\).
\end{corollary}
\begin{proof}
Let \(J\) be a proper \(U\)-invariant radical ideal. Noetherianity
gives finitely many minimal primes, and the group permutes them. The
stabilizer of each such prime \(P\) is closed: ideals in the complete
Noetherian local ring \(E_0\) are adically closed, and each of
\[
g(a)\in P,\quad g^{-1}(a)\in P\quad(a\in P)
\]
is closed by continuity of the action. Together these say \(gP=P\).
This closed stabilizer has finite index in \(U\), hence is open in
\(U\): the complement is a finite union of closed cosets. Since \(U\)
is open in \(A_n^\times\), so is the stabilizer.
Theorem~\ref{fin:primes} applies to every minimal prime of \(J\).
Their intersection is \(J\); since the standard primes form a chain,
this intersection is its smallest member. Hence \(J\) is standard.
An ideal invariant under \(G_n\) is invariant under \(A_n^\times\).
Conversely, each ideal in the list is invariant under the full
canonical action. For \(I_h\), this follows from its intrinsic
description as the locus where the universal formal group has height
at least \(h\), or from the successive height parameters of its
\(p\)-series. The zero and unit ideals are trivially invariant.
\end{proof}
Theorem~\ref{fin:primes} and Corollary~\ref{fin:radicals} prove
Theorem~\ref{thm:invariant-radicals}.
\subsection{Support and the spectral classification}
\label{fin:classification}
Write \(\mathcal D=\mathcal D_{(n,p)}\), and denote the thick tensor
ideal generated by an object \(X\) by \(\langle X\rangle_\otimes\).
For \(X\in\mathcal D\), put
\[
\mathcal F(X)=L_{K(n)}(E\wedge X),\qquad
M_X=\pi_0\mathcal F(X)\oplus\pi_1\mathcal F(X),\qquad
S(X)=\Supp_{E_0}(M_X).
\]
The two summands of \(M_X\) are the completed Morava \(E\)-homology
groups in degrees \(0\) and \(1\), which determine the remaining
degrees by periodicity.
\begin{proposition}[Established support comparison]\label{fin:support}
For every prime \(p\) and height \(n\ge 1\), the following hold.
\begin{enumerate}
\item The functor \(\mathcal F\colon\mathcal D\to\Perf(E)\) is exact
and symmetric monoidal (source with \(K(n)\)-localized product). It
detects tensor-nilpotence of morphisms.
\item The set \(S(X)\) is a closed \(G_n\)-invariant subset of
\(\Spec(E_0)\), and
\[
S(X\otimes Y)=S(X)\cap S(Y).
\]
For \(X,Y\in\mathcal D\),
\[
X\in\langle Y\rangle_\otimes
\Longleftrightarrow S(X)\subseteq S(Y).
\]
\item For any finite \(p\)-local spectrum \(F(k)\) of type \(k\),
\(0\le k\le n\),
\[
S(L_{K(n)}F(k))=V(I_k).
\]
Also \(S(0)=\varnothing\), \(S(L_{K(n)}S)=\Spec(E_0)\).
\end{enumerate}
\end{proposition}
\begin{proof}
We assemble established results and spell out the direction of support
descent. Base change is symmetric monoidal into the category of
\(K(n)\)-local \(E\)-modules; its dualizable objects are exactly the
ordinary perfect \(E\)-modules by \cite[Proposition~4.7]{BHN22}. So
\(\mathcal F\) has the asserted target even when \(X\) is dualizable
but not compact as a \(K(n)\)-local spectrum. Tensor-nilpotence
detection is \cite[Proposition~4.6]{BHN22}; neither statement has a
prime bound. These inputs use Morava-\(E\) descent and the
perfect-module comparison of Mathew
\cite[Propositions~3.27, 10.10--10.11]{Mathew16}.
The coefficient ring \(E_*\) is even-periodic, regular and Noetherian.
The comparison theorem of Dell'Ambrogio--Stanley
\cite[Theorem~1.1, Lemma~3.10]{DellAmbrogioStanley16}, applied to
Morava \(E\)-theory in \cite[Lemmas~4.3--4.4]{BHN22}, identifies
\[
\Spc(\Perf(E))\simeq\Spec(E_0)
\]
and identifies the tensor-triangular support of \(\mathcal F(X)\) with
\(S(X)\). The tensor-product identity for that support gives
\(S(X\otimes Y)=S(X)\cap S(Y)\). The module \(M_X\) is finite over
\(E_0\), so the support is closed. Its canonical semilinear
stabilizer action makes it invariant; equivalently use
\cite[Lemma~4.5]{BHN22}.
Tensor-nilpotence detection gives a surjection
\[
f\colon\Spc(\Perf(E))\to\Spc(\mathcal D)
\]
by Balmer's surjectivity theorem \cite[Theorem~1.3]{Balmer18}, in the
form \cite[Theorem~2.6, Corollary~4.8]{BHN22}. Writing
\(\operatorname{supp}\) for tensor-triangular support, functoriality
gives
\[
\operatorname{supp}(\mathcal F(X))
=f^{-1}\bigl(\operatorname{supp}(X)\bigr).
\]
If \(S(X)\subseteq S(Y)\), surjectivity of \(f\) therefore implies
\(\operatorname{supp}(X)\subseteq\operatorname{supp}(Y)\). Every
object of \(\mathcal D\) is dualizable, so its thick tensor ideals are
radical. Balmer's classification \cite[Theorem~4.10]{Balmer05}, in
the support-containment form of \cite[Corollary~2.4]{BHN22}, gives
\(X\in\langle Y\rangle_\otimes\). The converse follows from
exactness, tensor-compatibility, and the same criterion; this is also
the transfer in \cite[Corollary~2.7]{BHN22}.
For finite type-\(k\) objects the coefficient support is \(V(I_k)\), as
in \cite[Proposition~3.3 and proof of Theorem~4.9]{BHN22}: the usual
finite chromatic support calculation, unrestricted in \(p\). The two
remaining cases follow from \(\mathcal F(0)=0\) and
\(\mathcal F(L_{K(n)}S)=E\).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
By Corollary~\ref{fin:radicals} and Proposition~\ref{fin:support},
the support of every object of \(\mathcal D\) belongs to the finite
chain
\[
\Spec E_0=V(I_0)\supsetneq V(I_1)\supsetneq\cdots
\supsetneq V(I_n)\supsetneq\varnothing.
\]
Indeed, \(S(X)=V(\Ann_{E_0}M_X)\), whose defining radical ideal is
\(G_n\)-invariant. Empty support occurs only for the zero object,
by applying the support comparison with \(Y=0\).
Let \(\mathcal J\ne 0\) be any thick tensor ideal. The supports among
its objects form a nonempty subset of this finite chain and thus
have a greatest member attained by some \(Y\in\mathcal J\). For every
\(X\in\mathcal J\), support comparison gives
\(X\in\langle Y\rangle_\otimes\), and the reverse containment follows
from \(Y\in\mathcal J\). So
\(\mathcal J=\langle Y\rangle_\otimes\). No single-generator
assumption was needed; the attainment follows from finiteness of the
support chain.
Write \(S(Y)=V(I_k)\). The given object \(L_{K(n)}F(k)\) has the same
support, so applying Proposition~\ref{fin:support} in both directions
gives
\[
\mathcal J
=\langle L_{K(n)}F(k)\rangle_\otimes
=\mathcal D_k.
\]
The zero ideal is \(\mathcal D_{n+1}\). The same comparison proves
independence of the generator: another finite type-\(k\) spectrum has
support \(V(I_k)\), hence generates the same ideal. This also agrees
with the finite thick-subcategory theorem of Hopkins--Smith
\cite{HopkinsSmith98}. Strict inclusions of the displayed supports
imply \(\mathcal D_k\supsetneq\mathcal D_{k+1}\): the second generator's
support is not contained in \(V(I_{k+1})\). For \(k=n\),
\(V(I_n)\ne\varnothing\), so \(\mathcal D_n\ne 0\). Finally
\(V(I_0)=\Spec E_0\), which is the support of the unit, and \(\mathcal
D_0\) contains the unit with the same support; hence
\(\mathcal D_0=\mathcal D\). The list has exactly \(n+2\) entries.
\end{proof}
Every support in this deduction is realised by a finite spectrum
already listed in Theorem~\ref{thm:main}. No realisation of an
arbitrary Morava module -- hence no prime restriction from the
converse direction of the Hovey--Strickland--Chai comparison -- is
needed.
\subsection{The Balmer spectrum}
A proper thick tensor ideal \(\mathcal P\) is \emph{prime} if
\(X\otimes Y\in\mathcal P\) implies \(X\in\mathcal P\) or
\(Y\in\mathcal P\). The points of \(\Spc(\mathcal D)\) are these
ideals; the tensor-triangular support of \(X\) is
\(\operatorname{supp}(X)=\{\mathcal P:X\notin\mathcal P\}\). The
following identifies the spectrum, with the indexing of
\cite[Proposition~3.5]{BHN22}.
\begin{corollary}\label{fin:balmer}
For every prime \(p\) and every \(n\ge 1\),
\[
\Spc(\mathcal D_{(n,p)})
=\{\mathcal D_1,\ldots,\mathcal D_{n+1}\},\qquad
\overline{\{\mathcal D_j\}}
=\{\mathcal D_l:j\le l\le n+1\}.
\]
Thus \(\mathcal D_1\) is the generic point and \(\mathcal D_{n+1}=0\)
is the unique closed point. For \(0\le k\le n\),
\[
\operatorname{supp}(L_{K(n)}F(k))
=\{\mathcal D_j:k<j\le n+1\}.
\]
\end{corollary}
\begin{proof}
Possible coefficient supports form a total chain, including the empty
set. Hence \(S(X)\cap S(Y)\in\{S(X),S(Y)\}\). By the tensor-product
and containment statements in Proposition~\ref{fin:support}, if
\(X\otimes Y\) belongs to a proper \(\mathcal D_j\), one of \(X\),
\(Y\) does as well. This includes \(\mathcal D_{n+1}=0\): two
nonempty supports both contain \(V(I_n)\), so their intersection is
nonempty. Every proper member of the chain is therefore prime, and
Theorem~\ref{thm:main} leaves no other prime ideals.
In the Balmer topology, \(\mathcal Q\in\overline{\{\mathcal P\}}\)
precisely when \(\mathcal Q\subseteq\mathcal P\): every support
containing \(\mathcal P\) must contain \(\mathcal Q\), i.e.\
\(X\notin\mathcal P\) implies \(X\notin\mathcal Q\) for every \(X\).
The strict ideal chain gives the stated closure formula. Finally
\(L_{K(n)}F(k)\in\mathcal D_j\) exactly when \(k\ge j\), by the
coefficient-support comparison, which gives the support formula.
\end{proof}
The continuous surjection in Proposition~\ref{fin:support} has an
explicit description in terms of the height ideals. For
\(\mathfrak q\in\Spec(E_0)\), set
\[
h(\mathfrak q)=\max\{0\le h\le n:I_h\subseteq\mathfrak q\}.
\]
Then
\[
f\colon\Spec(E_0)\to\Spc(\mathcal D),\qquad
f(\mathfrak q)=\mathcal D_{h(\mathfrak q)+1}.
\]
Indeed, functoriality says \(X\in f(\mathfrak q)\) exactly when
\(\mathfrak q\notin S(X)\). For a nonzero object with \(S(X)=V(I_k)\),
this is equivalent to \(k>h(\mathfrak q)\), which by
Proposition~\ref{fin:support} is exactly the condition
\(X\in\mathcal D_{h(\mathfrak q)+1}\). The zero object belongs to
both ideals as well. So the fiber over \(\mathcal D_{h+1}\) is
\(V(I_h)\setminus V(I_{h+1})\) for \(0\le h<n\), and the fiber over
\(\mathcal D_{n+1}=0\) is \(V(I_n)\).
Here \(h(\mathfrak q)\) records the formal-group height stratum, not
the Krull height of the prime: all primes avoiding \(p\) map to the
generic Balmer point \(\mathcal D_1\), while the maximal ideal \(I_n\)
maps to the closed point \(0\).
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\end{document}