64c3814
==== TABLE OF CONTENTS ====
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//01-introduction.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//02-cohomology.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//03-completed-tower.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//04-modifications.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//05-degree-three.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//06-witt-transfer.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//07-detection.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//08-elementary-background.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//09-frobenius.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//10-oriented-projection.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//main.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//preamble.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//unicode-mappings.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//wedge-obstruction.tex
A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//references.bib
===========================
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//01-introduction.tex ====
\section{Introduction}
\label{sec:introduction}
Let \(S\) be the sphere spectrum, fix a prime \(p\), and write
\(S_p^\wedge\) for its \(p\)-completion. We use \(L_i\) for localization
at Johnson--Wilson theory \(E(i)\) and \(L_{K(i)}\) for localization at
Morava \(K\)-theory of height \(i\), all at \(p\). In particular, \(L_0\)
is rationalization. Chromatic splitting concerns the lower-height
overlap \(L_{n-1}L_{K(n)}S\) in the fracture square for \(L_nS\).
Hopkins' strong formulation, recorded by Hovey
\cite[Conjecture~4.2]{Hovey1995}, prescribes an assembly of this overlap
from localized spheres.
At height three, the proposed underlying spectrum is
\begin{equation}\label{eq:proposed-wedge}
\begin{aligned}
\mathcal W_3={}&L_2\bigl(S_p^\wedge\vee\Sigma^{-1}S_p^\wedge\bigr)\\
&{}\vee L_1\bigl(\Sigma^{-3}S_p^\wedge\vee\Sigma^{-4}S_p^\wedge\bigr)\\
&{}\vee L_0\bigl(\Sigma^{-5}S_p^\wedge\vee\Sigma^{-6}S_p^\wedge
\vee\Sigma^{-8}S_p^\wedge\vee\Sigma^{-9}S_p^\wedge\bigr).
\end{aligned}
\end{equation}
The strong prediction is an equivalence
\(L_2L_{K(3)}S\simeq\mathcal W_3\), with specified maps, including the
canonical unit on the first summand. The revised formulation of
Barthel--Beaudry retains this odd-prime prediction
\cite[Conjecture~6.3 and Remark~6.4]{BB2019}. We disprove even the
underlying equivalence of \(E(2)\)-local spectra.
The obstruction is a natural map. For every spectrum \(X\), the
\(K(2)\)-localization unit induces
\[
\tau_X:L_0X\longrightarrow L_0L_{K(2)}X.
\]
\begin{theorem}\label{thm:main}
For every prime \(p\geq5\), the canonical map
\begin{equation}\label{eq:main-map}
\tau_{L_{K(3)}S}:L_0L_{K(3)}S\longrightarrow
L_0L_{K(2)}L_{K(3)}S
\end{equation}
is nonzero on \(\pi_{-3}\).
\end{theorem}
\begin{corollary}\label{cor:strong}
For every prime \(p\geq5\), there is no equivalence of \(E(2)\)-local
spectra \(L_2L_{K(3)}S\simeq\mathcal W_3\), with \(\mathcal W_3\) as in
\eqref{eq:proposed-wedge}. Thus the height-three strong chromatic
splitting formula is false in this range, whether or not an equivalence
is required to preserve its specified unit summand.
\end{corollary}
\subsection{Why the canonical map matters}
The rational homotopy groups of the \(K(n)\)-local sphere are known
for every positive height \(n\) and every prime. Barthel--Schlank--Stapleton--Weinstein
identify their algebra with an exterior algebra over \(\Q_p\), with
primitive generators in degrees \(1-2i\), \(1\leq i\leq n\)
\cite[Theorem~A]{BSSW2025}. At height three the resulting degrees are
\[
0,-1,-3,-4,-5,-6,-8,-9,
\]
exactly the shifts appearing in \eqref{eq:proposed-wedge}. Thus the
rational degree pattern does not distinguish the proposed wedge from
the actual overlap.
The natural transformation \(\tau\) does distinguish them. In
\(\mathcal W_3\), a degree-\(-3\) class can occur only in the
lower-height part, which is \(K(2)\)-acyclic. The two top-height
summands have no rational homotopy in that degree. It follows that
\(\tau_{\mathcal W_3}\) vanishes on \(\pi_{-3}\). The formal argument
in \cref{sec:wedge-obstruction} also compares the maps for
\(L_{K(3)}S\) and \(L_2L_{K(3)}S\). Naturality makes the conclusion
valid for any hypothetical equivalence, with no assumption on its
individual summand maps.
At height two, Hovey records Hopkins' splitting
for \(p>3\), based on Shimomura--Yabe's calculations
\cite[\S4]{Hovey1995}, \cite[Theorem~2.4]{ShimomuraYabe1995}. Behrens later reorganized those calculations
and displayed the predicted overlap homotopy groups
\cite[Remark~7.8]{Behrens2012}. Goerss--Henn--Mahowald proved the
splitting at \(p=3\) \cite[Theorem~5.11]{GHM2014}. At \(p=2\),
Beaudry disproved the original formula \cite[Theorem~1.4]{Beaudry2017},
while Beaudry--Goerss--Henn subsequently proved a refined splitting
with additional Moore-spectrum summands and retained the weak unit
splitting \cite[Theorem~1.1.6]{BGH2022}.
The canonical map already controls the height-two argument of
Goerss--Henn--Mahowald. At \(p=3\), their proof identifies the rational
map from the \(K(2)\)-local sphere to its \(K(1)\)-localization with
projection onto the degrees zero and minus one, followed by canonical
localization; the fracture square then reconstructs the splitting
\cite[proof of Theorem~5.11, p.~1288]{GHM2014}.
Theorem~\ref{thm:main} finds a nonzero degree-minus-three component at
height three, which obstructs \(\mathcal W_3\). A finite filtration
with these cofibers may retain nonzero attaching maps that a wedge
would discard. The existence of a retraction of the canonical unit
in weak chromatic splitting remains a separate question.
\subsection{A detecting spectrum}
The proof of Theorem~\ref{thm:main} has two tasks: construct a map
from \(L_{K(3)}S\) to an ordinary \(K(2)\)-local spectrum, and prove
that this map preserves a nonzero rational degree-\(-3\) class. Locality
then forces the map to factor through the canonical localization unit.
The class is detected in degree-three stabilizer cohomology; the main
argument compares its height-three and height-two realizations.
Fix \(p\geq5\), put \(k=\F_{p^6}\), and choose \(C=\C_p\) with
compatible constant-field data. For \(n=2,3\), let \(\Gamma_n\) be the
height-\(n\) Honda formal group over \(k\), and put
\[
E_n=E(k,\Gamma_n),\qquad
P_n=\Aut_k(\Gamma_n)=\cO_{D_n}^{\times}.
\]
Here \(D_n\) is the division algebra over \(\Q_p\) of invariant
\(1/n\), and \(P_n\) is the nonextended stabilizer. Set
\[
G=P_3\times P_2,\qquad
\Delta=\Gal(k/\F_p),\qquad G_3=P_3\rtimes\Delta.
\]
The field \(k\) contains the endomorphism fields of both Honda groups.
On the height-exactly-two stratum \(k((u_2))\) of the height-three
deformation space, the Gross--Torii extension identifies the connected
height-two group with \(\Gamma_2\). Its valued-field completion
\(\widehat L\) carries the two framing actions, hence an action of
\(G\). The solid Lubin--Tate construction of
Barthel--Mann--Ray--Schlank--Senger--Weinstein--Zhou \cite{BMR2026}
gives the connected completed theory
\[
\widehat B^\square=E^\square(\widehat L,(\Gamma_2)_{\widehat L}),
\qquad \mathcal D=\bigl((\widehat B^\square)^{hG}\bigr)(*).
\]
The superscript \(\square\) records the solid structure and continuous
action; \((*)\) is evaluation in ordinary spectra. Fixed points are
formed before this evaluation.
Section~\ref{sec:completed-tower} constructs equivariant maps
\(E_3^\square\to\widehat B^\square\leftarrow E_2^\square\).
The first requires a section on oriented deformations, obtained by
adjoining the marked \'etale factor to the connected deformation.
The second is the canonical connected base-change map. Restriction
from \(G_3\) to \(P_3\), inflation along \(G\to P_3\), and the first
map define \(g:L_{K(3)}S\to\mathcal D\). Every profinite evaluation
of \(\widehat B^\square\) is \(K(2)\)-local, so its bar totalization
\(\mathcal D\) is too. Thus
\begin{equation}\label{eq:overview-detector}
L_{K(3)}S\xrightarrow{\eta}L_{K(2)}L_{K(3)}S
\xrightarrow{\overline g}\mathcal D,
\qquad g=\overline g\eta.
\end{equation}
The source identification and these actual maps are established in
\cref{prop:completed-theory,lem:local-detector}.
Write \(N_3:P_3\to\Z_p^\times\) for reduced norm and
\(P_3^1=N_3^{-1}(\mu_{p-1})\). The completed comparison
\[
E_2^\square\xrightarrow{\sim}(\widehat B^\square)^{hP_3^1}
\]
reduces the detector's coefficients to height two. We prove the
individual-coefficient comparison as well as this instance of
\cite[Theorem~5.2.6]{BMR2026}. The corresponding equivalence for the
uncompleted half sphere remains a separate conjecture
\cite[Conjectures~1.1.1 and~5.1.13]{BMR2026}, with a conditional
coheight-one consequence in \cite[Proposition~5.1.17]{BMR2026}.
The completed target in \eqref{eq:overview-detector} suffices here.
\subsection{Transporting the primitive}
The starting classes are nonzero elements
\(e_{P_n}\in H^3_{\cts}(P_n,\Q_p)\), for \(n=2,3\).
Under rational Lie comparison they are represented, up to nonzero
scalars, by
\[
(U,V,W)\longmapsto\tr_{\mathrm{red}}\bigl(U[V,W]\bigr).
\]
This is the classical cocycle attached to an invariant bilinear form
\cite[\S21]{ChevalleyEilenberg1948}, \cite[\S11]{Koszul1950}.
Section~\ref{sec:cohomology} constructs these classes and their linear
counterparts \(e_{J_n}\) for
\[
J_n=\{g\in\GL_n(\Q_p):|\det g|_p=1\}.
\]
The trace class restricts nontrivially from rank three to rank two;
the same compatibility holds for the relevant parabolic subgroup,
with determinant lattices retained. The low-rank calculation and the building argument are given
in \cref{app:background}.
To compare division-algebra and linear classes, we use a line
modification of a form of the rank-\(n\), degree-one Fargues--Fontaine
bundle \(\mathcal E_n^0=\cO(1/n)\), retaining a determinant lattice.
The stack \(\mathcal M_n\) of such modifications has two presentations:
\[
\mathcal M_n\simeq[\mathbb P_C^{n-1,\diamondsuit}/P_n]
\simeq[\Omega_C^{n-1,\diamondsuit}/J_n].
\]
Here projective space parametrizes quotient lines of the untilt fiber,
and Drinfeld space \(\Omega_C^{n-1}\) is the complement of the
\(\Q_p\)-rational hyperplanes. The relative bundle correspondence and
basic-stratum theorems of Fargues--Scholze provide the torsors in
families \cite[Theorem~II.2.19, Corollary~II.2.20, and
Theorems~III.2.4 and~III.4.5]{FS}. Keeping a lattice in the determinant
local system gives the structure group \(J_n\). These presentations
follow Faltings's Lubin--Tate/Drinfeld duality
\cite[Introduction]{Faltings2002}, in the perfectoid form of
Scholze--Weinstein with its exchanged period maps
\cite[Theorem~7.2.3]{SW2013}. Section~\ref{sec:modifications}
constructs the determinant-lattice comparison used here.
For the geometric cohomology below, write \(H^i_{\mathrm b}\) for
integral cohomology followed by inversion of \(p\). Integral cohomology
is the derived inverse limit of finite \(p\)-power coefficient complexes,
as specified in Section~\ref{sec:cohomology}.
The two presentations have complementary roles. The projective bundle
formula singles out a one-dimensional Galois-invariant line in
\(H^3_{\mathrm b}(\mathcal M_n)\), containing the nonzero image of \(e_{P_n}\).
Using the Galois weights in the integral Drinfeld cohomology theorem
of Colmez--Dospinescu--Nizioł \cite[Theorems~1.1 and~5.1]{CDN2021},
we prove that the image of \(e_{J_n}\) is also nonzero and belongs
to that line. Thus the two
classes are proportional.
The completed tower carries a surjection from the rank-three bundle
to the rank-two bundle. A common line modification turns this into
an exact sequence of rational local systems of ranks \(1,3,2\).
Parabolic restriction and projective-bundle injectivity then give
\[
e_{P_3}|_{Y_C}=a\,e_{P_2}|_{Y_C},\qquad a\in\Q_p^\times,
\quad
Y_C=[\Spd\widehat L/G]\times_{\Spd k}\Spd C.
\]
Section~\ref{sec:degree-three} proves this relation. Both classes on
\(Y_C\) could still vanish, so nonvanishing requires a further step.
Section~\ref{sec:witt-transfer} first reflects the relation from
\(Y_C\) to \(Y=[\Spd\widehat L/G]\) with Witt coefficients. The
maps from constants become equivalences after \(P_3^1\)-descent;
a bounded finite free complex proves injectivity under the change
of constants \(W(k)\to W(C^\flat)\). The continuous additive
retraction onto Witt constants of Barthel--Schlank--Stapleton--Weinstein
\cite[Proposition~2.5.1 and Corollary~2.5.9]{BSSW2025} then proves
nonvanishing of the height-two class in deformation coefficients.
If \(c_n\) is the image of \(e_{P_n}\) under projection from \(G\)
and the coefficient unit, the result is
\[
c_3=a\,c_2\ne0
\quad\text{in }H^3(G,\pi_0\widehat B^\square)[1/p].
\]
Finally, Section~\ref{sec:detection} uses the descent spectral
sequences for the actual map \(g\). A finite cohomological bound
produces a finite integral filtration in each ordinary homotopy
degree. After rationalization, central scalar automorphisms kill
all coefficient degrees \(t\ne0\). With \(s\) denoting group-cohomological
degree, the nonzero coefficient map in bidegree \((s,t)=(3,0)\) therefore survives to degree \(-3\) ordinary
homotopy. The factorization \eqref{eq:overview-detector} proves the
main theorem, and Section~\ref{sec:wedge-obstruction} gives its
wedge consequence.
Continuous invariants throughout the proof retain their solid
coefficient objects, and inversion of \(p\) follows integral cohomology.
This order matters for the noncompact spaces and groups that occur here.
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//01-introduction.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//02-cohomology.tex ====
\section{Integral cohomology and degree-three group classes}
\label{sec:cohomology}
Our goal is to construct the degree-three classes that will be transported
through the two-height tower, together with the exact restriction maps
between their linear realizations. We first fix the integral coefficient
convention and prove the finite-resolution statement that will also
control the later Galois-weight and ordinary-descent arguments.
\subsection{Coefficients and continuous descent}
Put $\Lambda_m=\Z/p^m$. For a small v-stack $X$ occurring below, put
\[
R\Gamma_{\mathrm{int}}(X)
=\Rlim_m R\Gamma_v(X,\Lambda_m),\qquad
H^i_{\mathrm{int}}(X)=H^iR\Gamma_{\mathrm{int}}(X),\qquad
H^i_{\mathrm b}(X)=H^i_{\mathrm{int}}(X)[1/p].
\]
For a locally profinite group $G$, define separately in $D(\mathrm{Ab})$
\begin{equation}\label{eq:integral-group-cochains}
\begin{gathered}
R\Gamma_{\mathrm{int}}(G)=\Rlim_m C_{\cts}^{\bullet}(G,\Lambda_m),
\\
H^i_{\mathrm{int}}(G)=H^iR\Gamma_{\mathrm{int}}(G),
\qquad H^i_{\mathrm b}(G)=H^i_{\mathrm{int}}(G)[1/p].
\end{gathered}
\end{equation}
Here $C_{\cts}^{\bullet}$ is the inhomogeneous continuous cochain complex,
with trivial discrete coefficients. The standard resolution in the
condensed classifying topos $BG_{\mathrm{pro\acute et}}$, the topos of
condensed sets with $G$-action, gives the natural comparison
\[
C_{\cts}^{\bullet}(G,\Lambda_m)\simeq
\RHom_{\Lambda_m[\underline G]\text{-}\mathrm{Cond}}
(\underline{\Lambda_m},\underline{\Lambda_m}).
\]
This is the external derived Hom; its continuous-cochain calculation for
these solid topological coefficients is the standard-resolution proof in
\cite[\S2, pp.~5--7, Lemmas~2.1--2.2]{Anschutz2020}.
We next construct the maps from this group complex to geometric quotients.
For a condensed $G$-complex $K$, write $R\Gamma(G,K)$ for its
\emph{internal} derived condensed invariants, and $R\Gamma(G,K)(*)$
for their point value, which is the external derived Hom
\cite[\S2, p.~8]{Anschutz2020}. Topological coefficients in this
notation are implicitly condensed. Reduction on finite cochains is
degreewise surjective: a function lifts by composing with a section of
the finite coefficient reduction. Compatible finite continuous
functions are exactly continuous $\Z_p$-valued functions. Hence
\begin{equation}\label{eq:group-cochain-bridge}
R\Gamma_{\mathrm{int}}(G)\simeq C_{\cts}^{\bullet}(G,\Z_p)
\simeq R\Gamma(G,\underline{\Z_p})(*).
\end{equation}
For the last comparison, $\Z_p$ with its $p$-adic topology is solid by
\cite[\S1, pp.~4--5]{Anschutz2020}, so the same standard-resolution
proof applies. We do not identify the internal invariant object with
the discrete condensation of the external cochain complex for a
noncompact group; see \cite[Lemma~2.5 and Remark~2.6]{Anschutz2020}.
The comparisons in \eqref{eq:group-cochain-bridge} are natural for
continuous group homomorphisms.
For compact $G$, continuous functions $G^r\to\Q_p$ are bounded, so
\[
C_{\cts}(G^r,\Z_p)[1/p]=C_{\cts}(G^r,\Q_p).
\]
Consequently $H^*_{\mathrm b}(G)$ agrees with ordinary continuous
$\Q_p$-cohomology for compact $G$. We use this last identification
only in that case.
Tate twists are taken at each finite level before taking the derived
limit. All maps of coefficients and all descent maps are formed
integrally. In particular, $H^i_{\mathrm b}(X)$ is not defined as
$H^i_v(X,\widehat\Z_p[1/p])$. Moving inversion of $p$ before
cohomology can change the result for a noncompact space or group.
For an analytic adic space in this paper, its \'etale site agrees with
the \'etale site of its diamond
\cite[Lemma~15.6]{ScholzeDiamonds}. On a locally spatial diamond,
bounded-below \'etale complexes have the same derived cohomology on
the diamond \'etale, quasi-pro-\'etale, and v-sites
\cite[Proposition~14.10]{ScholzeDiamonds}. We apply these statements
to finite constant coefficients, one level at a time. The integral
analytic pro-\'etale comparison used for Drinfeld space is the
separate comparison in \cite[\S5]{CDN2021}. Quotient stacks are
handled by descent for their covers. Only after the finite-level
comparisons do we take $\Rlim_m$.
Here is the map from group classes to geometric quotients. For a small
v-stack base $B$, put $B_BG=[B/\underline G]$, where $G$ acts trivially
on $B$. A $G$-space $X$ over $B$ gives a map $[X/G]\to B_BG$; a
$G$-torsor $T\to Z$ over $B$ gives its classifying map $Z\to B_BG$.
At each finite level define
\begin{equation}\label{eq:finite-constant-cochains}
\begin{split}
u_{X,G,m}:C_{\cts}^{\bullet}(G,\Lambda_m)
&\longrightarrow
\operatorname{Tot}_{a}R\Gamma_v
(X\times\underline{G^a},\Lambda_m)\\
&\simeq R\Gamma_v([X/G],\Lambda_m).
\end{split}
\end{equation}
In degree $a$, pull a locally constant function on $G^a$ back along
the projection from $X\times\underline{G^a}$. Its finitely many
clopen fibers define a section of the finite constant sheaf, followed
by the canonical degree-zero truncation map to derived sections.
These maps commute with
the faces and degeneracies of the action groupoid. The last
equivalence is its \v{C}ech descent. The case $X=B$ defines the
universal map into $B_BG$, and the general map is its pullback along
$[X/G]\to B_BG$. Likewise the map obtained from the \v{C}ech nerve
of $T\to Z$ is the pullback along its classifying map.
If $\phi:G'\to G$ is a continuous homomorphism and $f:X'\to X$ is
$\phi$-equivariant over a base map, this construction gives a
commutative square
\begin{equation}\label{eq:constant-cochain-naturality}
\begin{tikzcd}[column sep=large]
C_{\cts}^{\bullet}(G,\Lambda_m)
\arrow[r,"u_{X,G,m}"] \arrow[d,"\phi^*"']
&R\Gamma_v([X/G],\Lambda_m)\arrow[d,"{[f/\phi]^*}"]\\
C_{\cts}^{\bullet}(G',\Lambda_m)
\arrow[r,"u_{X',G',m}"']
&R\Gamma_v([X'/G'],\Lambda_m).
\end{tikzcd}
\end{equation}
It commutes already in each finite cosimplicial degree. The same
naturality holds for coefficient reduction, maps of finite coefficient
sheaves, and automorphisms of the base. Taking $\Rlim_m$ defines
$u_{X,G,\mathrm{int}}:R\Gamma_{\mathrm{int}}(G)\to
R\Gamma_{\mathrm{int}}([X/G])$; denote the induced maps on cohomology
after $[1/p]$ by $u_{X,G,\mathrm b}$. No acyclicity of $B$ or $X$
is used in this construction.
There is one restricted case in which the universal map is an
equivalence. Let $B_C=\Spd C$ for the algebraically closed perfectoid
field $C=\C_p$, and abbreviate $B_CG=[B_C/\underline G]$.
For every profinite set $S$, the actual constants map
is an equivalence
\begin{equation}\label{eq:geometric-point-constants}
C_{\cts}(S,\Lambda_m)[0]\xrightarrow{\ \sim\ }
R\Gamma_v(B_C\times\underline S,\Lambda_m).
\end{equation}
For the empty set both sides are zero. Otherwise write $S=\varprojlim_i S_i$
over its finite quotients, including the one-point quotient. The example
after \cite[Definition~7.8]{ScholzeDiamonds} constructs the affinoid
pro-\'etale limit $\operatorname{Spa}(C,\cO_C)\times S$ of the finite
unions indexed by $S_i$; the associated-diamond construction and
\cite[Example~11.12 and Lemma~15.2]{ScholzeDiamonds} identify its
diamond with $B_C\times\underline S$.
The point $\operatorname{Spa}(C,\cO_C)$ is strictly totally disconnected
by \cite[Definition~7.15 and Proposition~7.16]{ScholzeDiamonds}, so its
\'etale covers split and finite constant coefficients have no higher
cohomology there. The preceding \cite[Lemma~15.6]{ScholzeDiamonds}
comparison transfers this assertion to the finite unions of their
diamonds. The continuity statement
\cite[Proposition~14.9]{ScholzeDiamonds} computes the \'etale
cohomology on the limit as the filtered colimit of that on the finite
unions. Its degree-zero group is
$\varinjlim_i C(S_i,\Lambda_m)=C_{\cts}(S,\Lambda_m)$ and its higher
groups vanish. Proposition~14.10 of the same source, valid for the
coefficient ring $\Lambda_m$, transfers this calculation to the v-site.
Every comparison is induced by the constants map, and is natural in
$S$ and in automorphisms of $C$.
For compact $G$, apply \eqref{eq:geometric-point-constants} to
$S=G^a$ in every degree of \eqref{eq:finite-constant-cochains}.
It follows that $u_{B_C,G,m}$ is an equivalence. The same argument
applies to the finite \'etale fiber of $(\Z/p^m)(t)$ over $C$,
where $p$ is invertible, retaining its arithmetic action. The
compatible Tate line is finite free of rank one, so its tensor factor
commutes with the cochain terms and their derived coefficient limit.
Thus the integral comparison for
$B_CG$ is also compatible with Tate twists and arithmetic Galois
actions, without a choice of basis for the Tate fiber. This
geometric-point calculation is used below for the compact groups
$P_n$; it is not asserted over $\Spd k$.
For descent with geometric coefficients it is useful to retain
profinite parameters. Denote by $\mathscr C_{X,m}$ the condensed
derived complex whose derived value on a profinite set $S$ is
\[
\mathscr C_{X,m}(S)=R\Gamma_v(X\times\underline S,\Lambda_m),
\qquad \mathscr C_X=\Rlim_m\mathscr C_{X,m}.
\]
Descent for profinite covers gives these condensed complexes. The value
of $\mathscr C_X$ at the point is $R\Gamma_{\mathrm{int}}(X)$.
A group acting on $X$ acts continuously
on this complex. The coefficient complexes used below are
\emph{derived solid}: they lie in the derived category of solid
condensed modules. Here is a concrete check of this condition.
On an affinoid perfectoid cover, the characteristic-$p$ structure
sheaf $\cO^\flat$ has no higher cohomology, and its values with
profinite parameters are continuous-function modules. Their complete
linear topology expresses them as limits of discrete modules, so
they are solid. Perfectoid hypercover descent computes the general
$\cO^\flat$ complex by limits of these objects; this is also the
construction in \cite[Theorem~3.2.1(iii) and Remark~3.2.2]{BMR2026}.
The Artin--Schreier
sequence gives the $\F_p$ complex, the coefficient exact sequences
give the $\Z/p^m$ complexes, and $\Rlim_m$ gives $\mathscr C_X$.
Derived solidity is preserved by these limits and extensions
\cite[Theorem~4.4(ii)]{Tang2026}. This argument applies with the
group action and with all the profinite parameters retained. It
also gives derived $p$-completeness of $\mathscr C_X$.
For compact $G$, the condensed standard resolution with its derived
parameter values retained gives
\begin{equation}\label{eq:geometric-group-descent}
R\Gamma(G,\mathscr C_X)(*)\simeq
\Rlim_m\operatorname{Tot}_{a}R\Gamma_v
(X\times\underline{G^a},\Lambda_m)
\simeq R\Gamma_{\mathrm{int}}([X/G]).
\end{equation}
Indeed all $G^a$ are profinite. In the standard resolution, the
degree-$a$ derived Hom is precisely the derived parameter value on
$G^a$; this is the formal condensed construction in
\cite[\S2, pp.~5--6]{Anschutz2020}, before any acyclicity simplification.
The parameterized complex is bounded below, and the derived
totalization computes its point-valued invariants. The second
comparison is quotient \v{C}ech descent at each finite level; the
derived limit and totalization commute because both are limits.
This is the geometric descent comparison used here. On the same bar
terms, using \cite[Lemma~2.2]{Anschutz2020} for the source parameter
value, the finite unit $\underline{\Lambda_m}\to\mathscr C_{X,m}$
is exactly the map in \eqref{eq:finite-constant-cochains}. The complexes
$\mathscr C_{X,m}$ and $\mathscr C_X$ are concentrated in cohomological
degrees at least zero, so their units factor through their degree-zero
cohomology by the truncation triangle. Thus the derived-invariants map
agrees with $u_{X,G,\mathrm{int}}$ after the coefficient limit, and its
factorization through degree zero gives the constants bottom-row edge map.
The next lemma makes these descent groups computable from a finite
complex. Its final assertion will let us use arithmetic weights known
on ordinary cohomology groups while retaining the solid coefficients.
\begin{lemma}[Finite resolutions and scalar actions]
\label{lem:solid-resolutions}
Let $G$ be a compact $p$-adic analytic group without elements of order
$p$, and let $\Lambda_G=\Z_p[[G]]$. There is a finite resolution
$P_\bullet\to\Z_p$ by finitely generated projective profinite
$\Lambda_G$-modules. Its length can be taken to be $\dim G$.
For any derived-solid $\Z_p$-complex $K$ with a continuous $G$-action,
the point-valued derived invariants are computed by
\begin{equation}
\label{eq:finite-resolution}
R\Gamma(G,K)(*)\simeq
\Hom_{\Lambda_G}(P_\bullet,K).
\end{equation}
Here the right side means the finite total complex of morphisms in
solid modules. In particular, invariants of a module in the heart
have cohomology only in degrees $0,\ldots,\dim G$.
Suppose $K$ is bounded below and an operator $\gamma$ commutes with
$G$. If $\gamma$ acts by $\lambda_j\in\Q_p$ on $H^j(K)(*)[1/p]$,
then it acts by the same scalar on every term in the $j$th row of
the rationalized descent spectral sequence
\[
H^s\bigl(G,H^j(K)\bigr)(*)[1/p]
\ \Longrightarrow\ H^{s+j}\bigl(R\Gamma(G,K)(*)\bigr)[1/p].
\]
For $K=\mathscr C_X$, the abutment is $H^{s+j}_{\mathrm b}([X/G])$.
\end{lemma}
\begin{proof}
First construct the resolution in profinite modules. A compact $p$-adic analytic group has a noetherian completed group
ring; if it has no $p$-torsion, then
$\operatorname{pd}_{\Lambda_G}\Z_p=\operatorname{cd}_p(G)=\dim G$.
Here projective dimension is in profinite modules.
These results for compact groups, including the agreement with
abstract finitely generated modules, are recorded in
\cite[\S\S1.1--1.2, pp.~275--276]{Venjakob2002}.
The noetherianity input for analytic pro-$p$ groups is
\cite[V.2.2.4]{Lazard1965}; the compact-group statement uses the
finite-index passage recorded by Venjakob.
Choose successive finite free presentations. Noetherianity keeps
their kernels finitely generated, and the projective-dimension
bound makes the last syzygy projective. Their natural compact
topologies give the asserted profinite resolution.
To apply the resolution to the geometric coefficient complexes, we
pass from continuous actions to solid modules over the completed group
ring. Write
$A=\underline{\Z_p}[\underline G]$ for the free condensed group
ring. A continuous action on a complex means an object of
$D(A\text{-}\mathrm{Cond})$ with derived-solid underlying complex.
By \cite[Lemma~1.3]{Anschutz2020}, the group-ring analytic
structure has underived solidification; its normalized ring is
\[
A^\square\simeq\underline{\Z_p[[G]]}
=\underline{\varprojlim_{m,V}(\Z/p^m)[G/V]},
\]
where $V$ runs through open normal subgroups. The ring
identification, including multiplication, is
\cite[\S3.6, preceding Proposition~3.21]{Tang2026}.
Analyticity gives fully faithful embeddings of the \emph{unbounded}
derived category
of solid $G$-modules into both $D(A\text{-}\mathrm{Cond})$ and
$D(A^\square\text{-}\mathrm{Cond})$, as stated in
\cite[\S1, pp.~3--4]{Anschutz2020}.
The solid heart is characterized by solidity of the underlying
condensed module. The essential-image criterion of Clausen--Scholze's
analytic theory extends this characterization to unbounded complexes:
membership is detected on cohomology objects
\cite[Proposition~12.4 and Remark~12.5]{ScholzeAnalyticGeometry}.
Thus its objects are precisely the complexes with derived-solid
underlying coefficients. Consequently the external derived Hom
complexes satisfy
\[
R\Gamma(G,K)(*)
\simeq\RHom_{A\text{-}\mathrm{Cond}}(\underline{\Z_p},K)
\simeq\RHom_{G\text{-}\mathrm{Solid}}(\underline{\Z_p},K).
\]
The first identification is the external derived Hom for the condensed
classifying topos $BG_{\mathrm{pro\acute et}}$
\cite[\S2, pp.~5 and~8]{Anschutz2020}; the last category is that of
solid $\Z_p[[G]]$-modules by
\cite[Proposition~3.21]{Tang2026}.
They apply to the given complex $K$ itself, including when its
coefficients are neither profinite nor derived $p$-complete.
The profinite resolution stays exact and projective in solid
modules \cite[Theorems~3.2 and~3.14(i)]{Tang2026}.
Exactness can be seen by testing on extremally
disconnected profinite sets: maps from such a set lift through a
surjection of profinite spaces. Each $P_i$ is a retract of
$\Lambda_G^{r_i}$, and
\[
\Hom_{\Lambda_G}(\Lambda_G,M)=M(*).
\]
Evaluation at the point is exact, since the point is extremally
disconnected. This proves projectivity against an arbitrary solid
coefficient module $M$, even when $M$ is not profinite. A bounded
projective resolution computes derived Hom against any complex,
which proves the displayed calculation and the amplitude assertion.
Finally, $\Hom_{\Lambda_G}(P_i,H^j(K))$ is a retract of a finite
sum of $H^j(K)(*)$. The retract commutes with $\gamma$, because
$\gamma$ commutes with the $G$-action. After inverting $p$, the
entire finite complex computing the $j$th row therefore has scalar
action $\lambda_j$. Its cohomology has the same action.
This uses a scalar identity on point values only; it does not
deduce an identity of condensed sheaves from their point values.
Boundedness of the resolution gives a finite filtration in each
total degree, so exact rationalization preserves the spectral
sequence and its abutment.
\end{proof}
\subsection{Compact groups and the affine building}
Put
\[
J_n=\{g\in\GL_n(\Q_p):|\det g|_p=1\}\qquad(n=2,3).
\]
Let $D_n$ denote the division algebra of degree $n$ whose maximal
order has unit group $P_n$. We need only the following low-degree
part of the rational Lie algebra calculation.
\begin{proposition}[The group classes]
\label{prop:group-classes}
For $G=P_n$ or $J_n$, where $n=2,3$,
\[
H^1_{\mathrm b}(G)=\Q_p,
\qquad H^2_{\mathrm b}(G)=0,
\qquad H^3_{\mathrm b}(G)=\Q_p.
\]
For $n=2,3$ and every compact open subgroup $K\subset J_n$,
restriction gives an isomorphism in every degree, with the compact
comparison identifying its target:
\[
H^*_{\mathrm b}(J_n)\xrightarrow{\mathrm{res}}H^*_{\mathrm b}(K)
\cong H^*(\mathfrak{gl}_n(\Q_p),\Q_p).
\]
In degree three the block inclusion $J_2\hookrightarrow J_3$
induces a nonzero map. Fix arbitrary nonzero classes
\[
e_{P_n}\in H^3_{\mathrm b}(P_n),\qquad
e_{J_n}\in H^3_{\mathrm b}(J_n).
\]
\end{proposition}
\begin{proof}
For an inclusion \(K_1\subset K_2\) of compact open subgroups of
\(\GL_n(\Q_p)\), choose a \(K_2\)-stable lattice \(\Lambda\) and then
\(r\) large enough that
\[
U=1+p^r\End_{\Z_p}(\Lambda)\subset K_1.
\]
This is a uniform subgroup normal in \(K_2\). In the division-algebra
case one uses a sufficiently deep congruence subgroup of
\(\cO_{D_n}^{\times}\), which is normal in the relevant compact group.
For these small groups, Lazard's comparison is
\cite[Theorem~3.3.3 and Remark~3.3.4]{HKN2011} with trivial
\(\Z_p\)-coefficients, followed by inversion of \(p\).
Its coefficient naturality, rational agreement, and group naturality are
\cite[Theorem~3.1.1(1)--(3)]{HKN2011}, with numbering in the arXiv
revision specified in the bibliography. Equip the uniform groups with
their lower \(p\)-series. At these odd primes the proof of
Theorem~3.3.3 gives \(e=1\), and the associated graded target is generated
in degree one, as required by the group-naturality clause. Inclusions and
the block homomorphisms preserve these filtrations. The common subgroup
\(U\) therefore identifies restriction with the identity on the same
ambient Lie algebra cohomology.
Here is why passing back to a compact group is valid even for an index
divisible by \(p\). For \(U\triangleleft K\), the augmentation and norm
of the finite permutation module \(\Z[K/U]\) induce, by continuous
Shapiro, the identities
\[
\operatorname{cor}\operatorname{res}=[K:U],\qquad
\operatorname{res}\operatorname{cor}
=\sum_{q\in K/U}q^*
\]
at each finite coefficient level, compatibly with reduction in \(m\).
Shapiro here is the derived induction--restriction adjunction: induction
for an open subgroup is a direct sum on underlying condensed modules and
is exact by (AB4) \cite[Theorems~2.2 and~2.4]{Tang2026}; restriction is
also exact. Their adjunction therefore descends directly to derived
categories. Its naturality identifies the
permutation maps with the actual restrictions and conjugations in the
finite continuous bar complexes. Apply \(\Rlim_m\) to these identities
before taking cohomology and inverting \(p\). Only then divide by
\([K:U]\). This gives
\(H^*_{\mathrm b}(K)\xrightarrow{\sim}H^*_{\mathrm b}(U)^{K/U}\)
by restriction, with the compact
continuous-cochain interpretation of \eqref{eq:group-cochain-bridge}.
The finite quotient acts trivially in every Lie degree. The
Chevalley--Eilenberg complex is a finite-dimensional algebraic
\(\GL_n\)-representation after a splitting-field extension. Cartan's
formula \(\mathcal L_X=d\iota_X+\iota_Xd\) makes the derivative of its
action on cohomology zero. In characteristic zero, an algebraic action
of the connected group \(\GL_n\) with zero derivative is trivial.
Thus all inner automorphisms act trivially, in every degree; the same
conclusion descends for the division algebra. This proves the compact
restriction identification without an all-degree numerical calculation.
After a finite splitting-field extension, either Lie algebra is
$\mathfrak{gl}_n$. The finite Chevalley--Eilenberg calculation in
\Cref{app:background} gives its asserted groups in degrees one through
three, with degree-three representative
\[
(X,Y,Z)\longmapsto\tr\bigl(X[Y,Z]\bigr),
\]
or its reduced-trace version for $D_n$.
This is the invariant-form cocycle of
\cite[\S21]{ChevalleyEilenberg1948} and \cite[\S11]{Koszul1950};
the appendix records the normalization and restriction used here.
These representatives are defined over $\Q_p$ and invariant under
inner automorphisms, so the assertions descend from the splitting
field. Restriction of the displayed form to the upper-left or
lower-right $2\times2$ block is the corresponding form for
$\mathfrak{gl}_2$, whose class is nonzero. This proves the compact
calculations and their compatibility.
To pass to $J_n$, let $\mathcal B_n$ be the reduced affine building
of $\GL_n(\Q_p)$. It is contractible of dimension $n-1$.
The group $J_n$ preserves vertex types, its face stabilizers
$K_\sigma$ are compact open, and a chamber $\Delta$ is a
fundamental domain, including its faces. A proof of these building
properties by lattice norms is included in \Cref{app:background}.
Write its bounded augmented cellular resolution as
\[
P_r=\bigoplus_{\substack{\sigma\subset\Delta\\\dim\sigma=r}}
\Z[\underline{J_n/K_\sigma}].
\]
It is exact also as a complex of condensed modules. Indeed, on a
profinite parameter set a continuous section of a discrete cellular
chain group has finite image. Contractibility lifts each of the finitely
many cycles in that image; choosing those lifts on its clopen fibers
gives a continuous lift. The same exact derived induction--restriction
adjunction therefore supplies continuous Shapiro for each summand.
At coefficient level \(\Lambda_m\), applying derived Hom gives a finite
filtration with graded terms
\(\bigoplus_{\dim\sigma=r}C_{\cts}^{\bullet}(K_\sigma,\Lambda_m)[-r]\)
in \(D(\mathrm{Ab})\).
A cellular boundary is the literal signed projection of the corresponding
coset modules, so naturality of that adjunction makes each face map the
signed restriction between stabilizers, with no index multiplier.
All of these statements are compatible with coefficient reduction.
Take \(\Rlim_m\) of this finite filtration. There are finitely many
orbit terms in each of finitely many cellular degrees, so this step uses
only finite sums and cones in the cellular direction. Taking cohomology of
the resulting finite filtration gives the integral spectral sequence
\[
E_1^{r,s}=
\bigoplus_{\substack{\sigma\subset\Delta\\\dim\sigma=r}}
H^s_{\mathrm{int}}(K_\sigma)
\ \Longrightarrow\ H^{r+s}_{\mathrm{int}}(J_n),
\]
with precisely those signed restriction faces. It does not interchange
an infinite group-cochain totalization with the derived inverse limit.
Only after this construction do we invert \(p\); exact rationalization
preserves the finite filtration and its abutment.
Under the compact comparison, every restriction between face
stabilizers is the identity on the same Lie algebra cohomology;
any change of representatives contributes an inner conjugation,
which also acts trivially. Thus each rationalized row is the
cellular cochain complex of the simplex $\Delta$ with constant
coefficients $H^s(\mathfrak{gl}_n,\Q_p)$.
It has cohomology only in cellular degree zero.
The resulting edge map is restriction to a vertex stabilizer.
Restriction further to compact open subgroups, using intersections
and the compact comparison, proves the asserted canonical
identifications. The block restriction can now be tested on
compact open subgroups, where it is the trace calculation above.
\end{proof}
\subsection{The parabolic restriction}
The relevant subgroup remembers a lattice only in each determinant
line. In a basis adapted to a line it is
\[
D=\left\{
\begin{pmatrix}a&v\\0&A\end{pmatrix}:
a\in\Z_p^\times,\quad A\in J_2,\quad
v\in\Q_p^{1\times2}\right\}\subset J_3.
\]
Write $\rho_3:D\hookrightarrow J_3$ for the inclusion and
$\rho_2:D\to J_2$ for the quotient action.
\begin{lemma}[Unipotent acyclicity and restriction]
\label{lem:parabolic}
Inflation induces an isomorphism
\[
H^*_{\mathrm b}(\Z_p^\times\times J_2)
\xrightarrow{\ \sim\ }H^*_{\mathrm b}(D).
\]
There is a scalar $b\in\Q_p^\times$ such that
\[
\rho_3^*e_{J_3}=b\,\rho_2^*e_{J_2}.
\]
\end{lemma}
\begin{proof}
The kernel of $D\to\Z_p^\times\times J_2$ is
$N=\Q_p^2$, with its additive topology.
Fix $m$ and exhaust it by the compact open subgroups
$N_r=p^{-r}\Z_p^2$ for $r\geq0$. Their finite-coefficient
cohomology, computed by the two-variable Koszul resolution, is
\[
H^q(N_r,\Z/p^m)=\bigwedge^q(\Z/p^m)^2.
\]
The standard bases identify restriction from $N_{r+1}$ to $N_r$
with multiplication by $p^q$ in degree $q$: restriction multiplies
each degree-one homomorphism by $p$, and is compatible with cup
products. For $q>0$ this inverse system is pro-zero; for $q=0$
it is constant with identity transition maps.
This calculation computes the cohomology of $N$ without a missing
derived-limit term. Indeed a continuous cochain on $N$ is exactly
a compatible family of cochains on the $N_r$.
Restriction of cochains is surjective, since $N_r^a$ is clopen in
$N_{r+1}^a$ and a function extends by zero on its complement.
Thus the ordinary inverse limit of the cochain complexes realizes
their derived inverse limit. The Milnor exact sequence has
$\varprojlim^1_r=0$ in every degree: this holds both for a constant
tower with identity maps and for a pro-zero tower. It follows that
the constant map is an equivalence
\[
\Lambda_m\xrightarrow{\ \sim\ }C_{\cts}^{\bullet}(N,\Lambda_m).
\]
The same proof works with coefficients
$C_{\cts}(S,\Z/p^m)$ for every profinite parameter space $S$.
The finite Koszul complex gives finite sums of this module, the
positive-degree restrictions are again $p^q$, and extension by
zero is continuous on $N_r^a\times S$.
The condensed standard resolution evaluated on each such parameter
identifies this calculation with the internal equivalence
\[
\underline{\Lambda_m}\xrightarrow{\ \sim\ }
R\Gamma(N,\underline{\Lambda_m}).
\]
Thus it proves an equivalence of coefficient objects for extension
descent. The equivalence is the map of constants, so it is equivariant
under the Levi action.
Now take $\Rlim_m$. In degree zero the parameter modules have
surjective transition maps, by lifting locally constant values,
and their limit is $C_{\cts}(S,\Z_p)$.
Hence the internal coefficient limit satisfies
\[
\underline{\Z_p}\xrightarrow{\ \sim\ }
\Rlim_m R\Gamma(N,\underline{\Lambda_m}).
\]
At each finite level, internal descent for the split extension identifies
$R\Gamma(D,\underline{\Lambda_m})$ with
$R\Gamma(\Z_p^\times\times J_2,
R\Gamma(N,\underline{\Lambda_m}))$.
The Levi-equivariant constant equivalence therefore induces inflation.
After point evaluation, the standard-resolution comparison identifies
this map with finite continuous-cochain inflation. Taking $\Rlim_m$
and using \eqref{eq:integral-group-cochains} gives the claimed
isomorphism integrally before rationalization.
Finally, the compact resolution for $\Z_p^\times$ and the finite
building resolution for $J_2$ give the usual rational product
calculation. Since $H^*_{\mathrm b}(\Z_p^\times)$ is exterior
on one degree-one generator and $H^2_{\mathrm b}(J_2)=0$,
degree three of the Levi is exactly $H^3_{\mathrm b}(J_2)$.
Thus $\rho_3^*e_{J_3}$ is a scalar multiple of
$\rho_2^*e_{J_2}$. Restricting to the section
$A\mapsto\operatorname{diag}(1,A)$ shows that this scalar is
nonzero, by \cref{prop:group-classes}.
\end{proof}
The vanishing just proved is specific to our order of operations.
For example, continuous homomorphisms $\Q_p\to\Q_p$ give a
nonzero $H^1_{\cts}(\Q_p,\Q_p)$, whereas the preceding integral
calculation gives $H^1_{\mathrm b}(\Q_p)=0$.
\subsection{The integral Drinfeld theorem}
Let $\Omega_C^d$ be the base change to $C$ of Drinfeld's symmetric
space over $\Q_p$, the complement of the $\Q_p$-rational
hyperplanes in $\mathbf P^d$. The following precise input supplies
the weights used in \cref{prop:primitive-comparison}.
\begin{theorem}[Colmez--Dospinescu--Nizio\l]
\label{thm:drinfeld}
For $0\leq j\leq d$ there are compatible
$\GL_{d+1}(\Q_p)\times\Gal(\overline\Q_p/\Q_p)$-equivariant
isomorphisms of groups
\[
H^j_{\mathrm{int}}(\Omega_C^d)(j)
\simeq\operatorname{Sp}_j(\Z_p)^*,
\qquad
H^j_{\mathrm{\acute et}}(\Omega_C^d,\F_p(j))
\simeq\operatorname{Sp}_j(\F_p)^*.
\]
Here $\operatorname{Sp}_j$ is the generalized Steinberg
representation of \cite[\S4.1]{CDN2021}, the star denotes its
continuous coefficient-linear dual with weak topology, and
the right sides have trivial Galois action. These groups vanish
for $j>d$, and $H^0_{\mathrm{int}}(\Omega_C^d)=\Z_p$ by constants.
In particular, on the untwisted integral group in degree $j$,
an arithmetic Galois element $\gamma$ acts by
$\chi_{\mathrm{cyc}}(\gamma)^{-j}$.
\end{theorem}
The group calculation is \cite[Theorem~1.1]{CDN2021}, with its integral
cohomology convention. The comparison with
$H^j_{\mathrm{pro\acute et}}(\Omega_C^d,\widehat\Z_p(j))$
is made explicitly in the proof in \cite[\S5]{CDN2021}; finite
coefficient site comparison and $\Rlim_m$ identify it with the
convention above. Thus the scalar assertion holds on the
integral point groups themselves. To identify the stated degree-zero
map, choose a $C$-point of $\Omega_C^d$ by taking $d+1$ coordinates
linearly independent over $\Q_p$. By
\eqref{eq:geometric-point-constants}, evaluation there splits the
constants unit at each finite level and after $\Rlim_m$. The cited
calculation makes $H^0_{\mathrm{int}}(\Omega_C^d)$ a free rank-one
$\Z_p$-module, so this split inclusion from $\Z_p$ is an isomorphism.
This identifies the actual constants map locally from the degree-zero
calculation. Applying
\cref{lem:solid-resolutions} to a compact open uniform subgroup
of $J_{d+1}$ propagates it to the group-cohomology rows, without
any assumption about cohomology of a rationalized sheaf.
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//02-cohomology.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//03-completed-tower.tex ====
\section{The completed coheight-one tower}
\label{sec:completed-tower}
We now construct the geometric and spectral comparison objects used by
the detector. The spectral output is a pair of equivariant maps from the
two Morava theories to a completed height-two theory. We also need descent for each
coefficient module and ordinary \(K(2)\)-locality of every profinite
evaluation. These properties allow the later coefficient calculation to
detect the canonical localization map. The inputs
from Barthel--Mann--Ray--Schlank--Senger--Weinstein--Zhou include the
characteristic-\(p\) tower, its cohomology calculations, and the tensor
and completion constructions
in the version dated 26 August 2026 \cite{BMR2026}. We first establish
the connected target and its coefficient descent.
We then construct the map from height three by splitting a completed
connected--\'etale group. The detailed oriented deformation argument is
given in \cref{app:oriented-projection}.
\subsection{The characteristic-\(p\) tower}
Recall that $k=\F_{p^6}$ and that the height-$n$ Honda groups $\Gamma_n$ are over $k$,
for $n=2,3$. All their geometric endomorphisms are defined over $k$.
Let $D_n$ be the division algebra of invariant $1/n$ over $\Q_p$, with
the convention identifying
\[
P_n=\Aut_k(\Gamma_n)=\cO_{D_n}^{\times},\qquad
N_n:P_n\longrightarrow\Z_p^{\times}
\]
with its reduced norm. Thus $P_n$ is the nonextended stabilizer. Set
\[
P_n^1=N_n^{-1}(\mu_{p-1}),\qquad
\overline Z=P_3/P_3^1\simeq\Z_p,\qquad
G=P_3\times P_2,\qquad H=P_2\times\overline Z.
\]
The identification of $\overline Z$ with $\Z_p$ will not require a choice
of generator. We write $\mathcal E_n^0=\cO(1/n)$ for the corresponding
rank-$n$, degree-one Fargues--Fontaine bundle. A $P_n$-structured form
$\mathcal E_n$ means a form whose torsor of frames has been reduced from
$D_n^{\times}$ to $\cO_{D_n}^{\times}$. Equivalently, its determinant
framing is specified up to $\Z_p^{\times}$; this is determinant lattice
data, not a lattice in a rank-$n$ rational local system.
The height-exactly-two stratum of the height-three deformation space is
the Laurent field
\[
K=k((u_2)),\qquad p=u_1=0,\quad u_2\ne0.
\]
Let $L/K$ be the Gross--Torii extension classifying the isomorphisms
$(\Gamma_2)_L\simeq(\Gamma^{\mathrm{un}}_{3,L})^{\circ}$, and let
$\widehat L$ be its valued-field completion. The two change-of-framing
actions give the continuous $G$-action on $\widehat L$. Fix compatible
determinant identifications for the two Honda isocrystals, and hence an
identification
$\iota:\det\mathcal E_3^0\simeq\det\mathcal E_2^0$.
These choices exist over $k$: the determinant identifications in
\cite[\S2.7]{BMR2026} require $\F_{p^2}\subset k$.
\begin{proposition}[The completed tower]\label{prop:tower}
The field $\widehat L$ is perfectoid of characteristic $p$, and is
isomorphic, as a valued field over $k$, to the completed perfection of a
Laurent-series field:
\[
\widehat L\simeq
\left(\bigcup_{r\ge0}k((z^{1/p^r}))\right)^{\wedge}.
\]
There is a $G$-equivariant identification of v-sheaves over $\Spd k$
\[
\Spd\widehat L\simeq
\operatorname{Surj}(\mathcal E_3^0,\mathcal E_2^0).
\]
Consequently
\[
Y=[\Spd\widehat L/G]
\]
classifies surjections $\mathcal E_3\twoheadrightarrow\mathcal E_2$
between $P_n$-structured forms. In particular $Y$ carries the universal
surjection used below.
Let $S_3=\ker N_3$ and $T_3=P_3/S_3=\Z_p^{\times}$. Then
\[
[\Spd\widehat L/S_3]
\simeq\operatorname{BC}(-1/2)^*
\]
as v-sheaves, with the following residual $P_2\times T_3$-action.
On extension classes in $\operatorname{Ext}^1(\mathcal E_2^0,\cO)$,
write $\lambda_*$ for pushout by $\lambda$ on $\cO$ and $b^*$ for
pullback by $b$ on $\mathcal E_2^0$. For the framing action
$f\mapsto bfg^{-1}$ and $a=N_3(g)$, the action is
\[
(b,a)\cdot\xi
=\bigl(aN_2(b)^{-1}\bigr)_*(b^{-1})^*\xi.
\]
In particular, the $T_3$-action is pushout by $a$ on the kernel.
Only this restriction of the action on the extension chart is used below.
\end{proposition}
\begin{proof}
The imported result is \cite[Proposition 2.7.1(1)--(5)]{BMR2026}, whose
hypotheses are $h\ge2$, a perfect field containing
$\F_{p^{h(h-1)}}$, and the coheight-one Gross--Torii extension with the
specified determinant identifications. Here $h=3$ and
$k=\F_{p^6}$ satisfy those hypotheses exactly.
We recall why its formulation gives all surjections and the indicated
structure groups. The infinite-level two-tower theorem
\cite[Theorem 2.6.3]{BMR2026}, followed by forgetting the étale framing,
identifies $\Spd\widehat L$ with exact sequences
\[
0\longrightarrow\cO\longrightarrow\mathcal E_3^0
\longrightarrow\mathcal E_2^0\longrightarrow0
\]
whose determinant identification is $\iota$. Forgetting that framing is
the $\Z_p^{\times}$-torsor in the proof of
\cite[Proposition 2.7.1(2)]{BMR2026}. Given a surjection, its kernel is
the line
$\det\mathcal E_3^0\otimes(\det\mathcal E_2^0)^{-1}$; the fixed
$\iota$ trivializes this line and recovers the sequence. Thus no
additional condition is imposed on the surjection. Descent along the
two framing torsors gives exactly the asserted description of $Y$, also
in families.
Quotienting by $S_3$ instead allows the source bundle to vary while
retaining its determinant trivialization. It therefore classifies
extensions
\[
0\longrightarrow\cO\longrightarrow\mathcal E
\longrightarrow\mathcal E_2^0\longrightarrow0
\]
with middle term of slope $1/3$. By
\cite[Proposition 2.7.1(5)]{BMR2026}, these are precisely the extensions
that are nonsplit at every geometric point, which form
$\operatorname{BC}(-1/2)^*$.
To specify the joint action, let $i_f:\cO\to\mathcal E_3^0$ be the
kernel injection normalized by $\iota$. If $f'=bfg^{-1}$, the
determinant isomorphisms for the two exact sequences give
\[
g i_f=i_{f'}\,N_3(g)N_2(b)^{-1}.
\]
Indeed, $g$ acts on the source determinant by $N_3(g)$ and $b$ acts on
the target determinant by $N_2(b)$. The induced diagram of extensions
therefore has kernel map $N_3(g)N_2(b)^{-1}$ and quotient map $b$.
Functoriality of extensions by pushout and pullback proves the displayed
$P_2\times T_3$-action. This calculation is an identity of bundle maps
on every test object and commutes with base change. For $b=1$, it is
pushout by $a=N_3(g)$. On the extension group, $a_*=(aI_2)^*$,
so this is the action by inverse pullback of the central scalar $a^{-1}I_2$.
The underlying quotient and this $T_3$ restriction
are the parts of \cite[Proposition 2.7.1(5)]{BMR2026} used below; the
joint action includes the target determinant factor just computed.
The field assertions are parts (1) and (4) of the same proposition.
\end{proof}
\subsection{Acyclicity of constants}
Choose $C=\C_p$ and a map $\Spd C\to\Spd k$, and write $Y_C$ for
base change over $\Spd k$. Put
\[
\widetilde Y=\Spd\widehat L,\qquad
\widetilde Y_C=\widetilde Y\times_{\Spd k}\Spd C.
\]
The quotient torsors $\widetilde Y\to Y$ and $\widetilde Y_C\to Y_C$
give classifying maps
\[
Y\longrightarrow B_kG=[\Spd k/\underline G],\qquad
Y_C\longrightarrow B_CG=[\Spd C/\underline G].
\]
These are relative classifying stacks; the nonextended actions fix
their respective bases. If $q_n:G\to P_n$ is projection, the classes
denoted below by $e_{P_n}|_Y$ and $e_{P_n}|_{Y_C}$ mean respectively
$u_{\widetilde Y,G,\mathrm b}(q_n^*e_{P_n})$ and
$u_{\widetilde Y_C,G,\mathrm b}(q_n^*e_{P_n})$.
Their base-change compatibility is
\eqref{eq:constant-cochain-naturality} for
$\widetilde Y_C\to\widetilde Y$, already at each finite level. Introduce
\[
X=[\Spd\widehat L/P_3^1],\qquad
X_C=X\times_{\Spd k}\Spd C;
\quad Y=[X/H],\quad Y_C=[X_C/H].
\]
The notation $\cO^{\flat}$ denotes the characteristic-$p$ structure
sheaf. Cohomology with these coefficients is derived solid cohomology;
on the group quotients used here, quotient descent computes it as
continuous derived invariants of the parameterized geometric coefficient
complex. The comparisons
with integral constant coefficients will be made in
\Cref{prop:witt-comparison}.
\begin{proposition}[Acyclicity of the maps of constants]
\label{prop:constant-acyclicity}
The actual maps of constants induce equivalences
\begin{align}
k&\xrightarrow{\ \sim\ }R\Gamma(P_3^1,\widehat L),
\label{eq:finite-constant-map}\\
C^{\flat}&\xrightarrow{\ \sim\ }
R\Gamma_v(X_C,\cO^{\flat}).
\label{eq:geometric-constant-map}
\end{align}
These are equivalences of derived solid algebras with residual $H$-action.
The $H$-action on the left sides is trivial. The maps are natural for
the change of constants $k\to C^{\flat}$ and for the arithmetic actions
preserving the chosen base data.
\end{proposition}
\begin{proof}
Apply the underlying cohomology calculation of
\cite[Proposition~3.6.9]{BMR2026} to the quotient in \Cref{prop:tower},
retaining only its $T_3$-action. The map of constants and the vanishing
in the other degrees give a fiber sequence of derived solid $k$-modules
with $T_3$-action
\[
k\longrightarrow R\Gamma(S_3,\widehat L)
\longrightarrow k_{\chi_T}[-3].
\]
Here $\chi_T$ denotes the actual character on the line in degree three.
This is the restriction of the argument combining quotient descent and cohomology in
\cite[Theorem~3.6.10]{BMR2026} that is needed here. For $b=1$, the action
formula in \Cref{prop:tower} is pushout by $a\in T_3$ on the kernel.
On the extension group this equals pullback by $aI_2$, or the
action by inverse pullback of $a^{-1}I_2$. The reduced norm of $aI_2$
in $D_2^\times$ is $a^2$. Hence the subgroup
\[
P_3^1/S_3=\mu_{p-1}\subset T_3
\]
acts on the last term by $\alpha\mapsto\alpha^{2}$ or
$\alpha\mapsto\alpha^{-2}$; either convention gives the same vanishing
used below. The displayed exponent is also checked in
\cite[Lemma 5.2.5]{BMR2026}. This is a subgroup of the norm quotient;
it must not be replaced by the central $\mu_{p-1}$ in $P_3$.
Since $p\ge5$, this character is nontrivial. The averaging idempotent
for the finite group $\mu_{p-1}$ is defined in characteristic $p$, so its
derived invariants are exact and annihilate $k_{\chi_T}$.
Taking these invariants proves that the underlying map in
\eqref{eq:finite-constant-map} is an equivalence. The map itself is
$H$-equivariant: it is induced by constants from the original $G$-action
on $\widehat L$, followed by $P_3^1$-descent, and $G$ fixes $k$.
The forgetful functor from $H$-equivariant objects detects equivalences,
so this gives the required equivalence with its full residual action.
No splitting of the fiber sequence or joint character formula is needed.
For completeness, the finite-field hypothesis in this import is
substantive. Its proof uses the solid Frobenius comparison
\cite[Lemma 3.6.3 and Corollary 3.6.8]{BMR2026} for
$\bigl(\bigcup_{r\ge0}k((z^{1/p^r}))\bigr)^{\wedge}$
over a \emph{finite} field $k$.
The geometric base change of this point is a punctured perfectoid ball;
the proof solves coefficient Frobenius minus identity on its convergent functions
and then requires surjectivity in solid modules. For this precise
function ring, \Cref{lem:continuous-as-section} supplies a continuous
section and hence proves surjectivity on every profinite parameter space.
This justifies the solid enhancement directly, without applying a
linear open-mapping theorem to Frobenius minus identity. The resulting natural
comparison commutes with the $S_3$-action. Thus
\Cref{prop:tower} verifies the required hypothesis before finite-field
descent is applied.
For the geometric statement, take a quadratic extension $F/\Q_p$ that
is a maximal subfield of $D_2$. Proposition 3.1.2 of \cite{BMR2026}
identifies
\[
\bigl(\operatorname{BC}(-1/2)^*\bigr)_C
\simeq[\mathcal H^1_{F,C}/F],
\]
where $\mathcal H^1_{F,C}$ is the one-dimensional Drinfeld upper
half-plane for $F$ and the additive group $F$ acts by translations.
This identification is equivariant for the central $\Q_p^{\times}$.
It is distinct from the rank-$n$ Drinfeld space
$\Omega_C^{n-1}$ used in the modification argument.
Proposition 3.5.6 of \cite{BMR2026} computes the solid
$\cO^{\flat}$-cohomology of this quotient: it is $C^{\flat}$ in degree
zero, a character line in degree $[F:\Q_p]+1=3$, and zero otherwise.
By the $b=1$ action formula in \Cref{prop:tower}, this norm-quotient
$\mu_{p-1}$ acts by kernel scalars. Expressing those as central
pullbacks on the target gives reduced-norm character
$\alpha\mapsto\alpha^2$ or its inverse. Both are nontrivial, so the
top cohomology has no $\mu_{p-1}$-invariants; the choice of action or
dual convention does not affect the conclusion.
The degree-zero identification is the map of constants. Applying the
same averaging argument proves \eqref{eq:geometric-constant-map}.
Only central equivariance of the Drinfeld description was needed to
prove this vanishing. The constant map on $X_C$ itself has the full
residual $H$-equivariance, and an equivariant map is an equivalence when
its underlying map is. This proves the full assertion, without assuming
a $P_2$-equivariant splitting or a larger equivariance of that chart.
Finally, the actions of $P_2$ and $\overline Z$ are over the fixed base,
so they fix $k$ and $C^{\flat}$ pointwise. All arrows used above are
induced by structure maps and group descent; their stated naturality
follows.
\end{proof}
\subsection{Completed Lubin--Tate theory and coefficient descent}
Write $E_n=E(k,\Gamma_n)$ for ordinary Morava $E$-theory and
$E_n^{\square}=E^{\square}(k^{\delta},\Gamma_n)$ for its solid
enhancement, where $k^{\delta}$ is the discrete solid ring. For the
valued field $\widehat L$, its solid ring of parameters is
\[
\widehat L(T)=C_{\cts}(T,\widehat L)
\]
on profinite sets $T$. Define
\[
\widehat B^{\square}
=E^{\square}(\widehat L,(\Gamma_2)_{\widehat L}).
\]
Here $\pi_t$ of a solid spectrum is its solid homotopy module; $(*)$
will denote evaluation in ordinary spectra at the one-point set.
In this section and in \cref{sec:witt-transfer,sec:detection},
$H^s(A,M)$ denotes the point-valued continuous cohomology
$H^s((R\Gamma(A,M))(*))$ for a solid coefficient module $M$.
\begin{proposition}[The completed theory and its coefficients]
\label{prop:completed-theory}
There are continuous $G$-equivariant maps of solid $E_\infty$-rings
\begin{equation}\label{eq:completed-map}
E_3^{\square}\longrightarrow\widehat B^{\square}
\longleftarrow E_2^{\square},
\end{equation}
where $G$ acts through $P_3$ on the first source and through $P_2$ on
the second. The right arrow is canonical connected base change.
The target is even periodic, with
\[
\pi_0\widehat B^{\square}(*)\simeq W(\widehat L)[[u'_1]].
\]
Invariantly, its coefficients are complete base changes of the
height-two deformation ring and its invertible even coefficient lines.
The natural map $W(\widehat L)\to\pi_0\widehat B^{\square}$ is
$G$-equivariant. Every profinite evaluation
$\widehat B^{\square}(T)$ is an ordinary $K(2)$-local spectrum.
The second map in \eqref{eq:completed-map} induces an $H$-equivariant
equivalence
\[
E_2^{\square}\xrightarrow{\ \sim\ }
(\widehat B^{\square})^{hP_3^1},
\]
where $\overline Z$ acts trivially on the source. Moreover, for every
$t\in\Z$ it induces an equivalence on the individual coefficient
complexes
\[
\pi_tE_2^{\square}\xrightarrow{\ \sim\ }
R\Gamma(P_3^1,\pi_t\widehat B^{\square}).
\]
In particular, for $s\ge0$ and all $t$ there are natural isomorphisms
\begin{equation}\label{eq:coefficient-descent}
H^s(H,\pi_tE_2^{\square})
\xrightarrow{\ \sim\ }
H^s(G,\pi_t\widehat B^{\square}).
\end{equation}
These comparisons preserve the coefficient maps from Witt constants.
Finally, put $\Delta=\Gal(k/\F_p)$ and
$G_n=P_n\rtimes\Delta$. The ordinary stabilizer-descent identification
is
\[
\bigl((E_n^{\square})^{hG_n}\bigr)(*)\simeq L_{K(n)}S,
\qquad n=2,3,
\]
compatibly with the canonical maps to $E_n$.
\end{proposition}
The following subsections prove the proposition. Its coefficient
statements concern the connected
base-change map from \(E_2\). We prove them independently of the map
from \(E_3\); that map will be attached to the same endpoint by the
oriented splitting construction below.
\subsection{The connected endpoint and its coefficients}
For the Honda pairs over \(k\), the deformation ideals are
\((p,u_1,u_2)\) at height three and \((p,u'_1)\) at height two.
They are finite regular sequences in the respective deformation rings.
The connected completed pair uses the Noetherian pair
\((k,\Gamma_2)\) as a witness: every
\(C_{\cts}(T,\widehat L)\) is perfect, since inverse Frobenius is
continuous, and flat over \(k\). Thus the endpoint formulas of
\cite[Proposition~4.5.5]{BMR2026} apply through this fixed witness.
The finite Koszul argument below explains the reduction through
hypersheafification and totalization in these instances.
The pointwise coefficient formula and the coefficient-line base change
follow from \cite[Lemma 4.2.12 and Proposition 4.5.5]{BMR2026}, applied to
the constant connected height-two group. The formal parameter $u'_1$
is a coordinate inherited from a deformation coordinate for $E_2$; no
assertion that $P_2$ fixes this parameter is intended. Witt vectors and
power series carry their limit structures on profinite parameters.
Functoriality of the perfect-residue-ring deformation construction gives
the asserted map of Witt constants. The same pointwise description,
together with \cite[Proposition 4.4.2]{BMR2026}, proves $K(2)$-locality
of every evaluation.
The same formulas identify the condensed connected theory
\(E^{\mathrm{cond}}(\widehat L,(\Gamma_2)_{\widehat L})\)
with \(\widehat B^\square\). For a profinite set \(T\), its evaluation is
\[
E\bigl(C_{\cts}(T,\widehat L),
(\Gamma_2)_{C_{\cts}(T,\widehat L)}\bigr).
\]
The canonical connected
base-pair map defines
\[
c^\square:E_2^\square\longrightarrow\widehat B^\square.
\]
It is continuous \(G\)-equivariant by functoriality of the connected
base-pair construction: \(P_3\) acts through the base and \(P_2\)
changes the connected framing. In particular \(G\) acts on its
source through \(P_2\). The family naturality is checked in
\cref{app:oriented-projection}, before constructing the oriented section.
The coefficient and locality statements above apply to this endpoint
without using the full completed intermediate.
We give the coefficient argument separately from the spectral one.
Proposition~4.3.12 of \cite{BMR2026} states the following descent result:
if $(R,\Gamma)$ is a Noetherian Luriean pair and
$S^{-1}\to S^{\bullet}$ is an augmented cosimplicial diagram of flat,
relatively perfect $R$-algebras which is a limit diagram in derived
commutative rings, then both
\[
E(S^{-1},\Gamma)\longrightarrow
\operatorname{Tot} E(S^{\bullet},\Gamma)
\quad\text{and}\quad
\pi_{2j}E(S^{-1},\Gamma)\longrightarrow
\operatorname{Tot}\pi_{2j}E(S^{\bullet},\Gamma)
\]
are equivalences; the latter totalization is in $D(\Z)$.
We use this statement for fixed witnesses with finite regular deformation
ideals, and give the finite Koszul justification for their reduction
through totalization.
For any one witness $(R_0,\Gamma_0)$, put
$A_0=\pi_0E(R_0,\Gamma_0)$ and $I_0=\ker(A_0\to R_0)$.
Assume the indicated generators of $I_0$ form a finite regular sequence.
Thus $R_0=A_0/I_0$ has a finite free Koszul resolution and is a perfect
$A_0$-module. Each profinite hypercover diagram for that endpoint is
$A_0$-linear by functoriality from its fixed witness. For the final bar diagram this is
$A_0=W(k)[[u'_1]]$ and $I_0=(p,u'_1)$; its maps are $A_0$-linear
because $P_3^1$ acts trivially on the $E_2$ source.
Derived tensor with $R_0$ therefore commutes with all limits in
$D(A_0)$, including each of these totalizations.
By \cite[Lemma~4.2.12]{BMR2026}, the coefficient objects are derived
$I_0$-complete and their derived reductions are the characteristic-$p$
terms in the augmented diagram. The comparison cone is again derived
complete, since complete objects are closed under limits and cofibers.
Its reduction modulo $I_0$ vanishes by the characteristic-$p$ limit
diagram. If $I_0=(x_1,\ldots,x_c)$, each quotient by
$(x_1^m,\ldots,x_c^m)$ has a finite filtration with subquotients finite
free over $R_0$. The cone therefore vanishes modulo all these quotients.
Their ideals are cofinal with powers of $I_0$, so derived completeness
implies that the cone vanishes. Tensoring with an invertible even
coefficient line also preserves limits. This proves the required
individual-coefficient comparison for every even degree at every
endpoint; the odd coefficient modules vanish. The argument uses the
perfect Koszul quotient for the tensor--totalization interchange.
\paragraph{From coefficient descent to spectral descent.}
These coefficient comparisons also give the spectral comparisons in
ordinary spectra. They give descent for each Eilenberg--Mac Lane
layer, because the forgetful functor from $H\Z$-modules preserves limits;
exactness of totalization then gives descent for every finite Postnikov
interval. For an augmented cosimplicial spectrum
$Z^{-1}\to Z^\bullet$, fix $b$ and $q\leq b$. The canonical fiber sequence
\[
\tau_{[q-1,b]}Z\longrightarrow\tau_{\leq b}Z
\longrightarrow\tau_{\leq q-2}Z
\]
has a $(q-2)$-coconnective last term, and totalization preserves this
bound. Hence the first arrow induces an isomorphism on $\pi_q$ both on
the augmentation object and after totalization. The finite-interval
comparison therefore proves the degree-$q$ comparison for
$\tau_{\leq b}Z$. Degrees above $b$ vanish on both sides.
Thus $\tau_{\leq b}Z^{-1}\to\operatorname{Tot}\tau_{\leq b}Z^\bullet$
is an equivalence for every $b$. Taking the limit over $b$, using
Postnikov completeness of ordinary spectra and commutation of limits,
proves the spectral comparison without a bounded-below assumption.
\paragraph{Descent along the completed tower.}
Apply the descent result to \((R,\Gamma)=(k,\Gamma_2)\) and the
\(P_3^1\)-bar diagram of \(\widehat L\), augmented by the constants
\(k\); its map of theories is \(c^\square\). To justify the application
in the solid category, evaluate first on an extremally
disconnected profinite parameter set $T$. Such evaluation is exact on
condensed modules and preserves limits; moreover, the forgetful functor
from derived commutative algebras creates limits and detects equivalences.
Thus \eqref{eq:finite-constant-map} identifies
\[
C_{\cts}(T,k)\ \simeq\
\operatorname{Tot}
C_{\cts}\bigl((P_3^1)^{\bullet}\times T,\widehat L\bigr)
\]
in derived commutative rings. All displayed rings are perfect and
flat over $k$, hence relatively perfect. The restricted descent argument
therefore applies to this diagram. The endpoint formulas of
Proposition 4.5.5 and the cotensor comparison of Remark 4.5.6 in
\cite{BMR2026}, with the witness verification above, identify its
spectral and coefficient terms with the evaluations of the required
solid objects and their profinite cotensors. Extremally
disconnected parameter sets detect equivalences; thus this proves both
the spectral and individual-coefficient assertions for \(c^\square\).
Odd coefficient modules vanish on both sides. This also proves the
specialization of \cite[Theorem 5.2.6]{BMR2026}, whose numerical
hypothesis is exactly $p-1\nmid2$.
Every comparison is induced by the same augmented diagram and its
commuting actions. It therefore retains the residual \(H\)-action and
the maps from Witt constants; \(\overline Z\) acts trivially on the
\(E_2\) source.
\subsection{The map from height three}
The connected theory now has the coefficient descent and locality needed
for detection. To map the height-three sphere into it, we start with the
tensor construction and then pass from the full completed group to its
connected summand. The direction of this last map requires a section of
the deformation functor, rather than the forgetful transformation alone.
The uncompleted tensor maps are the starting point. Examples~4.6.2--4.6.3 of
\cite{BMR2026} use the pointwise constructions of Theorems~4.4.6 and~4.4.8,
followed by Construction~4.5.2, to give
\[
E_3^{\square}\longrightarrow L_{K(2)}^{\square}E_3^{\square}
\longrightarrow B^{\square}\longleftarrow E_2^{\square},
\]
where
\[
B^{\square}
=L_{K(2)}^{\square}
(E_3^{\square}\otimes_{SW(k)}^{\square}E_2^{\square})
\simeq E^{\square}(L,\Gamma^{\mathrm{un}}_{3,L}).
\]
Here \(SW(k)\) is the spherical Witt-vector ring and
\(L_{K(2)}^{\square}\) is localization in solid spectra. These maps have
the projection actions, as in \cite[(5.2.2)--(5.2.3)]{BMR2026}.
The tensor construction uses the constant pair \((k,\Gamma_2)\), whose
base is Noetherian and perfect and whose group is connected of height
two, as required in \cite[Construction~4.4.7]{BMR2026}. The fixed-witness
descent argument above identifies the required hypersheafifications once
the following regular quotient is checked.
For the localized $K$-pair and its uncompleted ind-\'{e}tale $L/K$
extension, the witness has classical deformation ring
\[
A_K=\bigl(W(k)[[u_1,u_2]][u_2^{-1}]\bigr)^{\wedge}_{(p,u_1)},
\qquad A_K/(p,u_1)=K.
\]
The sequence $(p,u_1)$ remains regular under localization and this
Noetherian completion. The localized witness is Noetherian Luriean
by \cite[Proposition~4.4.4 and Remark~4.2.2]{BMR2026}; the displayed
$p$-basis below also verifies that $K$ is $F$-finite. The profinite witness hypotheses are
\cite[Propositions~4.5.9, 4.5.10, and 4.5.12, Definition~4.5.11,
and Examples~4.6.1--4.6.3]{BMR2026}.
Concretely, $K(T)=C_{\cts}(T,K)$ is flat over the field $K$ and
relatively perfect: the decomposition
$K=\bigoplus_{j=0}^{p-1}u_2^jK^p$ has continuous coefficient-extraction
maps and hence applies to continuous functions on $T$.
Compactness of $T$ supplies a uniform denominator for inversion of
$u_2$. This verifies the local-field instance directly; no commutation
of arbitrary localization with totalization is required.
The ind-\'{e}tale $L$-extension retains the same witness as in
Proposition~4.5.12.
\begin{lemma}[The completed oriented projection]
\label{lem:completed-oriented-map}
Put \(F=\widehat L(*)\), regarded here as the ordinary completed field.
For a nonempty profinite set \(T\), put \(R_T=C_{\cts}(T,F)\) and let
\(\Gamma_T=\Gamma^{\mathrm{un}}_{3,R_T}\) be the full completed
\(p\)-divisible group. Let \(C_T\) be the base change of the connected
term in the uniquely split sequence over \(F\). It is identified with
\((\Gamma_2)_{R_T}\) by the fixed tautological Gross--Torii formal
isomorphism. There is a continuous adic \(E_\infty\)-map
\[
r_T:E(R_T,\Gamma_T)\longrightarrow E(R_T,C_T).
\]
The maps are natural for profinite pullbacks and every continuous
\(G=P_3\times P_2\) family of pair maps. With the terminal value at the
empty parameter set, pointwise application and hypersheafification give
a continuous \(G\)-equivariant map of condensed \(E_\infty\)-rings
\[
r^{\mathrm{cond}}:
E^{\mathrm{cond}}(\widehat L,\Gamma^{\mathrm{un}}_{3,\widehat L})
\longrightarrow E^{\mathrm{cond}}(\widehat L,C_{\widehat L}).
\]
Let \(b_T:E_2\to E(R_T,\Gamma_T)\) be the pointwise ordinary
\(E_2\) tensor-factor map followed by completion, and let
\(c_T:E_2\to E(R_T,C_T)\) be the canonical connected map induced by
\(k\to R_T\). Then \(r_Tb_T\simeq c_T\), naturally for the same
parameters and pair families. This homotopy preserves the connected
residue tag and orientation, and hence its coefficient and even-line maps.
After condensation and solid localization, it gives
\[
r^{\mathrm{cond}}b^\square\simeq c^\square
\]
as continuous \(G\)-equivariant maps of underlying solid spectra, where
\(b^\square\) is the tensor-factor map followed by completion and
\(c^\square\) is the canonical connected base-change map.
\end{lemma}
\begin{proof}[Construction; full proof in \cref{app:oriented-projection}]
Fix a parameter set \(T\). Write \(\mathscr D_\Gamma^{\mathrm{or}}\)
and \(\mathscr D_C^{\mathrm{or}}\) for oriented deformations with the
specified residue tags of the full and connected reference groups.
The connected--\'{e}tale sequence is uniquely split over the perfect
field \(F\). Let \(Q_T\) be the base change of its \'{e}tale quotient,
so \(\Gamma_T=C_T\oplus Q_T\). An oriented deformation \(C_A\)
over a complete adic test ring \(A\) has a
contractible space of lifts of the tagged \'{e}tale quotient \(Q_T\).
With such a marked lift \(Q_A\), send it to
\(C_A\oplus Q_A\), retaining its connected orientation. This constructs
\(\mathfrak s\) in the left diagram:
\begin{equation}\label{eq:oriented-section-diagram}
\begin{tikzcd}[column sep=large]
\mathscr D_C^{\mathrm{or}}
\arrow[r,bend left=18,"\mathfrak s"]
&\mathscr D_\Gamma^{\mathrm{or}}
\arrow[l,bend left=18,"\mathfrak f"]
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=large]
E(R_T,C_T)\arrow[r,bend left=18,"i_T"]
&E(R_T,\Gamma_T)\arrow[l,bend left=18,"r_T"] .
\end{tikzcd}
\end{equation}
Here \(\mathfrak f\) takes the connected deformation, and
\(\mathfrak f\mathfrak s\simeq\mathrm{id}\). Corepresentability
reverses the arrows and gives \(r_Ti_T\simeq\mathrm{id}\).
In particular the required map is \(r_T\), supplied by the split
section. The forgetful map alone supplies \(i_T\) in the other direction.
The tensor-factor identity keeps track of which connected map results.
The tensor construction compares the two oriented formal groups through
their common Quillen formal group. On the split deformation its comparison
is the inverse of the connected inclusion. The resulting connected tag
and orientation are therefore exactly those defining \(c_T\), so
\(r_Tb_T\simeq c_T\). The appendix verifies this cancellation on the
actual tags and proves its compatibility with adic refinement and every
continuous group-family map. It then upgrades the identity to the
continuous solid maps in the statement. This identifies the right-hand
tensor composite with the canonical map used in coefficient descent.
\end{proof}
Completion of the pair diagrams followed by the constructed map gives
\[
B^\square\longrightarrow
E^{\mathrm{cond}}(\widehat L,\Gamma^{\mathrm{un}}_{3,\widehat L})
\xrightarrow{r^{\mathrm{cond}}}\widehat B^\square.
\]
The completion map is \cite[Example~4.6.4]{BMR2026}; the full completed
pair uses the perfect-field witness verified in
\cref{app:oriented-projection}. The composite from \(E_3^\square\)
is therefore an actual continuous equivariant ring map. Take this
composite and \(c^\square\) as the arrows in \eqref{eq:completed-map}.
The later detection argument uses this unital equivariant map from
\(E_3^\square\) and the coefficient descent for \(c^\square\).
The identity \(r^{\mathrm{cond}}b^\square\simeq c^\square\) also
identifies the right arrow with the \(E_2\) tensor composite.
This construction uses no coefficient formula
or locality assertion for the nonconnected intermediate, and does not
assert that \(K\to\widehat L\) is relatively perfect.
\subsection{The ordinary sphere source}
It remains to identify the source and the sphere unit after evaluation.
We first spell out the effect of hypersheafification at the point. Let \(a_{\mathrm{hyp}}\) be
hypersheafification on profinite sets with the finite jointly surjective
topology. Every covering sieve of the one-point set is maximal: a
nonempty member of a covering family has a point and hence a section.
Consequently associated-sheaf formation for abelian presheaves leaves
their point value unchanged. The unit
\(P\to a_{\mathrm{hyp}}P\) for a spectrum-valued presheaf is an
isomorphism on associated homotopy sheaves. Evaluating these sheaves at
the point therefore gives an isomorphism on every ordinary homotopy
group, and hence
\[
P(*)\xrightarrow{\sim}(a_{\mathrm{hyp}}P)(*).
\]
This proof applies to the accessible presheaves on full profinite sets
used here and is natural in the unit. It is the full-profinite version of
\cite[Lemma~13.5 and Remark~13.6]{Clausen2025}; the discrete spectra are
solid by Example~13.4(1) there.
Definition~4.5.7 of \cite{BMR2026} computes condensed localization by
pointwise ordinary localization followed by hypersheafification, and its
restriction to solid spectra is solid localization. Discrete condensation
is the hypersheafification of the constant presheaf, so
\(V^\delta(*)\simeq V\). Thus, naturally in an ordinary spectrum \(V\)
and its maps,
\[
\bigl(L_{K(n)}^\square(V^\delta)\bigr)(*)\simeq L_{K(n)}V,
\]
with the evaluated localization unit equal to the ordinary unit.
In particular the input \(V=S\) is the ordinary sphere before any
completion. Corollary~4.6.8 of \cite{BMR2026} constructs
\(L_{K(n)}^\square S^\delta\to E_n^\square\) by applying this functor
to the ordinary map \(L_nS\to E_n\). The natural point comparison
identifies it with
\[
L_{K(n)}L_nS\longrightarrow L_{K(n)}E_n.
\]
Since \(S\to L_nS\) is a \(K(n)\)-equivalence and \(E_n\) is
\(K(n)\)-local, this is precisely the factor of the ordinary sphere
unit \(S\to E_n\) through \(L_{K(n)}S\). Lemma~4.5.8 and the endpoint
formula of Proposition~4.5.5 in that paper identify the target at the
point naturally with \(E_n\): the discrete-adjunction map used there is
adjoint at the point to the identity of \(E_n\).
Proposition~4.6.9 of \cite{BMR2026} identifies the \v Cech augmentation
of that same map with
\(L_{K(n)}^\square S^\delta\xrightarrow{\sim}(E_n^\square)^{hG_n}\).
Its composite to \(E_n^\square\) is the map just identified. Evaluation
at the point preserves limits, so this gives the asserted ordinary
fixed-point equivalence with its canonical map to \(E_n\).
The coefficient-field scope is also exact: Example~4.6.1 there allows an
algebraic field containing the endomorphism field. Here
\(k=\F_{p^6}\) is finite and contains both required fields. The Honda
groups descend to \(\F_p\), and all geometric endomorphisms are
\(k\)-rational. The descent identifications from the \(\F_p\)-models split
the pair-automorphism sequences, giving exactly
\(G_n=P_n\rtimes\Gal(k/\F_p)\). Thus the cited solid descent theorem
applies to this enlarged finite field, and the point comparison above
identifies its source and map for the uncompleted sphere \(S\).
\begin{proof}[Proof of \cref{prop:completed-theory}]
The connected endpoint calculation gives the coefficients and ordinary
locality of every evaluation. The tensor construction and the oriented
section supply the left equivariant ring map; the right map is the
canonical connected base change already constructed.
The constants calculation and finite Koszul
argument prove both fixed-point descent and descent for each coefficient
module, with their residual actions and maps from Witt constants.
Taking derived \(H\)-invariants in the extension
\(P_3^1\to G\to H\) gives \eqref{eq:coefficient-descent};
\(\overline Z\) acts trivially on the \(E_2\) source.
The preceding point-evaluation argument proves the ordinary
stabilizer-descent identification with its sphere unit.
\end{proof}
\begin{remark}
The completion in \eqref{eq:completed-map} is essential to the statements
proved here. The uncompleted half-sphere equivalence is
\cite[Conjectures 1.1.1 and 5.1.13]{BMR2026};
\cite[Proposition 5.1.17]{BMR2026} proves an implication from that
conjecture to coheight-one splitting. The completed theorem above
supplies an actual target for detection and does not assert an
equivalence from the uncompleted fixed points to that target.
\end{remark}
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//03-completed-tower.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//04-modifications.tex ====
\section{Two realizations of a modification}\label{sec:modifications}
We compare two descriptions of a line modification of a rank-$n$,
degree-one bundle. Framing the original bundle gives a projective
space with a division-algebra action; framing its slope-zero
modification gives a Drinfeld space with a linear-group action.
The common quotient will carry both degree-three classes. We then
apply the same modification to the universal rank-three-to-rank-two
surjection, obtaining the exact sequence that relates the two heights.
We work over $\Spd C$. For a perfectoid test object $S$ over this
base, write $\mathcal X_S^{\mathrm{FF}}$ for its relative
Fargues--Fontaine curve, with untilt divisor
$i_\infty:\infty\hookrightarrow\mathcal X_S^{\mathrm{FF}}$.
At the fixed geometric point determined by $C$, write
$\mathcal X_C^{\mathrm{FF}}$ for the corresponding curve.
For a locally profinite group $A$, write
$B_CA=[\Spd C/\underline A]$ for its relative classifying stack;
all classifying maps in this section are over this base.
Write $\mathcal E_n^0=\cO(1/n)$, of rank $n$ and degree one.
A \emph{determinant lattice} of a slope-zero rank-$n$ bundle means a
$\Z_p$-lattice in its rank-one determinant local system.
Its frame group is
\[
J_n=\{g\in\GL_n(\Q_p):|\det(g)|_p=1\}.
\]
This is the structure that will relate the two heights.
We use the following precise forms of the relative bundle theorems.
Let $\bar k$ be an algebraic closure of $k$. Since $C^\flat$ is
algebraically closed, fix an extension $\bar k\hookrightarrow C^\flat$
of the embedding $k\hookrightarrow C^\flat$ underlying the chosen map
$\Spd C\to\Spd k$, and hence a compatible map
$\Spd C\to\Spd\bar k$. Geometrically semistable bundles of slope zero
correspond to pro-\'etale $\Q_p$-local systems, by
\cite[Theorem II.2.19 and Corollary II.2.20]{FS}.
Pulling the geometrically trivial and basic strata of
\cite[Theorems III.2.4 and III.4.5]{FS} back along this map gives the
relative classifying descriptions $B_C\GL_n(\Q_p)$ and
$B_C\Aut(\mathcal E_n^0)$, respectively.
Thus their sheaves of frames are actual torsors in families.
The later arithmetic $G_F$-descent is established by the local
construction below. We will apply the bundle statements after proving
the required slope conditions. Geometric points alone are not being
used to identify arbitrary stacks.
\subsection{The lower and upper modifications}
\begin{lemma}\label{lem:lower-modification}
Let $\mathcal E_n$ be a form of $\mathcal E_n^0$, and let
$q:i_\infty^*\mathcal E_n\twoheadrightarrow\mathcal L$ be a quotient
line. The elementary lower modification
\[
0\longrightarrow\mathcal V_n\longrightarrow\mathcal E_n
\longrightarrow i_{\infty *}\mathcal L\longrightarrow0
\]
is geometrically semistable of slope zero. It therefore defines a
rank-$n$ rational local system.
\end{lemma}
\begin{proof}
The kernel is a vector bundle: locally at the Cartier divisor, the
quotient is a coordinate modulo a local equation of the divisor, and
the kernel has a basis obtained by multiplying that coordinate by the
local equation. At a geometric point,
$\rank(\mathcal V_n)=n$ and $\deg(\mathcal V_n)=0$.
If it had a positive-slope Harder--Narasimhan piece, its maximal
destabilizing subbundle $\mathcal F$ would have rank $b<n$ and integral
degree $d\geq1$. The injection $\mathcal F\hookrightarrow\mathcal E_n$
contradicts semistability of $\mathcal E_n$, because
\[
\mu(\mathcal F)=\frac d b\geq\frac1b>\frac1n
=\mu(\mathcal E_n).
\]
One may saturate its image in $\mathcal E_n$ without decreasing degree.
Hence $\mathcal V_n$ is semistable of degree zero at every geometric
point. The relative slope-zero equivalence stated above now applies.
\end{proof}
Conversely, an elementary upper modification of $\cO^n$ at $\infty$
is determined by a line of modification directions. Locally at a
geometric point, if $t$ is a uniformizer at $\infty$ and
$L_0=(B_{\mathrm{dR}}^+)^n$, its lattice is
\[
L_\ell=L_0+B_{\mathrm{dR}}^+t^{-1}\widetilde\ell
\subset t^{-1}L_0 ,
\]
where $\widetilde\ell$ lifts the direction line $\ell$.
Intrinsically the direction lies in
$(\cO(\infty)/\cO)^n$; tensoring every coordinate by the same
one-dimensional space identifies its projectivization with
$\mathbb P^{n-1}_C$.
\begin{lemma}\label{lem:drinfeld-locus}
The upper modification $\mathcal E_\ell$ is semistable of slope $1/n$
if and only if $\ell$ lies in no proper $\Q_p$-rational subspace.
Consequently its semistable locus is
\[
\Omega_C^{n-1}
=\mathbb P_C^{n-1}\setminus
\bigcup_{\substack{W\subset\Q_p^n\\ \dim W=n-1}}\mathbb P(W_C).
\]
On this locus $\mathcal E_\ell$ is a form of $\mathcal E_n^0$.
\end{lemma}
\begin{proof}
If $\ell\subset W_C$ for a proper rational subspace $W$, modify
$W\otimes_{\Q_p}\cO$ along $\ell$. The resulting rank-$\dim W$,
degree-one bundle embeds as a subbundle of $\mathcal E_\ell$; the
quotient is the unchanged trivial bundle associated to $\Q_p^n/W$.
Its slope is greater than $1/n$, so $\mathcal E_\ell$ is not semistable.
For the converse, take a saturated destabilizing subbundle
$\mathcal F\subset\mathcal E_\ell$ of rank $b<n$.
Its integral degree $d$ is at least one. Set
$\mathcal F'=\mathcal F\cap\cO^n$. The quotient
$\mathcal F/\mathcal F'$ embeds in $\mathcal E_\ell/\cO^n$ and has
length $\epsilon\leq1$. Since $\mathcal F'$ embeds in $\cO^n$,
semistability of the latter gives
\[
0\geq\deg(\mathcal F')=d-\epsilon\geq0.
\]
Thus $d=\epsilon=1$ and $\deg(\mathcal F')=0$.
The saturation of $\mathcal F'$ in $\cO^n$ cannot increase its degree:
a positive increase would contradict the same semistability inequality.
It is therefore already saturated. All its subbundles have
nonpositive degree, so it is semistable of slope zero. The
classification of bundles at a geometric point identifies it with
$W\otimes_{\Q_p}\cO$ for a $b$-dimensional rational subspace $W$;
here an inclusion between trivial bundles is a matrix over
$H^0(\mathcal X_C^{\mathrm{FF}},\cO)=\Q_p$.
The equality $\epsilon=1$ in the displayed local lattice description
then says precisely that $\ell\subset W_C$.
This proves the criterion at every geometric point. The classification
in \cite[Theorem II.2.14]{FS} gives $\cO(1/n)$ for a semistable
rank-$n$, degree-one bundle, and the
relative basic-stratum theorem gives the claimed family of forms.
\end{proof}
\subsection{A common torsor with a determinant lattice}
The lower and upper modification criteria identify the same geometric
objects. A full lattice in the slope-zero local system would impose
extra data on these objects. We retain only its determinant lattice,
which is already supplied by the $P_n$-structure, and construct the
common space of frames with that condition.
Put
\[
\Delta_n=\det(\mathcal E_n^0)(-\infty),\qquad
V(\Delta_n)=H^0(\mathcal X_C^{\mathrm{FF}},\Delta_n).
\]
The latter is a one-dimensional $\Q_p$-space. Choose a lattice
$\Lambda_n\subset V(\Delta_n)$.
The determinant action of
$\Aut(\mathcal E_n^0)=D_n^\times$ is the reduced norm.
The stabilizer of $\Lambda_n$ is
\[
\{g\in D_n^\times:v_p(N_n(g))=0\}
=\cO_{D_n}^{\times}=P_n.
\]
Accordingly a $P_n$-structure on a form $\mathcal E_n$ transports
$\Lambda_n$ to a determinant lattice on
$\det(\mathcal E_n)(-\infty)$.
These choices can be made with compatible arithmetic Galois action.
Take $F=W(k)[1/p]$, or a finite unramified extension of it.
The fixed bundle is obtained functorially from the Dieudonn\'e module
of $\Gamma_n$ over $k$, by extension to the period rings and Frobenius
descent; see \cite[Lemma 2.3.3 and the following paragraph,
pp.~23--24]{BMR2026}. Thus it is defined on perfectoid test objects
over $\Spd F$. Every endomorphism of $\Gamma_n$ is defined over $k$,
so this descent commutes with $P_n$. The untilt divisor also descends:
its ideal is the kernel of the functorial untilt map $\theta$.
It follows that $\Delta_n$ descends over $\Spd F$.
Its geometric slope is zero. Applying
\cite[Corollary II.2.20]{FS} on perfectoid test objects and descending
the resulting local systems therefore makes $V(\Delta_n)$ a
continuous rank-one representation of
$G_F=\Gal(\overline{\Q}_p/F)$.
Continuity is part of this conclusion: the descent automorphisms are
sections of the pro-\'etale $\Q_p^\times$-sheaf, hence continuous
functions on the profinite Galois parameters. The coefficient sheaf
here is the one identified by \cite[Proposition II.2.5(ii)]{FS}.
The image of its valuation character in $\Z$ is a compact subgroup,
hence zero. Every lattice $\Lambda_n$ is therefore $G_F$-stable.
We need no invariant choice of basis. The cyclotomic character of
$G_F$ still has open image.
Let $\mathcal Z_n$ be the v-sheaf of injections
\[
u:\cO^n\hookrightarrow\mathcal E_n^0
\]
whose cokernel is a line supported at $\infty$ and for which
\[
\det(u):\det(\cO^n)\xrightarrow{\ \sim\ }\Delta_n
\]
takes the standard determinant lattice onto $\Lambda_n$.
The commuting actions of $P_n$ and $J_n$ are postcomposition and
inverse precomposition.
Galois acts on a map $u$ by transport along the semilinear descent of
its source and target. On $\cO^n$ this descent fixes the standard
rational basis. It follows directly that $\gamma(guj^{-1})
=g\gamma(u)j^{-1}$ for $\gamma\in G_F$, $g\in P_n$, and $j\in J_n$.
The determinant condition is preserved by stability of $\Lambda_n$.
\begin{proposition}\label{prop:two-realizations}
Let $\mathcal M_n$ classify a form of $\mathcal E_n^0$ with
$P_n$-structure and a quotient line at $\infty$. There are
$G_F$-equivariant equivalences of v-stacks
\begin{equation}\label{eq:two-realizations}
\mathcal M_n
\simeq [\mathbb P(i_\infty^*\mathcal E_n^0)/P_n]
\simeq [\mathcal Z_n/(P_n\times J_n)]
\simeq [\Omega_C^{n-1}/J_n].
\end{equation}
Its framing torsor defines $\mathcal M_n\to B_CP_n$, and its universal
lower modification defines the indicated map $\mathcal M_n\to B_CJ_n$.
\end{proposition}
\begin{proof}
Fixing a quotient line of $i_\infty^*\mathcal E_n^0$ fixes its kernel
$\mathcal V_n$. By \cref{lem:lower-modification} its frames exist
pro-\'etale locally. Requiring a frame to preserve the determinant
lattice cuts its $\GL_n(\Q_p)$-torsor down to a $J_n$-torsor.
Thus
$\mathcal Z_n/J_n=\mathbb P(i_\infty^*\mathcal E_n^0)$
as v-sheaves.
In the other direction, a framed lower bundle gives an upper
modification $\cO^n\hookrightarrow\mathcal E_\ell$.
By \cref{lem:drinfeld-locus} its allowed directions are exactly
$\Omega_C^{n-1}$. Its determinant lattice is induced by the standard
one on $\cO^n$. The isomorphisms
$\mathcal E_\ell\simeq\mathcal E_n^0$ form a $D_n^\times$-torsor,
by the relative basic-stratum theorem. The norm valuation
$D_n^\times\to\Z$ is surjective, so this torsor has, v-locally,
isomorphisms carrying the induced lattice onto $\Lambda_n$.
Those isomorphisms form precisely a $P_n$-torsor.
It follows that $\mathcal Z_n/P_n=\Omega_C^{n-1}$.
Both identifications commute with base change and the remaining group
action. Taking the further quotients proves
\cref{eq:two-realizations}. They are Galois compatible because every
operation used---forming the kernel, determinant, lattice condition,
and sheaf of frames---is compatible with the descent just constructed.
More explicitly, an upper direction is a line in
$\Q_p^n\otimes_{\Q_p}(\cO(\infty)/\cO)|_\infty$.
The second factor is common to all coordinates, so its semilinear
Galois scalar cancels upon projectivization. The action on
$\Omega_C^{n-1}$ is consequently the standard arithmetic action,
commuting with $J_n$. The lower frames form a Galois-equivariant
$J_n$-torsor, which gives the equivariant map to $B_CJ_n$.
Here Galois acts on the $\Spd C$ base of $B_CJ_n$ and trivially on
the group $J_n$; the analogous statement holds for $B_CP_n$.
This assertion does not require
the torsor itself to have a Galois-fixed frame.
\end{proof}
The two frame-forgetting maps can be pictured as follows; the upper
arrows are torsors for the indicated groups.
\[
\begin{tikzcd}[column sep=large,row sep=large]
& \mathcal Z_n \arrow[dl,"J_n"'] \arrow[dr,"P_n"] & \\
\mathbb P(i_\infty^*\mathcal E_n^0)
\arrow[dr] && \Omega_C^{n-1}\arrow[dl]\\
& \mathcal M_n &
\end{tikzcd}
\]
Only a determinant lattice was retained. This explains the appearance
of $J_n$: its defining condition preserves the determinant lattice,
while the group as a whole preserves no full lattice in $\Q_p^n$.
\subsection{The filtration supplied by the two-height tower}
We have compared the two realizations at each rank. The remaining
construction relates those ranks using the universal surjection.
Return to the universal surjection
$f:\mathcal E_3\twoheadrightarrow\mathcal E_2$ over $Y_C$ from
\cref{prop:tower}; its all-surjections interpretation is
\cite[Proposition 2.7.1(3)]{BMR2026}.
Let
\[
\pi:\mathcal T=\mathbb P(i_\infty^*\mathcal E_2)\longrightarrow Y_C
\]
parametrize quotient lines. The linear $P_2$-descent on
$i_\infty^*\mathcal E_2$ gives an actual vector bundle on $Y_C$.
In particular $\mathcal T$ has its global tautological
$\cO_{\mathcal T}(1)$.
\begin{proposition}\label{prop:filtered-modification}
The tautological quotient of $\mathcal E_2$ and its composite with $f$
give maps $m_n:\mathcal T\to\mathcal M_n$ and an exact sequence
\begin{equation}\label{eq:filtered-modification}
0\longrightarrow\ker(f)\longrightarrow\mathcal V_3
\longrightarrow\mathcal V_2\longrightarrow0 .
\end{equation}
These are slope-zero bundles. Their corresponding rational local
systems form an exact sequence of ranks $1,3,2$.
With the induced determinant lattices this sequence is classified by
a map $d:\mathcal T\to B_CD$, where
\[
D=\left\{
\begin{pmatrix}a&v\\0&A\end{pmatrix}:
a\in\Z_p^\times,\quad A\in J_2,\quad v\in\Q_p^{1\times2}
\right\}
\simeq\Q_p^2\rtimes(\Z_p^\times\times J_2).
\]
The classifying maps of $\mathcal V_3$ and $\mathcal V_2$ are the
composites of $d$ with the relative maps induced by the inclusion
$\rho_3:D\hookrightarrow J_3$ and projection $\rho_2:D\to J_2$.
The $P_n$-classifying map of $m_n$ is the composite
$\mathcal T\xrightarrow{\pi}Y_C\to B_CP_n$ induced by
$q_n:G\to P_n$.
\end{proposition}
\begin{proof}
If $q:\mathcal E_2\twoheadrightarrow i_{\infty *}\mathcal L$
is the universal quotient, then
$\mathcal V_2=\ker(q)$ and $\mathcal V_3=\ker(qf)$.
Thus $\mathcal V_3=f^{-1}\mathcal V_2$, proving the exact sequence.
The two lower modifications are slope zero by
\cref{lem:lower-modification}. The kernel of $f$ has rank one and
degree $\deg(\mathcal E_3)-\deg(\mathcal E_2)=0$, so it too has slope
zero.
To check exactness on local systems, pass to a pro-\'etale cover
trivializing the three slope-zero bundles. Full faithfulness of the
relative correspondence identifies the maps with matrices of
continuous $\Q_p$-valued functions. The matrices have the indicated
ranks on every geometric fiber; on the open loci of invertible minors
they give exact, locally split sequences of rational local systems.
These loci cover the base, proving exactness before descent.
Transport $\Lambda_3$ and $\Lambda_2$ to the determinant local systems
of $\mathcal V_3$ and $\mathcal V_2$. The determinant isomorphism in
\cref{eq:filtered-modification} defines the lattice on the kernel as
their tensor quotient. Locally choose a lattice-compatible basis of
the kernel and determinant-compatible frames of the quotient, and
lift the latter to the middle local system. Such adapted frames form
a torsor; their change-of-frame matrices are exactly the displayed
matrices in $D$. The two classifying maps follow by forgetting the
filtration or taking its quotient. The construction of $m_n$ retains
the original $P_n$-structured form $\mathcal E_n$ on $Y_C$ and only
adds its quotient line. Its $P_n$-framing torsor is therefore the
pullback of the associated framing torsor on $Y_C$, proving the last
assertion as a statement of actual torsors.
\end{proof}
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//04-modifications.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//05-degree-three.tex ====
\section{Transport of the degree-three primitive}\label{sec:degree-three}
The projective presentation of $\mathcal M_n$ isolates a
one-dimensional Galois-invariant subspace of its degree-three
cohomology. The Drinfeld presentation shows that the linear trace
class has nonzero image in that subspace. This identifies the two
primitive classes up to a nonzero scalar; the rank-one filtration
then transports the identification from rank three to rank two.
All cohomology uses the integral complex followed by inversion of
$p$, as defined in \cref{sec:cohomology}.
\subsection{The integral projective bundle splitting}
\begin{lemma}\label{lem:projective-bundle}
Let $X$ be a small v-stack over $\Spd C$, and let $\mathcal W$ be a
rank-$(r+1)$ vector bundle on its untilt, with linear v-descent.
For $g:\mathbb P(\mathcal W)\to X$, the compatible classes
$h_m=c_1(\cO(1))\in H_v^2(\mathbb P(\mathcal W),(\Z/p^m)(1))$
give equivalences
\begin{equation}\label{eq:projective-finite}
\lambda_m:
\bigoplus_{j=0}^r(\Z/p^m)(-j)[-2j]
\xrightarrow{\ \sim\ } Rg_*(\Z/p^m).
\end{equation}
Consequently
\begin{equation}\label{eq:projective-integral}
\bigoplus_{j=0}^r
\Rlim_m R\Gamma_v(X,(\Z/p^m)(-j))[-2j]
\xrightarrow{\ \sim\ }
\Rlim_m R\Gamma_v(\mathbb P(\mathcal W),\Z/p^m).
\end{equation}
The term $j=0$ is the original pullback. Thus pullback is split
injective on $H^*_{\mathrm{int}}$ and on $H^*_{\mathrm b}$.
These statements are equivariant for actions on the vector bundle,
including arithmetic Galois actions.
\end{lemma}
\begin{proof}
Define $\lambda_m$ by the powers $h_m^j$, using adjunction and cup
product. It is therefore a global map of derived pushforward objects,
compatible with coefficient reduction and all descent data.
For arithmetic descent, the linear action on $\mathcal W$ induces
descent on the actual line bundle $\cO(1)$. Naturality of the Kummer
boundary makes $h_m$ invariant in its twisted coefficient group.
Over $\operatorname{Spa}(C,\cO_C)$, the finite-constant
projective-space calculation has basis
$1,h_1,\ldots,h_1^r$: apply
\cite[Construction~3.2.1 and Proposition~3.2.2]{LRZ2024}
with $\F_p$ coefficients, since $p$ is invertible on this base.
The first Chern class there is the Kummer boundary
\cite[Variant~3.1.8]{LRZ2024}, so this basis uses our actual classes.
To extend the calculation to arbitrary perfectoid test bases, apply
the integral Riemann--Hilbert functor
$\operatorname{RH}_{\Z_p}$ of \cite[Theorem~5.1.7]{ALM2026}.
Smooth proper primitive comparison
\cite[Proposition~5.2.2 and Example~5.2.3]{ALM2026} applies to
$\mathbb P_C^r\to\Spd C$ and the dualizable coefficient object
$\F_p$. Its mod-$p$ target is
\[
\mathcal D_{\mathrm{FF}}(-,\F_p)
=\mathcal D_{\widehat{\square}}(\cO^\flat)^{\varphi},
\]
by \cite[Remark~5.1.9 and the beginning of \S5.2]{ALM2026}.
The symmetric monoidal and pullback compatibility of the functor
carries the Kummer basis to its coefficient images. It therefore
gives the Frobenius-equivariant Chern-class equivalence with
$\cO^\flat$ coefficients over $C$.
After trivializing $\mathcal W$ on a perfectoid test object, its
projective bundle is a base change of $\mathbb P_C^r\to\Spd C$.
This projective morphism is $p$-bounded. Base change for solid
pushforward, \cite[Proposition~4.2.7(ii)]{ALM2026} with
$\Lambda=\F_p$, transports the equivalence to that test object.
The globally defined Chern-class map makes these local equivalences
compatible on overlaps, so they descend.
We identify the resulting solid calculation with ordinary v-sheaf
pushforward by testing on a perfectoid $T\to X$. Apply tensor-unit
Hom in the solid formalism on $T$. Adjunction and the tensor-unit
cohomology identification
\cite[Theorem~3.2.1(iii)--(iv)]{BMR2026} give
\[
\bigoplus_{j=0}^r
R\Gamma_v(T,\cO^\flat(-j))[-2j]
\xrightarrow{\ \sim\ }
R\Gamma_v(\mathbb P(\mathcal W)_T,\cO^\flat).
\]
This is natural in $T$ and induced by the same Chern classes.
It thus identifies their coefficient-image map with the equivalence
of ordinary derived v-sheaf pushforwards
\[
\bigoplus_{j=0}^r\cO^\flat_X(-j)[-2j]
\xrightarrow{\ \sim\ } Rg_*\cO^\flat.
\]
The displayed map is induced by the images of the actual mod-$p$
\'etale classes $h_1^j$. It therefore commutes with Frobenius on
$\cO^\flat$, including its descent data. Take the fiber of
$F-1$ on both sides. The Artin--Schreier sequence on the v-site
identifies this fiber with $\F_p$; with twists it identifies the
$j$th fiber with $\F_p(-j)[-2j]$.
Since derived pushforward preserves fibers, the resulting equivalence
is exactly $\lambda_1$.
This step uses the Frobenius compatibility of the Chern-class map.
The Kummer classes $h_m$ are compatible under reduction.
The coefficient sequences
\[
0\longrightarrow\Z/p^{m-1}\xrightarrow{\,p\,}\Z/p^m
\longrightarrow\F_p\longrightarrow0
\]
give compatible exact triangles on the source and target of
$\lambda_m$. Cup product with $h_m^j$ commutes with both coefficient
maps. Induction on $m$ therefore proves
\cref{eq:projective-finite}.
Apply global sections and then the derived inverse limit.
The finite direct sum commutes with both operations and gives
\cref{eq:projective-integral}. Projection to its zeroth summand
is a left inverse to pullback. Exact inversion of $p$ preserves that
left inverse.
\end{proof}
\subsection{Galois weights identify the primitives}
Recall from \cref{prop:group-classes} the nonzero classes
\[
e_{P_n}\in H^3_{\mathrm b}(P_n),\qquad
e_{J_n}\in H^3_{\mathrm b}(J_n),\qquad n=2,3.
\]
For a class $e\in H^i_{\mathrm b}(A)$ and a classifying map
$f:Z\to B_CA$, write
\[
e|_Z=f^*u_{B_C,A,\mathrm b}(e).
\]
Thus the class is first mapped from condensed group cohomology to the
relative geometric classifying stack by
\eqref{eq:finite-constant-cochains}, and then pulled back. For a group
homomorphism $\rho:D\to A$, naturality gives
\[
d^*u_{B_C,D,\mathrm b}(\rho^*e)
=(B_C\rho\circ d)^*u_{B_C,A,\mathrm b}(e).
\]
We abbreviate the left side as $d^*\rho^*e$. The associated framing
torsors on $Y_C$ give exactly the convention for $e_{P_n}|_{Y_C}$
fixed in \cref{sec:completed-tower}, by the same finite-level naturality.
\begin{proposition}\label{prop:primitive-comparison}
For $n=2,3$, the pullbacks of $e_{P_n}$ and $e_{J_n}$ to
$H^3_{\mathrm b}(\mathcal M_n)$ are nonzero.
There is a scalar $\kappa_n\in\Q_p^\times$ such that
\begin{equation}\label{eq:primitive-comparison}
e_{P_n}|_{\mathcal M_n}
=\kappa_n\,e_{J_n}|_{\mathcal M_n}.
\end{equation}
\end{proposition}
\begin{proof}
Use the common arithmetic action $G_F$ from
\cref{prop:two-realizations}.
The projective presentation is a projective bundle over $B_CP_n$.
Apply \cref{lem:projective-bundle} first to this relative geometric
base. Its two degree-three summands use the finite coefficients
$\Lambda_m$ and $\Lambda_m(-1)$ on $B_CP_n$, respectively, before
their derived limits; higher projective summands would have negative
cohomological degree and vanish. The compact geometric-point comparison
\eqref{eq:geometric-point-constants}, applied in the $P_n$ bar degrees
and with its Tate fiber, identifies these summands with the group
cohomology groups in the $G_F$-equivariant direct sum
\begin{equation}\label{eq:projective-degree-three}
H^3_{\mathrm b}(\mathcal M_n)
\cong H^3_{\mathrm b}(P_n)
\oplus H^1_{\mathrm b}(P_n)(-1).
\end{equation}
The first summand is the image of the actual constant-cochain pullback,
hence contains a nonzero $e_{P_n}$. The comparison is natural for the
Galois action on the $\Spd C$ base. It identifies the untwisted group
cochain factors with trivial Galois action because Galois fixes $P_n$
and the untwisted constants.
On the second summand it acts by $\chi_{\mathrm{cyc}}^{-1}$.
As $\chi_{\mathrm{cyc}}(G_F)$ is open, this summand has no invariants.
Thus
\begin{equation}\label{eq:invariant-line}
H^3_{\mathrm b}(\mathcal M_n)^{G_F}
=\Q_p\,e_{P_n}|_{\mathcal M_n}.
\end{equation}
It remains to prove that the pullback of $e_{J_n}$ is nonzero.
Choose a compact open uniform subgroup $U\subset J_n$.
Restriction sends $e_{J_n}$ to a nonzero element of
$H^3_{\mathrm b}(U)$ by \cref{prop:group-classes}.
We will detect it after the further pullback to $[\Omega_C^{n-1}/U]$.
Let $K_n=\mathscr C_{\Omega_C^{n-1}}$ be the derived solid integral
cohomology complex formed with profinite parameters.
Descent, as in \cref{lem:solid-resolutions}, gives the first-quadrant
spectral sequence
\begin{equation}\label{eq:drinfeld-descent}
E_2^{s,j}=H^s(U,H^j(K_n))(*)
\ \Longrightarrow\
H_{\mathrm{int}}^{s+j}([\Omega_C^{n-1}/U]).
\end{equation}
The complex $K_n$ is bounded below, and the group degree has the finite
bound supplied by the completed resolution. The sequence therefore
has a finite filtration in every degree and may be rationalized
exactly. The additional geometric row bound uses only point values:
\cref{thm:drinfeld} gives $H^j(K_n)(*)=0$ for $j>n-1$;
the finite Hom calculation below then makes every ordinary
point-valued $E_2^{s,j}$ in that row zero. No vanishing assertion for
the entire condensed module $H^j(K_n)$ is needed.
Choose the finite projective resolution
$P_\bullet\to\Z_p$ over $\Lambda=\Z_p[[U]]$ from
\cref{lem:solid-resolutions}. The unit
$\underline{\Z_p}\to K_n$ factors through $H^0(K_n)[0]$, since
$K_n$ is concentrated in cohomological degrees at least zero.
The degree-zero calculation and local constants argument following
\cref{thm:drinfeld} show that this unit is an isomorphism on point
values. Each $\Hom_\Lambda(P_i,-)$ is a retract of a finite sum of
point values, and the equivariant unit commutes with the idempotent
defining that retract. It follows that the unit gives an isomorphism
of the integral bottom-row computing complexes
\[
\Hom_\Lambda(P_\bullet,\underline{\Z_p})
\xrightarrow{\ \sim\ }
\Hom_\Lambda(P_\bullet,H^0(K_n)).
\]
After $[1/p]$, \eqref{eq:group-cochain-bridge} and
\eqref{eq:geometric-group-descent} identify the restricted group
class with the constants bottom-row edge class induced by
$u_{\Omega_C^{n-1},U,\mathrm b}$. This uses only the point-value
isomorphism, not an identification of the entire condensed module.
For a fixed $j$, the rationalized row is
computed by
\[
\Hom_\Lambda(P_\bullet,H^j(K_n))[1/p].
\]
Every term is a direct summand of a finite sum of
$H^j(K_n)(*)[1/p]$. By the integral Drinfeld theorem,
\cite[Theorem 1.1]{CDN2021}, a Galois element $\gamma$ acts on the latter
by the scalar $\chi_{\mathrm{cyc}}(\gamma)^{-j}$.
It commutes with $\Lambda$ and with the idempotents defining the
projective summands, so it has the same scalar action on the entire
row and on its cohomology. No identification of a solid module by
its point value is involved.
The constants row $j=0$ has trivial Galois action.
Choose $\gamma\in G_F$ with $\chi_{\mathrm{cyc}}(\gamma)$ of infinite
order. The only possible incoming differentials to bidegree $(3,0)$
are
\[
d_2:E_2^{1,1}\longrightarrow E_2^{3,0},
\qquad
d_3:E_3^{0,2}\longrightarrow E_3^{3,0}.
\]
The second source is absent when $n=2$.
After inversion of $p$, the respective source scalars are
$\chi_{\mathrm{cyc}}(\gamma)^{-1}$ and
$\chi_{\mathrm{cyc}}(\gamma)^{-2}$, whereas the target scalar is $1$.
Every differential commutes with $\gamma$, so both are zero.
There are no outgoing differentials from the bottom row.
Hence the nonzero bottom-row image of the restriction of $e_{J_n}$
survives in $H^3_{\mathrm b}([\Omega_C^{n-1}/U])$.
Its abutment image is
$u_{\Omega_C^{n-1},U,\mathrm b}(\operatorname{res}^{J_n}_U e_{J_n})$
by the unit calculation
above. Naturality \eqref{eq:constant-cochain-naturality} for
$U\hookrightarrow J_n$ identifies it with the pullback of
$u_{\Omega_C^{n-1},J_n,\mathrm b}(e_{J_n})$.
Thus the class on $[\Omega_C^{n-1}/J_n]=\mathcal M_n$ is nonzero.
Finally, that class is $G_F$-invariant. Its classifying map to
$B_CJ_n$ is equivariant for the action on the $\Spd C$ base, while
$G_F$ fixes $J_n$ and the untwisted finite coefficients. Naturality
of $u_{B_C,J_n,m}$ for this base action passes through the derived
coefficient limit and rationalization. It therefore sends the
invariant group class to an invariant geometric class.
Equation~\eqref{eq:invariant-line} therefore places both nonzero
classes on the same one-dimensional $\Q_p$-line, proving
\cref{eq:primitive-comparison}.
\end{proof}
\subsection{Transport along the rank-one filtration}
\begin{theorem}\label{thm:geometric-transport}
There is a scalar $a\in\Q_p^\times$ such that the classes from the two
stabilizer factors satisfy
\begin{equation}\label{eq:geometric-transport}
e_{P_3}|_{Y_C}=a\,e_{P_2}|_{Y_C}
\quad\text{in }H^3_{\mathrm b}(Y_C).
\end{equation}
\end{theorem}
\begin{proof}
By \cref{lem:parabolic}, for the inclusion $\rho_3:D\to J_3$ and
projection $\rho_2:D\to J_2$ there is a scalar $b\in\Q_p^\times$
with
\[
\rho_3^*e_{J_3}=b\,\rho_2^*e_{J_2}
\quad\text{in }H^3_{\mathrm b}(D).
\]
Apply $u_{B_C,D,\mathrm b}$ to this group-cohomology equality and
pull it back along $d:\mathcal T\to B_CD$.
The exact filtration in \cref{prop:filtered-modification} identifies
the two $J_n$-classifying maps with the composites induced by
$D\to J_n$, and \eqref{eq:constant-cochain-naturality} identifies
their pullbacks with these same group classes. The $P_n$-framing
torsors in that proposition give the first and last equalities below.
Consequently
\[
\begin{aligned}
\pi^*(e_{P_3}|_{Y_C})
&=m_3^*(e_{P_3}|_{\mathcal M_3})\\
&=\kappa_3\,d^*\rho_3^*e_{J_3}\\
&=\kappa_3b\,d^*\rho_2^*e_{J_2}\\
&=\frac{\kappa_3b}{\kappa_2}\,
m_2^*(e_{P_2}|_{\mathcal M_2})\\
&=\frac{\kappa_3b}{\kappa_2}\,\pi^*(e_{P_2}|_{Y_C}).
\end{aligned}
\]
All three scalars are nonzero. The projective morphism
$\pi:\mathcal T\to Y_C$ satisfies
\cref{lem:projective-bundle}, so $\pi^*$ is injective.
Taking $a=\kappa_3b/\kappa_2$ proves the theorem.
\end{proof}
This transports the primitive through a filtration of rational local
systems. The equality itself does not assert nonvanishing on $Y_C$.
The next section tests its coefficient image using the completed
tower and proves the required nonvanishing there.
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//05-degree-three.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//06-witt-transfer.tex ====
\section{Transfer through Witt coefficients}
\label{sec:witt-transfer}
Witt cohomology supplies an injective base-change map with which to
reflect the relation of \cref{thm:geometric-transport} from $Y_C$ to
$Y$. We first prove this injectivity by reducing to trivial Witt
coefficients for a compact group. We then carry the relation to
deformation coefficients and prove nonvanishing there using the
rank-two theory and the retraction onto its Witt constants.
Recall the notation
\[
G=P_3\times P_2,\qquad
X=[\Spd\widehat L/P_3^1],\qquad
H=P_2\times\overline Z,
\quad \overline Z=P_3/P_3^1.
\]
Thus $Y=X/H$, with the residual action supplied by quotient
descent. For any of these stacks $Z$, write
\[
R\Gamma_v(Z,W(\cO^\flat))
:=\Rlim_m R\Gamma_v(Z,W_m(\cO^\flat)),
\qquad
H_W^i(Z):=H^i\bigl((R\Gamma_v(Z,W(\cO^\flat)))(*)\bigr).
\]
The derived global sections are solid complexes; \((*)\) is the exact,
limit-preserving point evaluation, so \(H_W^i(Z)\) is an ordinary abelian
group. This is an integral definition. Inverting $p$ below always takes
place after this derived limit and after cohomology.
\subsection{Witt constants and base change}
In \cite[Remark~3.6.12(2)]{BMR2026}, Barthel and coauthors propose
studying mixed-characteristic Witt cohomology on the unquotiented
punctured Banach--Colmez space. Here the finite norm-unit quotient
$P_3^1/\ker N_3\simeq\mu_{p-1}$ has already removed the higher
characteristic-$p$ contribution in \cref{prop:constant-acyclicity}.
That constants equivalence lifts through the finite Witt levels,
giving the following more restricted comparison.
\begin{proposition}[Witt comparison]\label{prop:witt-comparison}
The natural constant maps give the commutative square of integral
cohomology groups
\begin{equation}\label{eq:witt-square}
\begin{tikzcd}[column sep=large,row sep=large]
H^i(H,W(k)) \arrow[r,"\sim"] \arrow[d]
& H_W^i(Y) \arrow[d]
\\
H^i(H,W(C^\flat)) \arrow[r,"\sim"]
& H_W^i(Y_C).
\end{tikzcd}
\end{equation}
The actions of $H$ on the two left-hand coefficient rings are trivial.
The right vertical map is injective after inverting $p$, for every $i$.
Moreover, quotient descent gives a natural identification
\begin{equation}\label{eq:witt-group-identification}
R\Gamma_v(Y,W(\cO^\flat))
\simeq R\Gamma(G,W(\widehat L)).
\end{equation}
\end{proposition}
\begin{proof}
For a characteristic-$p$ ring $R$, restriction of Witt components and
Verschiebung give an exact sequence of additive groups
\begin{equation}\label{eq:witt-devissage}
0\longrightarrow R\xrightarrow{\ V^{m-1}\ }W_m(R)
\longrightarrow W_{m-1}(R)\longrightarrow0
\qquad(m\geq2).
\end{equation}
The sequence is natural for ring maps and automorphisms. We use it as
a sequence of additive coefficient objects; no assertion of linearity
over $R$ is needed. It also gives an exact sequence of the associated
solid modules and of the corresponding sheaves on the v-site.
\Cref{prop:constant-acyclicity} identifies the actual constant maps
\[
k\longrightarrow R\Gamma(P_3^1,\widehat L),\qquad
C^\flat\longrightarrow R\Gamma_v(X_C,\cO^\flat)
\]
as equivalences with their residual $H$-actions. As proved there,
the characteristic-$p$ vanishing uses only the central scalar
restrictions of \cite[Propositions~3.5.6 and~3.6.9]{BMR2026}, identified
with $T_3$ by the determinant calculation in \cref{prop:tower}, and finite
norm-unit
invariants as in \cite[Lemma~5.2.5]{BMR2026}. Full $H$-equivariance
comes from the actual maps of constants and quotient descent.
Apply \eqref{eq:witt-devissage} to the source and target of each map.
Induction on $m$, using the resulting morphisms of exact triangles,
gives equivalences
\begin{align*}
W_m(k)&\xrightarrow{\sim}
R\Gamma(P_3^1,W_m(\widehat L)),\\
W_m(C^\flat)&\xrightarrow{\sim}
R\Gamma_v(X_C,W_m(\cO^\flat)).
\end{align*}
Every map used in this induction is the map induced by constants.
In particular the equivalences are $H$-equivariant, including the
$\overline Z$-action; $H$ fixes the constant rings.
Derived global sections and derived invariants preserve limits.
Passing to $\Rlim_m$ therefore gives the same equivalences with full
Witt coefficients. Here the ordinary Witt module is also the derived
limit of its truncations: in Witt coordinates, restriction has the
continuous set-theoretic section obtained by appending a zero.
Consequently restriction is surjective on continuous functions from
every profinite parameter space. The truncation towers have no higher
derived-limit term in the solid coefficient category. The argument
applies to $k$, $C^\flat$, and $\widehat L$ with their respective
topologies. Taking residual $H$-invariants gives the horizontal
isomorphisms of \eqref{eq:witt-square}. Naturality of the constant
maps under $k\to C^\flat$ gives its commutativity.
For \eqref{eq:witt-group-identification}, recall
$\widetilde Y=\Spd\widehat L$ from \cref{sec:completed-tower}.
This affinoid perfectoid space is acyclic for $\cO^\flat$, also with
profinite parameters. Applying \eqref{eq:witt-devissage} gives, for
every profinite set $S$, the finite equivalence induced by sections
\[
\alpha_{S,m}:C_{\cts}(S,W_m(\widehat L))[0]
\xrightarrow{\ \sim\ }
R\Gamma_v(\widetilde Y\times\underline S,W_m(\cO^\flat)).
\]
Here Witt coordinates identify $W_m(C_{\cts}(S,\widehat L))$ with
$C_{\cts}(S,W_m(\widehat L))$. The equivalence is natural in $S$
and in the $G$-action. Write $u^a_{\widetilde Y,G,m}$ for the
degree-$a$ map of \eqref{eq:finite-constant-cochains}. The actual
finite coefficient units give the commutative square
\begin{equation}\label{eq:finite-witt-constant-square}
\begin{tikzcd}[column sep=large]
C_{\cts}(G^a,\Lambda_m)[0]
\arrow[r,"u^a_{\widetilde Y,G,m}"] \arrow[d,"\iota_m"']
&R\Gamma_v(\widetilde Y\times\underline{G^a},\Lambda_m)
\arrow[d,"\omega_m"]\\
C_{\cts}(G^a,W_m(\widehat L))[0]
\arrow[r,"\alpha_{G^a,m}","\sim"']
&R\Gamma_v(\widetilde Y\times\underline{G^a},W_m(\cO^\flat)).
\end{tikzcd}
\end{equation}
The left map applies $\Lambda_m=W_m(\F_p)\to W_m(\widehat L)$
pointwise; the right map is induced by the corresponding sheaf unit.
Both composites are the same locally constant Witt section. These
squares commute with the bar faces, including the action face, and
with restriction in $m$. The same calculation with an extra
profinite parameter retains the solid structure. The condensed
standard resolution, quotient descent, and then $\Rlim_m$ therefore
give \eqref{eq:witt-group-identification}. The full Witt module is
the derived limit of its truncations as shown above. Under point
evaluation of this identification, the Witt image of
$u_{\widetilde Y,G,\mathrm{int}}$ is exactly the coefficient map
induced by $\Z_p\to W(\widehat L)$ and
\eqref{eq:group-cochain-bridge}.
It remains to prove injectivity, since an equivalence of the horizontal
groups alone would not imply it. The finite-resolution statement of
\cref{lem:solid-resolutions} applies to $H$. Indeed an element of
order $p$ in $P_2$ would have minimal polynomial of degree $p-1>2$
over $\Q_p$, whereas the reduced characteristic polynomial in $D_2$
has degree two. Thus $P_2$, and hence $H=P_2\times\Z_p$, has no
$p$-torsion.
Choose a bounded resolution $P_\bullet\to\Z_p$ by finitely generated
projective $\Z_p[[H]]$-modules. For a trivial solid coefficient module
$M$, the point-valued cochain complex is
\[
\Hom_{\Z_p[[H]]}(P_\bullet,M)
=\Hom_{\Z_p}(Q_\bullet,M),
\qquad
Q_\bullet=\Z_p\otimes_{\Z_p[[H]]}P_\bullet.
\]
Every $Q_j$ is finite free over $\Z_p$. Thus this complex is obtained
from a fixed bounded finite free $\Z_p$-complex by scalar extension
to the underlying module of $M$; its differentials are the same
$\Z_p$-linear matrices. This description is natural in $M$ and is
valid for the valued Witt topology as well as for the $p$-adic one,
by \cref{lem:solid-resolutions}.
Take $A=W(k)$ and $B=W(C^\flat)$. The map $A\to B$ is injective.
Since $A$ is a discrete valuation ring with uniformizer $p$ and $B$
is $p$-torsion-free, $B$ is flat over $A$. Applying the finite
complex just described gives
\begin{equation}\label{eq:witt-scalar-extension}
H^i(H,A)[1/p]\otimes_{A[1/p]}B[1/p]
\xrightarrow{\sim}H^i(H,B)[1/p].
\end{equation}
The field extension $A[1/p]\hookrightarrow B[1/p]$ is faithfully
flat, so the natural map from the first cohomology group to its
scalar extension is injective. The square now proves the assertion.
All infinite limits preceded this finite-complex argument.
\end{proof}
\subsection{Reflecting the geometric relation}
Via \eqref{eq:group-cochain-bridge}, let $c_n^W$ denote the image of
$e_{P_n}$ under projection from $G$ and the coefficient map
$\Z_p\to W(\widehat L)$:
\[
c_n^W\in H^3(G,W(\widehat L))[1/p]\qquad(n=2,3).
\]
Write $c_n$ for its further image under the natural equivariant map
$W(\widehat L)\to\pi_0\widehat B^{\square}$ of
\cref{prop:completed-theory}.
\begin{proposition}[Transport to deformation coefficients]
\label{prop:coefficient-transport}
For the nonzero scalar $a\in\Q_p^\times$ of
\cref{thm:geometric-transport}, one has
\begin{equation}\label{eq:witt-relation}
c_3^W=a\,c_2^W
\quad\hbox{in }H^3(G,W(\widehat L))[1/p],
\end{equation}
and hence
\begin{equation}\label{eq:deformation-relation}
c_3=a\,c_2
\quad\hbox{in }H^3(G,\pi_0\widehat B^{\square})[1/p].
\end{equation}
Under \eqref{eq:coefficient-descent}, the preimage of $c_2$ in
\[
H^3(H,\pi_0E_2^{\square})[1/p]
\]
is obtained by inflating $e_{P_2}$ and then including constants.
\end{proposition}
\begin{proof}
Let $\omega_{Z,m}$ be the map on cohomology complexes induced by
$\Lambda_m=W_m(\F_p)\to W_m(\cO^\flat)$ for $Z=Y$ or $Y_C$.
These maps are compatible with restriction in $m$. Their derived
inverse limit gives a map
\[
H^3_{\mathrm b}(Z)\longrightarrow H_W^3(Z)[1/p].
\]
Let $j:Y_C\to Y$ denote base change, and put
\[
\beta_m=\omega_{Y,m}u_{\widetilde Y,G,m},\qquad
\beta_{C,m}=\omega_{Y_C,m}u_{\widetilde Y_C,G,m}.
\]
The finite constant maps and the finite Witt units give
\begin{equation}\label{eq:finite-witt-base-change}
\begin{tikzcd}[column sep=large]
C_{\cts}^{\bullet}(G,\Lambda_m)
\arrow[r,"\beta_m"] \arrow[d,equal]
&R\Gamma_v(Y,W_m(\cO^\flat))\arrow[d,"j^*"]\\
C_{\cts}^{\bullet}(G,\Lambda_m)
\arrow[r,"\beta_{C,m}"']
&R\Gamma_v(Y_C,W_m(\cO^\flat)).
\end{tikzcd}
\end{equation}
This square commutes in each cosimplicial degree before totalization,
and is compatible with $q_n:G\to P_n$ and reduction in $m$.
Take the derived coefficient limit, cohomology, and then invert $p$.
The classifying-pullback convention and
\eqref{eq:constant-cochain-naturality} give
$j^*(e_{P_n}|_Y)=e_{P_n}|_{Y_C}$. Moreover,
\eqref{eq:finite-witt-constant-square} identifies the Witt image of
$e_{P_n}|_Y$ with $c_n^W$ under
\eqref{eq:witt-group-identification}. Thus the geometric relation says
that $c_3^W-a c_2^W$ maps to zero in $H_W^3(Y_C)[1/p]$.
The right vertical map of \eqref{eq:witt-square} is injective after
inverting $p$, so this difference is already zero on $Y$. This proves
\eqref{eq:witt-relation}. Applying the coefficient map
$W(\widehat L)\to\pi_0\widehat B^{\square}$ then proves
\eqref{eq:deformation-relation}.
Finally, \eqref{eq:coefficient-descent} is induced by the actual map
$E_2^{\square}\to\widehat B^{\square}$, by first taking
$P_3^1$-invariants and then $H$-invariants. Its compatibility with
constants is part of \cref{prop:completed-theory}, whose finite Koszul
argument proves the required coefficient instance of the descent
construction in \cite[Proposition~4.6.12]{BMR2026}.
The action of $\overline Z$ on the source is trivial. Thus
inflation from $P_2$ followed by this map is exactly the coefficient
map defining $c_2$. This proves the last assertion while retaining
the residual action.
\end{proof}
\subsection{Nonvanishing in the deformation ring}
The reflected relation reduces nonvanishing to $c_2$. Coefficient
descent identifies this class with constants in the rank-two theory;
the additive retraction onto Witt constants detects it there.
\begin{lemma}[Nonvanishing of constants]\label{lem:constant-nonvanishing}
The class $c_2$ is nonzero. Consequently $c_3$ is nonzero as well.
\end{lemma}
\begin{proof}
Set $A_k=\pi_0E_2^{\square}(*)$. The class corresponding to $c_2$
under \eqref{eq:coefficient-descent} restricts along $P_2\hookrightarrow H$
to the coefficient image of $e_{P_2}$ in $H^3(P_2,A_k)[1/p]$.
It suffices to prove that this image is nonzero.
Choose an algebraic closure $\overline k$ of $k$ and let
$A_{\overline k}$ be the deformation ring of the same height-two
formal group after this base extension. Deformation theory gives
a natural continuous $P_2$-equivariant map
\[
A_k\longrightarrow A_{\overline k}.
\]
In compatible coordinates these rings are $W(k)[[u'_1]]$ and
$W(\overline k)[[u'_1]]$, with their $(p,u'_1)$-adic topologies;
equivariance follows from deformation base change, without requiring
the coordinate to be fixed. Since $P_2$ is nonextended, it fixes
$W(\overline k)$ pointwise.
The splitting theorem of Barthel--Schlank--Stapleton--Weinstein
\cite[Proposition~2.5.1 and Corollary~2.5.9]{BSSW2025} gives a
continuous additive equivariant retraction
\[
r:A_{\overline k}\longrightarrow W(\overline k)
\]
of the inclusion of Witt constants. Restrict its equivariance to $P_2$.
Additivity and continuity make it $\Z_p$-linear. After condensation
it is a morphism of solid $P_2$-modules, and hence
induces a retraction on the continuous derived invariants used here.
The composite of the coefficient maps
\[
\Z_p\longrightarrow A_k\longrightarrow A_{\overline k}
\xrightarrow{r}W(\overline k)
\]
is the usual inclusion of constants. Applying the finite free
complex argument from \cref{prop:witt-comparison} with $P_2$ in
place of $H$ gives
\[
H^3(P_2,W(\overline k))[1/p]
\cong H^3_{\mathrm b}(P_2)
\otimes_{\Q_p}W(\overline k)[1/p].
\]
The image of $e_{P_2}$ here is $e_{P_2}\otimes1$, which is nonzero
by \cref{prop:group-classes}. If its image in
$H^3(P_2,A_k)[1/p]$ vanished, the displayed composite would also
send it to zero, a contradiction. Restriction to $P_2$ therefore
proves $c_2\ne0$, and \eqref{eq:deformation-relation} with $a\ne0$
proves $c_3\ne0$.
\end{proof}
We have thus identified the image of the degree-three group class
under the completed coefficient map. The remaining step is to
detect it in the homotopy of an ordinary $K(2)$-local spectrum.
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//06-witt-transfer.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//07-detection.tex ====
\section{Detection on ordinary spectra}
\label{sec:detection}
We now turn the coefficient relation into a statement about the natural
localization map. Keep $G=P_3\times P_2$ and
$H=P_2\times\overline Z$, and put
\[
\Delta=\Gal(k/\F_p),\qquad
G_3=P_3\rtimes\Delta,\qquad
\mathcal D=\bigl((\widehat B^{\square})^{hG}\bigr)(*).
\]
Thus $\Delta$ has order six. The symbol $\mathcal D$ denotes an ordinary spectrum:
the continuous fixed points are formed in solid spectra and then evaluated
at the point.
\begin{lemma}[The ordinary local detector]\label{lem:local-detector}
There is a natural map of ordinary spectra
\begin{equation}\label{eq:ordinary-detector}
g:L_{K(3)}S\longrightarrow\mathcal D.
\end{equation}
The spectrum $\mathcal D$ is $K(2)$-local, so $g$ factors through the canonical
localization unit:
\begin{equation}\label{eq:detector-factorization}
L_{K(3)}S\xrightarrow{\eta}L_{K(2)}L_{K(3)}S
\xrightarrow{\overline g}\mathcal D.
\end{equation}
\end{lemma}
\begin{proof}
Use the ordinary descent identification in
\cref{prop:completed-theory} to form the composite
\begin{equation}\label{eq:detector-construction}
\begin{split}
L_{K(3)}S
&\simeq\bigl((E_3^{\square})^{hG_3}\bigr)(*)
\longrightarrow\bigl((E_3^{\square})^{hP_3}\bigr)(*)\\
&\longrightarrow\bigl((E_3^{\square})^{hG}\bigr)(*)
\longrightarrow\bigl((\widehat B^{\square})^{hG}\bigr)(*).
\end{split}
\end{equation}
The first arrow restricts the group action, the second is inflation along
$G\to P_3$, and the last is induced by the equivariant map
\eqref{eq:completed-map}. In the middle term with group $G$, the action on
$E_3^{\square}$ is through $P_3$. Restriction and inflation are the
natural maps of continuous fixed points. The source equivalence and its ordinary unit are the point-valued
form of \cite[Proposition~4.6.9]{BMR2026} proved in
\cref{prop:completed-theory}; the last map uses the local split-section
completed map of \cref{lem:completed-oriented-map} and that proposition.
For every profinite set $T$, the evaluation
$\widehat B^{\square}(T)$ is the ordinary height-two Lubin--Tate theory
described in \cref{prop:completed-theory}, with the Noetherian witness
verified there; hence it is $K(2)$-local
\cite[Proposition~4.4.2]{BMR2026}.
The bar construction gives
\[
\mathcal D\simeq\operatorname{Tot}\bigl(\widehat B^{\square}(G^{\bullet})\bigr).
\]
Ordinary local spectra are closed under limits, so $\mathcal D$ is
$K(2)$-local. This also follows from the
pointwise criterion for solid locality
\cite[Definition~4.5.7 and the following paragraph]{BMR2026}.
The universal property of ordinary $K(2)$-localization now gives
\eqref{eq:detector-factorization}.
\end{proof}
\subsection{Convergence before rationalization}
We use integral continuous cohomology of the solid homotopy modules in
\cref{lem:solid-resolutions}. For such a module $M$, recall that
$H^s(A,M)$ denotes the point evaluation of $R^s\Gamma(A,M)$.
In particular, $[1/p]$ below is applied to this integral group after it
has been computed. For $M=\Z_p$, the natural comparison
\eqref{eq:group-cochain-bridge} identifies these groups with the
finite continuous-cochain derived-limit convention of
\eqref{eq:integral-group-cochains}.
The compact groups needed here have finite cohomological dimension in
this coefficient category. Indeed, if the degree-$n$ division algebra
defining $P_n$ contained an element of order $p$, it would contain the
field $\Q_p(\zeta_p)$: the nontrivial element has the irreducible
cyclotomic minimal polynomial of degree $p-1$. Its reduced
characteristic polynomial has degree $n$ and annihilates it.
Since $p-1>n$ for $n=2,3$ and $p\geq5$, this is impossible.
Thus neither $P_2$ nor $P_3$ has $p$-torsion. The same holds for
$G$ and $H$. An order-$p$ element of $G_3$ would project trivially to
$\Delta$, since $p\nmid6$, and hence would lie in $P_3$. Thus
\cref{lem:solid-resolutions} gives a uniform finite cohomological bound
for each of these groups. We only need the existence of a common bound
$d$, independent of the homotopy degree of the coefficient module.
\begin{lemma}[Finite-amplitude descent]\label{lem:descent-convergence}
For the continuous actions used in \eqref{eq:detector-construction},
there are natural, strongly convergent spectral sequences
\begin{equation}\label{eq:integral-descent-ss}
E_2^{s,t}=H^s(A,\pi_tX)
\ \Longrightarrow\ \pi_{t-s}\bigl((X^{hA})(*)\bigr),
\qquad 0\leq s\leq d.
\end{equation}
They give, after exact rationalization, natural strongly convergent
spectral sequences with $E_2$-terms $H^s(A,\pi_tX)[1/p]$ and abutments
$\pi_{t-s}((X^{hA})(*))[1/p]$.
This statement applies to the unbounded even-periodic theories here.
\end{lemma}
\begin{proof}
Let $F(X)=(X^{hA})(*)$. This is an exact functor preserving limits and
upper homotopy bounds. For the latter assertion, coconnective objects
remain coconnective under evaluation and limits. Although $X$ need not
be a module spectrum over $H\Z_p$, each Postnikov layer $HM$ is one.
The forgetful functor from module spectra preserves limits, so its
continuous fixed points agree with the derived module invariants.
On this Eilenberg--Mac Lane object, the finite
resolution in \cref{lem:solid-resolutions} gives
\[
\pi_{-s}F(HM)=H^s(A,M),\qquad
F(HM)\in\mathrm{Sp}_{[-d,0]}.
\]
The solid spectra are used with the hypersheaf convention of
\cite[Section~4.5, especially Proposition~4.5.5]{BMR2026}.
Truncations remain solid, since they retain or kill each solid homotopy
module. Evaluation on an extremally disconnected profinite set is
$t$-exact: homotopy sheafification evaluated on this projective object
agrees with pointwise homotopy. It therefore commutes with truncation.
These evaluations detect equivalences, so Postnikov completeness of
ordinary spectra gives
$X\simeq\varprojlim_n\tau_{\leq n}X$.
Put $T_n=F(\tau_{\leq n}X)$. Exactness identifies the successive fiber as
\[
\operatorname{fib}(T_{n+1}\longrightarrow T_n)
\simeq F\bigl(\Sigma^{n+1}H\pi_{n+1}X\bigr)
\in\mathrm{Sp}_{[n+1-d,n+1]}.
\]
Consequently $\pi_qT_{n+1}\to\pi_qT_n$ is an isomorphism for
$n\geq q+d$; the analogous tower in degree $q+1$ is also eventually
constant. The Milnor exact sequence therefore has zero derived-limit
term, and gives
\begin{equation}\label{eq:finite-postnikov-window}
\pi_qF(X)\cong\pi_qF(\tau_{\leq q+d}X).
\end{equation}
The infinitely negative tail causes no further issue: since
$F(\tau_{\leq q-1}X)$ is concentrated in degrees at most $q-1$, the
fiber sequence for this lower cutoff gives
\[
\pi_qF(\tau_{[q,q+d]}X)
\xrightarrow{\ \sim\ }\pi_qF(\tau_{\leq q+d}X).
\]
Hence each abutment degree is computed by a finite Postnikov interval.
The exact couple of this Postnikov construction has the $E_2$-term in
\eqref{eq:integral-descent-ss}. Its filtration in degree $q$ is
\[
\operatorname{Fil}^{s}\pi_qF(X)
=\ker\!\left(\pi_qF(X)\longrightarrow
\pi_qF(\tau_{\leq q+s-1}X)\right),
\quad 0\leq s\leq d+1.
\]
It starts with the whole group, ends in zero by
\eqref{eq:finite-postnikov-window}, and has associated graded
$E_\infty^{s,q+s}$. This proves the asserted strong convergence and its
naturality. Tensoring the exact couple and these finite filtrations with
$\Q$ is exact. It therefore gives the rational spectral sequence and
its stated abutment; no exchange of rationalization with an infinite
totalization is involved. The groups here are $\Z_p$-modules, so this
tensor operation agrees with $[1/p]$.
No uniform torsion exponent as $t$ varies is required: each fixed
homotopy degree has the finite filtration just constructed.
\end{proof}
\subsection{The surviving coefficient row}
\begin{lemma}[Central weights]\label{lem:central-weights}
For $t\ne0$ and every $s$, one has
\[
H^s(G_3,\pi_tE_3^{\square})[1/p]=0,
\qquad
H^s(G,\pi_t\widehat B^{\square})[1/p]=0.
\]
\end{lemma}
\begin{proof}
Odd homotopy modules vanish. For $t=2m$, scalar automorphisms
$a\in1+p\Z_p$ lift canonically to scalar automorphisms of the universal
deformation. They fix its deformation ring, and act by $a^m$ on its
$m$th invariant-differential line $\pi_{2m}E_n^{\square}$.
This is the central character used in
\cite[Lemma~2.6.1]{BSSW2025}; replacing the periodicity convention by its
dual replaces $m$ by $-m$ and has the same consequence.
Choose $b=1+p$. It is central in $G_3$, since the finite Galois action
fixes these scalars. A central group element acting on a coefficient
module induces the identity on its group cohomology. This follows
already from the bar resolution: its coefficient action is chain
homotopic to the action by conjugation, which is the identity for a
central element. Consequently $(b^m-1)$ annihilates
$H^s(G_3,\pi_{2m}E_3^{\square})$. For $m\ne0$ this is a nonzero
$p$-adic scalar, and is invertible after $p$ is inverted. This proves
the first vanishing.
For the target, use the natural coefficient descent
\eqref{eq:coefficient-descent} to identify its cohomology with
$H^s(H,\pi_tE_2^{\square})$. The factor $\overline Z$ acts trivially
on these coefficients, and the central scalars in $P_2$ act by the same
character $b^m$. The preceding argument applies to $H$, and proves
the second vanishing.
\end{proof}
\begin{proposition}\label{prop:nonzero-detector}
The ordinary map $g$ of \eqref{eq:ordinary-detector} is nonzero on
rational homotopy in degree $-3$:
\[
\pi_{-3}L_0g:\pi_{-3}L_0L_{K(3)}S
\longrightarrow\pi_{-3}L_0\mathcal D
\quad\text{is nonzero}.
\]
\end{proposition}
\begin{proof}
First extend the source primitive to the extended stabilizer.
By \cref{prop:group-classes}, $e_{P_3}$ is represented under rational
Lie comparison by a nonzero multiple of
\[
(X,Y,Z)\longmapsto\tr_{\mathrm{red}}\bigl(X[Y,Z]\bigr).
\]
In the standard presentation of the degree-three division algebra,
take its maximal unramified subfield $\Q_{p^3}$ and an element $\Pi$
such that
\[
\Pi x=\sigma(x)\Pi\quad(x\in\Q_{p^3}),\qquad \Pi^3=p,
\]
where $\sigma$ is arithmetic Frobenius of the unramified cubic
extension $\Q_{p^3}/\Q_p$, inducing $z\mapsto z^p$ on the residue
field. Its extension fixing $\Pi$ equals conjugation
$y\mapsto\Pi y\Pi^{-1}$, as is checked on the generators
$\Q_{p^3}$ and $\Pi$. It therefore preserves the reduced-trace form.
Its action on $P_3$ factors through the order-three quotient of
$\Delta$; the order-two kernel also fixes the $\Z_p$-coefficients.
Naturality of Lie comparison therefore makes $e_{P_3}$ invariant under
$\Delta$. Since $6$ is a unit in $\Z_p$, finite-group averaging and
extension descent give
\[
H^3(G_3,\Z_p)[1/p]
\xrightarrow{\ \sim\ }
\bigl(H^3(P_3,\Z_p)[1/p]\bigr)^\Delta.
\]
Let $\widetilde e_3$ be the class restricting to $e_{P_3}$, and let
$z_3$ be its image under the unit-coefficient map
$\Z_p\to\pi_0E_3^{\square}$.
Under \eqref{eq:group-cochain-bridge}, restriction from $G_3$ to $P_3$
and inflation along $G\to P_3$ on the constant classes are the
derived limits of their finite continuous-cochain pullbacks. These
are the group maps used in \eqref{eq:constant-cochain-naturality}.
All coefficient maps in the following square are the maps induced by
\eqref{eq:detector-construction}:
\[
\begin{tikzcd}[column sep=large]
H^3(G_3,\Z_p)[1/p]
\arrow[r] \arrow[d,"{\mathrm{res},\,\mathrm{inf}}"']
& H^3(G_3,\pi_0E_3^{\square})[1/p]\arrow[d]\\
H^3(G,\Z_p)[1/p]\arrow[r]
& H^3(G,\pi_0\widehat B^{\square})[1/p].
\end{tikzcd}
\]
On bar cochains, the right vertical map is restriction, pullback along
$G\to P_3$, and the coefficient map of
$E_3^{\square}\to\widehat B^{\square}$, in that order.
Its composite with the source coefficient unit is the target unit,
which is also $\Z_p\to W(\widehat L)\to\pi_0\widehat B^{\square}$.
Here the first map is the integral limit of the finite Witt units in
\eqref{eq:finite-witt-constant-square}; the map from full Witt vectors
to deformation coefficients is applied only after that limit.
The source descent identification used here is the canonical one in
\cref{prop:completed-theory}.
In particular the image of $z_3$ is exactly the class called $c_3$ in
\cref{prop:coefficient-transport}: it is obtained from $e_{P_3}$ by
projection and the actual constant-coefficient inclusion. That
proposition and \cref{lem:constant-nonvanishing} give
\[
c_3=a\,c_2\ne0,\qquad a\in\Q_p^{\times}.
\]
Thus the map between the rational descent $E_2$-terms is nonzero in
bidegree $(s,t)=(3,0)$.
Apply \cref{lem:descent-convergence} to the source and target of $g$.
By \cref{lem:central-weights}, only $t=0$ survives after rationalization.
The differential has bidegree $(r,r-1)$, so every rational differential
vanishes. For total homotopy degree $-3$, the finite filtration has
just one possibly nonzero graded piece, namely $(s,t)=(3,0)$.
The nonzero map on this piece is therefore a nonzero map on the
abutments. These abutments are the ordinary rational homotopy groups
asserted in the proposition.
\end{proof}
\begin{proof}[Proof of \cref{thm:main}]
Rationalize \eqref{eq:detector-factorization}. If the canonical map
$L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ were zero on $\pi_{-3}$,
its composite $L_0g$ would also be zero there. This contradicts
\cref{prop:nonzero-detector}. Every step above applies to each prime
$p\geq5$, proving the theorem.
\end{proof}
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//07-detection.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//08-elementary-background.tex ====
\appendix
\section{The low-rank cohomology and the building}
\label{app:background}
We give the two elementary ingredients in the proof of
\Cref{prop:group-classes}. The calculations are over characteristic
zero; no reduction modulo $p$ of the matrices below is used.
The degree-three trace class is the classical cocycle associated to an
invariant symmetric bilinear form
\cite[\S21]{ChevalleyEilenberg1948}, \cite[\S11]{Koszul1950}.
We give the low-rank calculation and block restriction in the
normalization used in this paper.
\subsection{A finite Chevalley--Eilenberg calculation}
\label{app:ce}
For $\mathfrak{sl}_n$, order the matrix units $E_{ij}$, $i\ne j$,
lexicographically, followed by $H_i=E_{ii}-E_{nn}$ for $1\le i<n$.
Order exterior bases by increasing tuples, lexicographically, starting
with index zero. The differential is determined by
\[
(d\omega)(X_0,\ldots,X_k)=
\sum_{i<j}(-1)^{i+j}
\omega([X_i,X_j],X_0,\ldots,\widehat X_i,\ldots,
\widehat X_j,\ldots,X_k).
\]
For $n=2$, the matrix of $d_1$ in these bases is
\[
\begin{pmatrix}0&0&-1\\2&0&0\\0&-2&0\end{pmatrix},
\]
with determinant $4$, and $d_2=0$. For $n=3$, the dimensions of
$C^0,\ldots,C^4$ are $1,8,28,56,70$. The following nonsingular
minors of $d_1,d_2,d_3$ provide a short exact-arithmetic certificate.
Their row and column indices refer to the bases just specified:
\begin{align*}
R_1={}&(1,2,3,5,7,9,15,16),&
C_1={}&(0,1,2,3,4,5,6,7),\\
\det(d_1[R_1,C_1])={}&-1.&&
\end{align*}
For the other two minors, use
\begin{align*}
R_2={}&(0,2,4,6,7,8,9,11,12,13,19,22,24,28,30,37,39,41,43,52),\\
C_2={}&(0,1,2,3,4,5,6,7,8,11,13,14,15,16,17,20,22,23,25,27),\\
\det(d_2[R_2,C_2])={}&2916,
\end{align*}
\begin{align*}
R_3={}&(0,1,2,3,4,5,6,7,9,10,11,12,15,16,17,18,19,\\
&\hspace{1em}20,21,22,23,25,26,27,31,35,36,37,39,40,41,45,55,56,61),\\
C_3={}&(0,1,2,3,5,6,7,8,9,10,11,12,13,14,15,16,17,18,\\
&\hspace{1em}20,21,22,24,25,28,31,32,36,37,38,40,42,43,44,45,52),\\
\det(d_3[R_3,C_3])={}&1024.
\end{align*}
These integer identities can be verified directly from the displayed
differential and the commutator formula
\[
[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}.
\]
The script \texttt{support/r-ce/ce\_certificate.py} generates these
integer matrices and evaluates the specified minors by fraction-free
elimination, using Python's standard library. To turn these rank bounds
into cohomology dimensions, we now prove that the trace cocycle is not
a boundary.
The alternating cochain $\omega(X,Y,Z)=\tr(X[Y,Z])$ is closed by
trace invariance and the Jacobi identity. Its restriction to the
$\mathfrak{sl}_2$ block satisfies
$\omega(E_{12},E_{21},E_{11}-E_{22})=2$. On this block every
three-coboundary is zero: substituting these three vectors in $d\eta$
gives
$-\eta(h,h)+\eta(-2e,f)-\eta(2f,e)=0$.
Thus the trace cochain is not a boundary, in either rank.
The first minor gives $\rank d_1=8$ for $\mathfrak{sl}_3$.
Since $d^2=0$, the second rank is at most $28-8=20$; its minor gives
equality. The kernel of $d_3$ contains this twenty-dimensional image
and the independent trace class, so its rank is at most $56-21=35$;
the last minor gives equality. Therefore, for $n=2,3$,
\[
H^1(\mathfrak{sl}_n)=H^2(\mathfrak{sl}_n)=0,
\qquad H^3(\mathfrak{sl}_n)=\Q_p\cdot[\omega].
\]
The direct sum $\mathfrak{gl}_n=\mathfrak{sl}_n\oplus\Q_p$ gives
exactly the low-degree groups used in \Cref{prop:group-classes}.
Block restriction of $\omega$ is the corresponding trace cochain.
For a central simple inner form, extend scalars to a finite splitting
field and use reduced trace. The finite-dimensional cochain complex
commutes with this faithfully flat extension, so the calculation descends.
We also need inner invariance in every degree for the cellular argument.
Cartan's identity $\mathcal L_X=d\iota_X+\iota_Xd$ shows that the Lie
algebra of inner automorphisms acts trivially on cohomology. After
splitting, the action is algebraic and the group $\GL_n$ is connected
in characteristic zero; hence its action on cohomology is trivial.
Faithfully flat descent gives the same assertion for the inner forms.
\subsection{An elementary model of the building}
\label{app:building}
Let $V=\Q_p^n$, $\cO=\Z_p$, and $|p|=p^{-1}$. The vertices of
$\mathcal B_n$ are homothety classes of $\cO$-lattices. A simplex is a
chain of distinct lattices between $L$ and $pL$, modulo homothety.
A maximal chain corresponds to a complete flag in $L/pL$, so its
realization has dimension $n-1$.
Here is a norm description, including contractibility. Every
nonarchimedean norm $\alpha$ on $V$ has an orthogonal basis.
Indeed, for a fixed maximum norm $N$, the triangle inequality gives
$\alpha\le CN$. Continuity and compactness of $N(v)=1$ give a positive
lower bound $cN\le\alpha$. Thus every positive-radius closed ball is
a lattice. Put $L=\{\alpha\le1\}$, so $pL=\{\alpha\le p^{-1}\}$.
On the compact set $L\setminus pL$ the norm is locally constant and
has finitely many values in $(p^{-1},1]$. Its balls modulo $pL$ form
a flag. Lift an adapted residue basis, choosing each lift in the
corresponding ball. For a linear combination with largest coefficient
norm one, use the largest basis weight among the unit coefficients.
Its nonzero residue in that flag quotient shows that the sum has
exactly this norm; nonunit coefficients contribute at most $p^{-1}$. Scaling
proves orthogonality.
Norms modulo positive real scaling identify with $|\mathcal B_n|$.
Explicitly, for an adapted basis $e_i$ put
$L_j=p\cO e_1+\cdots+p\cO e_j+\cO e_{j+1}+\cdots+\cO e_n$.
Barycentric coordinates $(\theta_0,\ldots,\theta_{n-1})$ on this chamber
give the split norm with weights $p^{a_i}$, where
\[
a_i=\sum_{j=i}^{n-1}\theta_j\quad(i<n),\qquad a_n=0.
\]
Conversely $\theta_0=1-a_1$ and $\theta_j=a_j-a_{j+1}$.
The periodic chain of norm balls and the ratios between its jumps recover
these data. Vanishing barycentric coordinates remove precisely the
corresponding jumps, and the endpoint identification is multiplication
of the lattice by $p$. This proves the claimed identification on faces
as well as chambers.
For two norms let $M(\alpha,\beta)$ and $m(\alpha,\beta)$ be the
maximum and minimum of $\alpha(v)/\beta(v)$ on $V\setminus0$.
They exist by compactness of $\mathbf P(V)$. The metric
$d([\alpha],[\beta])=\log M-\log m$ induces the simplicial topology.
To verify the only local issue, normalize $m(\alpha,\beta)=1$.
A bounded metric neighborhood satisfies
$\beta\le\alpha\le e^R\beta$. For $p^{-1}\le r\le1$ all its
lattices $\{\alpha\le r\}$ lie between the two fixed lattices
$\{\beta\le p^{-1}e^{-R}\}$ and $\{\beta\le1\}$.
There are finitely many such lattices; this radius interval represents
every ball homothety class. Thus the neighborhood meets finitely many
simplices. On each simplex the weight formula is continuous, and on
a finite union of closed simplices compactness gives the inverse
continuity. This proves the topology assertion.
With the same normalization define
\[
H_t(\alpha)(v)=\max\{t\alpha(v),\beta(v)\},\qquad 0\le t\le1.
\]
This is a norm, remains normalized, equals $\beta$ at $t=0$ and
$\alpha$ at $t=1$, and fixes $\beta$. In logarithmic ratios it is the
map $f\mapsto\max\{\log t+f,0\}$ for $t>0$.
Metric convergence of normalized norms is equivalent to uniform
convergence of these ratios. Near any fixed norm they are uniformly
bounded, so for sufficiently small $t$ the displayed maximum is
identically zero. This also proves continuity at $t=0$ and hence
contractibility.
Finally, the type of $[g\cO^n]$ is $v_p(\det g)$ modulo $n$.
Each chamber has every type once and $J_n$ preserves types.
Start a target chamber at its unique type-zero vertex. Scaling its
lattice by an integral power of $p$ makes its determinant valuation zero;
an adapted basis for its residue flag then carries the standard chamber
to it by an element of $J_n$. Thus $J_n$ is chamber-transitive.
A vertex stabilizer in $J_n$ actually preserves its lattice, since
$gL=aL$ implies $0=v_p(\det g)=n v_p(a)$. It is consequently a
conjugate of $\GL_n(\Z_p)$, compact open. Face stabilizers are finite
intersections of these groups and fix their vertices pointwise because
types are distinct. The closed chamber is therefore a fundamental
domain, including faces, with no inversions. These are exactly the
properties used by the finite cellular resolution in \cref{sec:cohomology}.
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//08-elementary-background.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//09-frobenius.tex ====
\section{A continuous section for the Frobenius comparison}
\label{app:frobenius}
The solid Frobenius comparison used in
\Cref{prop:constant-acyclicity} requires surjectivity on continuous
families, not only on point values. We supply the topological step for
its precise function ring. This avoids applying a linear open-mapping
theorem to Frobenius minus identity, which is not linear over the
algebraically closed coefficient field.
Let $C$ be an algebraically closed complete nonarchimedean field of
characteristic $p$, containing $k=\F_q$, and let
\[
R=H^0(\widetilde{\mathbb B}^{\,*}_C,\cO)
=\left\{\sum_{\alpha\in\Z[1/p]}c_\alpha t^\alpha:
\text{the series converges for every }0<|t|<1\right\}.
\]
Its Fr\'echet topology is given by the Gauss seminorms
$\|f\|_r=\sup_\alpha |c_\alpha|r^\alpha$, $0<r<1$.
Convergence means that these weighted coefficients form a $c_0$ family
for every such radius. Here $\phi_q$ acts on the $C$-coefficients
and fixes $t$; it is not the absolute Frobenius of $R$.
\begin{lemma}\label{lem:continuous-as-section}
The additive map $\phi_q-1:R\to R$ has a continuous section as a map
of sets. Its condensification is surjective, and its kernel, with the
induced topology, is the valued-field completion
$E=\bigl(\bigcup_{j\ge0}k((t^{1/p^j}))\bigr)^{\wedge}$.
Consequently the canonical map $E\to R^{h\phi_q}$ is an equivalence
of derived solid $\F_p$-modules.
\end{lemma}
\begin{proof}
For each coset of the open additive subgroup $C^{\circ\circ}$ in $C$,
choose a representative $a$ and a root $b_a$ of $b_a^q-b_a=a$.
Choose $a=b_a=0$ on the zero coset. Define
\[
s(a+\epsilon)=b_a-\sum_{j\ge0}\epsilon^{q^j},
\qquad |\epsilon|<1.
\]
The sum converges, and $s(x)^q-s(x)=x$.
If $|x-y|<1$, the points lie in the same coset and the first term
of the difference series dominates, so $|s(x)-s(y)|=|x-y|$.
If $|x-y|\ge1$, the equation for $s(x)-s(y)$ gives norm
$|x-y|^{1/q}$ (including norm one at the boundary).
Thus $s$ is continuous, $s(0)=0$, and
\[
|s(x)-s(y)|\le\max\{|x-y|,|x-y|^{1/q}\}.
\]
Define $S$ coefficientwise. For every $0<r<1$ this estimate gives
\[
\|S(f)-S(g)\|_r
\le\max\{\|f-g\|_r,\ \|f-g\|_{r^q}^{1/q}\}.
\]
It also gives the corresponding estimate on every coefficient tail.
Since $r^q$ is again in $(0,1)$, $S$ preserves convergence and is
continuous for the Fr\'echet topology. It is a section of $\phi_q-1$.
For any profinite space $T$, postcomposition with $S$ lifts every
continuous map $T\to R$. This proves the asserted surjectivity of
condensed groups, and therefore of solid modules.
The kernel consists of the same convergent series with coefficients in
$k$. Its nonzero coefficients all have norm one. The convergence
condition says that the exponents in its support form a left-finite
subset of $\Z[1/p]$: below any bound there are only finitely many.
Truncation therefore identifies it with the completed perfection $E$.
On this kernel every Gauss seminorm is $r^{v_t(f)}$; hence its induced
topology is exactly the valued-field topology of $E$.
The two-term complex $[R\xrightarrow{\phi_q-1}R]$ now has kernel
$E$ and zero cokernel in solid modules. It computes homotopy fixed
points for the infinite cyclic action, proving the last assertion.
\end{proof}
For the application, the quasi-Stein perfectoid calculation in
\cite[Lemma~3.6.2]{BMR2026} identifies the relevant solid cohomology
with this function ring. The lemma proves the enhancement needed in
\cite[Corollary~3.6.8]{BMR2026} directly for that ring.
The equivalence is the canonical map from constants; its construction
as a map is independent of the chosen section. It therefore commutes
with all actions and base changes already commuting with that map.
No equivariance of the auxiliary set-theoretic section is required.
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//09-frobenius.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//10-oriented-projection.tex ====
\section{The oriented projection over the completed field}
\label{app:oriented-projection}
We prove \cref{lem:completed-oriented-map}, including the continuous
identity that identifies the \(E_2\) tensor factor with connected base
change. The main construction is the split section in
\eqref{eq:oriented-section-diagram}. Here we verify its residue tags,
orientations, and family coherence on the parameter rings
\(R_T=C_{\cts}(T,F)\), where \(F=\widehat L(*)\) is the ordinary
completed field, and then pass to solid spectra.
\begin{proof}[Proof of \cref{lem:completed-oriented-map}]
The forgetful transformation in
\cite[Proposition~6.2.2 and Remark~6.2.3]{LurieEllipticII2018} goes from
deformations of a full group to deformations of the left term of a tagged
exact sequence with étale quotient. Corepresentability therefore gives a
map from the connected deformation ring to the full deformation ring.
We construct the reverse oriented map by a section of that transformation.
The field \(F\) is perfect of characteristic \(p\), and its inverse
Frobenius is continuous. Hence pointwise inverse Frobenius preserves
\(C_{\cts}(T,F)\), so \(R_T\) is perfect. For nonempty \(T\), the map
\(F\to R_T\) is flat, since \(F\) is a field, and relatively perfect, since
both Frobenius maps are isomorphisms. The ring \(R_T\) need not be a field
or Noetherian. The field \(F\) is Noetherian and \(F\)-finite.
More generally, for any perfect \(\F_p\)-algebra, every \(p\)-divisible group is nonstationary
and the absolute cotangent complex is almost perfect, by
\cite[Remark~3.4.2]{LurieEllipticII2018}. In particular the full group
over \(F\), without a connectedness assumption, is an eligible
Noetherian LKN witness for \((R_T,\Gamma_T)\), in the sense of
\cite[Definition~4.3.7 and Example~4.6.4]{BMR2026}. The connected pair has
the height-two witness \((k,\Gamma_2)\). At the empty parameter set we
assign the terminal presheaf value; no LKN assertion for the zero ring is
needed.
Over the perfect field \(F\), take the connected--étale sequence and its
unique splitting:
\[
\sigma_F:\quad
0\longrightarrow C_F\longrightarrow\Gamma_F
\longrightarrow Q_F\longrightarrow0,
\qquad \Gamma_F\simeq C_F\oplus Q_F .
\]
Here \(Q_F\) is étale of height one. Existence and uniqueness of the
splitting are
\cite[Proposition~2.5.20 and Remark~2.5.24]{LurieEllipticII2018}.
We apply the splitting remark only over the perfect field \(F\).
Write \(\iota_F:C_F\to\Gamma_F\) for the inclusion, and let
\[
\lambda_0:\Gamma_F^\circ\xrightarrow{\sim}(\Gamma_2)_F^\circ
\]
be exactly the tautological formal isomorphism of the isomorphism torsor
in \cite[Example~4.6.3]{BMR2026}. Over the discrete field \(F\), the
hypotheses of
\cite[Theorem~2.3.12 and Corollary~2.3.13]{LurieEllipticII2018} hold because
\(p=0\). Their fully faithful identity-component functor gives a unique
isomorphism
\(\lambda_F:(\Gamma_2)_F\xrightarrow{\sim}C_F\), characterized by
\[
\lambda_0\iota_F^\circ\lambda_F^\circ=\mathrm{id}.
\]
Thus our \(\lambda_F\) has the direction from \(\Gamma_2\) to the
connected summand. Define \(\sigma_T\), its splitting, \(\iota_T\), and
\(\lambda_T\) by base change to \(R_T\). In particular, \(C_T\) denotes
a \(p\)-divisible group, whereas the identity component of \(\Gamma_T\) denotes its formal group.
\paragraph{Naturality of the split reference pair.}
We check the family naturality of this splitting. If \(M\) is a finite
projective \(R_T\)-module, then
\[
M\longrightarrow
\prod_{t\in T}\bigl(M\otimes_{R_T,\operatorname{ev}_t}F\bigr)
\]
is injective: embed \(M\) as a direct summand of \(R_T^N\) and use
\(R_T\hookrightarrow\prod_tF\). It follows that maps between finite flat
group schemes over \(R_T\) are detected by these fibers, by applying the
same observation to the target of the corresponding coordinate-algebra
maps. It detects maps of \(p\)-divisible groups on each \(p^m\)-torsion
level. These are actual equalities of maps: finite flat algebras over the
ordinary ring \(R_T\) are ordinary finite projective algebras, and their
mapping spaces are discrete. The finite-flat functor description recalled
in the proof of \cite[Proposition~2.5.20]{LurieEllipticII2018} identifies
the compatible torsion-level maps with maps of \(p\)-divisible groups.
Let \(\rho:R_T\to R_U\) be a parameter pullback or a map furnished by a
continuous action family, and let
\(\gamma:\rho^*\Gamma_T\xrightarrow{\sim}\Gamma_U\) be the given pair
isomorphism. Both induced sequences are connected--étale: connectedness
is preserved by base change by the locally nilpotent augmentation
criterion of \cite[Remark~2.3.8]{LurieEllipticII2018}, and étaleness is
preserved by base change. The discrete characteristic-\(p\) ring \(R_U\)
is complete with ideal of definition zero. Thus
\cite[Theorem~2.5.13]{LurieEllipticII2018} gives the unique isomorphism of
these sequences extending \(\gamma\). The two resulting sections
\(Q_U\to\Gamma_U\) agree at each \(u\in U\), by uniqueness of the splitting
over the perfect field \(F\); the preceding detection argument makes them
equal over \(R_U\). Applying this with \(U=T\times G^a\), for every
\(a\geq0\), proves compatibility with all family, multiplication, and unit
maps. The same holds for \(\lambda_T\) by the fully faithful
identity-component functor. The input is the continuous action on the
full completed pair and its tautological formal trivialization from
\cite[Examples~4.6.3--4.6.4]{BMR2026}, restricted to the nonextended
\(P_3\times P_2\), which fixes \(k\). This argument uses the pulled-back
field splitting; it does not assert a splitting theorem for arbitrary
perfect rings.
\paragraph{The section on oriented deformations.}
We now construct the section on oriented deformations. Let \(A\) be any
complete adic \(E_\infty\)-ring, without a Noetherian or connectivity
assumption. For nonconnective \(A\), the notation \(\mathrm{BT}_p(A)\)
means \(\mathrm{BT}_p(\tau_{\geq0}A)\). For a finitely generated ideal of
definition \(I\), put \(B_I=\pi_0A/I\), write \(H(B_I)\) for its discrete
Eilenberg--Mac Lane \(E_\infty\)-ring, and use the canonical reduction
\[
\tau_{\geq0}A\longrightarrow H(B_I).
\]
A subscript \(I\) below denotes this reduction or the corresponding
ordinary base change of a reference group. No map \(A\to H(B_I)\) is
asserted. Orientations remain over
the full ring \(A\): their Bott map is an equivalence of \(A\)-modules.
They are not orientations over its connective cover; see
\cite[Warning~4.3.12]{LurieEllipticII2018}.
Represent a connected oriented deformation by
\((C_A,I,\mu,\alpha_C,e)\), where \(I\subset\pi_0A\) is a finitely
generated ideal of definition,
\[
\mu:R_T\longrightarrow\pi_0A/I,\qquad
\alpha_C:\mu^*C_T\xrightarrow{\sim}(C_A)_{\pi_0A/I},
\]
and \(e\) orients \(C_A^\circ\) over \(A\). Tags are identified only after
passage to a common larger finitely generated ideal of definition on
which both the residue maps and group isomorphisms agree, as in
\cite[Definitions~3.1.1 and~3.1.4]{LurieEllipticII2018}.
Since \(R_T\) has characteristic \(p\), the tag forces \(p\in I\).
Completeness for \(I\) therefore implies \((p)\)-completeness, and \(p\)
is topologically nilpotent, by
\cite[Remarks~0.0.10--0.0.12]{LurieEllipticII2018}. The tag makes \(C_A\)
formally connected. These facts meet the hypotheses of the
identity-component and connected--étale results used below; they do not
assert completeness of \(\tau_{\geq0}A\).
The reduction equivalence
\[
\mathrm{BT}^{\mathrm{et}}_p(A)\xrightarrow{\sim}
\mathrm{BT}^{\mathrm{et}}_p(\pi_0A/I)
\]
of \cite[Corollary~2.5.10]{LurieEllipticII2018} lifts \(\mu^*Q_T\) to an
étale group \(Q_A\), with a specified residual isomorphism \(\alpha_Q\).
The space of such lifts with this isomorphism is contractible; the
unmarked object \(Q_A\) itself may have automorphisms. More precisely, if
\(\mathscr D_{C,I}^{\mathrm{or}}(A)\) denotes the fixed-\(I\) stage, use
the homotopy pullback
\[
\mathscr X_I(A)=
\mathscr D_{C,I}^{\mathrm{or}}(A)
\mathop{\times}_{\mathrm{BT}^{\mathrm{et}}_p(B_I)^\simeq}
\mathrm{BT}^{\mathrm{et}}_p(A)^\simeq ,
\]
where the first map sends a tag to \(\mu^*Q_T\) and the second is the typed
reduction above. Its projection to
\(\mathscr D_{C,I}^{\mathrm{or}}(A)\) is an equivalence by
Corollary~2.5.10; its fiber includes the marking \(\alpha_Q\).
These projections and the split construction form natural diagrams on
the category of compatible data \((A,I,\mu)\), including adic maps,
residue maps after ideal refinement, and compatible maps of split base
sequences. For an adic map \(f:A\to A'\), start with a target ideal of
definition \(J_0\). Continuity gives \(f(I^n)\subseteq J_0\) for some
\(n\), so \(J_0+(f(I))\) is again a finitely generated ideal of definition
and contains \(f(I)\). Such choices admit common larger ideals. For
\(I\subseteq J\), the same \(Q_A\) with the reduced marking is used.
The fixed-\(I\) tagged categories for complete \(A\) are groupoidal by
the proof of \cite[Lemma~3.1.10]{LurieEllipticII2018}: a tagged morphism
is an equivalence modulo \(I\), and equivalence is detected on its finite
flat \(p\)-torsion level. Orientations are pulled back from
\(\mathrm{BT}_p(A)^\simeq\), independently of \(I\); filtered colimits
of spaces commute with this pullback. Hence the filtered colimit of the
fixed stages is precisely the oriented tagging anima of
Definition~3.1.4. The auxiliary projections are an objectwise
equivalence of diagrams; take their filtered colimits and invert that
equivalence in the functor category. This supplies all lift, refinement,
and base-map coherences. We do not replace that colimit by reduction
to the nilradical, which \cite[Warning~3.1.9]{LurieEllipticII2018} excludes
in this generality.
Form the split sequence
\[
0\longrightarrow C_A\longrightarrow C_A\oplus Q_A
\longrightarrow Q_A\longrightarrow0.
\]
It is an exact sequence of \(p\)-divisible groups by
\cite[Proposition~2.4.8]{LurieEllipticII2018}, applied to the split pushout
square. Tag the entire residual sequence using
\[
\mu^*\Gamma_T\simeq\mu^*C_T\oplus\mu^*Q_T
\xrightarrow{\alpha_C\oplus\alpha_Q}
(C_A\oplus Q_A)_{\pi_0A/I}.
\]
This retains the original full tag, including \(\mu\), and chooses no
basis or trivialization of the étale factor. Since \(A\) is
\((p)\)-complete, \(Q_A^\circ=0\) by
\cite[Proposition~2.5.8]{LurieEllipticII2018}. The identity-component fiber
sequence, as in the proof of
\cite[Proposition~2.5.17]{LurieEllipticII2018}, gives
\(C_A^\circ\xrightarrow{\sim}(C_A\oplus Q_A)^\circ\).
Transport the same orientation \(e\) through this equivalence over \(A\).
Write \(\mathscr D_\sigma^{\mathrm{or}}\),
\(\mathscr D_\Gamma^{\mathrm{or}}\), and
\(\mathscr D_C^{\mathrm{or}}\) for the resulting oriented deformation
functors. Let \(m:\mathscr D_\sigma^{\mathrm{or}}\to
\mathscr D_\Gamma^{\mathrm{or}}\) and
\(q:\mathscr D_\sigma^{\mathrm{or}}\to\mathscr D_C^{\mathrm{or}}\)
forget to the middle and left terms. Proposition~6.2.2 of
\cite{LurieEllipticII2018} makes the underlying \(m\) an equivalence.
It remains an equivalence with orientations: a tagged quotient is étale
by Remark~2.5.11 of that paper, and the left and middle identity formal
groups are naturally equivalent. Set \(\mathfrak f=qm^{-1}\), taking the
inverse in the functor category. The split construction gives
\(\widetilde{\mathfrak s}:\mathscr D_C^{\mathrm{or}}\to
\mathscr D_\sigma^{\mathrm{or}}\), whose left projection is the identity
on \((I,\mu,\alpha_C,e)\). Therefore
\[
\mathfrak f\mathfrak s\simeq\mathrm{id},
\qquad \mathfrak s=m\widetilde{\mathfrak s},
\]
naturally with all the preceding refinements and base maps.
\paragraph{Corepresentability and the reverse ring map.}
We justify the oriented mapping property at the possibly
non-Noetherian \(R_T\). The oriented mapping property in
\cite[Remark~6.0.7]{LurieEllipticII2018} is stated in the Noetherian
setting of its Theorem~6.0.3; the general statement of BMR
Remark~4.1.22 needs the following argument here. For either pair
\((R_T,\Gamma_T)\) or \((R_T,C_T)\), the characteristic is \(p\), and
\cite[Remark~3.4.2]{LurieEllipticII2018} supplies nonstationarity and an
almost perfect absolute cotangent complex. Theorem~3.4.1 of that paper
therefore gives a complete connective universal spectral deformation
ring \(R^{\mathrm{un}}\), a surjection
\(\pi_0R^{\mathrm{un}}\to R_T\), and a finitely generated kernel \(J\)
which is an ideal of definition.
Let \(E\) be the orientation classifier of its universal identity formal
group, as in \cite[Construction~4.1.17 and Definition~4.1.20]{BMR2026}.
The LKN witnesses and \cite[Definition~4.2.1 and Theorem~4.3.8]{BMR2026}
supply balancedness. By Remark~4.1.16 and Definitions~4.1.19--4.1.20
there, the map \(\pi_0R^{\mathrm{un}}\to\pi_0E\) is an isomorphism, the
odd homotopy groups of \(E\) vanish, and every even homotopy group is an
invertible \(\pi_0E\)-module. Here completeness of modules means
derived completeness. The homotopy-group completeness argument
in the proof of \cite[Corollary~6.0.5]{LurieEllipticII2018} now applies
without a Noetherian step. Specifically, the homotopy-group criterion
cited there as SAG Theorem~7.3.4.1 makes \(\pi_0R^{\mathrm{un}}\) \(J\)-complete because
\(R^{\mathrm{un}}\) is \(J\)-complete. An invertible module over this
ring is a direct summand of a finite free module, hence is also
\(J\)-complete. The same criterion then makes \(E\) \(J\)-complete.
Endow \(\pi_0E\) with the topology transported from
\(\pi_0R^{\mathrm{un}}\). An \(E_\infty\)-map \(E\to A\) is continuous
exactly when its restriction from \(R^{\mathrm{un}}\) is continuous,
because these \(\pi_0\) rings and topologies agree. The universal
deformation mapping property of Theorem~3.4.1, followed by the
orientation-classifier mapping property, therefore identifies the adic
mapping anima from \(E\) with the triples consisting of a deformation,
its strict tagging class, and an orientation over full \(A\). This
locally proves the use of BMR Remark~4.1.22 at \(R_T\).
It turns \(\mathfrak f\) and \(\mathfrak s\) into continuous adic maps
\[
i_T:E(R_T,C_T)\longrightarrow E(R_T,\Gamma_T),\qquad
r_T:E(R_T,\Gamma_T)\longrightarrow E(R_T,C_T),
\qquad r_Ti_T\simeq\mathrm{id}.
\]
They preserve the \(SW(k)\)-structure because they retain the restriction
of the same tag \(\mu\) to \(k\). The construction is natural in split
base sequences. The family argument above therefore makes the \(r_T\)
continuous \(G\)-natural maps.
For precision, this last assertion also survives condensation. Take the
category of split LKN sequences with both full and connected pairs LKN.
The preceding construction is a natural transformation from its full
\(E\)-functor to its connected \(E\)-functor. Extend these functors and
the transformation from the presheaf category and tensor with condensed
anima as in \cite[Construction~4.5.2]{BMR2026}. Hypersheafifying the raw
split parameter diagram gives a condensed object with its continuous
\(G\)-action; its full and connected images are the original pair
diagrams. The resulting enriched transformation is the pointwise \(r_T\)
followed by hypersheafification, giving \(r^{\mathrm{cond}}\).
\paragraph{The connected tensor factor.}
It remains to identify the \(E_2\) factor. For each \(T\), let
\(b_T:E_2\to E(R_T,\Gamma_T)\) be the ordinary tensor-factor map followed
by completion, and let \(c_T:E_2\to E(R_T,C_T)\) be the canonical
connected map using \(\lambda_T\). The pointwise tensor construction uses
the Noetherian perfect reference pair \((k,\Gamma_2)\), with connected
height-two group, and the localized height-two LKN pair as its other
input. These meet \cite[Construction~4.4.7]{BMR2026}. The proof of
Theorem~4.4.8 there identifies its mapping spaces on \(K(2)\)-local
complete adic \(SW(k)\)-algebra tests. Choose a common representative
ideal \(I\) for the two \(k\)-linear tag classes, using the strict
refinement relation. If \(e_1,e_2\) orient the formal groups of the two
deformations \(H_1,H_2\), let
\[
q_{e_i}:\Gamma_A^{Q}\xrightarrow{\sim}H_i^\circ,\qquad
j=q_{e_2}q_{e_1}^{-1}:H_1^\circ\xrightarrow{\sim}H_2^\circ
\]
be the isomorphisms from the same Quillen formal group and their
comparison. This is the natural orientation correspondence of
\cite[Remark~4.1.18]{BMR2026}, using Lurie's Propositions~4.3.21 and
4.3.23. The orientations are over full \(A\), while \(j\) is a formal
isomorphism over \(\tau_{\geq0}A\). Only \(j\), not an orientation, is
reduced along \(\tau_{\geq0}A\to H(B_I)\).
Let \(\alpha:\Gamma_{T,I}\xrightarrow{\sim}(H_1)_I\) be the full tag
after completion, and let \(\lambda_{0,I}\) be the base change of the
same torsor isomorphism \(\lambda_0\) to \(B_I\). The tuple repacking of
Theorem~4.4.8 first goes from the reference \(\Gamma_2^\circ\) through
\(\lambda_{0,I}^{-1}\) to the reference full formal group, then through
\(\alpha^\circ\) to \((H_1)_I^\circ\), and finally through \(j_I\).
Thus the formal tag of the second tensor factor is exactly
\[
\beta_{\mathrm{out}}^\circ
=j_I\,\alpha^\circ\,\lambda_{0,I}^{-1}.
\]
For the split section take
\[
H_1=C_A\oplus Q_A,\quad H_2=C_A,\quad
e_1=(i^\circ)_*e,\quad e_2=e,
\]
where \(i:C_A\to C_A\oplus Q_A\) is the inclusion. Naturality of the
Quillen correspondence gives \(q_{e_1}=i^\circ q_e\), so
\(j=(i^\circ)^{-1}\). In our direction for \(\lambda_T\), put
\[
\beta=\alpha_C\lambda_{T,I}:
(\Gamma_2)_{B_I}\xrightarrow{\sim}(C_A)_I .
\]
The defining relation for \(\lambda_F\) gives
\(\lambda_{0,I}^{-1}=\iota_{T,I}^\circ\lambda_{T,I}^\circ\), and the
definition of the full split tag gives
\(\alpha^\circ\iota_{T,I}^\circ=i_I^\circ\alpha_C^\circ\).
Substitution therefore cancels the precise torsor and orientation
isomorphisms:
\[
\begin{aligned}
\beta_{\mathrm{out}}^\circ
&=(i_I^\circ)^{-1}\alpha^\circ
\iota_{T,I}^\circ\lambda_{T,I}^\circ\\
&=(i_I^\circ)^{-1}i_I^\circ
\alpha_C^\circ\lambda_{T,I}^\circ
=\beta^\circ .
\end{aligned}
\]
Theorem~2.3.12 and Corollary~2.3.13 of \cite{LurieEllipticII2018}, over
\(A\) and over the discrete characteristic-\(p\) ring \(B_I\), identify
the formally connected output with \(C_A\) and the tag itself with
\(\beta\). Their hypotheses were checked above. The output orientation
is \(e\), since pushforward through \(i^\circ\) is undone through its
inverse. The output ring tag is exactly
\(k\to R_T\xrightarrow{\mu}B_I\). Completion only precomposes \(\mu\)
and base changes the same \(\lambda_0\), so the calculation respects the
\(SW(k)\)-linear tagging condition. There is no additional Bott unit or
automorphism normalization. It is natural under strict ideal refinement
and the parameter and family maps already treated. Hence, on the stated
local test category,
\[
(r_Tb_T)^*=b_T^*\mathfrak s_T\simeq c_T^*.
\]
The connected endpoint is \(K(2)\)-local by
\cite[Proposition~4.4.2]{BMR2026} for its height-two witness, and it is
complete by the argument above. The same statements hold for \(E_2\).
Thus both are \(K(2)\)-local complete adic \(SW(k)\)-algebras.
Yoneda on the stated test category, equivalently evaluation on the universal connected deformation at
\(A=E(R_T,C_T)\), gives \(r_Tb_T\simeq c_T\). This preserves the full
connected tag \((I,\mu,\alpha_C)\), its restricted reference-pair tag
\(\beta:(\Gamma_2)_{B_I}\xrightarrow{\sim}(C_A)_I\) along
\(k\to R_T\xrightarrow{\mu}B_I\), and the orientation \(e\); it uses
no locality of \(E(R_T,\Gamma_T)\).
\paragraph{The continuous solid identity.}
We identify the actual \(E_2\) tensor composite with \(c^\square\).
Let \(b^\square:E_2^\square\to
E^{\mathrm{cond}}(\widehat L,\Gamma^{\mathrm{un}}_{3,\widehat L})\)
be the tensor-factor map followed by completion. There are two points in
passing from the pointwise identity to solid theories. First,
the pointwise presheaf in the proof of
\cite[Proposition~4.5.12]{BMR2026} uses the constant ordinary \(E_2\), not \(E_2^\square(T)\) as the reference input
of Theorem~4.4.8. The preceding identity, natural in \(T\), therefore
identifies the two maps after precomposition from \(E_2^\delta\) and
hypersheafification. By \cite[Lemma~4.5.8]{BMR2026},
\[
\ell:E_2^\delta\longrightarrow E_2^\square
\simeq L^\square_{K(2)}E_2^\delta
\]
is the localization unit. The connected target is
\(L^\square_{K(2)}\)-local by its profinite evaluations, so its localization
mapping property identifies the two underlying maps from \(E_2^\square\).
Second, this does not give \(E_2^\delta\) a continuous \(P_2\)-action.
Put \(X=E_2^\square\), \(Y=\widehat B^\square\), and write \(Y^S\) for the
profinite cotensor, so \(Y^S(T)=Y(S\times T)\). For the underlying solid
spectra, the space of continuous \(G\)-equivariant maps is the bar
totalization with degree \(a\)
\[
\operatorname{Map}(X,Y^{G^a}).
\]
Every target cotensor is local. Restriction along \(\ell\) is therefore
an equivalence of mapping spaces in each degree; transport the bar cofaces
through these equivalences. This requires no action on \(E_2^\delta\).
To identify the transported source coface, let \(g:S\to P_2\) be a
continuous family. Since \(k\) is finite, compactness gives
\(C_{\cts}(S\times T,k)=C_{\cts}(T,C_{\cts}(S,k)_{\mathrm{disc}})\).
Thus the cotensor of \(k^\delta\) by \(S\) is
\((k^\delta(S))^\delta\), where \(k^\delta(S)=C_{\cts}(S,k)\).
The family gives a pair map
\[
\gamma_g:(k,\Gamma_2)\longrightarrow(k^\delta(S),\Gamma_2):
\]
at every torsion level its coefficients are locally constant, as required
by continuity. The naturality of Lemma~4.5.8 and the cotensor comparison
of \cite[Remark~4.5.6]{BMR2026} give the commuting square
\[
\begin{tikzcd}
E_2^\delta \arrow[r,"\ell"]
\arrow[d,"E(\gamma_g)^\delta"'] &
X \arrow[d,"a_{X,g}"]\\
E(k^\delta(S),\Gamma_2)^\delta \arrow[r,"\ell_S"'] & X^S .
\end{tikzcd}
\]
Here \(a_{X,g}\) is the given family action and \(\ell_S\) is the
corresponding localization unit under the cotensor identification.
Consequently the source coface after restriction is precomposition of the
connected tag by \(\gamma_g\). Target cofaces and interior cofaces are the
full-pair family maps and parameter pullbacks. The tuple identification
and \(\mathfrak f\mathfrak s\simeq\mathrm{id}\) are natural for exactly
these operations on every \(R_{G^a\times T}\). They give compatible
homotopies in all bar degrees, hence
\[
r^{\mathrm{cond}}b^\square\simeq c^\square
\]
as continuous \(G\)-equivariant maps of underlying solid spectra.
Both sides are independently maps of condensed \(E_\infty\)-rings, and
their source and target are solid. In particular they induce the same
coefficient and even-line maps. This uses the localization mapping
property degree by degree, not an interchange of localization with
totalization or fixed points.
\end{proof}
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//10-oriented-projection.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{A rational obstruction to strong chromatic splitting\\at height three}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
For every prime \(p\ge5\), the canonical map
\(L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S\) for the sphere spectrum \(S\)
is nonzero on \(\pi_{-3}\). Consequently, the height-three strong chromatic
splitting formula is false in this range, even as an equivalence of
underlying \(E(2)\)-local spectra without specified summand maps.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-cohomology}
\input{sections/03-completed-tower}
\input{sections/04-modifications}
\input{sections/05-degree-three}
\input{sections/06-witt-transfer}
\input{sections/07-detection}
\input{sections/wedge-obstruction}
\input{sections/08-elementary-background}
\input{sections/09-frobenius}
\input{sections/10-oriented-projection}
\begingroup
\small
\raggedright
\bibliographystyle{plain}
\bibliography{references}
\endgroup
\end{document}
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==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//wedge-obstruction.tex ====
\section{The formal wedge obstruction}
\label{sec:wedge-obstruction}
We finish by comparing the canonical map with the map forced by the
proposed wedge. The argument uses naturality and chromatic acyclicity;
it places no compatibility requirement on a hypothetical equivalence
of spectra.
\begin{lemma}\label{lem:wedge-obstruction}
Suppose \(Z\simeq A\vee B\), \(L_{K(2)}B=0\), and
\(\pi_dL_0A=0\). Then \(\pi_d(\tau_Z)=0\).
\end{lemma}
\begin{proof}
Localization is exact and preserves finite sums, so
\(L_{K(2)}(A\vee B)\simeq L_{K(2)}A\).
Naturality with the summand inclusions identifies
\(\tau_{A\vee B}\) with \(\tau_A\) on \(L_0A\) and the zero map on
\(L_0B\). It vanishes in degree \(d\). Naturality with an arbitrary
equivalence \(A\vee B\simeq Z\) transports this conclusion to \(Z\).
\end{proof}
\begin{proof}[Proof of \cref{cor:strong}]
In \eqref{eq:proposed-wedge}, let \(A\) be the first line and \(B\)
the sum of the last two lines. Smashing for \(L_1\) and the
Bousfield-class description of \(L_1S\) give \(L_{K(2)}B=0\)
\cite[Theorem~5.1 and Proposition~5.3]{HS1999}.
The \(p\)-complete sphere is connective. Each term of the Moore tower
\(S_p^\wedge=\varprojlim_m S/p^m\) is connective, and the degree-zero
transition maps are surjective; the possible negative Milnor term
therefore vanishes. Rationalization preserves connectivity. Since
\(L_0L_2\simeq L_0\), the two summands of \(L_0A\) have no homotopy
in degree \(-3\). Thus \cref{lem:wedge-obstruction} gives
\(\pi_{-3}\tau_{\mathcal W_3}=0\).
Put \(T=L_{K(3)}S\) and \(X=L_2T\). The cofiber of the localization
map \(T\to X\) is \(E(2)\)-acyclic. The identity
\[
\langle E(2)\rangle
=\langle K(0)\vee K(1)\vee K(2)\rangle
\]
makes it both rationally and \(K(2)\)-acyclic. The naturality square
\[
\begin{tikzcd}[column sep=large,row sep=large]
L_0T\arrow[r,"\tau_T"]\arrow[d,"\sim"']
&L_0L_{K(2)}T\arrow[d,"\sim"]\\
L_0X\arrow[r,"\tau_X"]&L_0L_{K(2)}X
\end{tikzcd}
\]
therefore has vertical equivalences. Any equivalence
\(X\simeq\mathcal W_3\) would force its lower horizontal map to vanish
on \(\pi_{-3}\), and hence force the same for \(\tau_T\).
This contradicts \cref{thm:main}. No step specifies the action of
the hypothetical equivalence on individual summands.
\end{proof}
==== END A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//wedge-obstruction.tex ====
==== BEGIN A-rational-obstruction-to-strong-chromatic-splitting-at-height-three//references.bib ====
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