64c3814
==== TABLE OF CONTENTS ====
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//01-introduction.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//02-background.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//03-characters.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//04-congruences.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//05-profinite.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//06-gluing.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//07-support.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//paper.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//preamble.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//references.tex
Cyclic-Length-and-Chromatic-Fixed-Point-Loss//unicode-map.tex
===========================
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//01-introduction.tex ====
\section{Introduction}\label{sec:intro}
The chromatic Smith problem asks how much information about fixed points at one subgroup is forced by chromatic acyclicity at another. Fix a prime \(p\), and let \(K(i)\) denote Morava \(K\)-theory at \(p\), with \(K(0)=H\Q\). For a finite \(p\)-group \(G\), a subgroup \(H\le G\), and an integer \(n\ge0\), define \(r_n(G,H)\) to be the least nonnegative integer \(r\) such that
\begin{equation}\label{eq:def-loss}
K(n+r)_*(\Phi^H X)=0
\quad\Longrightarrow\quad
K(n)_*(\Phi^G X)=0
\end{equation}
for every finite \(p\)-local genuine \(G\)-spectrum \(X\). Here \(\Phi^H X\) denotes the underlying spectrum of the geometric fixed points, and finite spectra include compact retracts.
Let \(\ell(G,H)\) be the least length of a subnormal chain
\[
H=G_0\trianglelefteq G_1\trianglelefteq\cdots
\trianglelefteq G_s=G
\]
whose successive quotients are cyclic. We set \(\ell(G,G)=0\). Such chains exist in finite \(p\)-groups. The cyclic chromatic Smith theorem, applied along a chain, gives \(r_n(G,H)\le\ell(G,H)\). Our result is the converse.
\begin{theorem}\label{thm:main}
For every prime \(p\), finite \(p\)-group \(G\), subgroup \(H\le G\), and integer \(n\ge0\),
\[
\boxed{r_n(G,H)=\ell(G,H).}
\]
\end{theorem}
Thus the optimal loss is independent of the height. For every integer \(0\le r<\ell(G,H)\), there is a finite genuine \(G\)-spectrum whose geometric \(H\)-fixed points are \(K(n+r)\)-acyclic and whose geometric \(G\)-fixed points have nonzero \(K(n)\)-homology. The theorem includes rational fixed-point detection at \(n=0\).
The embedding of \(H\) matters. For example, in
\(G=C_{p^2}\times C_p\), take
\(H_1=pC_{p^2}\times0\) and \(H_2=0\times C_p\).
Both subgroups are isomorphic to \(C_p\), but
\(G/H_1\cong C_p\times C_p\) and \(G/H_2\cong C_{p^2}\).
Chains above \(H_i\) correspond to chains in these quotients.
The first quotient requires two cyclic steps, while the second
requires one. Thus \(r_n(G,H_1)=2\) and \(r_n(G,H_2)=1\)
at every target height. A cyclic quotient of order \(p^2\)
still counts only once.
\subsection*{History and significance}
The classical starting point is Smith theory: for a finite complex with
an action of a finite \(p\)-group, mod-\(p\) acyclicity passes to fixed
points. Floyd strengthened this to an inequality between total
mod-\(p\) Betti numbers \cite[Theorem~4.4]{Floyd}; the finite
\(p\)-group statement follows by iteration from prime-order actions.
Chromatic Smith theory asks which implications survive when mod-\(p\)
homology is replaced by individual Morava theories.
Building on the nonequivariant thick subcategory theorem of Hopkins
and Smith \cite{HS}, Balmer and Sanders identified all primes of the
category of finite genuine equivariant spectra. They reduced its
topology to subgroup--height inclusions and computed the case
\(C_p\), crediting Strickland's earlier work
\cite[Introduction, Theorems~4.9 and~4.14, and Section~7]{BS}.
Barthel, Hausmann, Naumann, Nikolaus, Noel, and Stapleton determined
the finite abelian \(p\)-group case: the loss for \(H\le G\) is the minimum
number of generators of \(G/H\) \cite[Theorem~2.3]{BHNNNS}.
This replaced the original order-based prediction by a rank bound;
already \(C_{p^2}\) has loss one rather than two.
The sharp elementary abelian examples use the partition-complex work
of Arone, Dwyer, Lesh, and Mahowald
\cite[Theorems~2.2 and~4.1]{BHNNNS}. Balderrama and Kuhn later gave
an elementary proof of the abelian chromatic Smith theorem, including
target height zero \cite[Theorem~1.1 and Section~3.2]{BK}.
For nonabelian groups, a general lower bound is the rank of
the largest elementary abelian quotient of \(G\) in which \(H\)
maps to the identity. This can be smaller than the cyclic-chain upper
bound. Kuhn and Lloyd proved that chromatic acyclicity implications
are equivalent to corresponding inequalities between Morava homology
dimensions, allowing representation-theoretic constructions to give
stronger lower bounds \cite[Theorem~2.2]{KL}. They obtained equality
with \(\ell\) for every subgroup pair inside a central extension
\(C_2\to G\to E\) with \(E\) elementary abelian
\cite[Theorem~2.13]{KL}. In particular, a noncentral subgroup of order
two in \(D_8\) has loss two, although the elementary-quotient lower
bound is one. Their discussion of the general problem expresses
``cautious hope'' that \(r_n(G,H)=r_+(G,H)=\ell(G,H)\), and separately
asks whether the loss is independent of height
\cite[Section~2.2]{KL}. \Cref{thm:main} establishes the proposed
equality, and hence height independence, for all finite \(p\)-groups.
The remaining obstacle is sharpness for an arbitrary embedded
subgroup. The Hopkins--Kuhn--Ravenel characters are indexed by maps
from \(\Z_p^n\), whose images are abelian \cite[Theorem~C]{HKR}.
Thus they cannot directly supply characters with full nonabelian
image. General character rings with other source groups already
appear in Kuhn's framework \cite[Section~2]{KuhnCharacters}.
Here we select a profinite source from the finite groups detected by
telescopic cochains, derive constraints on all its open subgroups by
relative integration, and use those constraints to prove the required
fixed-point vanishing estimates.
Combined with the conjugacy and \(p\)-subnormal reductions of
Balmer--Sanders, the theorem determines the remaining finite-height
prime inclusions for arbitrary finite groups
\cite[Proposition~6.11]{BS}; \Cref{cor:general-groups} gives the
precise test. Their description of the infinite-height inclusions
and of Thomason subsets then determines the thick tensor ideals
\cite[Corollaries~8.19, 10.5, and~10.6]{BS}. The sharpness witnesses
in this paper are obtained by separation of primes.
\subsection*{The proof}
For a finite \(p\)-group \(P\) and a spectrum \(D\), write
\[
U_P(D)=\Phi^P\Bor_P(D),
\]
where \(\Bor_P(D)\) is the genuine cofree spectrum associated to the trivial \(P\)-action on \(D\). Let \(T(l)\) be a telescope at height \(l\). The first main step is the rational estimate
\begin{equation}\label{eq:intro-rational}
\bigl(U_P(L_{T(l)}S)\bigr)_{\Q}\ne0
\quad\Longrightarrow\quad \ell(P,1)\le l.
\end{equation}
To prove it at positive \(l\), we rationalize morphisms in the tensor category of dualizable \(T(l)\)-local spectra and split Burnside idempotents. A maximizing construction produces a pro-\(p\) group \(\Gamma\) mapping onto \(P\), with every finite section (a finite continuous quotient of a closed subgroup) generated by at most \(l\) elements. Cochains admit evaluation factors indexed by continuous homomorphisms out of \(\Gamma\). Height zero is handled separately by rational averaging.
The key additional information comes from relative integration over
finite groupoids. Strickland's Frobenius formulation of HKR character
theory already expresses such integration through automorphism-group
weights \cite[Section~10 and Theorem~10.2]{StricklandDuality}.
We derive the weights for the source \(\Gamma\) just constructed.
Applying them to groupoids of labeled finite sets, coinduced from an
open subgroup, gives congruences
\[
\sum_{\substack{V\le_o U\\{}[U:V]=p^h}}
\bigl(|X^V|-|X|\bigr)
\in p^{h+1}\Z_{(p)}
\]
for every open subgroup \(U\le\Gamma\), every finite continuous \(U\)-set \(X\), and every \(h\ge0\). The same construction gives
\(\lvert\Hom_{\cts}(\Gamma,F)\rvert/\lvert F\rvert\in\Z_{(p)}\)
for every finite group \(F\). Together with the finite-section bound,
these arithmetic conditions force a closed subnormal series with
\(\Z_p\)-factors and length at most \(l\).
The purely profinite argument uses shrinking semidirect products and an explicit \(p\)-adic limit of subgroup counts; it may be useful independently of the chromatic application.
Next, ring nilpotence and the generalized characters of Hopkins, Kuhn, and Ravenel \cite{HS,HKR} upgrade \eqref{eq:intro-rational} to
\begin{equation}\label{eq:intro-height}
T(k)\wedge U_P(L_{T(l)}S)\ne0
\quad\Longrightarrow\quad k+\ell(P,1)\le l.
\end{equation}
Chromatic ambidexterity and its torsion-nilpotence consequence \cite{CSY} allow this step to use arbitrary nonzero commutative local algebras. A product with increasingly large cyclic \(p\)-groups separates the height \(k\) from the cyclic length.
Finally, finite chromatic fracture converts \eqref{eq:intro-height}
into vanishing estimates for the right adjoints to subgroup geometric
fixed points. The support argument adapts the strategy of
Balmer--Sanders \cite[proof of Theorem~9.1, \textup{(B)}$\Rightarrow$\textup{(A)}]{BS}.
At a least height where the proposed bound fails, the estimates
isolate the offending subgroup component from the other components.
In the finite-height tensor-triangular spectrum, this forces the
support of a compact object to split into two nonempty disjoint
closed parts, although that support is the closure of a point and
hence irreducible. The contradiction supplies the lower bound for
every embedded subgroup.
Published inputs are recalled in \Cref{sec:background}; the character construction, congruences, profinite theorem, higher-height estimate, and support argument occupy \Cref{sec:characters,sec:congruences,sec:profinite,sec:gluing,sec:support}, respectively.
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//01-introduction.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//02-background.tex ====
\section{Chromatic and equivariant preliminaries}\label{sec:background}
We work \(p\)-locally. The category \(\Sp^G\) is the stable category of genuine \(G\)-spectra on a complete universe; its compact objects are retracts of \(p\)-local finite genuine \(G\)-CW spectra. All limits, tensor products, and fixed-point functors are derived. When a localized category is used, its tensor product is the corresponding localized smash product.
\subsection{Chromatic tests and finite localization}
For \(i>0\), choose a finite type-\(i\) spectrum with a periodic self-map and write \(T(i)\) for its telescope. Put \(T(0)=K(0)=H\Q\). The Bousfield class of \(T(i)\) is independent of the choice. For a finite spectrum \(V\),
\begin{equation}\label{eq:finite-tests}
T(i)\wedge V=0\quad\Longleftrightarrow\quad K(i)\wedge V=0.
\end{equation}
The type of a nonzero finite spectrum is its least nonvanishing Morava height. Nilpotence and periodicity give these facts and finite spectra of every type \cite{HS}.
Write \(L_m^f\) for localization at \(T(0)\vee\cdots\vee T(m)\). This is a smashing finite localization; its acyclic compact ideal is generated by a finite spectrum of type \(m+1\). Consequently an \(L_m^f\)-local spectrum is zero if and only if its smashes with \(T(0),\ldots,T(m)\) vanish. We use the telescopic fracture square, a homotopy pullback
\begin{equation}\label{eq:fracture}
\begin{tikzcd}[column sep=large,row sep=large]
L_m^fD \arrow[r]\arrow[d]&L_{T(m)}D\arrow[d]\\
L_{m-1}^fD\arrow[r]&L_{m-1}^fL_{T(m)}D
\end{tikzcd}
\end{equation}
for \(m>0\) and \(D\) already \(L_m^f\)-local; equivalently take \(D=L_m^fS\) for the sphere square used below. The square follows from the complementary localizations at the top telescope and the lower finite heights: \(L_{T(m)}L_{m-1}^f=0\), and the two localizations jointly detect \(L_m^f\)-local equivalences.
\subsection{Genuine fixed points and cofree spectra}
Inflation of an ordinary spectrum \(D\) will be denoted by \(\operatorname{inf}_G D\). It satisfies
\[
\Phi^H(\operatorname{inf}_G D)\simeq D
\qquad(H\le G).
\]
Geometric fixed points are exact and symmetric monoidal, preserve finite objects, and jointly detect equivalences when taken over all subgroups. For a normal subgroup, their genuine quotient versions are transitive.
The functor from genuine \(P\)-spectra to underlying \(P\)-diagrams has a cofree right adjoint. For the constant diagram with value \(D\), put
\[
\Bor_P(D)=F(EP_+,\operatorname{inf}_P D),
\qquad U_P(D)=\Phi^P\Bor_P(D).
\]
We also use \(\Bor_P\) for that right adjoint on a general \(P\)-diagram, specifying the diagram when its action is nontrivial. Its categorical fixed points on a constant diagram are \(D^{BP}=F(BP_+,D)\). The functor \(U_P\) is exact and lax symmetric monoidal. In particular it takes commutative algebras and their modules to commutative algebras and their modules.
If \(X\) is dualizable, adjunction gives the cofree projection formula
\begin{equation}\label{eq:cofree-projection}
X\otimes\Bor_P(M)\simeq
\Bor_P\bigl(\operatorname{res}X\otimes M\bigr),
\end{equation}
where \(\operatorname{res}X\) retains its underlying \(P\)-action. We use the usual restriction, coinduction, and normal geometric-localization formulas for genuine spectra \cite[Section~2]{BS}.
\begin{proposition}[Cyclic upper bound]\label{prop:cyclic-upper}
For every finite \(p\)-group \(G\), subgroup \(H\), and \(n\ge0\),
\(r_n(G,H)\le\ell(G,H)\).
\end{proposition}
\begin{proof}
The finite abelian chromatic Smith theorem gives loss at most one for every nontrivial cyclic \(p\)-group, including target height zero \cite[Theorem~2.1 and Corollary~3.12]{BHNNNS}; see also \cite[Theorem~5.1]{KL}. Along a length-\(s\) cyclic subnormal chain, apply it to the finite genuine \(G_i/G_{i-1}\)-spectrum obtained by restricting \(X\) to \(G_i\) and taking genuine geometric \(G_{i-1}\)-fixed points. Transitivity identifies the underlying and full fixed points with \(\Phi^{G_{i-1}}X\) and \(\Phi^{G_i}X\). Iterating at heights \(n+s-1,\ldots,n\) proves the assertion.
\end{proof}
\subsection{Ambidexterity and Burnside marks}
The relevant ambidexterity belongs to the development of chromatic
Tate vanishing and higher semiadditivity. Kuhn proved telescopic
finite-group Tate vanishing \cite[Theorem~1.5]{KuhnTate}; Hopkins
and Lurie established higher ambidexterity in the Morava-local
category \cite[Theorem~0.0.2]{HL}. The telescopic form below is due
to Carmeli, Schlank, and Yanovski. None of these uses identifies
telescopic localization with Morava localization on arbitrary spectra.
We use the published ambidexterity theorem for \(\Sp_{T(l)}\), \(\Sp_{K(l)}\), and \(K(l)\)-local \(E_l\)-modules \cite[Theorem~5.3.1 and Corollary~5.3.2]{CSY}. In these categories every finite groupoid is ambidextrous. Integration along maps of finite groupoids satisfies composition, homotopy base change, and the projection formula; for finite coverings it is the ordinary sum transfer \cite[Section~3.1]{CSY}. Cochains on a finite groupoid with values in the unit are dualizable \cite[Proposition~3.3.7]{CSY}. Tensoring preserves finite-groupoid limits, since it preserves their ambidextrous colimits \cite[Corollary~3.2.4]{CSY}. Thus the canonical exterior product of unit-cochains is an algebra equivalence
\[
\one^{BY}\otimes\one^{BZ}\simeq\one^{B(Y\times Z)}
\]
for finite groups \(Y,Z\), and similarly for finite groupoids.
For a finite group \(F\), its rational Burnside ring is the product of copies of \(\Q\) indexed by conjugacy classes of subgroups. The \(I\)-mark of a finite \(F\)-set \(X\) is \(|X^I|\). We denote by \(e_I^F\) the idempotent with mark one at the conjugacy class of \(I\) and zero at all others. A finite \(F\)-set \(X\) acts on cochains by multiplication by the transfer of \(1\) along \(X\mathbin{//}F\to BF\); for an orbit, this is restriction followed by transfer. This agrees with the genuine Burnside action. The full mark \(e_F^F\) will be used to isolate full geometric fixed points.
\subsection{Prime points and finite truncation}
For the rest of the paper, primes are indexed by the \emph{tested height}:
\[
\mathcal P_i^G(H)=
\{X\in(\Sp^G)^c:K(i)_*(\Phi^H X)=0\},
\qquad i\ge0.
\]
Thus \(\mathcal P_i^G(H)\) has index \(i+1\) in the convention of \cite{BS}. We suppress \(G\) when it is fixed. The support of a compact object \(X\) consists of the primes not containing it. In particular
\begin{equation}\label{eq:prime-closure}
q\in\overline{\{y\}}\quad\Longleftrightarrow\quad q\subseteq y.
\end{equation}
\begin{proposition}[Uniform finite localization]\label{prop:finite-local}
Let \(\mathcal C_m^G\) be the category of genuine \(G\)-spectra localized by the inflated sphere \(L_m^fS\). It is rigidly compactly generated. Its compact tensor-triangular spectrum is the finite subspace
\[
\{(H,i):H\le G\text{ up to conjugacy},\ 0\le i\le m\}
\]
of the equivariant spectrum, with \((H,i)\) representing \(\mathcal P_i^G(H)\). The image of a finite genuine spectrum is compact and dualizable. An object \(Z\in\mathcal C_m^G\) is zero if and only if
\[
T(i)\wedge\Phi^H Z=0
\qquad(H\le G,\ 0\le i\le m).
\]
For the image of a finite object, its support is tested equally by \(T(i)\) and \(K(i)\) at these heights.
For every compact object of \(\mathcal C_m^G\), membership in the support at
\((H,i)\) is equivalent to nonvanishing of \(K(i)\wedge\Phi^H(-)\).
\end{proposition}
\begin{proof}
Inflation of a finite type-\((m+1)\) spectrum generates the kernel as a localizing tensor ideal. Equivalently, use its products with the finite orbit generators. This is a finite localization, and its compact category is the idempotent completion of the corresponding compact quotient. The point classification and localization theorem of Balmer--Sanders identify its spectrum with the displayed subspace \cite[Theorems~4.9 and 4.14, Remark~5.3, and Proposition~5.24]{BS}. Indeed the inflated generator belongs precisely to the primes at test heights at most \(m\).
Geometric fixed points commute with this inflated localization. Each \(\Phi^H Z\) is therefore \(L_m^f\)-local; telescope detection at the retained heights, followed by genuine geometric detection, proves the zero criterion. For a finite representative, \eqref{eq:finite-tests} gives agreement of the two support tests.
For the assertion about all compact objects, the kernel of
\(K(i)\wedge\Phi^H(-)\) on compacts is a tensor-triangular prime by the
Morava \(K\)-theory field property. Its inverse image in the original compact
category is \(\mathcal P_i^G(H)\), since \(K(i)\wedge L_m^fS\simeq K(i)\)
for \(i\le m\). The injectivity of the spectrum map for finite localization
therefore identifies this kernel with the point labelled \((H,i)\).
\end{proof}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//02-background.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//03-characters.tex ====
\section{Rational characters from finite groups}\label{sec:characters}
Fix a prime $p$ and a positive height $l$, and put
$\one=L_{T(l)}S$. Starting from a finite group $P$ with
$(U_P(\one))_{\Q}\ne0$, we will construct a pro-$p$ group
$\Gamma\twoheadrightarrow P$ whose finite sections require at most $l$
generators. The same construction will describe rational cochain summands
by evaluation at continuous homomorphisms from $\Gamma$ to finite groups.
These evaluation coordinates will allow integration to impose the arithmetic conditions used in
\Cref{sec:profinite}. The construction uses finite-groupoid ambidexterity
and elementary abelian sharpness, without requiring $\Gamma$ to be abelian.
\subsection{The rational tensor category and occurrence}
Let $\mathcal C$ be the homotopy category of dualizable objects in
$\Sp_{T(l)}$, with its local tensor product. Write $\mathcal C_{\Q}$ for
the idempotent completion of the category with the same objects and
\[
\Hom_{\mathcal C_{\Q}}(X,Y)
=\Hom_{\mathcal C}(X,Y)\otimes\Q
\qquad (X,Y\in\mathcal C).
\]
The tensor product extends to this category; its algebras are algebra
objects in this ordinary tensor category. All algebra and module
identities below are inherited from the local category and then
rationalized. In particular, this operation is distinct from ordinary
rationalization of spectra, which we denote by $(-)_{\Q}$.
For a finite groupoid $Y$, put $A_Y=\one^Y$, and abbreviate $A_{BF}$ to
$A_F$. Finite-groupoid ambidexterity makes $A_Y$ dualizable and gives
natural algebra isomorphisms
\begin{equation}\label{eq:cochain-products}
A_{Y\times Z}\simeq A_Y\otimes A_Z,
\qquad
A_{Y\sqcup Z}\simeq A_Y\times A_Z.
\end{equation}
Here and throughout this section tensors are local tensors before passing
to $\mathcal C_{\Q}$. These facts follow from
\cite[Theorem~5.3.1 and Propositions~3.3.1, 3.3.7]{CSY}.
For a finite covering $f:Y\to Z$, the transfer $f_!:A_Y\to A_Z$ satisfies
base change and the projection formula.
Write $\mathbb A(F)$ for the Burnside ring of a finite group $F$. Sending
an $F$-set $S$ to the transfer of $1$ along $S//F\to BF$ gives a ring map
\[
\mathbb A(F)\longrightarrow\pi_0 A_F.
\]
The rational mark isomorphism
$\mathbb A(F)\otimes\Q\cong\prod_{(I\le F)}\Q$
has primitive idempotents $e_I^F$, characterized by the marks
$|(e_I^F)^J|=1$ if $J$ is conjugate to $I$ and $0$ otherwise.
Their images split $A_F$ into commutative algebra summands in
$\mathcal C_{\Q}$. A finite group $L$ \emph{occurs} if
\[
D_L:=e_L^L A_L\ne0.
\]
The unit of $D_L$ is the image of $e_L^L$ in
$\pi_0 A_L\otimes\Q$. Since its identity is multiplication by this
class, $D_L\ne0$ precisely when this unit class is nonzero.
\begin{lemma}\label{lem:occurrence-fixed-points}
For every finite group $L$ and ordinary commutative ring spectrum $R$,
there is an equivalence of rational commutative ring spectra
\begin{equation}\label{eq:general-full-mark-comparison}
e_L^L(R^{BL})_{\Q}\simeq\bigl(U_L(R)\bigr)_{\Q}.
\end{equation}
In particular,
\[
D_L\ne0\quad\Longleftrightarrow\quad
\bigl(U_L(\one)\bigr)_{\Q}\ne0.
\]
\end{lemma}
\begin{proof}
For any genuine $L$-spectrum $X$, categorical fixed points commute with
ordinary rationalization: their homotopy groups are represented by
suspensions of the compact genuine sphere. In the rational genuine
category, $e_L^L X_{\Q}$ restricts to zero at every proper subgroup, since
$e_L^L$ does. The isotropy separation sequence for the family of proper
subgroups consequently identifies
\[
(e_L^L X_{\Q})^L\simeq
\Phi^L(e_L^L X_{\Q})\simeq(\Phi^L X)_{\Q}.
\]
Apply this to the already formed cofree spectrum $X=\Bor_L(R)$, whose
categorical fixed points are $R^{BL}$. The comparison respects
multiplication and identifies the full idempotent unit with the unit
of $(U_L(R))_{\Q}$, proving
\eqref{eq:general-full-mark-comparison}. A unital ring spectrum is zero
exactly when its unit is zero. Taking $R=\one$ and using the preceding
unit criterion for $D_L$ proves the last assertion.
\end{proof}
\begin{proposition}\label{prop:occurrence-sections}
Every occurring finite group is a $p$-group. The class of occurring
groups is closed under subgroups and quotients. If $A$ is any section
of an occurring group, its minimum number of generators satisfies
$d(A)\le l$.
\end{proposition}
\begin{proof}
We first record how a subgroup summand detects occurrence. If $I\le F$,
the mark formula gives
\begin{equation}\label{eq:induced-top-mark}
\operatorname{ind}_I^F(e_I^I)
=|N_F(I):I|\,e_I^F.
\end{equation}
Indeed a subgroup has a nonzero mark on the left only when it is
conjugate to $I$, and the $I$-mark is $|N_F(I):I|$.
Covering transfer carries the full $I$-unit to the right side of
\eqref{eq:induced-top-mark}. Thus $e_I^F A_F\ne0$ implies $D_I\ne0$.
Let $S$ be a Sylow $p$-subgroup of $F$. The class $[F/S]\in\pi_0 A_F$
is a unit. To see this, regard it as an endomorphism of the constant
unit in $\Fun(BF,\Sp_{T(l)})$. Its value at the point is multiplication
by $[F:S]$, a $p$-local unit, and equivalences of diagrams are detected
at the point. If $F$ is not a $p$-group, the full $F$-mark of $[F/S]$
is zero. Hence $e_F^F[F/S]=0$, and invertibility forces $D_F=0$.
For a quotient $q:L\twoheadrightarrow Q$, pullback of $e_Q^Q$ has full
$L$-mark $1$. Its image therefore acts as the identity on $D_L$.
If the full $Q$-unit were zero, its pullback would be zero, contradicting
occurrence of $L$. This proves quotient closure.
For subgroup closure, let $I\le L$. Before any rationalization, there
is a unit-preserving map of spectra
\begin{equation}\label{eq:subgroup-unit-map}
U_I(\one)\longrightarrow U_L(\one).
\end{equation}
Here is its construction. The cofree adjunction gives a map
\[
N_I^L\Bor_I(\one)\longrightarrow\Bor_L(\one)
\]
whose adjoint on underlying spectra with $L$-action is indexed
multiplication
$\bigwedge_{L/I}\one\to\one$.
This uses the ordinary commutative ring spectrum underlying the local
unit; the displayed smash is the ordinary indexed smash. The
multiplication is $L$-equivariant and carries the indexed unit to the
unit. Apply geometric fixed points and the natural norm formula
$\Phi^L N_I^L\simeq\Phi^I$
\cite[Proposition~2.57]{HHR}. This gives
\eqref{eq:subgroup-unit-map}, with the stated unit preservation. If the
source unit were zero rationally, its image would be zero rationally.
\Cref{lem:occurrence-fixed-points} now proves subgroup closure.
It remains to bound generators. If an occurring elementary abelian
group $E=(C_p)^a$ had $a>l$, the elementary abelian sharpness theorem
of Arone, Dwyer, Lesh, and Mahowald, in the form
\cite[Theorems~2.2 and~4.1]{BHNNNS}, would give a finite genuine $E$-spectrum $X$
with underlying type $a$ and full geometric fixed points of type $0$.
Its underlying spectrum is $T(l)$-acyclic. Finite duality and the
cofree projection formula give
\[
X\wedge\Bor_E(\one)
\simeq\Bor_E\bigl((\operatorname{res}X)\wedge\one\bigr)=0.
\]
Here $\operatorname{res}X$ retains its underlying $E$-action. Its smash
with $\one$ is zero because this can be tested on the underlying
spectrum, and smashing a finite spectrum with the local unit computes
its localization. Taking geometric fixed points and
rationalizing yields
\[
(\Phi^E X)_{\Q}\wedge(U_E(\one))_{\Q}=0.
\]
The first factor is a nonzero finite rational spectrum, so it detects
nonzero rational spectra under smash. This contradicts occurrence of
$E$. Finally, every section $A$ of an occurring group occurs, and so
does its Frattini quotient $A/\Phi(A)\cong(C_p)^{d(A)}$. The elementary
abelian bound and the Burnside basis theorem give $d(A)\le l$.
\end{proof}
\subsection{A maximizing tower}
For a profinite group $\Gamma$ and a finite groupoid $Y$, let
\[
\mathcal X_\Gamma(Y)=\Fun_{\cts}(B\Gamma,Y).
\]
Continuity means that the homomorphism into the automorphism group of
the chosen object has open kernel. Thus
$\mathcal X_\Gamma(BF)=\Hom_{\cts}(\Gamma,F)//F$.
A finite section of $\Gamma$ means a finite continuous quotient of a
closed subgroup.
\begin{theorem}[Pro-character construction]\label{thm:pro-characters}
Suppose $P$ is a finite group with $(U_P(\one))_{\Q}\ne0$.
There is a tower of occurring finite $p$-groups
\[
\cdots\twoheadrightarrow L_2\twoheadrightarrow L_1
\twoheadrightarrow L_0=P,
\qquad \Gamma=\varprojlim_i L_i,
\]
with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every finite section $A$ of $\Gamma$ has $d(A)\le l$.
Every closed subgroup of $\Gamma$ is topologically generated by at most
$l$ elements; in particular $\Gamma$ is finitely generated and
$\Gamma\twoheadrightarrow P$.
\item For each finite collection of finite groupoids $Y$, at every
sufficiently advanced stage $L=L_i$ inflation is an equivalence
\begin{equation}\label{eq:mapping-groupoid-stabilization}
\Fun(BL,Y)\xrightarrow{\ \sim\ }\mathcal X_\Gamma(Y).
\end{equation}
At those stages the evaluation maps give an isomorphism of
$D_L$-algebras in $\mathcal C_{\Q}$,
\begin{equation}\label{eq:character-splitting}
D_L\otimes A_Y\xrightarrow{\ \sim\ }
\prod_{[\alpha]\in\pi_0\mathcal X_\Gamma(Y)}D_L.
\end{equation}
These isomorphisms are natural for pullback along maps of the selected
groupoids. In particular, a common stage can be chosen for a finite
diagram of finite groupoids, including both its bases and its total
groupoids.
\end{enumerate}
\end{theorem}
\begin{proof}
Enumerate the isomorphism classes of finite groups as $F_0,F_1,\ldots$.
Given an occurring group $L_i$, choose an occurring extension
$L_{i+1}\twoheadrightarrow L_i$ maximizing
\[
\bigl|\Hom(L_{i+1},F_i)/F_i\bigr|
\]
among all occurring extensions of $L_i$. The set of candidates is
nonempty, since $L_i$ is one. By
\Cref{prop:occurrence-sections}, every candidate has at most $l$
generators, so the displayed integer is at most $|F_i|^l$ and a maximum
is attained. Precomposition with a surjection injects conjugacy
classes of homomorphisms. Once a target $F_i$ has been maximized, every
later occurring extension is still an extension of $L_i$ and cannot
increase the maximum. Consequently, for every sufficiently advanced
stage $L$ and every occurring extension $M\twoheadrightarrow L$,
\begin{equation}\label{eq:maximal-hom-classes}
\Hom(L,F_i)/F_i\xrightarrow{\ \sim\ }\Hom(M,F_i)/F_i.
\end{equation}
Every continuous homomorphism $\Gamma\to F$ factors through some
$L_i$, since the stage kernels form a neighborhood basis of the
identity. After $F$ has been maximized, the preceding bijections imply
that every such homomorphism factors through each later stage. This
is actual factorization: a map conjugate to one factoring through a
quotient kills the same quotient kernel. Surjectivity of
$\Gamma\to L_i$ also identifies all conjugating elements. This proves
\eqref{eq:mapping-groupoid-stabilization} first for $Y=BF$, and then
for finite disjoint unions of such groupoids.
If $H$ is closed in $\Gamma$ and $H/K$ is finite, choose an open normal
subgroup $N\trianglelefteq_o\Gamma$ with $H\cap N\subseteq K$.
Then $H/K$ is a quotient of the subgroup $H/(H\cap N)$ of $\Gamma/N$.
The latter is a quotient of a tower stage, so
\Cref{prop:occurrence-sections} gives $d(H/K)\le l$.
For each open normal $K$ of $H$, the tuples in $H^l$ whose images
generate $H/K$ form a nonempty clopen subset. These subsets have the
finite intersection property, by applying the same bound to the
quotient by an intersection of finitely many such $K$. Compactness
gives a tuple generating every finite quotient, hence topologically
generating $H$. This proves \textup{(i)}.
We prove the splitting for $Y=BF$, at a stage $L$ satisfying
\eqref{eq:maximal-hom-classes} for $F$. We first show that every surviving
subgroup summand comes from a graph, and then identify each graph summand
by restriction. Put $G=L\times F$ and let
$\pi:G\to L$ be the first projection. By
\eqref{eq:cochain-products},
$D_L\otimes A_F$ is the summand $\pi^*(e_L^L)A_G$.
The marks of $\pi^*(e_L^L)$ are supported exactly on subgroups
$M\le G$ projecting onto $L$. If $e_M^G A_G\ne0$, then $M$ occurs by
\eqref{eq:induced-top-mark}. It is therefore an occurring extension of
$L$. Apply \eqref{eq:maximal-hom-classes} to its second projection
$M\to F$: this map is conjugate to $\alpha\circ\pi$ for some
$\alpha:L\to F$. Absorbing the conjugation into $\alpha$ shows that
$M$ is contained in its graph, and surjectivity onto $L$ gives
\[
M=M_\alpha:=\{(x,\alpha(x)):x\in L\}.
\]
Graphs are conjugate in $L\times F$ exactly when the corresponding
homomorphisms are conjugate in $F$: conjugation in the first factor is
already absorbed by conjugation by its image under $\alpha$.
We next identify each graph summand, keeping track of the map.
Let $B_\alpha=e_{M_\alpha}^G A_G$ and write
$j_\alpha:L\xrightarrow{\sim}M_\alpha\hookrightarrow G$.
Restriction along $j_\alpha$, followed by the full projection, gives
\[
r_\alpha:B_\alpha\longrightarrow D_L.
\]
It has a right inverse $s_\alpha$, obtained by pulling back along
$\pi$ and projecting to $B_\alpha$, because
$\pi j_\alpha=\id$ and
$j_\alpha^*(e_{M_\alpha}^G)=e_L^L$.
For the other inverse, let $t_\alpha:D_L\to B_\alpha$ be covering
transfer along $j_\alpha$, followed by projection. The projection
formula gives
\[
t_\alpha r_\alpha
= |(G/M_\alpha)^{M_\alpha}|\,\id_{B_\alpha}
= |C_F(\alpha(L))|\,\id_{B_\alpha}.
\]
Indeed the full graph mark of $[G/M_\alpha]$ is
$|N_G(M_\alpha):M_\alpha|=|C_F(\alpha(L))|$.
This positive integer is invertible in $\mathcal C_{\Q}$, so
$|C_F(\alpha(L))|^{-1}t_\alpha$ is a left inverse.
Thus $r_\alpha$ is an isomorphism, with inverse $s_\alpha$.
In particular every graph summand survives, since $D_L\ne0$.
Combining the graph summands proves
\eqref{eq:character-splitting} for $BF$. Its coordinate at
$\alpha:L\to F$ is explicitly the diagonal restriction
\begin{equation}\label{eq:character-evaluation}
D_L\otimes A_F\longrightarrow D_L,
\qquad d\otimes a\longmapsto d\,\bigl(e_L^L\alpha^*a\bigr).
\end{equation}
It is a map of $D_L$-algebras. Composition of homomorphisms composes
these restrictions, giving naturality. Applying the disjoint-union
identity in \eqref{eq:cochain-products} proves the assertion for every
finite groupoid. Only finitely many component automorphism groups
occur in any finite diagram, so they can all be maximized before
choosing the stage. This proves \textup{(ii)}.
\end{proof}
Naturality also controls homotopy fibers. For a map $Y\to Z$ of finite
groupoids, choose a stage satisfying
\eqref{eq:mapping-groupoid-stabilization} for both $Y$ and $Z$.
The resulting square of mapping groupoids is a square of equivalences
on its horizontal arrows. It therefore identifies homotopy fibers
over any chosen base label. For a groupoid bundle over $BQ$, the
targets to be stabilized include $Q$ and the automorphism groups of
the components of its \emph{total} groupoid. Stabilizing only the
quotient $Q$ defining the action would not ensure that all continuous
lifts of the base label are counted.
\begin{lemma}[Evaluation of covering transfers]\label{lem:cover-evaluation}
Let $f:Y\to Z$ be a finite covering of finite groupoids. Choose a
common stage $L$ as in \Cref{thm:pro-characters}, and a label
$\alpha:BL\to Z$. If the pullback of $f$ is the finite $L$-set $S$,
then the evaluation of $f_!(1)\in\pi_0 A_Z$ at $\alpha$ is
$|S^L|\,1_{D_L}$.
\end{lemma}
\begin{proof}
Base change identifies $\alpha^*f_!(1)$ with the Burnside class $[S]$
in $\pi_0 A_L$. Multiplication by $e_L^L$ selects its full mark
$|S^L|$. This is exactly the evaluation map
\eqref{eq:character-evaluation}.
\end{proof}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//03-characters.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//04-congruences.tex ====
\section{Integration and subgroup congruences}\label{sec:congruences}
Fix $l\geq1$ and retain the group $\Gamma$, the tower, and the notation of
\Cref{thm:pro-characters}. The purpose of this section is to extract arithmetic
information from the natural evaluation splitting. Integration determines the
weights of its coordinates. Integral cochains then force congruences between
weighted counts of finite sets with different actions.
For a finite groupoid $Y$, write
\[
A_Y=\one^Y,\qquad
\mathcal X_\Gamma(Y)=\Fun_{\cts}(B\Gamma,Y),\qquad
a_\alpha=|\Aut_{\mathcal X_\Gamma(Y)}(\alpha)|.
\]
We use finite groupoid models and continuous functors throughout.
The groupoid $\mathcal X_\Gamma(Y)$ is finite up to equivalence: on each
component $BF$ there are finitely many homomorphisms $\Gamma\to F$, since
$\Gamma$ is finitely generated. Its \emph{homotopy cardinality} is
\[
|\mathcal X_\Gamma(Y)|_{\mathrm h}
=\sum_{[\alpha]\in\pi_0\mathcal X_\Gamma(Y)}\frac1{a_\alpha}.
\]
All integrations below are first formed in $\Sp_{T(l)}$ and then transported
to $\mathcal C_{\Q}$. We use their composition, base-change, and module
identities, and their agreement with the sum transfer for finite coverings;
these follow from finite-groupoid ambidexterity
\cite[Theorem~5.3.1, Propositions~3.1.10, 3.1.14, and 3.3.1]{CSY}.
In particular, we do not form new cochain objects after rationalizing morphisms.
\subsection{Absolute and relative integration weights}
Choose a common sufficiently advanced stage $L$ for the finite groupoids under
consideration and put $D=D_L$. Under \Cref{thm:pro-characters} we have
\begin{equation}\label{eq:groupoid-coordinates}
D\otimes A_Y\simeq
\prod_{[\alpha]\in\pi_0\mathcal X_\Gamma(Y)}D.
\end{equation}
The coordinate maps are algebra maps. We denote evaluation at $\alpha$ by
$\chi_\alpha$, and write $e_\alpha$ for the corresponding coordinate idempotent.
Scalar values lie in the ring
$R_D=\Hom_{\mathcal C_{\Q}}(\one,D)$. We also write $\epsilon_Y$
and $f_!$ for their $D$-linear extensions; tensor products between
extended cochain algebras are taken over $D$.
For characters indexed by maps from $\Z_p^l$, the same automorphism weights
as in the following proposition occur in Strickland's Frobenius character
formulas \cite[Section~10 and Theorem~10.2]{StricklandDuality}.
We derive them for $\Gamma$ directly from the diagonal transfer and the
evaluation splitting of \Cref{thm:pro-characters}.
\begin{proposition}[Integration weights]\label{prop:relative-weights}
In the coordinates \eqref{eq:groupoid-coordinates}, the $D$-linear
extension of integration $\epsilon_Y:A_Y\to\one$ is the map
\begin{equation}\label{eq:absolute-weights}
(x_\alpha)_{[\alpha]}\longmapsto
\sum_{[\alpha]}\frac{x_\alpha}{a_\alpha}.
\end{equation}
For a map $f:Y\to Z$, its pushforward has coordinate formula
\begin{equation}\label{eq:relative-weights}
\bigl(f_!(x)\bigr)_\beta
=\sum_{[\alpha]:[f\alpha]=[\beta]}
\frac{a_\beta}{a_\alpha}\,x_\alpha.
\end{equation}
Consequently $\chi_\beta(f_!1)$ is the rational scalar
\begin{equation}\label{eq:relative-fiber-mass}
\left|
\mathcal X_\Gamma(Y)
\mathop{\times}^{h}_{\mathcal X_\Gamma(Z)}\{\beta\}
\right|_{\mathrm h}
\end{equation}
times the unit of $D$.
\end{proposition}
\begin{proof}
The diagonal $\Delta:Y\to Y\times Y$ is a finite covering: its homotopy fiber
at $(y,y')$ is the finite set $\operatorname{Iso}_Y(y,y')$.
By \Cref{lem:cover-evaluation}, its transferred unit evaluates at a pair
$(\alpha,\beta)$ as the number of isomorphisms between the two continuous
functors. Thus
\[
\Delta_!1=\sum_{[\alpha]}a_\alpha(e_\alpha\otimes e_\alpha)
\]
in the coordinates for $Y\times Y$. Composition of integrations, applied to
$\operatorname{pr}_1\Delta=\id_Y$, gives
\[
(\id\otimes\epsilon_Y)(\Delta_!1)=1.
\]
At the $\alpha$ coordinate this reads
$a_\alpha\epsilon_Y(e_\alpha)=1$. Positive integers are invertible in
$\mathcal C_{\Q}$, proving \eqref{eq:absolute-weights}.
Naturality of the evaluation splitting identifies $f^*$ with pullback along
$\pi_0\mathcal X_\Gamma(Y)\to\pi_0\mathcal X_\Gamma(Z)$.
The module identity for $f_!$ therefore forces $f_!(e_\alpha)$ to be supported
at $[f\alpha]$. If its value there is $c_\alpha$, composition with integration
to the point gives
\[
\frac{c_\alpha}{a_{f\alpha}}=\frac1{a_\alpha}.
\]
This proves \eqref{eq:relative-weights}.
For completeness, the homotopy fiber in \eqref{eq:relative-fiber-mass} has
objects $(\alpha,\eta)$ with $\eta:f\alpha\xrightarrow{\sim}\beta$.
The contribution of a fixed isomorphism class $[\alpha]$ is the cardinality
of the action groupoid
\[
\operatorname{Iso}(f\alpha,\beta)\mathbin{//}\Aut(\alpha),
\]
which is $a_\beta/a_\alpha$ if $f\alpha\simeq\beta$ and zero otherwise.
This computation includes automorphisms in the kernel of
$\Aut(\alpha)\to\Aut(f\alpha)$, so it applies to every map of finite
groupoids.
\end{proof}
\subsection{Integral scalar values}
The next argument concerns a cochain whose selected character value has been
computed as a rational scalar. It uses invertibility in the original local
category to constrain that scalar.
\begin{lemma}[Scalar integrality]\label{lem:scalar-integrality}
Let $F$ be finite, let $z\in\pi_0A_F$, and suppose that an evaluation at an
advanced stage satisfies $\chi_\alpha(z)=r\cdot1_D$ for $r\in\Q$.
Then $r\in\Z_{(p)}$. If the restriction of $z$ to the underlying point of
$BF$ is zero in $\pi_0\one$, then $r\in p\Z_{(p)}$.
\end{lemma}
\begin{proof}
We first show that $1+px$ is a unit in $\pi_0\one$ for every
$x\in\pi_0\one$. Put $M=S/p$. In the $p$-local stable category,
$M$-acyclic spectra have $p$ invertible and hence are rational. They are
therefore $T(l)$-acyclic for $l>0$. It follows that every $T(l)$-local
spectrum is $M$-local.
The cofiber triangle
\[
S\xrightarrow{p}S\xrightarrow{i}M\xrightarrow{\partial}\Sigma S
\]
gives $p^2\id_M=0$, including when $p=2$. Indeed,
$(p\id_M)i=ip=0$, so $p\id_M$ factors as $a\partial$ for a map
$a:\Sigma S\to M$. Hence
\[
p^2\id_M=p a\partial=a(\Sigma p)\partial=0.
\]
After smashing with $M$, the endomorphism $1+px$ of $\one$ thus has inverse
$1-px$. Its cofiber is both $M$-local and $M$-acyclic, so it is zero.
This proves the asserted invertibility.
An element of $\pi_0A_F$ is an endomorphism of the constant unit in
$\Fun(BF,\Sp_{T(l)})$. Evaluation at the unique object of $BF$ detects
equivalences of diagrams. Consequently a cochain whose point restriction
is a unit in $\pi_0\one$ is itself a unit in $\pi_0A_F$.
Suppose $r\notin\Z_{(p)}$. Then $r^{-1}\in p\Z_{(p)}$, and the
\emph{integral} cochain
\[
u=1-r^{-1}z\in\pi_0A_F
\]
has point restriction of the form $1+px$. It is a unit, whereas
$\chi_\alpha(u)=0$. An algebra map cannot send a unit to zero in the
nonzero algebra $D$, a contradiction. This proves $r\in\Z_{(p)}$.
If $z$ has zero point restriction and $r\notin p\Z_{(p)}$, then $r$ is a
unit of $\Z_{(p)}$; the same cochain $u$ now has point restriction exactly
$1$, giving the same contradiction.
\end{proof}
\begin{lemma}[Integrality of homomorphism counts]\label{lem:hom-integrality}
For every finite group $F$,
\begin{equation}\label{eq:hom-integrality}
\frac{|\Hom_{\cts}(\Gamma,F)|}{|F|}\in\Z_{(p)}.
\end{equation}
\end{lemma}
\begin{proof}
The integral $\epsilon_{BF}(1)$ is an actual element of $\pi_0\one$.
By \Cref{prop:relative-weights}, its image in a sufficiently advanced
nonzero $D$ is the scalar
\[
\sum_{[\alpha]\in\Hom_{\cts}(\Gamma,F)/F}
\frac1{|C_F(\im\alpha)|}
=\frac{|\Hom_{\cts}(\Gamma,F)|}{|F|},
\]
where the equality is orbit--stabilizer for conjugation by $F$.
Apply \Cref{lem:scalar-integrality} with the trivial group as its $F$.
\end{proof}
For the fixed source $\Z_p^l$, Ben-Moshe proves compatibility of
transchromatic characters with integration. His cardinality formula on
$BF$ gives
$|\Hom_{\cts}(\Z_p^l,F)|/|F|\in\Z_{(p)}$
\cite[Theorems~5.7 and~5.8]{BenMoshe}.
Here the source $\Gamma$ is selected by the occurrence construction of
\Cref{sec:characters}.
For finitely generated discrete groups, Gordon and Rodriguez-Villegas
characterize integrality of these homomorphism-count ratios for all finite
targets by infinite abelianization \cite[Corollary~3.3]{GRV}.
Here the homomorphisms are continuous and the conclusion is $p$-integrality;
cyclic targets will give the first $\Z_p$ quotient in \Cref{sec:profinite}.
\subsection{Labeled sets and open subgroups}
\begin{theorem}[Congruences for open subgroups]\label{thm:subgroup-congruence}
For every open subgroup $U\le_o\Gamma$, every finite continuous $U$-set
$X$, and every $h\geq0$,
\begin{equation}\label{eq:subgroup-congruence}
\sum_{\substack{V\le_o U\\{}[U:V]=p^h}}
\bigl(|X^V|-|X|\bigr)
\in p^{h+1}\Z_{(p)}.
\end{equation}
The sum is over all open subgroups of the indicated index.
\end{theorem}
\begin{proof}
We realize weighted counts of finite $U$-sets equivariantly labeled in
$X$ as character values of integral pushforwards. The logarithm of
their generating function isolates the transitive sets, whose
stabilizers give the open subgroups in \eqref{eq:subgroup-congruence}.
For $j\geq0$, let $\mathcal L_j(X)$ be the groupoid of continuous
$U$-sets $T$ of size $j$ equipped with a $U$-equivariant map $T\to X$.
Its arrows are the equivariant bijections preserving the labeling. Since
$U$ is finitely generated, it has only finitely many homomorphisms to
$\Sigma_j$, so this groupoid is finite up to equivalence. Set
\begin{equation}\label{eq:labeled-count}
b_j(X)=|\mathcal L_j(X)|_{\mathrm h}
=\sum_{[T\to X]}\frac1{|\Aut_U(T\to X)|}.
\end{equation}
Write $X_{\mathrm{triv}}$ for the same underlying set with trivial action.
We first prove
\begin{equation}\label{eq:labeled-integrality}
b_j(X)\in\Z_{(p)},\qquad
b_j(X)-b_j(X_{\mathrm{triv}})\in p\Z_{(p)}.
\end{equation}
Choose $N\trianglelefteq_o\Gamma$ contained in $U$ and in the kernel of
the action on $X$, and put $Q=\Gamma/N$ and $\overline U=U/N$.
The finite groupoid
\[
Y_j(X)=X^j\mathbin{//}\Sigma_j
\]
classifies $j$-element sets labeled in $X$; $\overline U$ acts by changing
the labels. Form its coinduction
\[
Z_j(X)=\operatorname{Coind}_{\overline U}^{Q}Y_j(X).
\]
This is a finite $Q$-groupoid whose underlying groupoid is a product of
$[\Gamma:U]$ copies of $Y_j(X)$. More explicitly, choosing representatives
$r_t$ for $t\in Q/\overline U$, with $r_{\overline U}=1$, gives this product
model; if
$a r_t=r_{at}u(a,t)$, then $a$ sends the $t$-coordinate to the
$at$-coordinate using the action of $u(a,t)\in\overline U$.
Let
\[
\pi_j:E_j(X)=Z_j(X)\mathbin{//}Q\longrightarrow BQ,
\qquad z_j(X)=(\pi_j)_!1\in\pi_0A_Q.
\]
These are a map of finite groupoids and an integral cochain.
Make the same construction for $X_{\mathrm{triv}}$, using the same $N$ and
$Q$. For this fixed $j$, choose a common stage $L$ sufficiently advanced
for $Q$ and every component automorphism group of both total groupoids
$E_j(X)$ and $E_j(X_{\mathrm{triv}})$. Then $D=D_L\ne0$, and inflation gives
equivalences
\[
\Fun(BL,E_j(X))\simeq\mathcal X_\Gamma(E_j(X)),
\qquad
\Fun(BL,BQ)\simeq\mathcal X_\Gamma(BQ),
\]
and likewise for $X_{\mathrm{triv}}$.
In particular, the quotient homomorphism $q:\Gamma\to Q$ factors through
$L$. Factorization gives essential surjectivity of these equivalences,
and surjectivity of $\Gamma\to L$ identifies the intertwining conditions
and gives full faithfulness. The equivalences therefore identify homotopy
fibers over $q$.
The quotient $Q$ specifies the original action, while $L$ also captures
actions on the labeled objects. Each coefficient uses finitely many total
groupoids, so such a common stage exists by \Cref{thm:pro-characters}.
We can now evaluate both cochains at the character indexed by $q$ over
this same $D$. By \Cref{prop:relative-weights}, the value of $z_j(X)$ is
the homotopy cardinality of
\begin{equation}\label{eq:coinduced-fiber}
\begin{aligned}
\mathcal X_\Gamma(E_j(X))
\mathop{\times}^{h}_{\mathcal X_\Gamma(BQ)}\{q\}
&\simeq Z_j(X)^{h\Gamma}\\
&\simeq Y_j(X)^{hU}
\simeq\mathcal L_j(X).
\end{aligned}
\end{equation}
Here $(-)^{h\Gamma}$ and $(-)^{hU}$ mean continuous equivariant objects,
including their coherence isomorphisms. The first equivalence identifies
a lift of $q$, together with its specified identification over $BQ$, with
an equivariant object of the pulled-back bundle.
For the second, evaluate an equivariant object at the identity coset.
To construct an inverse, choose lifts $g_t\in\Gamma$ of the representatives
$r_t$, with $g_U=1$, and place a copy of the $U$-equivariant object in each
coordinate.
The coherence map for $\gamma\in\Gamma$ from coordinate $t$ to coordinate
$\gamma t$ is supplied by the $U$-equivariance map for
\[
g_{\gamma t}^{-1}\gamma g_t\in U.
\]
Its image in $\overline U$ is $u(q\gamma,t)$, as required by the product
action above.
The cocycle identity proves coherence; the same construction on morphisms
shows that the two functors are inverse equivalences.
Continuity follows from continuity of the $U$-equivariance data and the
finiteness of $\Gamma/U$.
Finally, equivariance of a labeled $j$-element set is precisely a continuous
$U$-action on that set for which its map to $X$ is equivariant.
Thus $\chi_q(z_j(X))=b_j(X)\cdot1_D$.
The same argument gives
$\chi_q(z_j(X_{\mathrm{triv}}))=b_j(X_{\mathrm{triv}})\cdot1_D$.
After forgetting their $Q$-actions, the groupoids $Z_j(X)$ and
$Z_j(X_{\mathrm{triv}})$ are equivalent: their product factors have the same
underlying labeled-set groupoid. Base change along $*\to BQ$ therefore
gives the integral equality
\[
z_j(X)|_* =\epsilon_{Z_j(X)}(1)
=\epsilon_{Z_j(X_{\mathrm{triv}})}(1)
=z_j(X_{\mathrm{triv}})|_*
\quad\text{in }\pi_0\one.
\]
Apply \Cref{lem:scalar-integrality} to $z_j(X)$ and to the difference of the
two classes, evaluated over the same $D$. This proves
\eqref{eq:labeled-integrality}.
It remains to pass from all labeled sets to transitive ones. Define the
formal power series
\[
B_X(t)=\sum_{j\geq0}b_j(X)t^j\in1+t\Z_{(p)}[[t]].
\]
Every finite labeled $U$-set is a disjoint union of transitive labeled
$U$-sets. If a transitive isomorphism type $a$ has size $m_a$ and
automorphism group of order $c_a$, then $k$ copies have automorphism group
of order $c_a^k k!$. This orbit decomposition is the mechanism behind
Tamanoi's automorphism-weighted exponential formulas
\cite[Theorem~5-5 and equation~(5-7)]{Tamanoi2003}.
Consequently, degree by degree in $\Q[[t]]$,
\begin{equation}\label{eq:transitive-exponential}
B_X(t)=\prod_a\sum_{k\geq0}\frac{t^{k m_a}}{c_a^k k!},
\qquad
\log B_X(t)=\sum_a\frac{t^{m_a}}{c_a}.
\end{equation}
There are finitely many types in each degree, which justifies these formal
products and sums.
A transitive continuous $U$-set has size $p^h$ for some $h$, since $U$ is
pro-$p$. Pointing a transitive labeled set of size $p^h$ multiplies its
homotopy cardinality by $p^h$ and makes every automorphism trivial.
A pointed object is determined, up to unique isomorphism, by the stabilizer
$V\le_oU$ of its distinguished point and its label $x\in X^V$.
It follows that
\begin{equation}\label{eq:transitive-coefficient}
[t^{p^h}]\log B_X(t)
=\frac1{p^h}\sum_{\substack{V\le_oU\\{}[U:V]=p^h}}|X^V|.
\end{equation}
In particular this normalization counts actual subgroups, without passing
to conjugacy classes.
By \eqref{eq:labeled-integrality},
\[
\frac{B_X(t)}{B_{X_{\mathrm{triv}}}(t)}=1+pA(t)
\quad\text{for some }A(t)\in t\Z_{(p)}[[t]].
\]
Its formal logarithm lies in $p\Z_{(p)}[[t]]$, because
\[
\log(1+pA(t))
=\sum_{q\geq1}\frac{(-1)^{q+1}p^q}{q}A(t)^q,
\qquad q-v_p(q)\geq1.
\]
Only finitely many terms contribute to any coefficient, and the valuation
inequality holds also at $p=2$. Taking the coefficient of $t^{p^h}$,
using \eqref{eq:transitive-coefficient} for $X$ and $X_{\mathrm{triv}}$,
and multiplying by $p^h$ proves \eqref{eq:subgroup-congruence}.
\end{proof}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//04-congruences.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//05-profinite.tex ====
\section{From subgroup congruences to procyclic length}\label{sec:profinite}
We now isolate the group-theoretic consequence of the character calculations.
All homomorphisms between profinite groups are understood to be continuous, and a finite
section means a finite quotient of a closed subgroup. For a finite
$p$-group $F$, write $d(F)$ for its minimum number of generators.
\begin{theorem}\label{thm:procyclic}
Let $\Gamma$ be a finitely generated pro-$p$ group, and let $l\geq0$.
Suppose that $d(F)\leq l$ for every finite section $F$ of $\Gamma$, that
\begin{equation}\label{eq:profinite-hom}
\frac{|\Hom_{\cts}(\Gamma,F)|}{|F|}\in\Z_{(p)}
\qquad\text{for every finite group }F,
\end{equation}
and that, for every $U\leq_o\Gamma$, every finite continuous $U$-set $X$,
and every $h\geq0$,
\begin{equation}\label{eq:profinite-congruences}
\sum_{\substack{V\leq_o U\\{}[U:V]=p^h}}
\bigl(|X^V|-|X|\bigr)
\in p^{h+1}\Z_{(p)}.
\end{equation}
Then $\Gamma$ has a series of closed subgroups
\[
1=K_t\trianglelefteq K_{t-1}\trianglelefteq\cdots
\trianglelefteq K_0=\Gamma,
\qquad t\leq l,
\]
whose nontrivial factors are isomorphic to $\Z_p$.
In particular, every finite quotient $P$ of $\Gamma$ satisfies
$\ell(P,1)\leq l$.
\end{theorem}
The congruence in \eqref{eq:profinite-congruences} is summed over actual
open subgroups. These sums are finite: an open subgroup of a given
index is a point stabilizer in a permutation action of that degree,
and a finitely generated profinite group has only finitely many such
actions. We say that a finitely generated pro-$p$ group
\emph{satisfies the subgroup congruences} when this formula holds
for all its open subgroups. This property is inherited by open
subgroups. For nontrivial $\Gamma$, homomorphism-count integrality
supplies the first $\Z_p$ quotient. After each quotient is extracted,
shrinking its action replaces a semidirect product by a direct product
while preserving the subgroup congruences and section bound.
Subgroup counting then gives another $\Z_p$ quotient whenever the
remaining kernel is nontrivial.
\subsection{Finite quotients and shrinking actions}
We first record the elementary profinite facts needed to make that
limit precise.
\begin{lemma}\label{lem:profinite-elementary}
Let $E$ be a pro-$p$ group.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every finite section of $E$ is a section of a finite quotient of
$E$. If all finite sections of $E$ have generator number at most $l$,
then every closed subgroup of $E$ is topologically generated by at most
$l$ elements.
\item A procyclic pro-$p$ group is either trivial, finite cyclic, or
isomorphic to $\Z_p$. Every continuous epimorphism $K\twoheadrightarrow
\Z_p$ splits.
\item A finitely generated pro-$p$ group with infinite abelianization
has a continuous epimorphism onto $\Z_p$.
\end{enumerate}
\end{lemma}
\begin{proof}
For a closed subgroup $H\leq E$ and $N\trianglelefteq_o H$, choose
$M\trianglelefteq_o E$ with $H\cap M\subseteq N$. Then $H/N$ is a
quotient of the subgroup $H/(H\cap M)$ of $E/M$. Under the asserted
generator bound, for each $N\trianglelefteq_o H$ the set of tuples in
$H^l$ generating $H/N$ is nonempty and closed. These sets have the
finite intersection property, by replacing finitely many $N$ by their
intersection. Compactness supplies a tuple generating every finite
quotient of $H$, hence topologically generating $H$.
A topological generator gives a continuous epimorphism
$\Z_p\twoheadrightarrow Q$ onto any procyclic pro-$p$ group $Q$.
Every closed subgroup of $\Z_p$ is a $\Z_p$-submodule and is either
$0$ or $p^a\Z_p$, proving the classification. If
$K\twoheadrightarrow\Z_p$, choose $x\in K$ mapping to $1$.
The closed subgroup generated by $x$ maps isomorphically to $\Z_p$:
a nonzero closed kernel in a procyclic group would make the quotient
finite. This supplies a continuous section.
Finally, the abelianization of a finitely generated pro-$p$ group is a
finitely generated $\Z_p$-module. The elementary structure theorem over
the discrete valuation ring $\Z_p$ writes it as a finite group times
$\Z_p^b$. If it is infinite, then $b>0$.
\end{proof}
\begin{lemma}[Shrinking an action]\label{lem:shrinking-action}
Let $K$ be a pro-$p$ group and let $d,l\geq0$.
Suppose $K\times\Z_p^d$ satisfies the subgroup congruences and all its
finite sections have generator number at most $l$. If
$K\twoheadrightarrow\Z_p$ has kernel $N$, then
$N\times\Z_p^{d+1}$ has both properties.
\end{lemma}
\begin{proof}
By \Cref{lem:profinite-elementary}, write
$K=N\rtimes_\alpha\Z_p$, with $N$ finitely generated. For $s\geq0$,
put
\[
E_s=(N\rtimes_{\alpha_s}\Z_p)\times\Z_p^d,
\qquad \alpha_s(z)=\alpha(p^s z),
\qquad E_\infty=N\times\Z_p^{d+1}.
\]
The group $E_s$ is isomorphic to the open subgroup
$(N\rtimes p^s\Z_p)\times\Z_p^d$ of $K\times\Z_p^d$, and hence has
both stated properties.
We construct a common finite quotient which captures every permutation
action of a prescribed bounded degree. Fix $M\geq1$, and let
\[
O_M=\bigcap_{\rho\in\Hom_{\cts}(N,\Sigma_M)}\ker\rho.
\]
There are finitely many such homomorphisms, because $N$ is finitely
generated. Thus $O_M$ is open and characteristic in $N$. Choose
$a\geq0$ with $p^a\geq M$. Every $p$-element of $\Sigma_M$ has order
at most $p^a$, as its order is the largest length of one of its
$p$-power cycles. Therefore every homomorphism from $\Z_p$ to
$\Sigma_M$ kills $p^a\Z_p$.
The action $\alpha$ on the finite group $N/O_M$ is continuous, so
$\alpha(p^s\Z_p)$ acts trivially on it for all sufficiently large $s$.
For these $s$, both $E_s$ and $E_\infty$ have the quotient
\begin{equation}\label{eq:shared-permutation-quotient}
Q_M=(N/O_M)\times(\Z/p^a)^{d+1}.
\end{equation}
Every homomorphism from either group to $\Sigma_M$ factors through this
quotient: its restriction to $N$ kills $O_M$, and its restrictions to
the $d+1$ cyclic coordinates kill $p^a\Z_p$.
The same assertion holds for every smaller permutation degree, by
adding fixed points.
Now fix $U\leq_o E_\infty$, a finite continuous $U$-set $X$, and
$h\geq0$. If $X$ is empty, the required congruence is immediate.
Otherwise set $m=[E_\infty:U]$ and take
\[
M\geq\max\{m,\,m|X|,\,mp^h\}.
\]
The coset action on $E_\infty/U$ factors through $Q_M$, so $U$ is the
inverse image of a subgroup $\overline U\leq Q_M$. The action on the
induced finite set $E_\infty\times_U X$ also factors through $Q_M$.
In particular, the quotient kernel acts trivially on its fiber $X$
over $U$, making $X$ a $\overline U$-set. Let $U_s\leq E_s$ be the
inverse image of $\overline U$, and give the same set $X$ the resulting
$U_s$-action.
Every open subgroup $V\leq U$ of index $p^h$ has index $mp^h$ in
$E_\infty$. Its coset action therefore factors through $Q_M$, so $V$
contains the quotient kernel. Exactly the same argument applies to
every open subgroup of index $p^h$ in $U_s$. Thus the two collections
of subgroups are both in bijection with
\[
\{\overline V\leq\overline U:
[\overline U:\overline V]=p^h\}.
\]
The corresponding fixed-point counts on $X$ agree term by term.
The congruence for $U_s$ consequently proves the congruence for $U$.
This proves the required assertion for every open subgroup of
$E_\infty$ and every test on it.
For the generator bound, let $F$ be a finite quotient of $E_\infty$.
Its regular permutation action shows, using $M\geq|F|$, that the
quotient map factors through \eqref{eq:shared-permutation-quotient}.
Thus $F$ is also a quotient of $E_s$ for sufficiently large $s$.
Every finite section of $E_\infty$ is a section of such an $F$ by
\Cref{lem:profinite-elementary}; it therefore has generator number at
most $l$.
\end{proof}
\subsection{Detecting the remaining kernel}
The subgroup-counting formula below is classical; see
Tamanoi \cite[Lemma~5-5]{Tamanoi2001}. We include its proof and record
the \(p\)-adic limit needed for the kernel argument.
\begin{lemma}\label{lem:lattice-limit}
For $d\geq1$, let $a_j(d)$ be the number of open subgroups of index
$p^j$ in $\Z_p^d$. Then
\begin{equation}\label{eq:lattice-generating-function}
\begin{aligned}
\sum_{j\geq0}a_j(d)t^j
&=\prod_{u=0}^{d-1}(1-p^u t)^{-1},\\
\lim_{j\to\infty}a_j(d)
&=c_d:=\prod_{u=1}^{d-1}(1-p^u)^{-1}\ne0
\quad\text{in }\Z_p.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
For $d=1$ there is exactly one subgroup of each index.
For $d>1$, project a sublattice of
$\Z_p\times\Z_p^{d-1}$ onto the first coordinate. If its image is
$p^a\Z_p$ and its intersection with the remaining coordinates has
index $p^b$, it is determined by that intersection and a lift of
$p^a$ modulo it. There are $p^b$ choices of lift, and the total index
is $p^{a+b}$. Consequently
\[
a_j(d)=\sum_{a+b=j}p^b a_b(d-1),
\]
which gives the displayed generating function by induction.
Write
\[
\prod_{u=1}^{d-1}(1-p^u t)^{-1}
=\sum_{m\geq0}b_m t^m.
\]
Each $b_m$ is divisible by $p^m$, and the generating function gives
$a_j(d)=\sum_{m=0}^j b_m$. These partial sums converge $p$-adically.
The convergent geometric series in the finite product identify their
limit with $c_d$. Every factor $1-p^u$, $u\geq1$, is a $p$-adic unit,
so $c_d$ is nonzero. The empty product for $d=1$ is $1$.
\end{proof}
\begin{lemma}\label{lem:kernel-abelianization}
Let $K$ be a nontrivial finitely generated pro-$p$ group and $d\geq1$.
If $K\times\Z_p^d$ satisfies the subgroup congruences, then $K$ has
infinite abelianization.
\end{lemma}
\begin{proof}
Let $\Phi(K)$ be the intersection of the kernels of homomorphisms
$K\to C_p$. There are finitely many such homomorphisms, so
$F=K/\Phi(K)$ is a finite elementary abelian group. It is nontrivial:
a nontrivial pro-$p$ group has a nontrivial finite $p$-group quotient,
and every nontrivial finite $p$-group has a quotient $C_p$.
We will use the following precise property of $F$:
\begin{equation}\label{eq:frattini-detector}
J\leq K\text{ closed and }J\twoheadrightarrow F
\quad\Longrightarrow\quad J=K.
\end{equation}
Indeed, if $J$ were proper, its image would be proper in some finite
quotient of $K$. A maximal subgroup containing that image is normal
of index $p$, as in every finite $p$-group. Its inverse image is the
kernel of a map $K\twoheadrightarrow C_p$ containing $J$, contradicting
surjectivity onto $F$.
The rational full-mark idempotent $e_F^F$ is a rational virtual $F$-set
with mark $1$ at $F$ and mark $0$ at every proper subgroup. Since
$F\ne1$, its underlying virtual cardinality is zero. Choose a positive
integer $D$ clearing its denominators and inflate $De_F^F$ to
$E=K\times\Z_p^d$ through $E\to K\to F$.
For a closed subgroup of $E$, its projection to $K$ is closed.
By \eqref{eq:frattini-detector}, the inflated virtual set has mark $D$
exactly when that projection is all of $K$, and mark zero otherwise.
Let $c_h$ be the number of open subgroups of index $p^h$ in $E$
projecting onto $K$. The congruences extend by additivity to integral
virtual sets. Applying them to $De_F^F$ at $U=E$ gives
\begin{equation}\label{eq:surjective-count-vanishes}
Dc_h\in p^{h+1}\Z_{(p)},
\qquad v_p(c_h)\geq h+1-v_p(D).
\end{equation}
In particular, $c_h\to0$ in $\Q_p$.
We compute this count in a second way. Set $A=\Z_p^d$.
For an open subgroup $H\leq K\times A$ projecting onto $K$, put
$V=H\cap K$ and $W=\operatorname{pr}_A(H)$.
Then $V\trianglelefteq_o K$: a lift in $H$ of any element of $K$
normalizes $V$. Also $W\leq_o A$, and $H/V$ is the graph of a unique
continuous surjection
\[
f:W\twoheadrightarrow K/V.
\]
Conversely, every such triple $(V,W,f)$ gives one subgroup $H$.
Its index is $[K:V][A:W]$. Every open subgroup $W$ of $A$ is
isomorphic to $A$, and hence
\begin{equation}\label{eq:graph-subgroup-count}
c_h=
\sum_{\substack{V\trianglelefteq_o K\\{}[K:V]=p^i,\ i\leq h}}
a_{h-i}(d)\,|\Sur_{\cts}(\Z_p^d,K/V)|.
\end{equation}
There is no automorphism denominator: the actual graph map is uniquely
determined by the actual subgroup.
Suppose now that $K^{\mathrm{ab}}$ were finite. Any $K/V$ appearing
with a nonzero summand in \eqref{eq:graph-subgroup-count} is abelian,
so $V$ contains the closed commutator subgroup. There are therefore
only finitely many contributing $V$. By \Cref{lem:lattice-limit},
\[
\lim_{h\to\infty}c_h
=c_d\sum_{V\trianglelefteq_o K}
|\Sur_{\cts}(\Z_p^d,K/V)|\ne0,
\]
where the sum has only finitely many nonzero terms and is a positive
integer: $V=K$ contributes $1$. This contradicts
\eqref{eq:surjective-count-vanishes}, proving the lemma.
\end{proof}
\subsection{The series and the rational gluing estimate}
\begin{proof}[Proof of \Cref{thm:procyclic}]
If $\Gamma=1$, the empty series suffices. Otherwise $\Gamma$ has
infinite abelianization. For if $\Gamma^{\mathrm{ab}}$ were finite
of order $p^b$, then for every sufficiently large $r$,
\[
|\Hom_{\cts}(\Gamma,C_{p^r})|=p^b.
\]
This follows by decomposing the finite abelianization into cyclic
factors, each of which has as many maps to $C_{p^r}$ as it has
elements once $p^r$ is divisible by its exponent. Equation
\eqref{eq:profinite-hom} would then assert
$p^{b-r}\in\Z_{(p)}$ for arbitrarily large $r$, a contradiction.
Thus \Cref{lem:profinite-elementary} supplies an epimorphism
$K_0=\Gamma\twoheadrightarrow\Z_p$ with kernel $K_1$.
By \Cref{lem:shrinking-action}, $K_1\times\Z_p$ satisfies the
subgroup congruences and has the same finite-section bound. More
generally, after $d$ steps we obtain $K_d\times\Z_p^d$ with these
properties. If $K_d\ne1$, it is finitely generated by
\Cref{lem:profinite-elementary}, and for $d\geq1$,
\Cref{lem:kernel-abelianization} supplies an epimorphism
$K_d\twoheadrightarrow\Z_p$. Take its kernel $K_{d+1}$ and apply
\Cref{lem:shrinking-action} again.
If $K_d\ne1$, it has a quotient $C_p$, so
$K_d\times\Z_p^d$ has a quotient $(C_p)^{d+1}$. The section bound
therefore forces $d+1\leq l$. The construction must terminate with
$K_t=1$ for some $t\leq l$, and
$K_{i-1}/K_i\cong\Z_p$ at every step. Taking the images of these
subgroups in a finite quotient $P$ gives a subnormal series with
cyclic factors. Deleting repetitions proves $\ell(P,1)\leq l$.
\end{proof}
\begin{corollary}[Rational gluing estimate]\label{cor:rational-gluing}
For every finite $p$-group $P$ and every $l\geq0$,
\[
\bigl(U_P(L_{T(l)}S)\bigr)_{\Q}\ne0
\quad\Longrightarrow\quad \ell(P,1)\leq l.
\]
\end{corollary}
\begin{proof}
For $l\geq1$, \Cref{thm:pro-characters} constructs a pro-$p$ group
$\Gamma\twoheadrightarrow P$ with the required finite-section bound.
\Cref{lem:hom-integrality,thm:subgroup-congruence} give
\eqref{eq:profinite-hom} and \eqref{eq:profinite-congruences}.
The conclusion follows from \Cref{thm:procyclic}.
For $l=0$, let $P\ne1$. Rational group cohomology, computed by
averaging, gives $(H\Q)^{BP}\simeq H\Q$.
The transfer class of $P/1$ acts on these cochains as multiplication
by $|P|$, which is invertible, whereas its full $P$-mark is zero.
The full-mark summand is consequently zero. The rational
categorical-to-geometric fixed-point comparison of
\Cref{lem:occurrence-fixed-points} gives
$\bigl(U_P(H\Q)\bigr)_{\Q}=0$.
Cofree completion and geometric fixed points preserve rational
objects, since multiplication by every nonzero integer remains an
equivalence. Hence $U_P(H\Q)=0$.
Thus nonvanishing at $l=0$ forces $P=1$, as required.
\end{proof}
The next elementary consequence of inverse limits will subtract a
free abelian rank in the higher-height argument.
\begin{lemma}\label{lem:product-series}
Let $P$ be a finite $p$-group, and let $k,l\geq0$.
If
\[
\ell\bigl(P\times(C_{p^r})^k,1\bigr)\leq l
\qquad\text{for every }r\geq1,
\]
then $k+\ell(P,1)\leq l$.
\end{lemma}
\begin{proof}
Write $G_r=P\times(C_{p^r})^k$.
At each level let $\mathcal S_r$ be the finite nonempty set of all
cyclic-quotient subnormal chains from $1$ to $G_r$ of length exactly
$l$, allowing repetitions. Projection induces maps
$\mathcal S_{r+1}\to\mathcal S_r$.
There is a compatible choice of chains: in the compact product of
the finite sets $\mathcal S_r$, any finite collection of compatibility
conditions is satisfied by projecting a chain from the largest level
mentioned. The finite intersection property applies even though the
maps between the sets of chains need not be surjective.
Denote the selected chain at level $r$ by $(H_{i,r})_{0\leq i\leq l}$.
Compatibility means that every map $H_{i,r+1}\to H_{i,r}$ is
surjective. The inverse limits give closed subgroups
\[
1=H_0\trianglelefteq H_1\trianglelefteq\cdots
\trianglelefteq H_l=P\times\Z_p^k.
\]
Moreover,
\[
H_i/H_{i-1}\cong
\varprojlim_r H_{i,r}/H_{i-1,r}.
\]
To see surjectivity of this comparison, take a compatible tuple of
cosets. Its nonempty finite sets of representatives at the finite
levels have compatible choices by compactness, giving a representative
in $H_i$. The kernel is $H_{i-1}$.
Each displayed inverse limit is procyclic: the surjective transition
maps of its cyclic finite quotients send generators to generators,
so compatible generators exist by the same compactness argument,
and their resulting closed cyclic subgroup surjects to every level.
Let $A_i$ be the image of $H_i$ in $\Z_p^k$, and set
$J_i=H_i\cap P$. These subgroups are closed, and there is an exact
sequence
\begin{equation}\label{eq:product-series-exact}
1\longrightarrow J_i/J_{i-1}
\longrightarrow H_i/H_{i-1}
\longrightarrow A_i/A_{i-1}\longrightarrow1.
\end{equation}
For the kernel assertion, an element of $H_i$ projecting into
$A_{i-1}$ can be multiplied by an element of $H_{i-1}$ with the same
projection; the resulting element belongs to $J_i$, and
$J_i\cap H_{i-1}=J_{i-1}$.
Each $A_i$ is a $\Z_p$-submodule of $\Z_p^k$.
The procyclic quotient $A_i/A_{i-1}$ has rank at most one, and
the rank increments add to $k$. Exactly $k$ steps therefore increase
rank. At any such step, $H_i/H_{i-1}$ is an infinite procyclic
pro-$p$ group, hence $\Z_p$ by \Cref{lem:profinite-elementary}.
Its finite subgroup $J_i/J_{i-1}$ is trivial. Thus
$J_i=J_{i-1}$ at each of these $k$ steps.
The $J_i$ form a subnormal chain from $1$ to $P$.
Their factors embed in the procyclic factors of the $H_i$ and,
being finite, are cyclic. Deleting the $k$ repetitions just found
leaves a cyclic-quotient chain of length at most $l-k$, proving the
claim.
\end{proof}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//05-profinite.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//06-gluing.tex ====
\section{Gluing at every chromatic height}\label{sec:gluing}
We now strengthen the rational estimate of \Cref{cor:rational-gluing}.
A nonzero height-\(k\) part of a cofree fixed-point algebra will produce
rational nonvanishing after adjoining \(k\) cyclic factors of arbitrarily
large order. The rank subtraction in \Cref{lem:product-series} then
accounts for the additional \(k\) in the estimate.
\subsection{Detecting a nonzero local algebra}
\begin{lemma}[Ring detection]\label{lem:ring-detection}
Let \(R\) be a unital associative ring spectrum and \(k\ge0\). Then
\[
T(k)\wedge R\ne0\quad\Longrightarrow\quad K(k)\wedge R\ne0.
\]
\end{lemma}
\begin{proof}
The assertion at \(k=0\) is immediate. For \(k>0\), choose a
positive-degree periodic self-map \(v:\Sigma^dV\to V\) of a finite
type-\(k\) spectrum whose telescope represents \(T(k)\). Suppose that
\(K(k)\wedge R=0\). In the associative ring spectrum
\[
A=\End(V)\wedge R
\]
consider the element \(x=v\otimes1\in\pi_dA\). Its image in
\(K(j)_*A\) is nilpotent at every finite height \(j\ne k\), since the
corresponding image of \(v\) is already nilpotent. At height \(k\) it is
zero by hypothesis. The mod-\(p\) Hurewicz image of \(v\) is nilpotent
as well: it has positive degree in the bounded graded ring
\(H_*(\End(V);\F_p)\). This remains true after mapping to
\(H_*(A;\F_p)\).
The ring form of the nilpotence theorem therefore gives \(x^N=0\) for
some \(N\) \cite[Theorem~3(i)]{HS}. Here the tests include rational
homology and mod-\(p\) homology, and the theorem imposes no boundedness
condition on \(A\). Acting on \(V\wedge R\) shows that
\((v\wedge1_R)^N\) is null. Its telescope is consequently zero, so
\(T(k)\wedge R=0\), as required.
\end{proof}
\begin{lemma}[The rational unit]\label{lem:rational-unit}
If \(k>0\) and \(B\) is a nonzero commutative algebra in
\(\Sp_{K(k)}\), then its unit has nonzero image in
\(\pi_0B\otimes\Q\).
\end{lemma}
\begin{proof}
The category \(\Sp_{K(k)}\) is stable, \(p\)-local, and
\(1\)-semiadditively symmetric monoidal
\cite[Corollary~5.3.2]{CSY}. Apply the torsion-nilpotence theorem
\cite[Theorem~4.3.5]{CSY} with its coalgebra equal to the unit and its
commutative algebra equal to \(B\). It says that every torsion element
of \(\pi_0B\) is nilpotent. If the unit were torsion, it would be zero.
The unitality identity would then make \(\id_B\) zero, contradicting
\(B\ne0\).
\end{proof}
\subsection{An abelian full-mark trace}
Fix \(k>0\), and let \(E=E_k\) be Lubin--Tate theory of height \(k\)
at \(p\). Thus \(E\) is an even periodic, \(K(k)\)-local
commutative ring with
\[
E_0=W(\F_{p^k})[[u_1,\ldots,u_{k-1}]].
\]
We use its commutative realization from the Goerss--Hopkins--Miller
theorem \cite[Section~7]{GH}.
We write \(\widehat\otimes_E\) for tensor product in the category of
\(K(k)\)-local \(E\)-modules. In particular, the hat denotes the
local tensor product, not an assumption of flatness on coefficients.
\begin{proposition}[Surjective characters survive base change]
\label{prop:abelian-full-mark}
Let \(B\) be a nonzero commutative \(E\)-algebra in
\(\Sp_{K(k)}\). For every \(r\ge1\), put \(C=(C_{p^r})^k\).
Then
\[
\bigl(U_C(B)\bigr)_{\Q}\ne0.
\]
\end{proposition}
\begin{proof}
By \Cref{lem:occurrence-fixed-points}, it suffices to show that the
full-mark projector on \(B^{BC}\) is nonzero after rationalizing
morphisms. We will detect it by computing its trace over \(E\) and
then extending scalars to \(B\).
With degree-zero coordinates for the formal
group, the complex-oriented cyclic calculation gives
\[
E^0(BC)=E_0[[x_1,\ldots,x_k]]/
([p^r](x_1),\ldots,[p^r](x_k)).
\]
It is finite free of rank \(p^{rk^2}\) over \(E_0\), and
\(E^*(BC)\) is concentrated in even degrees
\cite[Corollaries~5.8 and~5.10]{HKR}. Choosing representatives for a
degree-zero basis gives an equivalence
\[
E^{BC}\simeq E^{\oplus p^{rk^2}}
\]
of \(E\)-modules. Thus the categorical trace of an endomorphism of
\(E^{BC}\) is its ordinary matrix trace over \(E_0\).
Lubin--Tate theory satisfies the complete-local and finite-height
hypotheses of the generalized character theorem
\cite[Theorem~C and the examples following it]{HKR}. After a faithfully
flat extension of \(E_0[1/p]\), the character isomorphism identifies
the base change of \(E^0(BC)\) with functions on
\(\Hom_{\cts}(\Z_p^k,C)\). A finite \(C\)-set \(S\) acts at a
character \(\alpha\) by \(|S^{\im\alpha}|\), by the transfer formula
\cite[Theorem~D and Example~6.16]{HKR}. Consequently the full mark
\(e_C^C\) is the projector retaining precisely the surjective
characters. Its trace is
\begin{equation}\label{eq:full-mark-trace}
\tr_E(e_C^C\mid E^{BC})
=N_{k,r}\cdot1,
\qquad
N_{k,r}=p^{k^2(r-1)}\prod_{i=0}^{k-1}(p^k-p^i)>0
\end{equation}
in \(E_0\otimes\Q\). Indeed every homomorphism to \(C\) factors
through \((\Z/p^r)^k\); it is onto precisely when its matrix reduces
to an invertible matrix over \(\F_p\). Each such matrix has
\(p^{k^2(r-1)}\) lifts. Faithful flatness descends this trace equality
from the character extension to \(E_0[1/p]\).
Now apply extension of scalars
\[
B\widehat\otimes_E(-):
\Mod_E(\Sp_{K(k)})\longrightarrow\Mod_B(\Sp_{K(k)}).
\]
Both module categories are \(1\)-semiadditive: their finite-groupoid
norms are computed after forgetting the module structure, where they
are equivalences. Extension of scalars preserves colimits and hence
finite-groupoid limits \cite[Corollary~3.2.4]{CSY}. We obtain the
canonical equivalence
\begin{equation}\label{eq:local-cochain-base-change}
B\widehat\otimes_E E^{BC}\simeq B^{BC}.
\end{equation}
It respects the cochain algebra structures, restrictions, and
finite-cover transfers; compatibility of the canonical norms with
such functors gives the transfer assertion
\cite[Theorem~3.2.3 and Corollary~3.2.7]{CSY}. In particular, it carries the Burnside
projector over \(E\) to that over \(B\).
This symmetric monoidal functor preserves categorical traces of
dualizable objects. After tensoring morphism groups with \(\Q\),
\eqref{eq:full-mark-trace} therefore becomes
\[
\tr_B(e_C^C\mid B^{BC})=N_{k,r}\cdot1_B
\quad\text{in }\pi_0B\otimes\Q.
\]
This is nonzero by \Cref{lem:rational-unit}. The projector is thus a
nonzero endomorphism in the rationalized module category. It acts by
multiplication by
\(a=e_C^C\cdot1\in\pi_0(B^{BC})\otimes\Q\), so \(a\ne0\).
The rational full-mark comparison of
\Cref{lem:occurrence-fixed-points} identifies this element with the
unit of \((U_C(B))_{\Q}\), which proves the proposition.
All tensor products and groupoid limits in
\eqref{eq:local-cochain-base-change} were formed before rationalizing
morphisms. The last step concerns the ordinary rational homotopy of
the already formed cofree spectrum.
\end{proof}
\subsection{The product comparison and the height bound}
\begin{lemma}[Coherent product comparison]\label{lem:product-comparison}
For finite groups \(P,C\) and an ordinary commutative ring spectrum
\(D\), there is a natural unit-preserving map
\[
U_{P\times C}(D)\longrightarrow U_C(U_P(D)),
\]
where each displayed cofree construction uses the trivial action on
its coefficient algebra.
\end{lemma}
\begin{proof}
Write \(G=P\times C\). For a genuine \(G\)-spectrum \(X\), let
\(\operatorname{res}_{\mathrm{mix}}X\in\Fun(BC,\Sp^P)\) be its
restriction to a complete \(P\)-universe with trivial \(C\)-action.
Let \(\operatorname{res}_C\) take a genuine \(C\)-spectrum to its
underlying coherent \(C\)-diagram. We first justify the natural comparison
\begin{equation}\label{eq:geometric-mixed-comparison}
\operatorname{res}_C(\Phi^P X)
\simeq\Phi^P(\operatorname{res}_{\mathrm{mix}}X).
\end{equation}
On the left, geometric fixed points retain genuine quotient symmetry;
on the right, they are applied pointwise in \(\Fun(BC,\Sp^P)\).
Let \(\rho_P,\rho_C\) be the real regular representations, and let
\(\overline\rho_P\) be the reduced regular representation of \(P\).
Put \(V=\overline\rho_P\otimes1_C\) and
\(W=\overline\rho_P\otimes\rho_C\). For every \(H\le G\),
\[
\dim W^H=[C:\operatorname{pr}_C H]\bigl([P:H\cap P]-1\bigr).
\]
Thus \(S^{\infty W}\) is the localization sphere for the family of
subgroups not containing \(P\). The invariant line
\(1_C\to\rho_C\) induces a unital localization comparison
\(S^{\infty V}\to S^{\infty W}\) which is a \(P\)-equivalence.
After restriction to \(\Fun(BC,\Sp^P)\), this map is an equivalence
and compares the two isotropy-separation formulas. For the right side of
\eqref{eq:geometric-mixed-comparison}, smash with \(S^{\infty V}\)
and take derived categorical \(P\)-fixed points pointwise. For the left
side, use \(S^{\infty W}\) and evaluate the normal geometric-fixed-point
formula on trivial \(C\)-representations. The comparison of spheres
therefore gives \eqref{eq:geometric-mixed-comparison}, retaining the
coherent \(C\)-action and the commutative algebra structures.
Now set \(Q=\Phi^P\Bor_G(D)\), retaining its genuine
\(C\)-structure. Model the cofree algebra as
\[
\Bor_G(D)=F((EP\times EC)_+,\operatorname{inf}_G D).
\]
Based-space cotensors are computed levelwise, so restriction to the
chosen \(P\)-universe identifies its mixed restriction with
\(F_P((EP\times EC)_+,\operatorname{inf}_P D)\), including the
coherent \(C\)-action. Projection \(EP\times EC\to EP\) induces
a map of commutative algebras
\[
\operatorname{const}_C\Bor_P(D)
\longrightarrow\operatorname{res}_{\mathrm{mix}}\Bor_G(D).
\]
Forgetting \(C\), the projection is a \(P\)-equivalence: the spaces
are contractible at the trivial subgroup and have empty fixed points
at every nontrivial subgroup of \(P\). The displayed map is therefore
an equivalence in \(\Fun(BC,\Sp^P)\). Applying pointwise geometric
\(P\)-fixed points and \eqref{eq:geometric-mixed-comparison} gives
an equivalence of commutative algebras
\[
\operatorname{res}_C Q\simeq\operatorname{const}_C U_P(D)
\quad\text{in }\Fun(BC,\Sp).
\]
Now apply \(\Phi^C\) to the unit
\(Q\to\Bor_C(\operatorname{res}_C Q)\) of the underlying/cofree
adjunction. Transitivity of geometric fixed points identifies its
source with \(U_{P\times C}(D)\) and its target with
\(U_C(U_P(D))\). The construction takes place in commutative
algebras, so the map preserves units.
\end{proof}
\begin{theorem}[Height gluing]\label{thm:height-gluing}
For a finite \(p\)-group \(P\) and integers \(k,l\ge0\),
\begin{equation}\label{eq:height-gluing}
T(k)\wedge U_P(L_{T(l)}S)\ne0
\quad\Longrightarrow\quad k+\ell(P,1)\le l.
\end{equation}
\end{theorem}
\begin{proof}
The case \(k=0\) is \Cref{cor:rational-gluing}. For \(l=0\),
the cofree rational calculation gives \(U_P(H\Q)=0\) for
\(P\ne1\), while \(U_1(H\Q)=H\Q\); the latter is killed by
every positive-height telescope. Hence we may assume \(k,l>0\).
Put \(R=U_P(L_{T(l)}S)\). This is an ordinary commutative ring
spectrum, and \Cref{lem:ring-detection} gives
\(K(k)\wedge R\ne0\). Set
\[
B=L_{K(k)}(E_k\wedge R).
\]
The algebra \(B\) is nonzero. To check this without any commutativity
assumption on Morava \(K\)-theory, use the graded-field module
decomposition \cite[Proposition~1.4]{HS}
\[
K(k)\wedge E_k\simeq\bigvee_{a\in I}\Sigma^{d_a}K(k).
\]
Here \(I\ne\varnothing\), since \(E_k\) is nonzero and
\(K(k)\)-local. Smashing with \(R\) leaves a nonempty wedge of
nonzero spectra, so \(K(k)\wedge E_k\wedge R\ne0\). Localization
preserves this homology. Moreover, \(B\) is a commutative
\(E_k\)-algebra in the local symmetric monoidal category, and the
natural map \(R\to B\) is unital.
For each \(r\ge1\), take \(C=(C_{p^r})^k\).
\Cref{lem:product-comparison} and the map \(R\to B\) give unital
maps
\[
U_{P\times C}(L_{T(l)}S)
\longrightarrow U_C(R)\longrightarrow U_C(B).
\]
The target has a nonzero rational unit by
\Cref{prop:abelian-full-mark}. The source unit therefore survives
rationalization as well. Applying \Cref{cor:rational-gluing} to
\(P\times C\), we obtain
\[
\ell(P\times(C_{p^r})^k,1)\le l\qquad(r\ge1).
\]
By \Cref{lem:product-series}, these inequalities imply
\(k+\ell(P,1)\le l\).
\end{proof}
\subsection{Finite chromatic localization}
\begin{corollary}[Finite gluing]\label{cor:finite-gluing}
For every finite \(p\)-group \(P\) and all \(m,k\ge0\),
\begin{equation}\label{eq:finite-gluing}
m<k+\ell(P,1)\quad\Longrightarrow\quad
T(k)\wedge U_P(L_m^fS)=0.
\end{equation}
\end{corollary}
\begin{proof}
Fix \(k\ge0\) and write \(d=\ell(P,1)\). We induct on \(m\),
retaining all values of \(k\), including \(k>m\). At \(m=0\),
the calculation \(U_P(H\Q)=0\) for \(P\ne1\) proves the
assertion when \(d>0\). If \(P=1\), the required cases have
\(k>0\), and \(T(k)\wedge H\Q=0\).
Let \(m>0\) and assume \(m<k+d\). Apply the exact functor
\(U_P\) and then smash with \(T(k)\) in the sphere fracture square
\eqref{eq:fracture}. The corner \(U_P(L_{m-1}^fS)\) vanishes after
this test by induction. The corner \(U_P(L_{T(m)}S)\) vanishes by
\Cref{thm:height-gluing}.
For the overlap, smashing finite localization makes
\(L_{m-1}^fL_{T(m)}S\) a module over \(L_{m-1}^fS\).
Lax monoidality therefore makes its image under \(U_P\) a unital
module over \(U_P(L_{m-1}^fS)\). If \(T\wedge A=0\) and \(M\)
is any unital \(A\)-module, the unit and action exhibit
\[
M\longrightarrow A\wedge M\longrightarrow M
\]
as a retraction, proving \(T\wedge M=0\). Thus the overlap also
vanishes after smashing with \(T(k)\). Three corners of the
resulting pullback square are zero, so its remaining corner is zero.
This completes the induction.
\end{proof}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//06-gluing.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//07-support.tex ====
\section{Separation of subgroup supports}\label{sec:support}
\Cref{cor:finite-gluing} bounds the interaction between a subgroup and a
normal quotient. We now use those bounds to control specialization for every
embedded subgroup. We follow the finite-localization and support-splitting
strategy of Balmer--Sanders \cite[proof of Theorem~9.1, \textup{(B)}
$\Rightarrow$\textup{(A)}]{BS}: a specialization violating the desired bound
would disconnect the irreducible support of a compact object. Here the
cyclic-length estimate isolates an individual subgroup component by
controlling its interactions with normal conjugates above and below it.
\subsection{Subgroup layers and their right adjoints}
Fix a finite \(p\)-group \(G\) and an integer \(m\ge0\). Put
\[
E=L_m^fS,\qquad e=\operatorname{inf}_G E,
\qquad \mathcal C=\mathcal C_m^G.
\]
For \(J\le G\), write \(N_J=N_G(J)\), \(W_J=N_J/J\), and
\[
\mathcal D_J=\Fun(BW_J,\Sp_{L_m^f}).
\]
Let \(\phi_J:\mathcal C\to\mathcal D_J\) be geometric \(J\)-fixed points,
retaining the underlying coherent \(W_J\)-action. We use one representative
of each conjugacy class when indexing these functors. The notation
\(J\sim K\) means conjugacy in \(G\).
\begin{lemma}[Layer adjunctions]\label{lem:layer-adjunction}
The functor \(\phi_J\) is strong symmetric monoidal and has a fully faithful
right adjoint \(F_J\). Write
\(\mathcal A_J=F_J(\one_{\mathcal D_J})\). Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The underlying geometric fixed points of its unit value are
\begin{equation}\label{eq:layer-unit}
\Phi^K \mathcal A_J\simeq
\bigoplus_{\substack{J'\sim J\\J'\trianglelefteq K}}
U_{K/J'}(E).
\end{equation}
For arbitrary \(M\in\mathcal D_J\), choose, for each normal conjugate
\(J'=gJg^{-1}\trianglelefteq K\), the diagram \(M_g\) obtained by
restricting \(M\) along
\[
K/J'\longrightarrow W_J,\qquad xJ'\longmapsto g^{-1}xgJ.
\]
The formula for \(\Phi^K F_JM\) has the same index set, with summand
\(\Phi^{K/J'}\Bor_{K/J'}(M_g)\). In particular,
\(\phi_KF_JM=0\) if no conjugate of \(J\) is normal in \(K\).
\item For every dualizable \(z\in\mathcal C\) and every
\(M\in\mathcal D_J\), the projection map is an equivalence
\begin{equation}\label{eq:layer-projection}
z\otimes F_JM\simeq F_J(\phi_Jz\otimes M).
\end{equation}
\item The object \(\mathcal A_J\) is a commutative algebra, and \(F_JM\) is a
unital \(\mathcal A_J\)-module.
\end{enumerate}
\end{lemma}
\begin{proof}
First work before chromatic localization. With \(N=N_J\), \(W=W_J\),
factor the functor as
\[
\Sp^G\xrightarrow{\operatorname{Res}_N^G}\Sp^N
\xrightarrow{\Phi_N^J}\Sp^W
\longrightarrow\Fun(BW,\Sp).
\]
Here the middle functor retains genuine quotient symmetry. Let
\(\mathscr F_J=\{L\le N:J\not\le L\}\), and let
\(\widetilde E\mathscr F_J\) be the cofiber of
\((E\mathscr F_J)_+\to S\). Normal geometric localization identifies
\(\Sp^W\) with the full subcategory of genuine \(N\)-spectra whose
geometric fixed points vanish at subgroups not containing \(J\)
\cite[Section~2\textup{(H)}]{BS}. Its fully faithful right adjoint is
\[
\rho_J(Y)=\widetilde E\mathscr F_J\otimes\pi^*Y,
\qquad \pi:N\longrightarrow W,
\]
where \(\pi^*\) is genuine inflation. Consequently the required right
adjoint before chromatic localization is the explicit composite
\begin{equation}\label{eq:layer-construction}
F_JM=\operatorname{Coind}_{N}^{G}\bigl(\rho_J\Bor_W(M)\bigr).
\end{equation}
For completeness, compute its geometric \(K\)-fixed points. Coinduction
equals induction for finite groups. The fixed-orbit formula therefore gives
one term for each coset \(gN\in(G/N)^K\). Such a coset satisfies
\(g^{-1}Kg\le N\). The factor \(\widetilde E\mathscr F_J\) kills this
term unless \(J\le g^{-1}Kg\). Thus the surviving cosets correspond
bijectively to conjugates \(J'=gJg^{-1}\) normal in \(K\). Above such a
subgroup, normal geometric inflation has the quotient fixed-point formula;
restriction of a cofree spectrum is again cofree. Its coefficient diagram
is the restriction \(M_g\) specified in the statement, so the resulting
term is \(\Phi^{K/J'}\Bor_{K/J'}(M_g)\). For the constant diagram
\(E\), this is exactly \(U_{K/J'}(E)\), proving
\eqref{eq:layer-unit}. If the index set is empty, the fixed points vanish
for every \(M\).
All these constructions retain the residual actions. More explicitly,
\(N_G(K)\) acts on the surviving cosets by left translation, or equivalently
on the normal conjugates \(J'\) by conjugation. It transports each term by
the conjugation and restriction maps in \eqref{eq:layer-construction}; the
action factors coherently through \(W_K\). Formula
\eqref{eq:layer-unit} records the underlying spectrum of this diagram.
At \(K=J\), a surviving conjugate \(J'\le J\) has the same order as
\(J\), so \(J'=J\). Only the identity coset of \(N\) remains. The
counit \(\phi_JF_JM\to M\) then identifies with the cofree counit on
underlying \(W_J\)-diagrams and is an equivalence. This proves full
faithfulness, including when \(J\) is not normal in \(G\).
The unlocalized right adjoint already takes \(E\)-local diagrams to
\(\mathcal C\). Indeed, for such an \(M\) and every genuine \(G\)-spectrum
\(X\), adjunction gives
\begin{align*}
\Map(e\otimes X,F_JM)
&\simeq\Map(E\otimes\phi_JX,M)\\
&\simeq\Map(\phi_JX,M)
\simeq\Map(X,F_JM).
\end{align*}
These equivalences identify precomposition with the localization map
\(X\to e\otimes X\); this is exactly the criterion for \(F_JM\) to be
local. Hence the adjunction and its formulas restrict to the categories
under consideration.
In the restricted adjunction, duality gives, for \(X\in\mathcal C\),
\begin{align*}
\Map(X,z\otimes F_JM)
&\simeq\Map(z^\vee\otimes X,F_JM)\\
&\simeq\Map(\phi_J(z^\vee)\otimes\phi_JX,M)\\
&\simeq\Map(\phi_JX,\phi_Jz\otimes M).
\end{align*}
This proves \eqref{eq:layer-projection}. It uses dualizability in
\(\mathcal C\), and places no finiteness condition on \(M\).
The projection equivalence also identifies the unit on \(z\otimes w\)
with \(z\) tensored with the unit on \(w\): their adjoints agree by the
triangle identity
\(\varepsilon_{\phi_Jw}\circ\phi_J(\eta_w)=\id_{\phi_Jw}\).
Finally,
the right adjoint of a strong symmetric monoidal functor is lax symmetric
monoidal; its unit value is therefore a commutative algebra and each of its
values is a unital module over that algebra.
\end{proof}
\subsection{A finite splitting argument}
For a dualizable object \(z\) and a subgroup \(B\le G\), the unit
\(z\to F_B\phi_Bz\) becomes an equivalence after applying \(\phi_B\).
To split this map, we will confine its target to that layer and show
that its fiber remains zero there while the other layers are removed.
The two hypotheses below provide these separate controls through the
projection formula.
\begin{lemma}[Splitting an isolated layer]\label{lem:layer-splitting}
Let \(z\in\mathcal C\) be dualizable and let \(B\le G\). Suppose that
\begin{align}
\phi_K(z\otimes \mathcal A_B)&=0 &&(K\not\sim B),\label{eq:outgoing-orthogonality}\\
\phi_B(z\otimes \mathcal A_J)&=0 &&(J\not\sim B).\label{eq:incoming-orthogonality}
\end{align}
Then the adjunction unit splits as
\[
z\simeq X\oplus Q,\qquad Q=F_B\phi_Bz\simeq z\otimes \mathcal A_B,
\]
with \(\phi_BX=0\), \(\phi_KQ=0\) for \(K\not\sim B\), and
\(X\otimes Q=0\). If \(z\) is compact, both summands are compact.
\end{lemma}
\begin{proof}
Let \(X\) be the fiber of \(z\to Q\). By the projection formula,
\[
X=z\otimes w_0,\qquad
w_0=\operatorname{fib}(\one_{\mathcal C}\to \mathcal A_B).
\]
Full faithfulness of \(F_B\) gives \(\phi_BX=0\).
List the other subgroup conjugacy classes as \(J_1,\ldots,J_t\), in
nondecreasing order of subgroup size. Set \(X_0=X\) and inductively form
fiber sequences
\begin{equation}\label{eq:layer-stripping}
X_s\longrightarrow X_{s-1}\longrightarrow
Y_s:=F_{J_s}\phi_{J_s}X_{s-1}
\qquad(1\le s\le t).
\end{equation}
We verify that the process preserves three properties:
\(X_s=z\otimes w_s\) for some \(w_s\), its \(B\)-component is zero,
and its \(J_1,\ldots,J_s\)-components are zero.
If \(X_{s-1}=z\otimes w_{s-1}\), the projection formula identifies
\[
Y_s\simeq z\otimes F_{J_s}\phi_{J_s}w_{s-1}.
\]
It also identifies the map to \(Y_s\) with \(z\) tensored with the
adjunction unit on \(w_{s-1}\). Thus its fiber has the same form.
The \(J_s\)-component of the fiber vanishes by full faithfulness.
A previously removed subgroup has smaller order, or the same order in a
different conjugacy class. It cannot contain a conjugate of \(J_s\), so
\Cref{lem:layer-adjunction}\textup{(i)} shows that its component of
\(Y_s\) is zero. Its vanishing therefore persists in the fiber.
To check the \(B\)-component, put
\(M_s=F_{J_s}\phi_{J_s}w_{s-1}\). This is an \(\mathcal A_{J_s}\)-module.
For any unital \(A\)-module \(M\), the unit and action exhibit
\(M\) as a retract of \(A\otimes M\). Consequently
\(\phi_Bz\otimes\phi_BM_s\) is a retract of
\[
(\phi_Bz\otimes\phi_B\mathcal A_{J_s})\otimes\phi_BM_s=0
\]
by \eqref{eq:incoming-orthogonality}. Hence \(\phi_BY_s=0\), and
\(\phi_BX_s=0\) follows. This argument applies to the possibly
noncompact objects \(w_s\) and \(Y_s\).
At the end every subgroup component of \(X_t\) is zero; geometric
detection gives \(X_t=0\). On the other hand,
\eqref{eq:outgoing-orthogonality} and adjunction give, for each \(s\),
\[
\Map(Q,Y_s)
\simeq\Map(\phi_{J_s}Q,\phi_{J_s}X_{s-1})=0.
\]
Here \(\Map\) denotes the mapping spectrum, so this vanishing holds in
every degree. Applying \(\Map(Q,-)\) to
\eqref{eq:layer-stripping} and descending from \(X_t=0\) gives
\(\Map(Q,X)=0\). In particular the boundary \(Q\to\Sigma X\)
in the original fiber sequence is zero, proving the splitting.
The stated component vanishings follow from construction and
\eqref{eq:outgoing-orthogonality}. They imply
\(\Phi^K(X\otimes Q)=0\) for every \(K\), hence \(X\otimes Q=0\).
When \(z\) is compact, its retracts \(X\) and \(Q\) are compact as well.
\end{proof}
\subsection{Minimal-height separation}
\begin{lemma}[Chromatic orthogonality]\label{lem:layer-orthogonality}
Let \(z\in\mathcal C_m^G\) be the image of a finite genuine
\(G\)-spectrum, and let \(n\le m\). Suppose that \(B\le G\) satisfies
\[
m-n<\ell(G,B),
\]
and that every \((K,k)\in\supp(z)\) with \(k<m\) satisfies
\[
\ell(G,K)\le k-n.
\]
Then \(z\) and \(B\) satisfy both orthogonality conditions in
\Cref{lem:layer-splitting}.
\end{lemma}
\begin{proof}
There is no point \((B,k)\in\supp(z)\) with \(k<m\): otherwise the
second hypothesis would contradict the first. Thus only the test at height
\(m\) can be nonzero on \(\Phi^Bz\).
Fix \(K\not\sim B\). In a summand of \(\Phi^K\mathcal A_B\), let
\(B'\sim B\) be normal in \(K\), and put
\(d=\ell(K,B')=\ell(K/B',1)\). The equality follows by the subgroup
correspondence for \(K\to K/B'\). Since \(K\not\sim B\), this quotient
is nontrivial and \(d>0\). Consider a height
\(0\le k\le m\) with \((K,k)\in\supp(z)\). If \(k=m\), then
\(m<k+d\). If \(k<m\), concatenate cyclic subnormal chains and use
conjugacy invariance to obtain
\[
m-n<\ell(G,B')
\le\ell(K,B')+\ell(G,K)
\le d+k-n.
\]
Again \(m<k+d\), so \Cref{cor:finite-gluing} gives
\(T(k)\wedge U_{K/B'}(E)=0\).
At a height outside \(\supp(z)\), the finite representative of \(z\)
instead gives \(T(k)\wedge\Phi^Kz=0\), by
\Cref{prop:finite-local}. Thus every retained telescope kills
\(\Phi^K(z\otimes \mathcal A_B)\). This spectrum is \(L_m^f\)-local, so it is
zero. Forgetting a coherent action detects zero, which proves
\eqref{eq:outgoing-orthogonality}.
For \(J\not\sim B\), every summand of \(\Phi^B\mathcal A_J\) comes from a
proper normal subgroup \(J'\sim J\) of \(B\). Since
\(\ell(B/J',1)>0\), \Cref{cor:finite-gluing} at \(k=m\) kills that
summand after smashing with \(T(m)\). All smaller-height tests are killed
by the \(z\)-factor. The same detection argument proves
\eqref{eq:incoming-orthogonality}.
\end{proof}
\begin{theorem}[Relative specialization bound]\label{thm:specialization}
For a finite \(p\)-group \(G\), a subgroup \(H\le G\), and
\(m,n\ge0\),
\[
(H,m)\in\overline{\{(G,n)\}}
\quad\Longrightarrow\quad
m\ge n+\ell(G,H).
\]
The closure is taken in the tensor-triangular spectrum of finite genuine
\(p\)-local \(G\)-spectra.
\end{theorem}
\begin{proof}
Fix \(G,n\) and put \(y=(G,n)\). Any specialization \((H,m)\) of
\(y\) has \(m\ge n\). Indeed, if \(m<n\), inflate a finite spectrum
of exact type \(m+1\). It belongs to \(\mathcal P_m(H)\) and not to
\(\mathcal P_n(G)\), contradicting \eqref{eq:prime-closure}.
Suppose that the theorem fails. Choose the least violating height \(m\)
and a violating subgroup \(B\) at that height. Thus
\[
(B,m)\in\overline{\{y\}},\qquad m-n<\ell(G,B),
\]
and \(B\ne G\). By \Cref{prop:finite-local}, the spectrum
\(X_m=\Spc((\mathcal C_m^G)^c)\) is a finite subspace containing
\(y\) and \((B,m)\). Let \(Z=\overline{\{y\}}\cap X_m\), the
closure of \(y\) in that subspace.
We construct one compact object with support exactly \(Z\), keeping a
finite genuine representative. For every \(q\in X_m\setminus Z\),
reverse prime inclusion gives \(q\not\subseteq y\). Choose a finite
genuine spectrum \(a_q\in q\setminus y\). Its support contains \(y\),
hence all of \(Z\), and excludes \(q\). The finite tensor product
\[
a=\bigotimes_{q\in X_m\setminus Z}a_q,\qquad z=e\otimes a
\]
(with the unit as empty product) satisfies \(\supp(z)=Z\).
It is compact and dualizable, and its displayed finite representative
allows us to apply \Cref{lem:layer-orthogonality}.
Minimality of \(m\) shows that every \((K,k)\in Z\) with \(k<m\)
satisfies \(\ell(G,K)\le k-n\). Therefore
\Cref{lem:layer-orthogonality,lem:layer-splitting} give
\[
z\simeq X\oplus Q,\qquad
Q=F_B\phi_Bz,\qquad X\otimes Q=0,
\]
with both summands compact, \(\phi_BX=0\), and
\(\phi_KQ=0\) for \(K\not\sim B\). Their compact supports are
disjoint closed subsets whose union is \(Z\). Both are nonempty:
\((B,m)\in\supp(Q)\), since \(\phi_BQ\simeq\phi_Bz\), and
\(y\in\supp(X)\), since \(B\ne G\) and \(\phi_GQ=0\).
This contradicts irreducibility of the closure of a point.
\end{proof}
\begin{proof}[Proof of \Cref{thm:main}]
For an integer \(r\ge0\), the universal implication defining
\(r_n(G,H)\) is precisely
\[
\mathcal P_{n+r}^G(H)\subseteq\mathcal P_n^G(G).
\]
By \eqref{eq:prime-closure} and \Cref{thm:specialization}, this inclusion
forces \(r\ge\ell(G,H)\). Equivalently, for every smaller \(r\), its
failure supplies a finite genuine spectrum in the first prime and outside
the second, witnessing failure of the proposed implication.
\Cref{prop:cyclic-upper} supplies the opposite inequality. Thus
\(r_n(G,H)=\ell(G,H)\) for all the stated groups, subgroups, and heights.
The rational height \(n=0\) has been retained throughout, and when
\(H=G\) both sides are zero.
\end{proof}
\subsection{The general finite-group consequence}
For a finite group \(J\), let \(O^p(J)\) denote its smallest normal
subgroup of \(p\)-power index. We continue to use tested-height
indices and work in the \(p\)-local genuine category.
\begin{corollary}\label{cor:general-groups}
Let \(G\) be any finite group, \(p\) a prime, \(H,K\le G\), and
\(m,n\ge0\). Then
\[
\mathcal P_m^G(K)\subseteq\mathcal P_n^G(H)
\]
if and only if there is a conjugate \(K^g=gKg^{-1}\le H\) such that
\[
N=O^p(H)=O^p(K^g),\qquad
m\ge n+\ell(H/N,K^g/N).
\]
In particular, the numerical test is applied to a qualifying
conjugate in the finite \(p\)-group \(H/N\).
\end{corollary}
\begin{proof}
Balmer--Sanders \cite[Proposition~6.11]{BS} reduce the displayed
inclusion to the existence of such a conjugate and the inclusion
\(\mathcal P_m^{H/N}(K^g/N)\subseteq
\mathcal P_n^{H/N}(H/N)\). Their prime indices are \(m+1,n+1\).
The common \(p\)-residual condition is equivalent to
\(p\)-subnormality of \(K^g\) in \(H\), and \(H/N\) is a
\(p\)-group. The relative specialization bound and the cyclic upper
bound give precisely the stated inequality. When \(m<n\), an
inflated finite spectrum of type \(m+1\) rules out the reduced
inclusion, as in the proof of \Cref{thm:specialization}.
For the upper direction at \(m>n+\ell(H/N,K^g/N)\), use also
the nesting of the Morava-acyclic ideals of finite spectra.
\end{proof}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//07-support.tex ====
==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Cyclic length and chromatic fixed-point loss}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
For a finite \(p\)-group \(G\) and a subgroup \(H\), we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from \(H\) to \(G\) with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.
\end{abstract}
\input{sections/01-introduction}
\input{sections/02-background}
\input{sections/03-characters}
\input{sections/04-congruences}
\input{sections/05-profinite}
\input{sections/06-gluing}
\input{sections/07-support}
\input{references}
\end{document}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//paper.tex ====
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==== BEGIN Cyclic-Length-and-Chromatic-Fixed-Point-Loss//references.tex ====
\begin{thebibliography}{99}
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\end{thebibliography}
==== END Cyclic-Length-and-Chromatic-Fixed-Point-Loss//references.tex ====
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