CHROMATIC SPLITTING AND SPHERE KERNELS
A reader's guide to the paper:
"Counterexamples to weak chromatic splitting: sphere kernels and descent
exponents"
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HOW TO READ THIS
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This is a research paper in algebraic topology. The paper proves that a
particular guess about how a certain object (the "p-completed sphere
spectrum") decomposes into coloured layers is FALSE at every prime
p >= 5 (at height p) and at every prime p >= 7 (at height p+1). It also
gives a second, independent proof of the failure at height p using a
different tool: descent exponents.
The original is 13 LaTeX sections plus an appendix. Its proofs are
complete but written in compressed graduate notation. This version
keeps the same mathematical content. Every named theorem, lemma, and
proposition of the original appears here, stated plainly, with its
full proof in plain prose. Nothing is left as "sketch" or "idea" or
"the reader can check"; where a step needs a computation (a cochain
homotopy, a filtration degree count, an inverse-limit argument), the
computation is written out.
You are expected to know what a group, a ring, a homomorphism, and a
field are. Everything else (spheres, spectra, chromatic layers,
localization, formal groups) is introduced as we go. The later sections
use undergraduate-accessible algebra, some compact-group theory, and
Morava E-theory. Those last items are genuinely graduate-level. Every
time one appears, we say plainly what job it does and why the proof
needs it. If you follow the storyline, you have understood the paper.
The one sentence the whole paper is about:
The p-completed sphere's canonical "height-(p-1) = lower-bounded
overlap with height p" map has NO homotopy retraction, for every
prime p >= 5. The same failure happens at height p+1 for every
prime p >= 7. The evidence is a specific sphere class
z_p = (beta_1)^{(p-1)^2} whose image at the lower height survives
but whose image at the upper height is ZERO.
Every chunk of the proof is a step toward making that sentence a
theorem.
There is a glossary at the end. Read the paper once for the story, then
use the glossary for words.
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PART A. BACKGROUND: WHAT ARE ALL THESE OBJECTS?
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A.1 The sphere spectrum and stable homotopy
Ordinary spheres are S^0 (two points), S^1 (a circle), S^2 (an ordinary
sphere), S^3, and so on. For two spheres S^m and S^n, we can ask: how
many genuinely different continuous maps are there from S^m to S^n, up
to the relation of continuous deformation (homotopy)? We write
pi_m(S^n) = set of homotopy classes of maps S^m -> S^n.
Examples: pi_n(S^n) = Z (the degree of a map); pi_3(S^2) = Z (Hopf);
pi_6(S^3) = Z/12; and so on.
There is a natural operation called suspension: from a map S^m -> S^n,
build a new map S^{m+1} -> S^{n+1} by joining a pole. This gives
compatible maps
pi_m(S^n) -> pi_{m+1}(S^{n+1})
which pass to a stable system: for fixed k = n - m, the groups pi_{m}(S^n)
for varying m and n eventually agree. Their common value is
pi_k(S), the k-th STABLE HOMOTOPY GROUP OF SPHERES.
The hidden object that produces these groups is the sphere spectrum S.
You do not need to construct S explicitly. It suffices to know that S has
homotopy groups pi_k(S), it has a commutative ring structure (smash
product), and it is the "unit" of a symmetric monoidal category of
spectra.
The p-completed sphere S_p^ is defined as:
S_p^ = holim_{j >= 1} S / p^j.
Here S / p^j is the cofiber of multiplication by p^j on S. The map
p: S -> S is null up to finite stages, so the limit succeeds. All
finiteness in the paper happens inside S_p^; nothing depends on
intermediate p-torsion.
Two operations on spectra will be used throughout:
smash X ^ Y - takes products at the level of spectra
function spectrum F(X,Y) - takes "maps from X to Y" as a spectrum
Inside spectra these behave like tensor product and internal hom.
Homotopy groups of the smash product involve a Tor-type tower, not just
the naive product of homotopy groups; this is where differentials and
multiplicative structure live.
A NOTE ON CONVENTION: unless a localized tensor is explicitly written,
every smash product and function spectrum in this document is ordinary
(unlocalized). All homotopy groups are those of the underlying ordinary
spectra.
A.2 Chromatic filtration: seeing the sphere layer by layer
The sphere spectrum S is a "complicated" object: its homotopy groups
are infinite-dimensional, containing integer classes (from the identity
map), classes from complex cobordism, and further periodic families
that assemble into a self-similar structure. Quillen, Hopkins, and
Devinatz showed that this structure organizes by HEIGHT.
A height-a theory is one that sees a piece of the sphere whose homotopy
groups are of dimension one over a ring, concentrated in even
degrees, and periodic. Heights are labelled a = 0, 1, 2, 3, ...
height 0: rational cohomology H Q
height 1: complex K-theory
height 2: elliptic cohomology at a chosen elliptic curve
height a: Morava K-theory K(a), constructed below
The chromatic ATTEMPT at reconstructing S goes as follows. For each
integer r >= 0, define L_r to be the localization of spectra with
respect to the direct sum K(0) v K(1) v ... v K(r). Then:
L_r X = the part of X that lives at heights at most r.
Localizations form a tower:
X = L_infinity X -> ... -> L_2 X -> L_1 X -> L_0 X = X ^ H Q
Each L_r kills one more height below the top. The associated graded
pieces are the "monochromatic" slices:
M_r X = fib(L_r X -> L_{r-1} X)
= the pure height-r part of X.
For X = S the tower is (conjecturally and in low cases provably) a
reconstruction:
S ~ product over r of M_r S at the chromatic level of formal
groups plus explicit products coming from Galois, deformation, and
automorphism structure.
Devinatz-Hopkins' nilpotence theorem, plus Hopkins-Miller's theorem on
the uniqueness of Morava E-theory, make this reconstruction rigorous
for the localized sphere at each height and work coherently across
heights. The paper uses exactly this infrastructure.
A.3 Morava K-theory and Morava E-theory
The height-a objects the paper needs are defined as follows.
Let p be a fixed prime. Let k_a = F_{p^a}. Let W(k_a) be the Witt vectors
of k_a (the unique unramified extension of Z_p with residue field k_a).
Let H_a be the Honda formal group of height a over k_a; its moduli
deformation ring O_a is
O_a = W(k_a) [[ u_1, ..., u_{a-1} ]],
complete with respect to the maximal ideal
m_a = (p, u_1, ..., u_{a-1}).
Define MORAVA E-THEORY E_a by
(E_a)_* = O_a [ u^{+1} ], |u| = -2.
It is a commutative ring spectrum, complex oriented, even periodic, and
finite-free of rank a over K(a)_* in a suitable sense. The unit u is a
periodicity class.
MORAVA K-THEORY is the residue:
K(a) = E_a / m_a.
It is a field spectrum in the sense of homotopy theory: everything mod
m_a is displayed by the multiplicative group of a finite field. That
is why K(a) is simpler than E_a and why it appears first.
For a = 0, we set K(0) = H Q (rational cohomology), E_0 = H Q[u^{+1}],
and there is nothing extra.
The group of automorphisms of the pair (k_a, H_a) is the EXTENDED
STABILIZER
G_a = S_a >| Gal( k_a / F_p ).
The pure stabilizer S_a (S for "stabilizer", not sphere) is the group of
automorphisms of H_a over the fixed residue field k_a; the Galois
quotient exchanges residue-field roots. For a = 1, G_1 is Z_p^\times;
for a = 2 (modulo finite-automorphism exceptional cases), G_2 is
modelled on the formal group of a supersingular elliptic curve. In
general G_a is a compact p-adic analytic group of dimension a^2.
The pair (E_a, G_a) is a Morava E-theory with action by G_a. Precisely,
E_a has a coherent action of G_a by commutative ring automorphisms, and
the special fiber of this action is the prescribed Honda stabilizer
action on K(a). The exact coherence is Goerss-Hopkins' 2004 theorem
(gh2004 section 7). The paper uses this coherence whenever it wants to
compare E_a and K(a).
A.4 What a "localization unit" means
For a spectrum X and a height-a theory (either K(a) or E_a), the
K(a)-LOCALIZATION UNIT is a natural map
eta_X^{(a)}: X -> L_{K(a)} X
landing in the "K(a)-local" world. L_{K(a)} is a functor that kills the
complement of K(a)-local information; for X = S, the resulting
spectrum L_{K(a)} S = B_a is the "height-a sphere." Its homotopy groups
are computed by the Morava descent spectral sequence
E_2^{s,q} = H^s_{cts}( G_a ; (E_a)_q ) => pi_{q-s} B_a.
Here "cts" means CONTINUOUS cohomology: the group G_a is profinite and
the coefficients are given the m_a-adic topology. The spectral sequence
converges strongly.
The pair (i_{n,X}, kernel) that the paper studies is built from these
units by stacking one more localization. Details are in Part A.6.
A.5 The overlap map i_{n,X}
Fix heights n > 0 and a spectrum X. Apply the height-n unit, then
localize down to heights at most n-1:
L_{n-1}X --(i_{n,X})--> L_{n-1} L_{K(n)} X.
Precisely:
i_{n,X} = L_{n-1}( eta_X^{(n)} ),
eta_X^{(n)}: X -> L_{K(n)}X.
This is a map of spectra. Its source L_{n-1}X is the part of X below
height n. Its target takes X's height-n piece and then forgets about
height n and above, keeping only heights 0, ..., n-1. The result is a
"shadow": what the height-n piece of X looks like when viewed by the
lower-height rulers.
This is the OVERLAP map. It measures exactly how much of the height-n
color is secretly made of lower colors.
Two features worth noting:
* i_{n,X} is always a natural transformation, because eta and L are.
* If X = S (the sphere) and n is odd, some authors call the target
the "chromatic overlap" and denote it X_{n,p} = L_{n-1} L_{K(n)} S.
The paper mostly writes just the target.
The question the paper asks is whether i_{n,X} admits a HOMOTOPY
RETRACTION: a map r: target -> source with r i_{n,X} = id_{source} up
to homotopy. A retraction need NOT be natural in X; it needs only to
exist as a map of spectra. Weak chromatic splitting conjectures exactly
existence of such an r, for every finite X, with the correct convention
on completion. (See A.6.)
A.6 The chromatic splitting conjecture (Hopkins' original)
Hopkins considered the following guess (recorded by Hovey, 1995). Fix a
prime p and height n. Suppose X is a finite spectrum and complete it at
p, obtaining X = X_p^. Then the FULL chromatic overlap object
L_{n-1} L_{K(n)} X
should decompose into a direct sum of explicit lower-height terms:
L_{n-1} L_{K(n)} X ~ L_{n-1} X v Sigma^{?} L_{K(n-1)} X_v ...
with known summands coming from the Brown-Comenetz dual and higher real
K-theory pieces. Hovey (1995) recorded this as Conjecture 4.2, parts
(i)-(v), in his paper on Bousfield localization.
The full decomposition is speculative and hard to check. What matters
here is the WEAK form, the part about retraction:
WEAK CSC: For any finite X and any 1 <= n < infinity, the overlap
map i_{n,X}: L_{n-1}X -> L_{n-1}L_{K(n)}X admits a homotopy
retraction.
The weak form is implied by the strong form, because the strong form's
canonical summand inclusion comes with a section. The weak form
disproves with a SINGLE counterexample; the strong form needs one on
each specific finite X. So the weak form is the right target if you
want to refute the entire programme with a small example.
Beaudry (2017) disproved the strong conjecture at (n = p = 2). Beaudry-
Goerss-Henn (2022) revised it with Moore terms, keeping the canonical
weak retraction intact. Their revised conjectures (1.1.10 and 1.1.11
in their paper) distinguish strong from weak. Minami (2003) records
Devinatz's counterexample for a BP analogue with COMPLETED BP. All of
these live at specific small primes and in specific formulations.
The paper proves:
THEOREM (main). Weak CSC is FALSE for X = S_p^ (the p-completed
sphere) at height p for every prime p >= 5, and at height p+1 for
every prime p >= 7.
Two independent proofs are given:
proof one (Part C-G): a specific sphere class z_p in pi_* S survives
to the lower height but dies at the upper height; the same class
gives a kernel element of i.
proof two (Part H): the descent exponents of the two constructions
differ, and weak CSC would force them to agree.
A.7 Why the sphere is the right test case
Choose X = S_p^, the p-completed sphere.
Reasons:
* S_p^ is finite in the chromatic sense: each L_r S_p^ is generated
by finitely many monochromatic pieces of the sphere at fixed height.
So the finite hypothesis of the weak CSC is satisfied.
* S_p^ is the RESIDUAL BASE POINT: every other finite X_p^ is built
from S_p^ by finitely many cofiber constructions. If the canonical
map for X = S_p^ fails retraction, every attempt to build a
natural-in-X retraction fails.
* The sphere has very specific named homotopy classes. In particular,
for primes p >= 5 there are first beta element classes
beta_1 in pi_{d_p}(S), and the named power z_p = beta_1^{(p-1)^2}
has good upper and lower behavior. This makes the proof CONCRETE.
Why p >= 5 (and p >= 7 at the next height)? The proofs use three
inequalities that hold automatically in that range:
2(p-1)^2 - (p^2 + 1) = p^2 - 4p + 1 > 0 (p >= 5)
2(p-1)^2 - ((p+1)^2 + 1) = p(p - 6) > 0 (p >= 7)
2(p-1)^2 - ((p-1)^2 + 2) = p^2 - 4p + 1 > 0 (p >= 5)
These correspond to: a certain class lying low enough in the upper
filtration that it must vanish (Part F), and at the next height one
more filtration step is used (Part F again, one height higher).
A.8 The overall strategy of the paper
Two INDEPENDENT proofs of the main theorem are given. You can read
either alone.
PROOF ONE (kernel): Part C-G.
* build a completion lemma (kernel criterion for S -> X)
* hit the sphere with a lower detector (Part D) that sees a
specific power z_p = beta_1^{(p-1)^2}
* prove z_p is killed at the upper height via a filtration
argument (Part F)
* conclude z_p gives a nonzero kernel of i_{n,X}
PROOF TWO (exponents): Part H.
* fix the finite subgroup H = C_p >| C_{(p-1)^2} of the height
p-1 pure stabilizer
* compute the H-exponent of E_{p-1}: > p^2 + 2 (Part H.2)
* compute the H-exponent of T(B_p) where T is the exact transfer
functor T(V) = L_{K(p-1)}( E_{p-1} ^ V ): <= p^2 + 2 (Part H.3.c)
* argue that a retraction of i_{p,X} would force
exp_H(E_{p-1}) <= exp_H(T(B_p)), contradicting the two bounds.
The reader can check either proof independently. Both are numbered to
make cross-references easy from the paper.
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PART B. THE MAIN THEOREMS, STATED
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B.1 Notation and constants
Throughout the paper, fix a prime p and put:
d_p = 2p(p-1) - 2 (a positive even integer)
k_p = (p-1)^2 (a positive integer)
z_p = beta_1^{k_p} (a sphere class of degree d_p k_p)
Here beta_1 in pi_{d_p}(S) is the first beta element of the stable
homotopy of spheres. Its existence and its detection behavior at all
odd primes are classical (Ravenel 1978, Lee-Ravenel 1994). For
concreteness: at p = 5, d_p = 38 and k_p = 16, so z_p = beta_1^{16} in
pi_{608}(S). This is the "prime-five class" mentioned in the abstract.
We use S = S_{(p)}^0 for the p-local sphere, X = S_p^ = S_p^0 for its
p-completion, and B_a = L_{K(a)} S for the height-a localized sphere.
B.2 Theorem 1 (the height-p kernel)
THEOREM 1. Let p >= 5 be prime. With the notation above, the image
of z_p in
pi_{d_p k_p} L_{p-1} X
is NONZERO, and is KILLED by the map
pi_{d_p k_p}( i_{p,X} ): pi_{d_p k_p} L_{p-1} X
-> pi_{d_p k_p} L_{p-1} L_{K(p)} X.
Equivalently, z_p defines a nonzero element of the kernel of
i_{p,X}. Consequently i_{p,X} has no homotopy retraction.
Words:
* "image of z_p in L_{p-1} X" means: apply the lower localization
L_{p-1} to the map z_p: S^{d_p k_p} -> S, then identify with the
corresponding class.
* "is killed by pi_(i_{p,X})" means: apply i_{p,X}, get zero in
the target.
* The nonzero-kernel statement immediately excludes retraction:
if r i_{p,X} = id and i_{p,X}(z_p) = 0, then z_p = r(0) = 0,
contradiction.
B.3 Theorem 2 (the same kernel at height p+1)
THEOREM 2. Let p >= 7 be prime. The image of the SAME class z_p
in
pi_{d_p k_p} L_p X
is NONZERO, and is KILLED by
pi_{d_p k_p}( i_{p+1,X} ): pi_{d_p k_p} L_p X
-> pi_{d_p k_p} L_p L_{K(p+1)} X.
Therefore i_{p+1,X} has no homotopy retraction.
At p = 7 this is beta_1^{36} in degree 82*36 = 2952. Same class, one
height up. The clean statement is: one sphere power produces kernels at
two different heights, at two overlapping prime ranges.
B.4 Corollary 1 (the p = 5 case, simplest example)
COROLLARY. At p = 5, the image of beta_1^{16} in pi_608 L_4 X is
nonzero and belongs to the kernel of pi_608( i_{5,X} ).
pi_608( i_{5,X} ): pi_608(L_4 X) -> pi_608(L_4 L_{K(5)} X).
This is Theorem 1 evaluated at p=5. It is the smallest case that can
be stated in ordinary integers without saying "for all p".
B.5 Theorem 3 (independent descent-exponent obstruction)
This is the second, unrelated proof of the same height-p nonretraction.
THEOREM 3. Let p >= 5 be prime. Put t = p-1, let
H = C_p >| C_{(p-1)^2}
be the finite subgroup of the height-t pure stabilizer, acting on
E_t by its coherent Morava action. In D_H = Fun(BH, Sp):
* put A_H = F(H_+, S) with translation action,
* put J_H = fib(S -> A_H) (the "augmentation fiber"),
* define the H-exponent exp_H(V) = the least b >= 1 such that
J_H^{^ b} ^ V -> V is null, else infinity,
* let T(V) = L_{K(t)}(E_t ^ V), with H acting on the first factor.
Then with m = p^2 + 2,
exp_H( E_{p-1} ) > m,
exp_H( T(B_p) ) <= m.
A homotopy retraction of i_{p,X} would make E_{p-1} a retract of
T(B_p) in D_H, contradicting the two bounds.
The key phrase is D_H = Fun(BH, Sp): spectra equipped with a coherent
action of H. "Coherent" is stronger than "an action up to homotopy";
it means an actual algebra-over-the-group-ring structure in the
symmetric monoidal category of spectra. The paper uses the finite
group H, so all constructions are in the stable world and stay
finite at every stage.
B.6 How the two proofs use different tools
PROOF ONE: sphere classes and multiplicative stability in the
cohomology of the Morava stabilizer.
PROOF TWO: H-equivariant descent exponents and the coherent trace.
Neither gives the other. The paper includes both because the range of
applicability of each is slightly different (proof one requires p >= 5
at height p and p >= 7 at height p+1; proof two currently only covers
height p, but with a much wider conjecture in mind).
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PART C. THE COMPLETION LEMMA AND THE KERNEL CRITERION
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Two ingredients are combined in every proof of the paper. The first is
a way to pass between working with the finite sphere S and working
with its p-completion X = S_p^. The second is a general principle: a
class detected by a lower theory but killed by an upper one lands in
the kernel of the overlap map.
C.1 What p-completion does to K(a)-theory
Whereas S is a p-local finite spectrum, X = S_p^ is the derived
completion. These two are not equivalent: X has different homotopy
groups in positive and negative degrees (X is connected but not finite).
What saves the argument is that K(a)-locally, they are the same.
LEMMA C.1 (completion lemma). Let a >= 1. The p-completion map
c: S -> X is a K(a)-equivalence. That is, after applying K(a)_*,
the map c becomes an isomorphism.
Consequently, if Y is a K(t)-local spectrum for some 1 <= t <= r,
every map S -> Y factors as S -> X -> L_r X -> Y.
Proof. Take the fiber of the completion map. From the definition
of X as holim_{j >= 1} S / p^j, the cofiber sequences S -p^j-> S -> S/p^j
compose into
fib(c) = holim_{j >= 1} ( S -p^j-> S )
~ F( S[1/p] , S ),
the p-complete function spectrum out of formal inversion S[1/p]. The
map p: S[1/p] -> S[1/p] is an equivalence, so multiplying by p is an
equivalence on F(S[1/p], S). It remains an equivalence after smashing
with K(a). On the other hand, K(a)_* is an F_p-algebra, so p acts by
ZERO on any K(a)_*-module; the groups M = pi_*(K(a) ^ F(S[1/p], S))
are such modules. Multiplying by p on M is therefore simultaneously an
equivalence (from the previous paragraph) and the zero map. A group on
which p is both an isomorphism and the zero map must be trivial. Hence
K(a) ^ fib(c) is zero.
That gives the first claim. For the factorization, any map S -> Y with
Y K(t)-local extends across c because the fiber is K(t)-acyclic. Then
L_r-localization of Y is the identity (Y is K(t)-local with t <= r, and
K(t)-acyclic spectra are contained in L_r-acyclic spectra). Diagram:
S --c--> X --lambda_X--> L_r X
\ |
\ v
------ eta_S -----> Y
(dashed arrow = the factorization)
The commutation of the square uses naturality of eta; the dashed arrow
is the composition.
C.2 The kernel criterion
PROPOSITION C.2 (kernel criterion). Let n >= 2, let 1 <= t <= n-1,
and let z in pi_q S be a sphere class. Suppose:
* there exists a map S -> Y with Y a K(t)-local spectrum that
carries z to a NONZERO element of pi_q Y,
* the unit S -> L_{K(n)} S carries z to ZERO.
Then the image of z in pi_q L_{n-1} X is NONZERO and belongs to the
kernel of the overlap map i_{n,X}:
i_{n,X}: L_{n-1}X -> L_{n-1} L_{K(n)} X.
Proof. Write c: S -> X for completion, eta_Z: Z -> L_{K(n)}Z
for the height-n unit, and lambda_Z: Z -> L_{n-1}Z for lower
localization. Naturality gives the two commuting squares:
S --eta_S--> L_{K(n)} S X --eta_X--> L_{K(n)} X
| | | |
c L_{K(n)}(c) lambda_X lambda_{L_{K(n)}X}
v v v v
X --eta_X--> L_{K(n)} X L_{n-1}X --i_{n,X}--> L_{n-1}L_{K(n)}X
The left square uses c as the vertical map on the left; the right square
uses lambda as the vertical map on the left. BOTH commute by naturality.
By Lemma C.1, the map S -> X is a K(t)-equivalence for t <= n-1, so
the detecting map S -> Y factors through L_{n-1}X, proving the image of
z is nonzero there. Applying the height-n unit sends z to zero (second
hypothesis) and then applying lower localization preserves zero; by the
right square, i_{n,X}(z) = 0 in the target. Combined: z's image lies in
the kernel.
If a retraction r of i_{n,X} existed, then the nonzero class z-bar in
pi_q L_{n-1}X (the image of z) would satisfy r(i_{n,X}(z-bar)) = r(0) =
0 on the one hand, but r(i_{n,X}(z-bar)) = z-bar (retraction identity)
on the other. Hence z-bar = 0, contradicting that z-bar is nonzero.
C.3 Remarks on the criterion
The kernel criterion is entirely elementary in the sense of not
needing any Morava coherence theory: it is just the naturality of
localization units, plus the completion lemma. Everything hard is
pushed into supplying the two test hypotheses:
(a) a lower K(t)-theory Y detecting z, and
(b) the upper unit S -> L_{K(n)} S killing z.
Part D does (a). Part E-F does (b).
One more wrinkle. The "detecting" map S -> Y is NOT completed: it lands
in an honest K(t)-local spectrum Y, and the class z has a nonzero image
in pi_*Y. This is a statement about the sphere, not about X. The
completion lemma then infers survival in L_{n-1}X. This is why the
paper can work with named sphere classes (beta powers) and not with
their completions.
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PART D. THE LOWER DETECTOR: HOW z_p SURVIVES AT HEIGHT p-1
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The paper uses z_p = beta_1^{(p-1)^2}. This section shows it survives
a map from S to a height-(p-1)-local spectrum Y. Section F shows the
same power dies at the upper height. Together with Proposition C.2,
this proves Theorems 1 and 2.
D.1 The Morava E-theory detector
Let h = p-1 (so h is one less than the target height the paper is
denying a retraction for). Let E_h be Morava E-theory of height h over
F_{p^h}. Write S_h for the pure stabilizer subgroup of G_h and
G_h = S_h >| Gal(F_{p^h}/F_p) for the extended stabilizer.
Choose a finite subgroup G of G_h with:
G ∩ S_h = C_p >| C_{h^2} (finite intersection with
the pure stabilizer),
image of G in Gal(F_{p^h}/F_p) = all of Gal(F_{p^h}/F_p) (onto),
as constructed in Heard 2015, Section 2. This is a finite group of order
p * h^2 * h. Call it G. Put
Y = E_h^{hG},
the ordinary homotopy fixed point spectrum of G acting on E_h. This Y
is the lower detector.
The spectrum Y is a commutative ring spectrum with a unit map
eta: S -> Y. Devinatz-Hopkins (2004, Theorems 2(ii) and 3) identify
its homotopy fixed point spectral sequence with the LOCAL ADAMS
SPECTRAL SEQUENCE for the K(h)-local sphere. Both converge strongly,
and the E_2 page is:
E_2^{s,t} = H^s( G ; pi_t E_h ) ==> pi_{t-s} Y.
The decreasing homotopy filtration F^s pi_* Y is therefore separated,
with graded pieces E_infinity.
D.2 Heard's calculation: beta_1 is up in the E_2 page
The Hopkins-Miller calculation gives the POSITIVE-FILTRATION part of
the E_2 page. For the finite group G above, this is:
E_2^{s,*} (positive filtration) =
F_p[ alpha, beta, Delta^{+1} ] / ( alpha^2 ),
with bidegrees:
|alpha| = ( 1 , 2h )
|beta| = ( 2 , 2ph )
|Delta| = ( 0 , 2ph^2 )
(Heard 2015, Proposition 5.5; also HMS 2017, Proposition 2.7.)
Here alpha is a height-one class, beta a height-two class, Delta a
height-zero but h^2-multiple class. In particular, beta has filtration
two and internal degree 2ph.
Heard identifies the stable sphere class beta_1 with the class beta in
this E_2 page:
eta_*( beta_1 ) in F^2 pi_{d} Y (for d = 2p(p-1) - 2),
class of eta_*( beta_1 ) mod F^3 = c * beta
for some nonzero scalar c in F_p^*.
That is, the unit eta: S -> Y carries the named stable beta element to
a specific class in filtration two whose associated-graded symbol is a
nonzero scalar times beta. The symmetry factor c depends on choices of
cohomology generator but is nonzero for every odd prime.
D.3 The long power remains nonzero
Heard's E_2 calculation continues: in positive even filtration, the
E_infinity page is
E_infinity^{even,*} = F_p[ Delta^{+p} ][ beta ] / ( beta^{h^2+1} )
where "h^2+1" is the nilpotence order of beta (Heard 2015, Lemma 5.8;
see also HMS 2017). In particular, beta^{h^2} is NONZERO on E_infinity.
Now multiply: from D.2, [eta_*(beta_1)] = c * beta at E_2. Therefore
the class of eta_*(beta_1)^{h^2} mod the next filtration is
c^{h^2} * beta^{h^2} != 0,
nonzero because c has an h^2-th power in F_p^* and beta^{h^2} is
nonzero.
To go from E_infinity to pi_*Y, we need PRODUCTS on the spectral
sequence to agree with the actual multiplication on pi_*Y. That is the
content of section D.4.
D.4 The ordinary multiplicative structure on the bar tower
LEMMA D.4 (ordinary filtered product). Let Q be a finite group
acting coherently on a commutative ring spectrum R. The ordinary
homotopy fixed point spectral sequence
E_2^{s,t} = H^s( Q ; pi_t R )
has the cup product on E_2 and the derivation rule for its
differentials. Whenever this spectral sequence converges to
pi_* R^{hQ} with a separated homotopy filtration whose successive
quotients are E_infinity, its product on E_infinity is the
associated graded product for that actual homotopy filtration.
Proof. Set EQ_bullet = the homogeneous free bar construction
Q_0 = Q, Q_1 = Q x Q, Q_2 = Q x Q x Q, ... with the usual face and
degeneracy maps. Let EQ^{(s)} be its s-skeleton. The homotopy fixed
point tower is V_s = F_Q( EQ^{(s)}_+ , R ), with holim_s V_s = R^{hQ}.
To define a product on V_s, one needs a "diagonal" map EQ -> EQ x EQ
compatible with the bar faces. Basterra-Mandell's "prismatic" bar
diagonal (their 2013 paper, proof of Theorem 7.1) supplies exactly
this: for barycentric coordinates t = (t_0, ..., t_q) on Delta^q,
define C_{-1} = 0, C_i = t_0 + ... + t_i, and split each vertex into
two barycentric vectors:
ell_i(t) = 2 ( min(C_i, 1/2) - min(C_{i-1}, 1/2) )
rho_i(t) = 2 ( max(C_i, 1/2) - max(C_{i-1}, 1/2) )
These are the lengths of the parts of [C_{i-1}, C_i] on each side of
1/2, doubled. Set
delta[s, t] = ( [s, ell(t)], [s, rho(t)] )
for a bar simplex (s, t) in EQ_q x Delta^q. This is compatible with
faces and degeneracies (each θ: [q] -> [q'] acts on ell and rho by the
pushforward theta_*), Q-equivariant (the group action changes s but
not t), and homotopic to the ordinary diagonal via the linear
interpolation
[s, t] \mapsto ( [s, (1-tau)t + tau ell(t)], [s, (1-tau)t + tau rho(t)] ).
Skeletal comparison: delta maps the s-skeleton EQ^{(s)} into the union
over i+j <= s of EQ^{(i)} x EQ^{(j)}. On the s-th region, delta
identifies the fundamental cellular chain with the Alexander-Whitney
diagonal
[s] |-> sum_{k=0}^{q} [s|_{[0,...,k]}] tensor [s|_{[k,...,q]}].
Cofiber sequence comparison: for a, b >= 0, delta passes to a
Q-equivariant relative map
EQ / EQ^{(a+b-1)} -> (EQ / EQ^{(a-1)}) ^ (EQ / EQ^{(b-1)}).
Multiplying in R and precomposing into V = R^{hQ} gives a product
F_Q( EQ/EQ^{(a-1)}, R ) ^ F_Q( EQ/EQ^{(b-1)}, R )
-> F_Q( EQ/EQ^{(a+b-1)}, R ).
This is compatible in a, b, giving products on the derived exact couples.
On cellular cochains the chain formula is the ordinary cup product; the
derivation rule for the differentials follows from the cellular boundary
formula with Koszul signs.
Associativity for iterated products follows by the same reasoning with
the m-fold diagonal: split [0,1] into m successive intervals of positive
lengths, split each [C_{i-1}, C_i] proportionally, and the same
support-and-contractibility argument applies. The positive interval
lengths form a convex simplex, so interpolation gives filtered
associativity up to homotopy. Thus the geometric product on V_s is the
product of the associated-graded filtration.
Cup-product-sign check. In inhomogeneous cochains, the cup product is
(v ^ w)(g_1, ..., g_{a+b})
= v(g_1, ..., g_a) . (g_1 ... g_a) w(g_{a+1}, ..., g_{a+b}),
with the "twisted" middle factor g_1 ... g_a of the pure action. That
is: the second coset sits modulo the product of the first a. This is
exactly what the ordinary group cohomology cup product gives.
D.5 The lower detector statement
PROPOSITION D.5 (the actual multiplication in the detector). The
unit eta: S -> Y carries z_p = beta_1^{h^2} to a NONZERO class in
pi_{d h^2} Y, with d = 2p(p-1) - 2 and h = p-1.
Proof. Apply Lemma D.4 to the homotopy fixed point spectral sequence of
G acting on E_h. The class of eta_*(beta_1)^{h^2} in
F^{2 h^2} / F^{2 h^2 + 1} is c^{h^2} beta^{h^2} (Part D.3), which is
nonzero in E_infinity. Hence the actual class survives to pi_{d h^2} Y.
Since eta is a ring map, eta_*(beta_1^{h^2}) agrees with
(eta_*(beta_1))^{h^2}: the sphere power z_p passes to the h^2-th power
of the detected class.
--------------------------------------------------------------------------
PART E. UPPER VANISHING: HOW z_p DIES AT THE UPPER HEIGHT
--------------------------------------------------------------------------
This is the crux of the kernel proof. To show z_p dies under the upper
unit S -> L_{K(n)} S (with n = p or p+1), we compute the Morava descent
spectral sequence of S at that height and prove that the class lies in
a filtration high enough to vanish. The vanishing is driven by an
upper bound on continuous cohomology of the Morava stabilizer group.
E.1 The cohomological line
PROPOSITION E.1 (the cohomological vanishing). Let a >= 1 be an
integer satisfying one of:
(a = p, p >= 5), or
(a = p+1, p >= 7).
Let V_0 be a finite p-local spectrum, let M = pi_q( E_a ^ D V_0 )
(where D V_0 = F(V_0, S)) with the m_a-adic topology and the
prescribed Morava action of G_a. Then
H^s_{cts}( G_a ; M ) = 0 for all s > a^2 + 1.
The cases a = p and a = p+1 are handled by the same strategy; the
difference is only that at a = p we use a semilinear trace contraction
for the Galois factor, and at a = p+1 we use that the Galois factor has
order coprime to p so the FULL group has the same dimension as the pure
subgroup.
Here is what each ingredient contributes.
E.2 Ingredient one: p-adic analytic dimension
The pure stabilizer S_a of Morava E-theory of height a is the
maximal-order unit group in the central division algebra D over Q_p of
dimension a^2 (invariant 1/a). Morava 1985 (Sections 2.1-2.2) gives the
compactness and analyticity structure of this group. It follows that
S_a is a compact p-adic analytic group of dimension a^2.
We need to know its p-COHOMOLOGICAL DIMENSION for discrete p-primary
modules. Lazard's theorem on which compact p-adic analytic groups have
finite cohomological dimension and Serre's 1965 contribution give:
S_a is torsion-free mod p iff cd_p(S_a) = a^2.
So we must exclude elements of order p from S_a. Here is the
exclusion:
Suppose g in S_a is nontrivial of order p. Write phi_p(X) =
(X^p - 1)/(X - 1) for the p-th cyclotomic. In D_x:
(g - 1) phi_p(g) = 0.
Since D_x is a division algebra and g != 1, this gives
phi_p(g) = 0.
Now phi_p(X + 1) is an Eisenstein polynomial over Q_p; its roots
generate the degree-(p-1) subfield Q_p( zeta_p ). So g generates
Q_p( zeta_p ) as a subfield of D.
But D is a division algebra of degree a over its centre with D
containing Q_p( zeta_p ) as a subfield, so this subfield must embed
in D; a degree-(p-1) subfield embeds only if (p-1) | a^2.
* For a = p, this forces (p-1) | 1, false for p >= 3.
* For a = p+1, this forces (p-1) | 4, so p-1 in {1, 2, 4}, giving
p in {2, 3, 5}. For p >= 7, false.
Contradiction in both cases: S_a has no element of order p.
Therefore cd_p(S_a) = a^2 for these heights when the coefficients are
discrete p-primary modules.
E.3 Ingredient two: the semilinear trace-one contraction
At height a = p, the Galois factor Gal(F_{p^p}/F_p) has order p (equal
to the characteristic). So the extra +1 in the cohomological bound
cannot come from a dimension argument: we must handle the Galois
factor algebraically.
LEMMA E.3 (trace-one contraction). Let Q be a finite group acting
on a commutative ring W. Suppose w in W has group trace
Sum_{g in Q} g(w) = 1 in W.
Then for any SEMILINEAR W-module P (P is a W-module, and each
g in Q acts by w -> g(w) * (linear action on P)), the group
cohomology H^i(Q, P) = 0 for all i > 0,
with no finiteness or flatness hypothesis on P.
Proof. Use homogeneous group cochains: a degree-q cochain is an
equivariant function f: Q^{q+1} -> P, with differential being the
alternating deletion of coordinates. Define the contraction h:
(hf)(g_0, ..., g_{q-1}) = Sum_{gamma in Q} gamma(w) * f(gamma, g_0, ..., g_{q-1}).
Semilinearity says that replacing gamma by g*gamma linearises:
g*( gamma(w) * f(gamma, ...) ) = (g*gamma)(w) * g*(f) = g*(gamma)(w) * g*(f).
Because (g*gamma)(w) = g(gamma(w)) and the cochain condition gives
g*(f(gamma, gamma_0, ...)) = f(g*gamma, g*gamma_0, ...), this is
precisely the equivariance of hf. In h*d + d*h, the "delete one of the
g_i" terms cancel in pairs by alternating signs. The remaining term,
the deletion of the inserted coordinate gamma, gives Sum_gamma
gamma(w) * f(g_0, ..., g_{q-1}) = 1 * f = f. Thus hd + dh = id in
positive degrees, proving acyclicity.
For W = W(F_{p^a}) (Witt vectors) with Q = Gal(F_{p^a}/F_p), the
finite-field trace is onto F_p. Lift a residue trace-one element w_0 in
W. Its Witt trace lies in Z_p and is congruent to 1 mod p, so it is a
unit. Dividing w_0 by this invariant unit supplies the required w. We
never divide by |Q| (so this works even when the characteristic
divides the group order, which is the whole point here).
For a = p (the case Q = Z/p), this lemma makes the Galois cohomology
vanish in positive degrees. For a = p+1 with p >= 7, the group order is
p+1 which is coprime to p, so the ordinary trace 1/(p+1) * Sum g of
g = 1 / (p+1) already gives trace one. In both cases the finite
pure-quotient cohomology vanishes above the pure dimension a^2.
More precisely, for any finite quotient module M_j = M / m_a^j M with
surjective transition maps, the continuous Hochschild-Serre spectral
sequence at the discrete coefficient level gives
H^i( Q ; H^b_{cts}( S_a ; M_j ) ) ==> H^{i+b}_{cts}( G_a ; M_j ),
where Q = Gal(F_{p^a}/F_p) = G_a / S_a. Lemma E.3 kills all positive
Q-columns, giving
H^s_{cts}( G_a ; M_j ) = H^s_{cts}( S_a ; M_j )^Q = 0 for s > a^2.
E.4 The inverse-limit step: why one extra degree survives
The full coefficient module M = pi_q( E_a ^ D V_0 ) has the
maximal-ideal (m_a)-adic topology and is a countable inverse limit of
its finite quotients M_j. Continuous cochains into M are compatible
cochains into the M_j with degreewise surjective transition maps. This
gives a Milnor exact sequence for every s:
0 -> lim^1_j H^{s-1}_{cts}(G_a; M_j) -> H^s_{cts}(G_a; M) -> lim_j H^s_{cts}(G_a; M_j) -> 0.
For s > a^2 + 1, both the lim^1 and the lim terms live in degree > a^2
of the finite-quotient theory, hence vanish. The resulting bound is
precisely the claimed
H^s_{cts}( G_a ; M ) = 0 for all s > a^2 + 1.
The inverse limit is exactly why we get a^2 + 1 rather than a^2. The
+1 is not a pollution; it is genuine (there may be a surviving lim^1
contribution in degree a^2 + 1 in general).
E.5 The zero tail of the upper filtration
The Morava descent spectral sequence reads
E_2^{s,q} = H^s_{cts}( G_a ; (E_a)_q )
==> pi_{q-s} B_a.
Devinatz-Hopkins (2004, Theorems 1(iii)-(iv)) prove strong convergence
and identify this spectral sequence with the local E_a-Adams spectral
sequence. Its homotopy filtration F^s pi_* B_a needs two proofs before
it is usable: (i) any two relative E_a-Adams resolutions give the SAME
stage-s images, and (ii) the tensor powers of the augmentation fiber
I_a = fib(B_a -> E_a) form one such resolution. We give both, then the
zero-tail follows.
LEMMA E.5a (independence of Adams resolution). Let E be a unital
algebra in a stable symmetric monoidal infinity-category with
exact tensor. Consider two relative E-Adams resolutions of the
same object V: triangles T_{s+1} -> T_s -> Q_s and T'_{s+1} ->
T'_s -> Q'_s, each with Q_s, Q'_s E-injective and the maps
E ^ q_s, E ^ q'_s split monic. Then the stage-s images
im([W, T_s] -> [W, V]) and im([W, T'_s] -> [W, V]) agree for
every object W and every s >= 0.
Proof. By symmetry, it suffices to construct compatible maps
f_s: T_s -> T'_s over V.
Suppose f_s is given. Put g_s = q'_s f_s in [T_s, Q'_s]. Choose a
retract Q'_s ->(iota_s) E ^ Z_s ->(rho_s) Q'_s with
rho_s iota_s = id, and a splitting sigma_s: E ^ Q_s -> E ^ T_s
with sigma_s (1 ^ q_s) = id. Define bar-g_s: Q_s -> Q'_s by
bar-g_s = rho_s (mu ^ id) (1 ^ iota_s g_s) sigma_s eta,
where eta: Q_s -> E ^ Q_s is the E-unit and mu: E ^ E -> E the
multiplication on E. Then
bar-g_s q_s
= rho_s (mu ^ id) (1 ^ iota_s g_s) sigma_s (1 ^ q_s) eta_{T_s}
= rho_s (mu ^ id) (1 ^ iota_s g_s) eta_{T_s}
= rho_s iota_s g_s
= g_s,
using unit naturality at Q_s and at T_s, then the module unit
law. Complete the square to a map of fiber triangles to obtain
f_{s+1}: T_{s+1} -> T'_{s+1}. Starting at f_0 = id_V, we get a
map over V at each finite stage, so im([W, T_s] -> [W, V])
contains im([W, T'_s] -> [W, V]). Repeating in the other
direction gives the reverse inclusion. Only finite stages are
used; no equivalence of towers is asserted.
LEMMA E.5b (Adams filtration = tensor-power filtration). Work in
C_a = Sp_{K(a)}, with B_a = L_{K(a)}S, E_a the height-a
E-theory, I_a = fib(B_a -> E_a), and product on C_a given by
U ^ V = L_{K(a)}(U ^ V). The decreasing homotopy filtration of
Lemma E.5a, for the tensor-power resolution, is
F^s pi_* B_a = im( pi_* I_a^{^ s} -> pi_* B_a ),
I_a^{^ 0} = B_a.
It is separated, its successive quotients are the E_infinity^{s,*}
of the descent SS, and F^r F^s is contained in F^{r+s}.
Proof. Devinatz-Hopkins (2004, Theorems 1(iii)-(iv)) give the
spectral sequence with strong convergence and the full
maximal-ideal-adic topology (Remark 1.3 there). Their ideal
(p, v_1, ..., v_{a-1}) agrees with m_a = (p, u_1, ..., u_{a-1}),
since v_i = u_i u^{1 - p^i} and u is invertible.
The tensor powers T_s = I_a^{^ s} fit the triangles
T_{s+1} -> T_s ->(q_s) E_a ^ T_s.
The right term E_a ^ T_s is E_a-injective (multiplication on E_a
retracts it back). The map E_a ^ q_s splits via multiplication on
E_a. Therefore T_s is a relative E_a-Adams resolution of B_a.
Lemma E.5a identifies its stage-s images with the stage-s images
of the Devinatz-Hopkins local resolution, which are F^s pi_* B_a
of the descent SS. Strong convergence supplies separation and the
associated-graded identification with E_infinity^{s,*}.
For multiplicativity, a map from a suspended sphere to a local
object is equivalently a map from a suspended B_a. Multiplying a
lift through I_a^{^ r} with a lift through I_a^{^ s} gives a lift
of the product through I_a^{^ (r+s)}, so F^r F^s is contained in
F^{r+s}. Monoidal localization carries the sphere unit and its
products to the local unit, so these are the products of the
actual sphere classes.
PROPOSITION E.5 (the zero upper tail). Let a = p (p >= 5), or
a = p+1 (p >= 7). Then
F^s pi_* B_a = 0 for all s >= a^2 + 2.
Proof. Proposition E.1 with the finite test V_0 = S gives
H^s_{cts}( G_a ; (E_a)_q ) = 0 for s > a^2 + 1.
Therefore E_2^{s,q} = 0 for s >= a^2 + 2, hence E_infinity^{s,q} = 0
for s >= a^2 + 2, hence the successive quotients of the filtration
vanish from degree a^2 + 2 onward. A separated homotopy filtration with
zero quotients in a tail is constant in that tail, and E.1's
separation makes the constant value zero.
Everything in this proposition is about homotopy images, not map
nullity. Section H will require a separate argument that a specific
map is NULL (not just zero on tests), and the paper is careful not to
confuse those.
--------------------------------------------------------------------------
PART F. PUSHING THE NAMED PRODUCT PAST THE ZERO TAIL
--------------------------------------------------------------------------
We now take the specific sphere class z_p = beta_1^{h^2} (h = p-1) and
show it dies at the upper height. The argument combines three
ingredients: (i) positive sphere stems are torsion; (ii) Morava E-theory
has only even-degree homotopy, so the first filtration step of a
torsion class vanishes automatically; (iii) multiplicativity of the
Adams filtration then pushes the h^2-th power up to filtration 2h^2,
which exceeds the upper tail.
F.1 Positive sphere stems are torsion
FACT F.1. Every class x in pi_d S with d > 0 is torsion: it
generates a finite (cyclic or abelian torsion) subgroup of pi_d S.
This is Serre 1951, Chapter V, Section 3, Proposition 3. The proof
suspends an odd-dimensional sphere (which is a finite spectrum) into
the class's stable range, so x is detected in a reduced homology group
of a finite complex; a finiteness argument then shows x TORSION.
LEMMA F.1'. Let x in pi_d S with d > 0. Then the unit image
x_a in pi_d B_a belongs to F^1 pi_d B_a = ker( pi_d B_a -> pi_d E_a ).
Proof. The map S -> E_a of ring spectra factors the unit through the
localization S -> B_a (because E_a is K(a)-local). Consequently the
composite pi_d S -> pi_d B_a -> pi_d E_a equals the unit pi_d S ->
pi_d E_a. But pi_d E_a for d even is the free rank-one O_a-module
O_a . u^{-d/2}, hence TORSION-FREE. So the image of a torsion class
is zero. This says x_a is killed by pi_d B_a -> pi_d E_a, i.e., it
lies in the F^1 step of the Adams filtration.
F.2 The first filtration step vanishes on even total degree
FACT F.2. For any positive height a, on even total degree d,
F^1 pi_d B_a / F^2 pi_d B_a = 0.
Proof. This quotient is E_infinity^{1, d+1} in the descent spectral
sequence, i.e., a piece of H^1_{cts}(G_a; (E_a)_{d+1}). But E_a is
even-periodic: pi_{d+1} E_a = 0 for d even (since d+1 is odd). A
coefficient module in odd degree, integrated against a continuous
cochain, is zero. Hence the E_2 page is zero in bidegree (1, d+1) and
therefore so is E_infinity.
Combined with F.1': the unit image x_a of a positive-degree x lies at
least in F^2.
F.3 Long powers land in F^{2h^2}
FACT F.3. If x_a in F^2 pi_d B_a, then x_a^{k} in F^{2k} pi_{dk} B_a.
Proof. F^r F^s is contained in F^{r+s} by multiplicativity
(Lemma E.5b, proven above).
For x = beta_1, d = d_p = 2p(p-1) - 2, so the unit image x_a^{h^2}
of z_p = beta_1^{h^2} lies in F^{2h^2} pi_{d h^2} B_a.
F.4 The numerical margin
At height a = p (p >= 5):
2h^2 = 2(p-1)^2 = 2p^2 - 4p + 2,
a^2 + 2 = p^2 + 2.
Difference:
2(p-1)^2 - (p^2 + 1) = p^2 - 4p + 1 > 0 for p >= 5.
Since 2h^2 > a^2+1 and both are integers, 2h^2 >= a^2 + 2.
At height a = p+1 (p >= 7):
2h^2 = 2(p-1)^2 = 2p^2 - 4p + 2,
a^2 + 2 = (p+1)^2 + 2 = p^2 + 2p + 3.
Difference:
2(p-1)^2 - ((p+1)^2 + 1) = p(p - 6) > 0 for p >= 7.
Again 2h^2 >= a^2 + 2.
F.5 The upper vanishing statement
PROPOSITION F.5 (upper vanishing of the same power). Let p >= 5 be
prime, x = beta_1 in pi_d S with d = 2p(p-1) - 2, and h = p-1.
Then:
* the unit image of x^{h^2} is zero in pi_{d h^2} B_p,
* the unit image of x^{h^2} is zero in pi_{d h^2} B_{p+1},
provided p >= 7.
Moreover: these are the images of the ORIGINAL SPHERE PRODUCT under
the units S -> B_p and S -> B_{p+1}, by the multiplicativity of the
filtration (Lemma E.5b, multiplicative part).
Proof. Fix a = p in the first bullet, a = p+1 in the second (the
second requires p >= 7). Let h = p-1, and let x_a in pi_d B_a be the
unit image of x = beta_1 under S -> B_a.
Step 1 (F^1). x = beta_1 has d > 0, so by F.1' its unit image x_a
lies in F^1 pi_d B_a.
Step 2 (F^2). d is even, so by F.2 the quotient F^1 pi_d B_a /
F^2 pi_d B_a vanishes. Hence F^1 pi_d B_a = F^2 pi_d B_a, and x_a
lies in F^2.
Step 3 (F^{2h^2}). The multiplicativity of the descent filtration
(Lemma E.5b) puts F^r . F^s inside F^{r+s}. Since x_a is in F^2, we
get x_a^{h^2} in F^{2h^2} pi_{d h^2} B_a.
Step 4 (tail). E.5 gives F^{a^2+2} pi_* B_a = 0 for a = p with p >= 5
or a = p+1 with p >= 7.
Step 5 (numerics). F.4 states 2h^2 > a^2+1 in each range: at a = p
the difference is p^2-4p+1 > 0 for p >= 5, and at a = p+1 it is
p(p-6) > 0 for p >= 7. Since both sides are integers and 2h^2 > a^2+1,
we have 2h^2 >= a^2+2. Therefore
F^{2h^2} pi_{d h^2} B_a is contained in F^{a^2+2} pi_{d h^2} B_a = 0.
So x_a^{h^2} = 0.
Step 6 (sphere lift). The unit S -> B_a is a ring map, so x_a^{h^2}
is the unit image of x^{h^2}. Combined with Step 5, the sphere product
x^{h^2} dies under each unit. Done.
F.6 The p = 5 case spelled out
At p = 5:
d = 38, h = 4, h^2 = 16,
d h^2 = 608.
a = p = 5, a^2 + 2 = 27.
The upper vanishing (Part F.5) applies at height 5. The unit image of
x^{16} has filtration 2*16 = 32, and 32 >= 27, so the class is ZERO in
pi_608 B_5.
Compare at height 6 (a = p+1 = 6). The zero tail there starts at
a^2 + 2 = 38, NOT at 27. Filtration 32 < 38, so this argument does NOT
show vanishing at height 6 for p = 5. This is exactly why the height-6
statement is not part of Corollary 1: the argument for Theorem 2 needs
p >= 7, and at p = 5 the class x^{16} does not stay in the zero tail
one height higher.
--------------------------------------------------------------------------
PART G. ASSEMBLING THE KERNEL THEOREMS
--------------------------------------------------------------------------
We now have everything to prove Theorems 1 and 2 and the p = 5
corollary.
G.1 Proof of Theorem 1 (height-p kernel, p >= 5)
Set h = p-1, n = p, t = p-1 = h, d = d_p = 2p(p-1) - 2, and k = k_p =
h^2. Take z = z_p = beta_1^{h^2} in pi_{d h^2} S.
Hypotheses of the kernel criterion (Proposition C.2):
(a) LOWER detector. Proposition D.5 gives a K(h)-local spectrum
Y = E_h^{hG} (with G the finite subgroup of G_h), and a unit
eta: S -> Y that carries z to a nonzero element of
pi_{d h^2} Y.
(b) UPPER vanishing. Proposition F.5 (height a = p) shows that the
unit S -> L_{K(p)} S = B_p kills z: the image of z in
pi_{d h^2} B_p is zero.
Apply Proposition C.2 with t = p-1 and n = p. This gives: the image of
z in pi_{d h^2} L_{p-1} X is nonzero and is killed by i_{p,X}. Hence
i_{p,X} has no retraction.
G.2 Proof of Theorem 2 (height-(p+1) kernel, p >= 7)
Same z = z_p. Same Y = E_h^{hG}, still detected (D.5 works at every odd
prime). Set n = p+1 instead of p.
UPPER vanishing. Proposition F.5 (height a = p+1, requires p >= 7)
shows the unit S -> L_{K(p+1)} S = B_{p+1} kills z.
Apply Proposition C.2 with t = p-1 and n = p+1. This gives: the image
of z in pi_{d h^2} L_p X is nonzero and is killed by i_{p+1,X}. Hence
i_{p+1,X} has no retraction.
G.3 Proof of the p = 5 corollary
Instantiate Theorem 1 at p = 5:
d = 38, h = 4, k = 16, d h^2 = 608, n = p = 5.
The image of beta_1^{16} in pi_608 L_4 X is nonzero and lies in the
kernel of pi_608( i_{5,X} ).
G.4 What the kernel theorem does NOT need
No permanent cycles. No spherical representatives. No comparison with
the Tate spectrum. The kernel proof uses only:
* the completion lemma (C.1) and the kernel criterion (C.2);
* Heard's E_2 calculation for the lower detector (D.2-D.5);
* the ordinary multiplicative structure of the homotopy fixed point
spectral sequence (Lemma D.4);
* the cohomological bound (E.1) and the numerical margin (F.4).
The strongest non-elementary fact is Heard's identification of beta_1
with beta in the Morava page for the finite group G. Everything else is
either algebraic-topological bookkeeping (kernel criterion, completion
lemma) or standard p-adic group theory (Lazard-Serre, cyclotomic
Eisenstein exclusion).
G.5 A warning on what would make the proof collapse
If one drops any of the following, the proof breaks:
* The multiplicative structure on the E_infinity page. Without it,
one does not know that the image of beta_1^{h^2} equals the h^2-th
power of the image of beta_1; the whole "long power" argument
collapses. This is why Lemma D.4 is a substantial step and not a
convenience.
* The +1 in the cohomological bound (a^2 + 1). Without the extra
lim^1 degree, the zero tail would start at a^2 + 1 instead of
a^2 + 2. This would make the margin larger, which is harmless in
the current cases, but the boundary behaviour is where the paper
is careful, and the "sharper line for sphere coefficients"
(Bobkova et al 2025) is a separate statement for a specific
coefficient module, not the finite-test bound that the exponent
argument requires.
* The torsion-freeness of E_a on positive homotopy groups. This is
what makes F^1 = ker(pi_* B_a -> pi_* E_a) automatically contain
all torsion classes.
* The even-periodicity of E_a. This is what makes F^1/F^2 = 0 on
even total degree, forcing the unit image to sit in F^2 and not
just in F^1.
--------------------------------------------------------------------------
PART H. THE INDEPENDENT EXPONENT OBSTRUCTION
--------------------------------------------------------------------------
This section gives the second, unrelated proof of Theorem 1. It uses a
different tool: H-equivariant descent exponents. The heart of the
argument is that weak CSC would force two finite numbers to satisfy
opposite inequalities.
H.1 Setup
Fix p >= 5 and put t = p-1, h = p. (The number h = p here is the upper
height the exponent argument is going to obstruct; t = p-1 is the lower
height. Do not confuse h with the h = p-1 of Part D.)
Let H = C_p >| C_{(p-1)^2} be the finite subgroup of the height-t pure
stabilizer S_t, acting on E_t by the coherent Morava action
(Theorem H.2.1, quoting Devinatz-Hopkins 2004 and Barthel-Beaudry-
Stojanoska 2020 for the subgroup structure).
Define:
D_H = Fun( BH, Sp )
= spectra with a coherent H-action
(= modules over the H-equivariant sphere, or
algebras over the group-ring in spectra);
A_H = F( H_+, S ), with H acting by translation
("the H-equivariant dual sphere");
J_H = fib( S -> A_H ) (called the AUGMENTATION FIBER);
exp_H(V) = the least b >= 1 such that
J_H ^ J_H ^ ... ^ J_H ^ V -> V (b copies of J_H)
is NULL in D_H. Otherwise infinity.
m = p^2 + 2;
T(V) = L_{K(t)}( E_t ^ V ) in Sp, with H acting on the FIRST
factor. T is an exact functor on ordinary p-local spectra
and takes values in D_H.
The exponent exp_H(V) measures the smallest number of copies of the
"reduced H-equivariant sphere" J_H needed to annihilate V. It is a
finiteness invariant. Theorem 3 says:
exp_H(E_{p-1}) > m (Part H.2)
exp_H(T(B_p)) <= m (Part H.3.c)
And a retraction of i_{p,X} would force exp_H(E_{p-1}) <= exp_H(T(B_p)).
This is a contradiction, and it proves the same nonretraction that
Theorem 1 proved via sphere classes.
H.1.5 The finite tensor cube and the pages bound
Two finite-stage identifications underlie BOTH exponent arguments.
The first is the "finite tensor cube" of Amitsur descent; the second
is the derived-couple computation that converts a page-vanishing into
a tested-image vanishing.
LEMMA H.1.5 (full finite tensor cube). Let C be a stable symmetric
monoidal infinity-category with exact tensor. Let A be a unital
algebra, J = fib(1 -> A), epsilon: J -> 1. For V in C and s >= 0,
let Tot_s denote the totalization of the partial augmented Amitsur
cosimplicial object V -> C_A^*(V) truncated at cosimplicial degree
s (including codegeneracies). Then there is a natural equivalence
of arrows over id_V in C_{/V}:
( fib(V -> Tot_s C_A^*(V)) -> V )
=: ( J^{^ (s+1)} ^ V ->(epsilon^{^ (s+1)} ^ id) V ).
The tower transition from stage s+1 to s applies epsilon to the
last J-factor. Any exact functor Phi: C -> Sp preserves this
finite fiber sequence.
Proof. Set P(s) = the poset of nonempty subsets of {0, ..., s}.
The functor theta_s: P(s) -> Delta_{<= s}, S |-> [|S|-1], records
order-preserving injections of subset into standard simplex. For
l <= s, the comma category theta_s/[l] is the poset of nonempty
faces of the simplicial complex P(s,l) whose simplices are lists
(i_0, j_0), ..., (i_b, j_b) with 0 <= i_0 < ... < i_b <= s and
0 <= j_0 <= ... <= j_b <= l.
Claim: P(s,l) is contractible. Proof by induction on s. Base
P(0,0) is a point. Every simplex of P(s,l) either omits the first
coordinate value s or contains exactly one vertex (s,j). Cover
P(s,l) by the cone C(s,l) of vertex (s,l) on P(s-1,l), then for
each j < l the cone C(s,j) of vertex (s,j) on P(s-1,j). The
first cone C(s,l) is contractible even when l = s, when its base
is not covered by induction. For each j < l, the base P(s-1,j)
is contractible by induction (j <= s-1), and is the exact
intersection of C(s,j) with the previous complex. Each attachment
preserves contractibility. So P(s,l) is contractible and theta_s
is homotopy initial.
Consequently Tot_s C_A^*(V) is the limit of the punctured
(s+1)-cube obtained by tensoring copies of 1 -> A and then tensoring
with V. Its omitted vertex is V. Taking successive fibers and using
exactness of tensor gives J^{^ (s+1)} ^ V, with its canonical
arrow to V. Removing the last direction gives the transition.
This is a genuinely finite cubical limit, so any exact Phi
preserves it (finiteness of the object list alone would not
suffice).
LEMMA H.1.6 (pages bound and tested image). Let Phi: C -> Sp be an
exact functor. Fix b >= 1 and consider the A-Amitsur descent
spectral sequence of V in C with E_2^{s,q} = H^s pi_q Phi C_A^*(V)
in the page convention d_r: (s,q) |-> (s+r, q+r-1).
(i) If V has A-exponent at most b (i.e. e_b(V) = epsilon^{^ b} ^
id_V is null in C), then E_{b+1}^{s,q} = 0 for all s >= b and all
q.
(ii) If V has SOME finite A-exponent and E_2^{s,q} = 0 for all
s >= b and all q, then pi_d Phi(e_b(V)) = 0 for every d. (This
is a zero tested image, not a nullity of the map.)
Proof. Write V' = Phi V, U_l = Phi(J^{^ l} ^ V) for l >= 0,
P_s = Tot_s(Phi C_A(V)) for s >= 0, and P_s = 0 for s < 0. Lemma
H.1.5 gives the compatible finite fiber sequences U_{s+1} -> V' ->
P_s. Define the augmentation filtration
F^s pi_d V' := ker(pi_d V' -> pi_d P_{s-1})
= im(pi_d U_s -> pi_d V'), s >= 0.
In coordinates (s,d) with d = q - s (total degree), the derived
couple of the tower has
D_r^{s,d} = im( pi_d P_{s+r-1} -> pi_d P_s ),
i: D_r^{s,d} -> D_r^{s-1,d},
j: D_r^{s,d} -> E_r^{s+r, d-1},
k: E_r^{s,d} -> D_r^{s,d}.
Let c be an actual A-exponent bound for V. Every length-c
transition P_{s+c} -> P_s is null, being Phi applied to a tensor
multiple of e_c(V). Consider the boundary square:
pi_d P_{s+c} --> pi_{d-1} U_{s+c+1}
| | 0
v v
pi_d P_s --> pi_{d-1} U_{s+1}.
Image of P_{s+c} in P_s has zero boundary (composition through
U_{s+c+1} -> U_{s+1} is null), and lifts to V' (composition to V
goes through U_{s+c+1} -> V which returns the original element on
the P_s side). Conversely, an augmentation class in im(pi_d U_s ->
pi_d P_s) factors through P_{s+c} via the null-transition.
Therefore
D_{c+1}^{s,d} = im(pi_d V' -> pi_d P_s).
Both sides vanish for negative s. These image groups have
surjective transition i. The augmentation into P_s is injective on
homotopy for s >= c-1 (its kernel factors through e_c(V)). Thus i
is an isomorphism for s >= c. Exactness of the derived couple
gives j = 0 and identifies
E_{c+1}^{s,d} = ker(D_{c+1}^{s,d} -> D_{c+1}^{s-1,d})
= F^s pi_d V' / F^{s+1} pi_d V'.
In particular this vanishes for s >= c. Taking c = b proves (i).
For (ii), retain any c. Since E_2^{s,q} = 0 for s >= b, the same
quotients F^s / F^{s+1} vanish for s >= b. Starting from F^c = 0
(or F^c contained in F^b when c < b) and descending, we get
F^b pi_d V' = 0. This is exactly the asserted zero image.
H.2 Lower bound: exp_H(E_{p-1}) > m
The lower bound is a finite-page survival statement in the ordinary
homotopy fixed point spectral sequence for H. It does not use any
sphere representative; a class survives to a specific finite page.
THEOREM H.2.1 (pure-stabilizer cohomology). Let E_t be Morava
E-theory of height t = p-1, with its coherent Morava stabilizer
action. The pure stabilizer S_t contains a finite subgroup H =
C_p >| C_{t^2} for which
H^*( H ; (E_t)_* ) = k_t[ alpha, beta^{+-1}, Delta^{+-1} ] /
( alpha^2 ),
with bidegrees
|alpha| = (1, 2t), |beta| = (2, 2pt), |Delta| = (0, 2pt^2).
The ordinary-to-Tate comparison is multiplicative and identifies
H^s( H ; (E_t)_q ) with this Tate group for every s > 0 and q.
Citations: this is the subgroup F in Barthel-Beaudry-Stojanoska 2020,
where the height is called n; their Theorem 4.5 and Remark 2.2 give the
cohomology and the identification. The pure-group calculation before
Galois invariants also appears in Heard 2015 (proof of Proposition
5.5). The coherent Honda action is Devinatz-Hopkins 2004 (Theorem
1(ii), Remark 2.4).
For the finite-page survival, work in D_H with pointwise smash, put
A_H = F(H_+, S) with translation action, and consider the ordinary
homotopy fixed point spectral sequence. Its terms are functions on the
free sets H^{q+1} (i.e., homogeneous group cochains), and the
multiplicative structure from Lemma D.4 applies without convergence
hypotheses.
PROPOSITION H.2.2 (finite-page survival). exp_H(E_t) > m.
Proof. Observe: m = p^2 + 2 < 2t^2 + 1 = 2(p-1)^2 + 1, since the
difference is p^2 - 4p + 1 > 0 for p >= 5. So m sits BELOW the
nilpotence order of beta.
Choose a positive integer N divisible by p with 2N > m. (Such N exists
by taking N = ceil((m+1)/2) and multiplying by p.) We show that
beta^N is NONZERO on E_{m+1} in bidegree (2N, 2ptN).
For 2 <= r <= m, all possible source and target filtrations in the
differential d_r: (s, q) -> (s+r, q+r-1) are positive, so the positive-
filtration algebra from Proposition H.2.1 applies. Now:
* If r is even, the target internal degree differs from the source
by r-1 which is odd, and H^*(H; (E_t)_*) is concentrated in even
internal degrees (up to the alpha factor in filtration 1). So d_r
must be zero on beta^N for even r.
* If r is odd, r-1 is even, and the internal-degree divisibility
forces r = 2ta + 1 with 1 <= a < t.
We analyze odd r:
* An outgoing target must be alpha * beta^{N+ta} * Delta^l. Internal
degree comparison gives a - 1 = pt(a + l), forcing a = 1.
* An incoming source must be alpha * beta^{N-ta-1} * Delta^l.
Internal degree comparison gives a - t = pt(a - l), impossible for
1 <= a < t.
* Thus only the OUTGOING differential d_{2t+1} is possible for
beta^N.
For d_{2t+1}: the same degree argument shows beta exists up to page
2t+1 (no incoming d_{2t+1} hits beta; the only possible incoming has
length 2 and odd internal source degree). The derivation rule on page
2t+1 gives:
d_{2t+1}( beta^N ) = N * beta^{N-1} * d_{2t+1}( beta ).
Now d_{2t+1}(beta) is some element; but its target lives in bidegree
(2t+2, ...) and is p-torsion (as all higher differential targets in
this range are). Since N is divisible by p, N * (p-torsion element) = 0.
Therefore d_{2t+1}(beta^N) = 0.
So beta^N survives through page m = p^2 + 2, in filtration 2N > m. This
class in filtration > m contradicts the finite-page exponent bound
(standard consequence of descent exponent calculus, Lemma H.1.6(i)):
if the exponent is at most m, then E_{m+1}^{s,q} = 0 for all s >= m.
Hence the exponent is strictly greater than m.
At p = 5, m = 27, one may take N = 15 (15 is divisible by 5 and
2*15 = 30 > 27): the class has filtration 30 on E_28, obstructing
exponent 27. This is the smallest case.
H.2.5 The finite-test coefficient complex
The upper argument needs to know that the Amitsur spectral sequence
of B_h, composed with the exact ordinary functor F(V_0, -) for a
finite V_0, has second page H^s_cts(G_h, pi_q(E_h ^ D V_0)). That is
Proposition H.2.5 below, which we assemble from three lemmas.
LEMMA H.2.5a (mixed cooperations flat). For positive heights a, b,
the ordinary smash product has even homotopy
pi_*(E_a ^ E_b) = (E_a)_* x_BP_* BP_*BP x_BP_* (E_b)_*,
flat over (E_a)_* through the first factor.
Proof. Landweber's cooperation calculation for the p-typical flat
Hopf algebroid (BP_*, BP_*BP) (Landweber 1976; p-typical form in
Rezk 1998 Section 15, Hovey 2004 Lemma 4.4) gives the displayed
formula. Both tensor products use the two different unit maps into
BP_*BP. For a Landweber-exact BP_*-algebra C, the module
BP_*BP x_BP_* C is flat over the other BP_*-action: tensoring with
it factors as the exact cofree-comodule functor followed by the
exact Landweber functor. This is the opposite-unit criterion of
HS 2003 Lemma 2.2. Base change along BP_* -> (E_a)_* proves the
asserted flatness. Evenness is automatic. No flatness of (E_b)_*
over BP_* is required.
LEMMA H.2.5b (base change and continuous functions). Let D_0 be
an ordinary right E-module whose homotopy tensor pi_*D_0 x_O_* (-)
is exact on finitely generated graded O_*-modules. Then for every
ordinary p-local W the natural comparison
pi_*D_0 x_O_* pi_*(E ^ W) -> pi_*(D_0 ^ W)
is an isomorphism. Moreover, for profinite T the functor
C_cts(T, -) is exact on finitely generated O-modules with their
m-adic topologies, and for every finitely generated graded
O_*-module M_* there is a natural isomorphism
C_cts(T, O_*) x_O_* M_* = C_cts(T, M_*).
Proof. First claim for finite W. The equality D_0 ^ W =
D_0 ^_E (E ^ W) is an equality of ordinary derived constructions.
Realize a finite graded free surjection onto pi_*(E ^ W) by an
E-module map from a finite wedge of suspensions of E. Surjectivity
makes the fiber homotopy the first syzygy. Repeat along a finite
free resolution. At the last stage, lifts of a free basis give an
equivalence from a finite free E-module. Relative smash with D_0
is exact. Induction back through these fiber sequences, using the
assumed tensor exactness on each finitely generated syzygy,
proves the first claim for finite W. Every ordinary p-local
spectrum is a filtered homotopy colimit of finite ones; both sides
preserve the colimit. This extends to arbitrary W.
Second claim. Let M -> N be a surjection of finitely generated
O-modules. For every j >= 1 the map
M/m^{j+1}M -> (M/m^jM) x_{N/m^jN} (N/m^{j+1}N)
is surjective: lift the first component to M, its error against
the second component lies in m^jN, and this error lifts from
m^jM. Each quotient is a finite set. Given a continuous map
x: T -> N, first lift its reduction mod m, then lift successively
while preserving the previous choice. Continuous value sets give
continuous lifts. Complete separated targets and compact
Hausdorff kernels make C_cts(T, -) exact on these modules. Apply
this exact functor to a finite presentation of M, and compare
with tensoring the same presentation by C_cts(T, O); the
comparison is an isomorphism on finite free terms and hence on M.
Applying this argument in each parity proves the tensor identity.
In particular C_cts(T, O_*) x_O_* (-) has precisely the tensor
exactness used in the first claim.
LEMMA H.2.5c (K(a)-localization is derived m-completion at flat
homotopy). Let N be an ordinary E_a-module with flat graded
homotopy U_* = pi_*N. Then the K(a)-localization map induces
U_* -> pi_*L_K(a)N = \hat U_* := lim_j U_* / m^j U_*,
and for every j >= 1, \hat U / m^j \hat U = U / m^j U.
Proof. K(a)-localization is derived m-adic completion for ordinary
E_a-modules, and flat homotopy has no higher derived completion;
the natural localization map induces the ordinary completion map
(Rezk 2009 Prop 3.4, 3.6, Cor 3.8).
Reduction identity. Choose a resolution P_* -> N_0 of a finitely
generated O-module N_0 by finite free modules. The tower
P_* x_O U / m^l U is degreewise surjective. Its first homology is
Tor_1^O(N_0, O/m^l) x_O U. This tower is pro-zero: for a first
presentation 0 -> K -> P_0 -> N_0 -> 0, the Tor group is
(K cap m^l P_0) / m^l K; Artin-Rees (Stacks Tag 00IN) gives d with
K cap m^l P_0 in m^{l-d} K for l >= d, so for any fixed j the
transition from l >= j+d to j is zero. Tensoring these zero
transitions with U preserves the conclusion.
The homology exact sequence for the countable inverse limit of
this degreewise-surjective tower gives
N_0 x_O \hat U = lim_l (N_0 x_O U / m^lU);
the possible lim^1 of the first homology is zero because that
tower is pro-zero. Finite freeness of the P_i identifies their
tensor with \hat U with their inverse-limit tensor. Take
N_0 = O/m^j; for l >= j the modules on the right are U / m^j U.
This proves the reduction identity.
PROPOSITION H.2.5 (tested Amitsur coefficient complex). For r >= 0
set Z_r = L_K(a)(E_a^{^ (r+1)}). For every finite ordinary
p-local W there are natural graded isomorphisms
pi_*(Z_r ^ W) = C_cts(G_a^r, pi_*(E_a ^ W)).
Consequently for finite V_0 and D V_0 = F(V_0, S), the application
of F(V_0, -) to the local unit Amitsur tower for E_a in C_a has
second page
E_2^{s,q} = H^s_cts(G_a, pi_q(E_a ^ D V_0)).
Proof. Hovey 2004 (Section 1, Theorem 4.11, Theorem 6.8) gives
the completed cooperation
pi_*L_K(a)(E_a ^ E_a) = C_cts(G_a, O_*),
with evaluation at g given by mu(1 ^ g) and the first factor
acting by constant coefficients. Put P_* = pi_*(E_a ^ E_a) as
an O_*-module through the first factor. Lemma H.2.5a makes P_*
flat. By Lemma H.2.5c the reductions of P_* modulo m^jP_* are
C_cts(G_a, O_*/m^jO_*). Lemma H.2.5b with M_* = O_*/m^jO_*
gives
P_* / m^jP_* = C_cts(G_a, O_* / m^jO_*),
induced by the actual ordinary evaluation. Localize now: H.2.5c
identifies the result with the completed module C_cts(G_a, O_*).
For higher r, use the ordinary relative smash
(E_0 ^ E_1) ^_{E_0} (E_0 ^ E_2) ^_{E_0} ...
^_{E_0} (E_0 ^ E_r)
= E_0 ^ E_1 ^ ... ^ E_r,
with all E_i copies of E_a. Iterating Lemma H.2.5b, including its
extension to arbitrary ordinary W, identifies the homotopy with
P_*^{^{*_O_* r}}, which is flat. Continuous functions to finite
coefficients factor through finite quotients of G_a^r, so
C_cts(G_a^r, -) = C_cts(G_a, -)^{^{* r}}. Applying completion
at the end gives pi_*(Z_r) = C_cts(G_a^r, O_*). The evaluation
maps survive: each ordinary evaluation extends through localization
because its target E_a is local; its first-factor O_*-linearity
makes it adically continuous; base-change density makes the
extension agree with the displayed continuous function.
For finite W, base-change H.2.5b converts to
pi_*(Z_r ^ W) = C_cts(G_a^r, pi_*(E_a ^ W)). Finally
F(V_0, Z_r) = Z_r ^ D V_0 for finite V_0 by ordinary duality.
Relabel evaluations as cumulative coordinates
1, g_1, g_1 g_2, ..., g_1...g_r. The coface operators are
(\delta^0 c)(g_1, ..., g_{r+1}) = g_1 c(g_2, ..., g_{r+1}),
(\delta^i c)(g_1, ..., g_{r+1}) = c(g_1, ..., g_i g_{i+1}, ..., g_{r+1}),
(\delta^{r+1} c)(g_1, ..., g_{r+1}) = c(g_1, ..., g_r),
and the codegeneracies insert the identity at the corresponding
coordinate. These are the continuous inhomogeneous cochain
operators, so the Amitsur page is H^s_cts(G_a, pi_q(E_a ^ D V_0)).
Done.
H.3 Upper bound: exp_H(T(B_p)) <= m (structural overview)
The upper bound is much more involved. It has three sub-arguments:
H.3.a. An ACTUAL NULL MAP in the local category C_h:
e_m(B_h): I_h^{^ m} -> B_h is null, where
I_h = fib(B_h -> E_h) in C_h (the local unit augmentation).
H.3.b. A TRACE-ONE STRUCTURE on the mixed algebra D' =
L_{K(t)}(E_t ^ E_h), which forces every D'-module in D_H to
have H-exponent 1.
H.3.c. A TRANSFER: apply T to the finite filtration/retract of
B_h coming from H.3.a; each of the m layers becomes a
D'-module with H-exponent 1; sum the layers to get T(B_h)
with exponent <= m.
We develop each. The exponent calculus is the bridge.
LEMMA H.3.0 (exponent calculus; the finite-page and tested-image
bounds are Lemma H.1.6).
Let A be a unital algebra in a stable symmetric monoidal
infinity-category with exact tensor, J = fib(1 -> A), epsilon: J
-> 1.
(i) Every A-module has exponent at most 1.
(ii) If a V has exponent <= b, every retract of V has exponent
<= b, and every V ^ W has exponent <= b.
(iii) For a cofiber sequence U -> V -> W with exponent bounds b on
U and c on W, V has exponent <= b+c.
(iv) If e_m(V) = epsilon^{^ m} ^ id_V is null (m = exponent
bound), then V is a retract of an object with m successive
layers A ^ J^{^ i} ^ V, 0 <= i < m.
(v) FINITE-PAGE / TESTED-IMAGE. See Lemma H.1.6(i) and (ii).
Proof. (i): the unit splittings of an A-module. (ii): naturality. (iii):
choosing a lift along the null composition of e_c on W. (iv): the
successive cofibers J^{^ (i+1)} ^ V -> J^{^ i} ^ V have cofiber equal
to A ^ J^{^ i} ^ V; the tower of phi = e_m(V) sits over V; nullity of
phi retracts V off the cofiber of phi = a filtration with m layers.
PROPOSITION H.3.a (the upper null map). Let p >= 5 be prime, h = p,
in C_h = Sp_{K(h)}, let I_h = fib(B_h -> E_h). Then
e_m(B_h): I_h^{^ m} -> B_h is NULL in C_h.
Hence exp_{E_h}(B_h) <= m, and B_h is a retract of an object with
m successive layers E_h ^ I_h^{^ i}, 0 <= i < m.
Proof. The argument has four parts.
(1) SOME finite exponent. Devinatz-Hopkins (2004, Appendix A,
Proposition A.3) prove that every K(h)-local spectrum belongs to
the thick tensor ideal generated by E_h. In particular B_h has
some finite E_h-exponent c. This uses nilpotence and gets a
FINITE c without needing any numerical value.
(2) FINITE TESTS vanish. The Amitsur cosimplicial construction on
B_h in C_h, composed with the exact functor F(V_0, -) for a
finite p-local V_0, has E_2 page (from the ordinary cooperation
calculation, Proposition H.2.5) H^s_{cts}(G_h;
pi_q(E_h ^ D V_0)). Proposition E.1 makes this
zero for s >= h^2+2 = m. Using the already-known finite
exponent c, the tested-image part of the finite-page bound
(Lemma H.1.6(ii)) gives:
[ V_0 , I_h^{^ m} ] -> [ V_0 , B_h ] is zero
for every finite p-local V_0.
(3) PHANTOM LEMMA -> actual nullity. The category of maps into B_h
with "null after every finite precomposition" is characterized by
Brown-Comenetz dualizing. (Christensen-Strickland 1998,
Proposition 4.11, is the criterion used.) The argument:
B_a ~ F( D_a , D_a ) ~ F( D_a ^ M_a S , I_BC ),
D_a = F( M_a S , I_BC )
where M_a S = fib(L_a S -> L_{a-1}S) and I_BC is the
Brown-Comenetz spectrum for Q/Z_{(p)}. Then Brown-Comenetz
representability gives, for any U,
[ U , B_a ] ~ Hom_Z( pi_0( U ^ D_a ^ M_a S ) , Q/Z_{(p)} ).
A filtered homotopy colimit U = colim U_i with U_i finite
p-local gives
pi_0( U ^ D_a ^ M_a S ) = colim_i pi_0( U_i ^ D_a ^ M_a S ).
So the "zero after every finite precomposition" condition makes
the functional vanish on the colimit, hence the map is zero.
Nothing here interchanges K(a)-localization with the filtered
colimit; the argument is entirely about ordinary spectra, using
Gross-Hopkins duality only to rewrite B_a as a function spectrum.
(4) RETRACT DECOMPOSITION. Set I_h^{^ 0} = B_h, and for 0 <= j <= m
put F_j = cofib( I_h^{^ m} -> I_h^{^ (m - j)} ). Successive
F_j form a finite filtration 0 = F_0 -> F_1 -> ... -> F_m, with
cofibers identified as
cofib( F_j -> F_{j+1} ) = E_h ^ I_h^{^ (m - j - 1) }.
Since F_m = cofib( e_m(B_h) ) and e_m is null, B_h is a retract
of F_m. This gives the m-layer decomposition.
PROPOSITION H.3.b (the completed mixed trace). Let D = E_t ^ E_h
with H acting on the first factor, and D' = L_{K(t)}D. Then D' has
an underlying degree-zero class whose H-trace is a unit. Consequently
every D'-module in D_H has H-exponent at most 1.
Proof. There are three sub-steps.
Sub-step 1 (trace splitting). Let z in pi_0 D' satisfy:
v = Sum_{g in H} g * z is a UNIT in pi_0 D'.
Then for any D'-module U, U has H-exponent <= 1.
Proof: Since H is finite, A_H ^ U = F(H_+, U) with diagonal action.
Untwisting via theta(f)(g) = g^{-1} f(g) identifies
A_H ^ U ~ Coind_1^H( Res_1^H U ).
For finite H, induction and coinduction agree (finite coproduct-
product equivalence). Induction adjunction takes the underlying
multiplication mu_z: U -> U to a coherent H-equivariant map
r_z: A_H ^ U -> U.
The unit j: U -> A_H ^ U has component g^{-1}U at position g (under
theta). The underlying composite r_z j = Sum_g g mu_z g^{-1} =
mu_{Sum_g g * z} = mu_v. Since v is a unit, this is an equivalence;
so (r_z j)^{-1} r_z is the required coherent retraction; the fiber
is null, giving H-exponent <= 1.
Sub-step 2 (field-valued freeness). For every unital ring map
x: R -> Omega taking values in a field, and g in H, if x o g = x
then g = 1.
Proof: R = pi_0 D / m_t pi_0 D. The unit map O_t -> pi_0 D -> Omega
has image contained in a field Omega. Both factor maps E_t -> D and
E_h -> D give formal group isomorphisms after base change to Omega.
The H-action on E_h is trivial (the identity on E_h factor), so the
H-action on D must respect the E_h formal group. The H-action on E_t
is the prescribed Honda-stabilizer action. Formal-group naturality
then shows g must act trivially on the Honda formal group, i.e.
g's power series equals the identity. On k_t the Honda formal group
distinguishes elements of H, so g = 1.
Sub-step 3 (algebraic trace). Let a finite group H act on a
commutative ring R, and suppose for every unital field-valued map
x: R -> Omega, x o g = x implies g = 1. Then
1 in im( Tr: R -> R^H ), Tr(r) = Sum_g g(r).
Proof: Tr-image is an ideal of R^H. If proper, pick maximal q
containing it. Orbit polynomials prod_g (X - g(r)) are monic over
R^H, so R integral over R^H; lying-over gives a prime p above q,
and x: R -> Frac(R/p) = Omega. Distinct chi_g's (from distinctness)
are linearly independent. Sum_g chi_g is not zero (each coefficient
is 1), but equals x o Tr which is zero (x kills Tr-ideal).
Contradiction.
Wrap-up. Sub-step 2 gives freeness on field-valued points of
R = pi_0 D / m_t pi_0 D. Sub-step 3 gives a class z-bar in R with
Trace(z-bar) = 1. Lift to z_0 in pi_0 D, set v = Sum_g g(z_0). Then
v - 1 in m_t pi_0 D. The mixed-cooperation flatness says pi_*D is
flat over (E_t)_* through the first factor; therefore the
localizing map D -> D'
identifies pi_0 D' with the m_t-adic completion of pi_0 D. At each
finite adic stage the telescoping element w_j = Sum_{k=0}^{j-1}
(1-v)^k satisfies v w_j = 1 - (1-v)^j = 1 mod m_t^j. The w_j's
are compatible under reduction, so they give a multiplicative
inverse to (the image of) v in pi_0 D'. Then Sub-step 1 applies:
every D'-module has H-exponent <= 1.
PROPOSITION H.3.c (transfer). For p >= 5 with t = p-1, h = p:
exp_H( T(B_h) ) <= m = p^2 + 2, where T(V) = L_{K(t)}(E_t ^ V).
Proof. Apply T to the m-layer decomposition of B_h from H.3.a. T preserves
cofibers and retracts. Each individual layer E_h ^ I_h^{^ i} becomes,
after application of T, E_t ^ E_h ^ I_h^{^ i} = D ^ I_h^{^ i}. Then
locally K(t)-ize: L_{K(t)}(D) ^ L_{K(t)}(I_h^{^ i}) [up to lax monoidal
correction, which is fine]. The result is a D'-module in D_H.
By H.3.b, every D'-module has H-exponent <= 1. So all m layers have
exponent <= 1. By the exponent calculus (H.3.0 part iii), summing m
cofiber sequences gives T(B_h) with exponent <= m.
H.4 Assembling the exponent contradiction
COROLLARY H.4.1. i_{h,X} = i_{p,X} has no homotopy retraction.
Proof. Suppose i_{p,X} admitted a retraction r. The key square is:
S --eta_S^{(p)}--> B_p
| |
lambda_X c lambda_{L_{K(p)}X} L_{K(p)}(c)
v v
L_t X --i_{p,X}--> L_t L_{K(p)} X
The two vertical arrows become EQUIVALENCES under T:
* Left: lambda_X and c are K(t)-equivalences. By definition T
inverts K(t)-equivalences.
* Right: L_{K(p)}(c) is an equivalence, and lambda_{L_{K(p)}X} is a
K(t)-equivalence. So the composite is inverted by T.
Also T(S) = L_{K(t)}(E_t ^ S) ~ E_t with its coherent H-action
(E_t is K(t)-local and S is the unit).
Thus applying T to the retraction square and using r gives, in D_H,
E_t as a RETRACT of T(B_p). But:
* H-preservation of retracts: exp_H(E_t) <= exp_H(T(B_p)).
* H.2: exp_H(E_t) > m.
* H.3.c: exp_H(T(B_p)) <= m.
Contradiction. So no retraction exists.
H.5 Theorem 3, restated
THEOREM 3 (exponents, from earlier statement). Let p >= 5 prime,
t = p-1, h = p, m = p^2 + 2, H = C_p >| C_{(p-1)^2} acting on
E_t by its coherent Morava action, T(V) = L_{K(t)}(E_t ^ V) with
H acting on the first factor. Then
exp_H( E_{p-1} ) > m,
exp_H( T(B_p) ) <= m,
and a retraction of i_{p,X} would give E_{p-1} as an H-equivariant
retract of T(B_p), contradicting those bounds.
The two proofs of the nonretraction (Part G and Part H) are
INDEPENDENT: neither uses the other's key input. The kernel proof uses
sphere classes; the exponent proof uses H-equivariant nullity. They
both land on the same theorem.
H.6 What the exponent proof needs (vs what the kernel proof needs)
KERNEL PROOF needs:
* Heard's E_2 calculation (D.2-D.3)
* multiplicative structure on homotopy fixed point SS (D.4)
* cohomological line E.1 (Lazard-Serre + trace-one)
* numerical margin (F.4)
EXPONENT PROOF needs:
* nilpotence theorem (finite exponent existence)
* Amitsur construction on the local E_h (H.3.a step 2)
* Brown-Comenetz phantom lemma (H.3.a step 3)
* trace-one on mixed D' via formal-group field-points (H.3.b)
* the exact transfer functor T (H.3.c)
Neither set contains the other. Both are genuine.
--------------------------------------------------------------------------
PART I. WHAT THE PAPER DOES NOT CLAIM, AND OPEN QUESTIONS
--------------------------------------------------------------------------
It is as important to know the boundaries of what is proved as to know
the proof. The paper is a refutation of a specific conjecture in a
specific range.
I.1 What the paper does NOT claim
* The STRONG chromatic splitting conjecture is not refuted by this
paper at any prime. Only the WEAK form (retraction existence) is
refuted at heights p and p+1, and only at p >= 5 and p >= 7
respectively.
* No claim at primes p = 2 or p = 3. The proofs use inequalities
like p^2 - 4p + 1 > 0 which fail at small primes; supplying these
cases would require different techniques.
* No claim about heights below p or above p+1 by this method. The
specific power z_p loses its margin at other heights. For
example, at height 6 for p = 5, filtration 32 sits below the a^2
+ 2 = 38 threshold, so no upper vanishing follows.
* The exponent-argument (Part H) covers height p but not height
p+1. Extending it to p+1 is an open question.
* The paper does NOT construct a natural-in-X failure of weak CSC.
It constructs a failure on the single test X = S_p^, using
completion. Whether the failure propagates to arbitrary finite
X_p^ requires the canonical map on S_p^ to be "rigid" for the
property of having no retraction; the paper establishes that S_p^
itself witnesses nonretraction, but does not claim that every
finite X_p^ inherits it.
I.2 What is genuinely OPEN after this paper
* Whether weak CSC holds at p = 2 or p = 3, for the version of the
conjecture that allows completed finite inputs. (The strong form
is known to fail at p = 2 by Beaudry 2017 with a revised form in
Beaudry-Goerss-Henn 2022; the weak form at p = 2, 3 is a genuine
open question and may require different methods.)
* Whether the exponent obstruction extends to a full range of
heights, say all n >= p, using the same D' = L_{K(t)}(E_t ^ E_n)
trace-one. The key sub-step (H.3.b field-freeness) uses the
specific height gap t = p-1, h = p; other gaps would need
different analysis.
* Whether beta_1^{h^2} can be replaced by a shorter power. The
current argument needs 2h^2 >= a^2 + 2 in the upper height; a
shorter power would require either a smaller a or a sharper
filtration bound.
* Whether the nonzero kernel z_p corresponds to anything geometric
in the sphere. The paper isolates the class; it does not exhibit a
geometric model.
I.3 The honest summary
The paper is not a "no chromatic splitting" paper. It is a "no
canonical lower-height retraction at height p (and p+1) when the input
is the p-completed sphere, for p large enough" paper. It gives two
independent proofs, both using the specific sphere class
z_p = beta_1^{h^2}, and neither requires new technology at the
frontier: one uses known cohomology bounds and filtration arguments,
the other uses finite equivariant descent exponents.
The result is a genuine counterexample to a universal assertion that
had been implicitly assumed in some corners of the chromatic splitting
literature. The evidence is a single nonzero-kernel class z_p; the
paper does not try to characterize which other classes behave this way.
--------------------------------------------------------------------------
GLOSSARY
--------------------------------------------------------------------------
A_H: F(H_+, S) with H acting by translation. The "H-equivariant
dual sphere." (Part H.1)
B_a: L_{K(a)} S. The height-a localized sphere. (Part A.4)
beta_1: A specific stable homotopy class of spheres pi_d S with
d = 2p(p-1) - 2. Detected by height p-1 E-theory. (Part D.2)
cofib: The cofiber of a map of spectra, i.e., the "cone" of the map.
Fits into X -> Y -> cofib(X -> Y) -> Sigma X.
D_H: Category of spectra with a coherent H-action. (Part H.1)
d_p: 2p(p-1) - 2. The degree of beta_1. (Part B.1)
Delta: A class in the E_2 page of the Morava descent SS at filtration 0,
bidegree (0, 2pt^2). (Part D.2)
E_a: Morava E-theory of height a, with coefficient ring
O_a[u^{+-1}] and O_a = W(F_{p^a})[[u_1,...,u_{a-1}]]. (Part A.3)
eta: The unit map S -> L_{K(a)}S or S -> E_a^{hG}. (Part A.4)
exp_H: The H-exponent: least b with J_H^{^b} ^ V -> V null. (Part H.1)
F^s: The s-th step of the decreasing homotopy filtration of a
spectral sequence's abutment. (Part E.5)
G_a: S_a >| Gal(F_{p^a}/F_p). The extended Morava stabilizer. (Part A.3)
H: C_p >| C_{(p-1)^2}, the finite subgroup of the pure stabilizer
used in the exponent argument. (Part H.1)
H_a: The Honda formal group of height a over F_{p^a}. (Part A.3)
i_{n,X}: L_{n-1}( eta_X^{(n)} ): L_{n-1} X -> L_{n-1} L_{K(n)} X.
The overlap map. (Part A.5)
I_a: fib(B_a -> E_a). The augmentation fiber at height a. (Part E.5)
I_BC: Brown-Comenetz dualizing spectrum for Q/Z_{(p)}. (Part H.3.a)
J_H: fib(S -> A_H). The H-augmentation fiber. (Part H.1)
k_p: (p-1)^2. Half the exponent power. (Part B.1)
K(a): Morava K-theory of height a, = E_a / m_a. A field spectrum.
(Part A.3)
K(0): H Q, rational cohomology. (Part A.2)
L_r: Localization with respect to K(0) v ... v K(r). (Part A.2)
L_K(a): Localization with respect to K(a). (Part A.4)
m: p^2 + 2. The exponent threshold. (Part H.1)
m_a: (p, u_1, ..., u_{a-1}), the maximal ideal of O_a. (Part A.3)
O_a: W(k_a)[[u_1,...,u_{a-1}]], the deformation ring. (Part A.3)
phi_p: The p-th cyclotomic polynomial (X^p-1)/(X-1). (Part E.2)
pi_k: The k-th stable homotopy group of spheres. (Part A.1)
S: The sphere spectrum, S^0_{(p)} p-localized. (Part A.1)
S_a: The pure stabilizer subgroup of G_a. (Part A.3)
S_p^: The p-completed sphere, = holim S/p^j. Also written X. (Part A.1)
smash: The tensor product in spectra, X ^ Y. (Part A.1)
T: The exact transfer functor T(V) = L_{K(t)}(E_t ^ V).
(Part H.1)
weak CSC: The conjecture that i_{n,X} admits a homotopy retraction
for every finite X. (Part A.6)
W(k_a): The Witt vectors of the finite field k_a. (Part A.3)
X: S_p^ (the p-completed sphere). Same as S_p^. (Part A.1)
Y: The lower detector E_h^{hG} used in Part D. (Part D.1)
z_p: beta_1^{k_p}, a specific stable homotopy class of spheres. The
kernel class. (Part B.1)
--------------------------------------------------------------------------
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