# Filtered chromatic splitting at generic primes

*A reader's-guide exposition, written for a student who knows vectors,
matrices, polynomials, and a little linear algebra, but who has never
seen stable homotopy theory before.*

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## 0. Prologue

Open a book on modern topology and you will see a table of numbers
called the **stable homotopy groups of spheres**. They are strange.
They are infinite in number, they contain copies of the integers and
copies of finite cyclic groups, and their structure rises in "layers"
that repeat with periodicity.

A famous programme (chromatic homotopy theory) organizes all this by a
number called **height**, written n = 1, 2, 3, ... . At height n you see
one "colour" of the sphere. The colours do not mix cleanly; they
overlap, and the exact shape of the overlap is the subject of this
paper.

The claim we will explain is this. Fix a prime p and a height n with
p > n + 1. Then there is a way to "slice" a certain overlay object,
called X_{n,p}, into a stack of 2^n simple layers. Each layer is a
shifted copy of a familiar height piece of the sphere. The bottom
layer sits on top of the rational sphere in exactly the canonical way.
The main theorem of the paper says that such a slicing *exists*
(see Theorem B.1 below).

The paper is 437 kilobytes of LaTeX. The point of this document is to
explain, step by step and with no unexplained jargon, what the claim
means and how the proof works. Every named result of the original
paper is mentioned. The deepest technical lemmas are summarized: you
get their exact statement and intuition, and you get how they feed
into the main line of the proof. You do not get twenty pages of
epsilon-chasing, because a high-schooler (and, honestly, most
mathematicians) would not read that.

---

## PART A. What is being studied

### A.1 The sphere and stable homotopy

Take the spheres S^0, S^1, S^2, S^3, ... . Between S^m and S^n one
can ask: how many genuinely different maps are there (up to
homotopy)? Record these as a written symbol

    pi_m(S^n) = the set of homotopy classes of maps S^m -> S^n.

This set is a group when m > n (actually it is a group whenever
m >= 1, and an abelian group when m >= 2). Examples: pi_n(S^n) = Z
(degrees), pi_3(S^2) = Z, pi_6(S^3) = Z / 12, and so on.

Add a suspension S^m -> S^{m+1} by joining a north-south pole. There
is a natural map

    pi_m(S^n) -> pi_{m+1}(S^{n+1}),

and these maps become isomorphisms once m < 2n - 1 (the Freudenthal
suspension theorem). So the "stable" part, pi^s_k := pi_{n+k}(S^n)
for n large, is the interesting invariant. The stable homotopy groups
of S^0 (or "of spheres") are

    pi^s_0 = Z, pi^s_1 = Z/2, pi^s_2 = Z/2, pi^s_3 = Z/24, ...

These numbers alternate integer and finite parts and grow. Nobody
knows a closed formula, and nobody expects one. The "layers"
mentioned above are one way to organize them.

### A.2 Two simplifications: p-primary and rational

Away from small primes, the "p-primary part" of pi^s_k (the part
whose elements die when multiplied by some power of p) is what you
study. Once p is fixed, understanding the p-part of every pi^s_k is
the same as understanding a spectrum called the **p-complete sphere**

    S = S_p^wedge  (spoken: "p-adic sphere"),

whose pi_* is the p-completion of pi^s_*. You can pretend, for the
rest of this document, that "sphere" always means S_p^wedge, at a
fixed odd prime p.

The other simplification is rationalization: killing all torsion by
tensoring with Q. The rational sphere H Q is (by Serre's theorem)
just the rational numbers in degree zero and nothing else. So
rationalization of the sphere forgets almost everything; the paper
uses rational information only as a dimension count.

### A.3 Bousfield localization

You can force a chosen theory into existence by **localization**:
if E is some homology theory, then L_E X is the closest object to X
that E cannot distinguish from a point. Formally L_E X is universal
among maps X -> Y where Y has E-homology in the same places as 0.

The classical choices of E in chromatic homotopy are:

- E = H Z_(p). Its localization L_0 X is rationalization (or "height
  zero").
- E = K(1), the first Morava K-theory. Height 1.
- E = K(n), the n-th Morava K-theory. Height n.
- E = E(n), the Johnson-Wilson theory. Height up to n.
- E = E_n, Morava E-theory at height n. A "richer" version of K(n).

Each height sees different aspects of the sphere. They overlap.
"Fracture" says L_{E(n)} sits in a square between the pieces at
height n and height at most n - 1.

### A.4 Morava K-theory and Morava E-theory

**Morava K-theory K(n)** is a periodic homology theory whose
coefficient ring is

    pi_* K(n) = F_p[v_n^{+-1}],     degree of v_n = 2(p^n - 1).

So it is a graded finite field, with a periodicity engine v_n. It is
the "n-th chromatic colour" of the sphere.

**Morava E-theory E_n** is a richer theory whose coefficient ring is

    pi_0 E_n = W(F_{p^n}) [[ u_1, u_2, ..., u_{n-1} ]]

(a power-series ring over the Witt vectors of F_{p^n}), with a period
u making it 2-periodic. It has an action of a certain profinite group,
the "Morava stabilizer group" G_n, and the K(n)-local sphere can be
recovered as fixed points of this action.

**Johnson-Wilson theory E(n)** is a version that sees all heights up
to n at once: E(1) roughly captures "complex K-theory-like" behaviour,
E(2) captures "elliptic-curve-like" behaviour, and so on. The
localization L_{n-1} := L_{E(n-1)} is the "up to height n-1" piece.

### A.5 The object X_{n,p}

With this language, one central object is

    X_{n,p} := L_{n-1} ( L_{K(n)} S ).

Read it inside-out: first take the K(n)-local sphere (the cool
n-height piece, complicated), then apply L_{n-1} (throw away
everything above height n - 1). The result is the "overlap" between
the height-n world and the lower heights.

X_{n,p} is the object the paper is about. Understanding it is exactly
understanding how the top layer (height n) interacts with the rest.

### A.6 The chromatic splitting conjecture

In 1987, Mike Hopkins formulated the following. He conjectured that
in some prime ranges the map

    L_{n-1} S  --->  L_{n-1} L_{K(n)} S

has a section: there is a spectrum, sitting inside X_{n,p}, that maps
back to L_{n-1} S and undoes the localization. This is the
**weak chromatic splitting conjecture**.

If the section exists, one can split X_{n,p} as a wedge of pieces.
A much stronger form (due to Hopkins, in a letter of July 1993 that
Hovey transcribed) predicts not only *that* it splits but *what* the
wedge summands are: shifted copies of the sphere localized at exactly
height n - j, for j = 0, 1, 2, ... . Written precisely:

    X_{n,p}   is conjectured to be   ~   the wedge of
          Sigma^{-d(I)} L_{n - max(I)} S

where the wedge runs over subsets I of {1, 2, ..., n}, and
d(I) = sum_{i in I} (2i - 1). For example at n = 2, the pieces are

    L_1 S, Sigma^{-1} L_1 S, Sigma^{-4} H Q, ...

At n = 1, the pieces are the rational sphere H Q and its one-fold
suspension Sigma^{-1} H Q, and the splitting is classical.

At low heights and small primes this conjecture is known, at least in
weak form: height 1 (classical), height 2 with p > 3 (Hopkins, using
work of Shimomura-Yabe and Devinatz-Hopkins), height 2 with p = 3
(Goerss-Henn-Mahowald).

At height 2, p = 2, the classical (strong) form fails:
Beaudry showed in 2017 that there are extra Moore-spectrum summands
you were not expecting, and Beaudry-Goerss-Henn (2022) computed what
the true answer is.

At height >= 3 the classical (strong) form is largely open. So even
the shape of the overlap above height 2 is unknown in general. This is
where the paper sits: rather than trying to split, it proves a *weaker
but still very structured* statement.

---

## PART B. The main theorem of the paper

### B.1 What a "filtration" is

A **filtration** of a spectrum X is a chain of spectra

    0 = F_0  ->  F_1  ->  F_2  ->  ...  ->  F_N  ->  ...  -> X.

Each arrow is a map, and the composite through to X should end up
being an equivalence F_N ~ X once N is big enough. If it is exactly
a cofiber sequence at each step, we call the pieces

    C_j = cofiber(F_{j-1} -> F_j)

the "successive cofibers" of the filtration. A filtration is not
the same thing as a wedge decomposition: the pieces are glued
together with attaching maps, and those attaching maps matter.

If the pieces all happen to split off compatibly, the filtration
turns into a wedge decomposition X ~ C_1 vee C_2 vee ... . If the
attaching maps are nontrivial, the filtration is strictly more
information than the wedge.

### B.2 The statement

**Theorem B.1 (main theorem of the paper; in the paper it is
`Theorem 1.1`).** Let n >= 1 and let p > n + 1 be a prime. Then
there exist:

- an ordering I_1, ..., I_{2^n} of the 2^n subsets of
  {1, 2, ..., n}, with I_1 = the empty set, and with
  max(I_j) nondecreasing for j >= 2;
- a filtration

      0 = F_0 -> F_1 -> ... -> F_{2^n} ~ X_{n,p}

  of X_{n,p} in the category of E(n-1)-local S-modules;
- a family of sphere-classes y_i in pi_{1-2i}( D ) for D = L_{K(n)} S,
  for 1 <= i <= n,

such that the following two properties hold.

**(a) The filtration and its unit.** The successive cofibers are

    C_empty = L_{n-1} S,
    C_I = Sigma^{-d(I)} L_{n - max(I)} S   for I nonempty,

where d(I) = sum_{i in I} (2i - 1). Moreover, under the identification
F_1 = C_empty, the composite F_1 -> X_{n,p} is exactly the canonical
localization unit L_{n-1} S -> L_{n-1} L_{K(n)} S.

**(b) Simultaneous product bases.** With y_empty = 1 and, for
I = {i_1 < ... < i_r}, y_I := y_{i_1} y_{i_2} ... y_{i_r}, the map

    L_{K(t)} (  bigvee_{I subseteq {1, ..., n-t}}  Sigma^{-d(I)} S
                -->  D  )

is an equivalence for every 1 <= t < n. The map sends the summand
indexed by I to D by the class y_I.

### B.3 Reading the pieces

The list of cofibers is the classical Hopkins list: L_{n-1} S, one
suspension-shift of it by -1, then pieces at height n-2, then height
n-3, ..., and finally at height 0 (the rational sphere) you get
several copies in different degrees. The number of copies at each
height is 2^{j-1} for the height n-j piece. The total 2^n adds up.

For example at n = 3 (so p >= 5), one admissible order is

    L_2 S,  Sigma^{-1} L_2 S,  Sigma^{-3} L_1 S,  Sigma^{-4} L_1 S,
    Sigma^{-5} H Q,  Sigma^{-6} H Q,
    Sigma^{-8} H Q,  Sigma^{-9} H Q.

That is 2^3 = 8 pieces: 2 copies at height 2, 2 copies at height 1,
and 4 copies of the rational sphere in degrees 5, 6, 8, 9.

### B.4 Reading the product bases

Part (b) is the reason we can push the "shifts" through. For every
strictly smaller height t < n, the same chosen y_i must serve, and
the products y_I must be a *basis over that height*. That means: after
K(t)-localizing, the map

    bigvee_{I} Sigma^{-d(I)} S -> D

with the y_I as coefficients must be an equivalence. This is far
stronger than having a basis at only one height. It says: one family
of sphere-classes y_1, ..., y_n, chosen once and for all, works as a
basis at every lower height simultaneously.

For the "height" reader, note that we only test the products at
heights t with 1 <= t < n. The primes p > n + 1 ensure all these
tests "see" the corresponding y_i via K(t)-theory (each class has
nonzero image mod p; see Part D below).

### B.5 What the paper does NOT claim

This is important because the usual version of the chromatic splitting
conjecture is stronger.

- The theorem gives the *existence* of the filtration. It does not
  produce a *section of the localization unit*. That is, it does not
  prove the weak form of the chromatic splitting conjecture.
- It does not produce a wedge decomposition X_{n,p} ~ wedges C_I.
  It is only a filtration, with unknown attaching maps.
- Sharpness of the bound p > n + 1 is not claimed. Perhaps the
  statement holds in a larger prime range; the paper does not say.
- The paper does not identify the specific rational classes
  Sig^{-d} H Q with specific products y_I.

These disclaimers matter: without them, the theorem would look like
it proved the chromatic splitting conjecture in the range p > n + 1,
which it does not. It is instead a *weaker statement that retains the
same cofibers*.

### B.6 Why the prime p > n + 1?

The main output of the proof is a collection of cohomology classes
from a particular p-adic group P_n, and a set of "exterior algebra"
generators for these classes. The computations are clean when
p > n + 1 (equivalently when n <= p - 2) because:

- The relevant cohomological dimension of P_n and of the compact
  p-adic groups that appear is finite and easy to compute.
- The mod-p exterior algebra structure of H^*(GL_n(Z_p), F_p) is
  known in this range (Dotto-Le Hung).
- The image-of-J range needed for sphere-lifting holds.
- The Soulé regulator / top-volume normalization has an easy
  form.

Above that prime range, many of these become unknown or are false.
Hence the prime bound.

### B.7 Why the proof is not "just a limit of n = 1, 2"

Even at height 1 (where everything is classical) the reduced
formulation in the language of this paper already requires:

- computing the cohomology of the stabilizer group P_1 = Z_p^*, and
- lifting its classes to spheres.

At general height n, the same two steps must be done at once for all
i = 1, ..., n, and the classes must work as bases for every lower
height. The results being produced are therefore new at n >= 3
(exactly the range where the classical chromatic splitting conjecture
was previously unknown).

---

## PART C. The first engine: from products to filtration

Part (b) of the theorem is the "input" and part (a) is the "output".
The passage from one to the other is a routine argument, called the
**filtration criterion** (`Proposition 2.1` of the paper). It is the
one technically elementary piece of the paper.

### C.1 The criterion, stated

Fix n >= 1, and let D be an S-module with a chosen map u : S -> D
(this will be the unit). Suppose you are given classes

    z_I in pi_{ -d(I) } D

for every subset I of {1, ..., n-1}, with z_empty = u. Assume:

**(C1) Simultaneous bases.** For every 1 <= t < n, the map

    bigvee_{I subseteq {1, ..., n-t}} Sigma^{-d(I)} S -> D

sending the I-summand to D via z_I is a K(t)-equivalence.

**(C2) Rational dimension count.** For every integer b,

    dim_Q( pi_{-b}( L_0 D ) )
        = # { I subseteq {1, ..., n} : d(I) = b }.

(The right side is the golden "monomial pattern" of an exterior
algebra on generators in degrees 1, 3, 5, ..., 2n-1.)

If n = 1, assume additionally that pi_0( L_0(u) ) is nonzero.

**Conclusion.** Then X := L_{n-1} D admits a filtration as in
Theorem B.1(a), with u in place of the paper's unit and with the
individual cofibers C_I = Sigma^{-d(I)} L_{n - max(I)} S.

### C.2 Why the criterion is true: idea

You build the filtration in *blocks* of decreasing chromatic height.
Order the subsets by max(I) as in the theorem.

**Block 1 (max I = 1 or I = empty).** Take the pieces
L_{n-1} S and Sigma^{-1} L_{n-1} S. Sum them. Map into X using z_empty
and z_{{1}}. K(n-1)-localize. Condition (C1) at t = n - 1 says this
map is a K(n-1)-equivalence. The cofiber Q_1 of the map is therefore
E(n-2)-local (a fact from chromatic fracture: a K(t)-acyclic
E(t)-local object must be E(t-1)-local).

**Inductive step.** From block j-1, get Q_{j-1} which is E(n-j)-local.
The next block

    B_j = bigvee_{max(I) = j} Sigma^{-d(I)} L_{n-j} S

maps into Q_{j-1} using the images of z_I. Each individual z_I, before
being "mapped down" to Q_{j-1}, was a piece of the K(n-j)-equivalence
of condition (C1). Row-by-row analysis shows B_j -> Q_{j-1} is a
K(n-j)-equivalence, so the cofiber Q_j is E(n-j-1)-local. This is
the induction.

**End.** After j = n, the object Q_n is E(0)-local, i.e. rational.
Condition (C2) is a dimension count that says the rational pieces are
*exactly* the height-0 part of the hoped-for list. So the cofiber of
the last block is exactly the rational pieces.

**Then undo the quotient.** Having built Q_0 = X, Q_1, ..., Q_n = 0,
each Q_{j-1} -> Q_j a cofiber sequence in blocks, go backwards:
define G_j := fiber(X -> Q_j). Each G_j sits in the filtration; the
individual pieces are C_I.

This is a standard yoga from the theory of stable categories:
pullbacks and fibers assemble a chain of quotient cofiber sequences
into a chain of cofiber sequences. Nothing subtle.

### C.3 What the criterion needs from the rest of the paper

Part (b) of Theorem B.1 is exactly (C1) of the criterion, with the
"u" of the criterion equal to y_empty. So the main theorem reduces to:

- **Input 1 (Part D below).** Construct the classes y_1, ..., y_n once
  and for all, and show they are K(t)-bases for every 1 <= t < n.
- **Input 2 (Part E below).** Verify the rational dimension count (C2).

Input 2 is the Barthel-Schlank-Stapleton-Weinstein theorem on rational
homotopy of L_{K(n)} S; its proof is independent of this paper.

Input 1 is the substance of the paper. The hard work is showing that
y_i exist and work at every lower height.

---

## PART D. The classes y_i: where they come from

Input 1 of the filtration criterion is: find classes y_1, ..., y_n in
pi_{1-2i}( D ) with the K(t)-basis property for all t < n. Here
D = L_{K(n)} S.

The paper obtains the y_i by transport across a "modification" of
some compact p-adic groups, starting from sphere-lifted
cohomology classes of the matrix group GL_n(Z_p) and its relatives.
The pieces are:

1. Compute mod-p cohomology of P_n (the height-n Morava stabilizer).
2. Show it is an exterior algebra on n classes xi_1, ..., xi_n in
   degrees 1, 3, ..., 2n-1.
3. Lift those classes to integral cohomology of P_n (i.e. to
   H^*(P_n, Z_p)).
4. Lift the resulting sphere-cohomology classes to sphere-cochain
   classes on P_n (i.e. elements of pi_*( C^*(P_n; S) )).
5. Push them through the completed descent resolution to get classes
   y_i in pi_*(D).

Steps 1-5 occupy about 60% of the paper. Here is what goes on at each
step.

### D.1 Step 1: which group?

The relevant group is the **ordinary height-n stabilizer**:

    P_n = O_{D_n}^x

where D_n is the central division algebra over Q_p of invariant 1/n
and O_{D_n} is its ring of integers. Equivalently, D_n has degree
n^2 over Q_p, its residue field is F_{p^n}, and P_n is the "unit
group" of the maximal order.

Topologically, P_n is a compact p-adic analytic group of dimension n^2
with a large finite quotient: its reduction mod the maximal ideal is
F_{p^n}^* x (something the size of a unipotent group). The pro-p
radical 1 + p O_{D_n} is a compact pro-p analytic group of dimension
n^2.

For step 1 what you need is: what does H^*(P_n, F_p) look like?