CHROMATIC OVERLAP AT THE PRIME THREE
A plain-language guide to the paper
"Failure of finite assembly for a chromatic overlap at the prime three"
--------------------------------------------------------------------------
HOW TO READ THIS, AND WHO IT IS FOR
--------------------------------------------------------------------------
The original is a research paper in algebraic topology, the branch of
mathematics that studies shapes and gluing in a very abstract, flexible
way. As written it assumes years of graduate study. This version keeps the
mathematics honest and keeps the actual result, but it builds every idea
from the ground up in ordinary words.
Some of the tools near the end (p-divisible groups, perfectoid spaces,
Cartier duality, Morava E-theory) really are graduate-level. You are not
expected to master them. Every time one appears we say plainly what job it
does and why the proof needs it. If you can follow the main storyline in
the later sections, you have understood the paper.
There is a glossary at the very end. Read the paper once for the story, then
use the glossary for the words.
The single sentence the whole paper is about:
At the prime 3, and at "height" 3, one special object can NOT be
built out of three simpler objects by any finite amount of the usual
gluing, even though each gluing step is allowed to be as clever and
non-straight as you like.
That answers, in the negative, a natural weakened form of a famous old
guess about how these objects fit together. The rest of this document
explains every word of that sentence.
--------------------------------------------------------------------------
1. THE BIG PICTURE: SPLITTING LIGHT WITH A PRISM
--------------------------------------------------------------------------
"Chromatic" literally means "having to do with color", and that is the
right picture. When white light passes through a prism it splits into
separate colors. Homotopy theorists found that a very complicated object
called the sphere splits, in a precise sense, into separate "colors" called
heights. Height 0 is the flattest, simplest color. Height 1 is more
intricate. Height 2 more so, and so on forever.
Each color is simpler than the whole, and the whole can be rebuilt from its
colors. That is the good news. The subtle question is this: when you look
at one color, how much of the LOWER colors is still tangled up inside it?
This leftover tangle is called an "overlap", and the whole paper is about
one overlap at one specific color.
So the paper has three parts:
(a) a way to measure whether something is "simple" or not, a kind of
detector;
(b) a hard calculation showing that the overlap at one prime and one
height carries a mark that simple things can never carry;
(c) a careful argument that the mark survives all the way into the
real object, so the overlap really is NOT simple.
Parts (a), (b), (c) are Sections 2, 6-7, and 8-10 of the original. Here we
tell the same story in the same order.
--------------------------------------------------------------------------
2. THE SPHERE: THE ATOM OF SHAPE
--------------------------------------------------------------------------
In everyday topology you study shapes like circles and spheres. In
homotopy theory you do something stranger. You allow yourself to bend,
stretch, collapse and glue, and you stop caring about exact distance or
size. What survives this abuse is "holes and how they interlock".
The basic object is the sphere S. There is also a "stable" version of the
whole subject, where you never fix dimensions and you can freely suspend
(raise dimension) and desuspend (lower it). In this stable world there is a
special object, still written S and called the sphere spectrum, that plays
the role of the number 1. It is the atom.
Here is the key point for us. In the stable world, EVERY object is built
out of the atom by gluing. You take copies of S, and copies shifted up or
down in dimension (written S^n), and you glue them together with maps
between them. Infinitely many copies and infinitely weird gluing are
allowed in general. When mathematicians say "the sphere is complicated"
they mean that figuring out all the possible gluings, summarized by the
"stable homotopy groups of spheres", is one of the hardest problems in the
subject.
So the sphere is simple to state and terrifying to compute. The chromatic
program is the most successful known way to tame it: break it into colors.
--------------------------------------------------------------------------
3. PRIMES: LOOKING AT ONE FREQUENCY AT A TIME
--------------------------------------------------------------------------
In ordinary arithmetic, every positive integer is a product of primes, and
a hard question about all integers often becomes a manageable question
about one prime at a time. Homotopy theory works the same way.
The stable homotopy groups of spheres have a hidden arithmetic. For any
prime p (2, 3, 5, 7, ...) you can isolate the part whose arithmetic is
governed by p. The cleaned-up object is written S_p and called the
p-complete sphere: the sphere with only its p-power information kept.
Because the different primes do not interfere, you can study them one at a
time and reassemble.
This paper lives entirely at one prime:
p = 3.
Almost every "3" you see below is that prime. It is not an exponent and not
a dimension. It is the arithmetic prime the whole paper is parked at.
--------------------------------------------------------------------------
4. LOCALIZATION: ZOOMING IN WITH A MICROSCOPE
--------------------------------------------------------------------------
To study one color you need a microscope that shows only that color and
blurs out everything else. In homotopy theory such a microscope is called a
localization. It is a rule that takes any object X and replaces it by the
closest object that the chosen microscope can see, written L X.
Two facts make localizations useful:
- Zooming in twice is the same as zooming in once. (L L X = L X.)
- Zooming is "exact": if A -> B -> C is a gluing triangle, then
L A -> L B -> L C is still a gluing triangle. Exactness is what lets
localizations respect all the constructions later.
A gluing triangle, mentioned above, is the fundamental move of the stable
world. If f: A -> B is a map, its "cofiber" or "cone" is the object that
records what is left over after gluing along f. We write it C. The three
objects fit into a triangle
A --f--> B --> C --> (back to A, shifted)
Cofiber, finite sum, shifting up and down, and taking "retracts"
(pieces cut out by gluing), these are the allowed moves. Anything you can
produce from a starting set of objects by finitely many of these moves is
said to lie in their "thick subcategory", written
Thick{ X_1, X_2, ... }.
Read "Thick" as "everything finitely buildable from". It is exactly the
notion of "finite assembly" that the title refers to. When we ask whether
some object can be assembled from some building blocks in finitely many
steps, we are literally asking whether it lies in a Thick{ }.
Different microscopes see different things:
L_0 = rationalization, the height-0 microscope (a very gentle blur);
L_n = the microscope that sees everything of height at most n;
L_{K(n)} = the microscope tuned to exactly height n and nothing else.
The paper needs L_0, L_1, L_2 (heights 0, 1 and 2) and the very sharp
L_{K(3)} (exactly height 3). The sharp microscopes are named after Morava
K-theory K(n); the blunt ones after Johnson-Wilson theory E(n).
--------------------------------------------------------------------------
5. HEIGHTS: WHAT "COLOR" ACTUALLY MEANS
--------------------------------------------------------------------------
You may want a little more concrete picture of a height, since it is the
central word.
A height is a number attached to a formal group law. A formal group law is
a power series F(X,Y) that pretends to be the addition rule of a group but
only near zero, with no worry about convergence. Formal group laws are the
algebraic shadow of the way a geometric object (like a curve, or a
p-divisible group) adds points. Over characteristic p they come in
"heights": a number n >= 0 measuring how thoroughly the p-power map
scrambles the rule. Height 0 is the ordinary additive rule; height 1 is
like the multiplicative rule (as with roots of unity); height 2, 3, and up
are genuinely higher-dimensional.
The Morava microscopes match these heights. Height 0 keeps only the smooth,
rational information: the rational sphere is essentially the ordinary
rational number line wearing a disguise. Height 1 keeps the information
linked to complex K-theory and elliptic curves. Height 2 keeps what is
linked to two-dimensional formal groups, and so on.
In the paper, "height 3" objects are governed by 3-dimensional formal
groups, and the overlap we test lives in height 3 while only remembering
height 2 and below. The technical name for the object under test will be
L_2 L_{K(3)} S, explained next.
(Read on. The rest of this file explains the conjecture, the theorem, and
the proof. Nothing above is optional background; every idea returns.)
--------------------------------------------------------------------------
6. THE CHROMATIC SPLITTING CONJECTURE
--------------------------------------------------------------------------
Now the conjecture the paper pushes against.
If you look at the height-n color with the exact height-n microscope,
written L_{K(n)} S, you do not get something new and isolated. You get
something that still remembers a shadow of the lower colors. That shadow
is the object
L_{n-1} L_{K(n)} S
which you read as: "first take the height-n piece, then forget everything
of height n and above, keeping only heights up to n-1." What is left is the
overlap: the part of the height-n color that is secretly made of lower
colors.
Hopkins made a beautiful and precise guess about this overlap, called the
chromatic splitting conjecture: that it decomposes cleanly into the lower
exact-height pieces, roughly
L_{n-1} L_{K(n)} S ~= (a direct sum of the L_{K(i)} S for i < n),
with extra explicit terms in low characteristic. "Splits" means the total
breaks apart as a sum of independent pieces with no leftover gluing.
The conjecture has been checked in some small cases and REFUTED in others.
The history matters because it is exactly the trap this paper avoids:
- height 1: known for a long time (Hopkins, and others);
- height 2, odd primes p >= 5: computed by Shimomura-Yabe, corrected
by Behrens;
- height 2, prime 3: a clean splitting holds (Goerss-Henn-Mahowald);
- height 2, prime 2: the ORIGINAL strong splitting is FALSE (Beaudry),
but the object is still what you could call "nicely built": it is the
canonical piece plus a little extra (Beaudry-Goerss-Henn). Those extra
pieces are of a kind our finite-building rules can assemble.
That last bullet is the crucial lesson. Even when the pretty splitting
fails, the object might still be buildable by our finite rules. So "the
strong splitting is false" does not by itself prove "the object cannot be
built at all". This paper asks the strictly weaker, harder-to-kill question.
--------------------------------------------------------------------------
7. THE FINITE-ASSEMBLY QUESTION AND THE MAIN THEOREM
--------------------------------------------------------------------------
Here is the exact question. At prime p and height n, let the building
blocks be the lower local spheres
L_0 S, L_1 S, ..., L_{n-1} S.
We do NOT demand that they combine into a clean direct sum. We allow any
finite gluing:
finite sums, dimension shifts up and down, cofibers (cones), and
retracts, in the category of E(n-1)-local modules over S.
That is precisely the thick subcategory Thick{ L_0 S, ..., L_{n-1} S }.
The finite-assembly question is:
Is L_{n-1} L_{K(n)} S in Thick{ L_0 S, L_1 S, ..., L_{n-1} S }
for every prime p and height n?
Notice how permissive this is. The gluing maps may be as nasty as you
want. Nonsplit extensions are allowed. Moore corrections (cofibers of
multiplication by a power of p) are allowed. You are not required to make
the canonical map retract. The only real restriction is FINITENESS: at
most finitely many building blocks and finitely many gluing steps. The
allowed number of cofibers and their attaching maps may even depend on p
and n.
THE MAIN THEOREM OF THE PAPER. At p = n = 3, with S = S_3 (the 3-complete
sphere),
L_2 L_{K(3)} S is NOT in Thick{ L_0 S, L_1 S, L_2 S }
in the category of E(2)-local S-modules.
So finite assembly FAILS at the first height-three odd-prime case. This is
a genuine strengthening over "the pretty splitting is false": it says even
the wild, non-straight, finite gluing of the three lower spheres cannot
reach the overlap. It is stronger than the earlier failures because it
uses no proposed list of summands and no bound on cofibers. And the
obstruction is genuinely height-three-odd-prime: the paper also points out
that height 2 at prime 3 DOES split (GHM), while this fails one step up;
and that at prime 2 the Beaudry-Goerss-Henn extra terms are exactly the
kind the finite rules allow, which is why prime 2 is not a counterexample
to THIS question.
So the paper closes a real gap: from "the clean form is false somewhere" to
"even the wild form is false at p = 3, n = 3".
--------------------------------------------------------------------------
8. THE OVERALL STRATEGY OF THE PROOF
--------------------------------------------------------------------------
Every proof of a "cannot build" statement needs a measuring device. The
modern tool is the thick subcategory plus an exact functor. The plan:
STEP A. Find an exact functor (a "detector") T that sends the three
building blocks to awkwardly simple values, so that anything
buildable from them MUST have a "nice" T-value.
STEP B. Compute, honestly, the actual T-value of the overlap. Show its
value is NOT nice. Then no finite assembly exists.
"Exact" means T turns gluing triangles into exact sequences. That is the
hinge: exactness + "the allowed operations preserve niceness" gives
"everything buildable is nice". Let us state it as a clean principle.
If a class C of values is closed under the operations that the
building blocks generate (sums, shifts, kernels, cokernels, extensions)
and if T sends each building block into C, then T sends every object of
Thick{ blocks } into C.
Then, the actual TEST changes height before measuring. Take the overlap
object X, apply the height-2 localization, and only then apply T. Two facts
let us replace X by something manageable:
exactness of the height-2 localization,
the known arithmetic relations L_{K(2)} L_i S = 0 for i < 2, and
L_{K(2)} L_2 S = L_{K(2)} S =: 1_2 (the height-2 unit).
Under L_{K(2)}, two of the three building blocks die and the third becomes
the height-2 unit 1_2, while the overlap becomes the mixed object
L_{K(2)} L_{K(3)} S. Therefore:
IF the overlap were in Thick{L_0 S, L_1 S, L_2 S},
THEN L_{K(2)} L_{K(3)} S would be in Thick{ 1_2 }.
So it suffices to prove that L_{K(2)} L_{K(3)} S is NOT in the thick
subcategory of the height-2 unit. This reduces a question about three
building blocks to a question about ONE building block (the height-2
unit) and one test object. Much easier to attack, and that is exactly what
the detector of Section 9 does.
The proof is now about TWO different heights living at once: height 3
(where the overlap lives) measured by a height-2 instrument. Keeping track
of what happens across the height jump is the technical heart.
The next sections build the detector (Section 9), compute the value on the
overlap (Sections 10-11), and transport the bad value into real homotopy
(Sections 12-14). Sections 15-16 finish and say why only this prime/height
combination is attacked.
--------------------------------------------------------------------------
9. THE DETECTOR: A HEIGHT-TWO INSTRUMENT
--------------------------------------------------------------------------
We now build the measuring device T. Recall we have reduced to showing
that the height-2 unit 1_2 cannot reach the mixed object. The trick is to
measure with a height-2 E-theory, i.e. with Morava E-theory of height 2.
THE INSTRUMENT ITSELF. For any object X, define
T(X) = L_{K(2)} ( E_2 smash X ) / p.
Two things happen here.
1. We smash with E_2, the height-2 Morava E-theory. E_2 is a ring
spectrum whose homotopy is a power-series ring W(k)[[x]], where W(k)
is the Witt vectors of a field k and x is the "deformation
parameter", the thing that lets the height-2 formal group wobble.
2. We take the height-2 localization and then reduce modulo the prime p.
Reducing mod p destroys a lot of noise and leaves a clean object.
Next, look at the homotopy groups of T(X), and then throw away the
p-adic/subtle part by inverting x:
V_d(X) = ( the degree-d part of T(X) ) tensor_{k[[x]]} k((x)).
You read this as: "after the test, we live over the field of Laurent series
k((x)); a thing is simple if its module over that field is built from the
periodicity in even degrees."
The instrument is EXACT: it turns gluing triangles into long exact sequences
of these V_d's. That is the technical reason we chose it: exactness is what
makes the "everything buildable is nice" principle bite.
WHY A HEIGHT-2 INSTRUMENT ON A HEIGHT-3 OBJECT? Because the object we test
is L_{K(2)} L_{K(3)} S. Its ordinary top piece "sits at height 3", but when
we smash with E_2 (height 2) we are using the lower height to interrogate
the higher one. It is like using a lower-frequency tuning fork to detect a
high-frequency tone: the taller height leaves a tell-tale response.
Now a subtlety. The coefficient field k must be big enough. Choose k to be
a finite field containing F_{3^6} (in particular containing F_9). The
height-2 Morava E-theory carries a whole stabilizer group G_F acting on its
coefficients, and the V_d(X) are semilinear representations of G_F:
an element g of G_F not just moves the vectors but also moves the scalars
through the "deformation parameter" x. That semilinearity, the fact the
group acts on the numbers too, is what makes the obstruction possible.
Specifically, F is an unramified quadratic extension of Q_3 (think of
Q_3(sqrt of something inert) ). Its ring of integers is O_F, its units form
G_F = O_F^*.
Because F/Q_3 has degree 2, G_F contains a copy of the 3-adic units and a
copy of the F_9^* of the residue field. The residue-field piece carries a
small but crucial sign-like character.
THE NICE CLASS. Inside all semilinear representations over k((x)), we pick
out the class C consisting of those whose "simple pieces" (simple
constituents, in the sense of sub-quotients) are all
(periodicity line) omega^{ tensor b}, for any integer b,
where omega = pi_2 E_2 is the single line in which E_2 repeats every 2
degrees. Because Class C is defined via constituents, it is a SERRE
SUBCATEGORY: closed under subobjects, quotients, and extensions. That
closure is exactly what the "everything buildable is nice" principle needs,
because sub/quotient/extension are what long exact sequences hand you.
On the building block it is wonderfully simple. For the height-2 unit 1_2,
V_{even}(1_2) = omega^{ tensor (d/2)}, V_{odd}(1_2) = 0.
So the unit lands entirely in Class C. Exactness of the detector, plus the
Serre closure, gives the DETECTOR PROPOSITION (Proposition 2.3 of the
original):
If X is in Thick{ 1_2 }, then every V_d(X) is finite-dimensional over
k((x)), and every simple constituent of V_d(X) is a periodicity power
omega^{ tensor b}.
That is the "must look nice" statement. To contradict it we need to compute
V_d of the actual overlap and find something in it that is NOT a
periodicity power. That is exactly what the paper does.
CONVENIENT NAMING. From here we write
Q_x = k((x)), omega-bar = omega / p,
so "omega-bar^{ tensor b}" means the reduced periodicity line. Class C is
the class of things built from these. The detector says finite assembly
forces every V_d to be built from omega-bar powers.
--------------------------------------------------------------------------
10. STEP B, FIRST HALF: WHAT THE CALCULATION PRODUCES
--------------------------------------------------------------------------
The computation of the overlap's V_d-value happens in two very different
worlds, and the paper spends most of its length making the two talk.
WORLD 1 (coefficients): explicit continuous cohomology, where we
compute actual function spaces and their group actions. This is where
the "norm line" is born.
WORLD 2 (homotopy): actual spectra and their descent spectral
sequences. This is where we must show the line computed in World 1
survives into a real, honest V_d of the real overlap.
WORLD 1, the answer. Using the geometric theory of p-divisible groups
(a p-divisible group is the p-power part of an abelian variety; its
"deformation" is how it bends as a parameter moves) and the associated
towers of markings, the paper computes the ordinary continuous
cohomology of the algebraic marking algebra L with respect to the
height-3 stabilizer group P. The answer is the COEFFICIENT THEOREM:
H^0 and H^1 of L for P are each a copy of k;
H^3 and H^4 of L for P are each a single line carrying a special
character;
H^a = 0 for all other a.
Those four "rows" in degrees 0, 1, 3, 4 are the whole coefficient story.
THE SPECIAL LINE. The upper two lines are the interesting ones. Their
character is a map of the unit group G_F of the unramified quadratic F into
the small field F_3:
delta( g ) = the residue norm of g, reduced mod 3.
"NORM" means: take the F/Q_3 field norm (F has degree 2 over Q_3), and then
look at it modulo 3 (in the field-with-9-elements, whose units map onto the
units of F_3). That little sign is called THE NORM CHARACTER delta.
WHY delta IS NONTRIYIAL. The residue field of F is F_9. The ordinary norm
map from F_9^* (9-1=8 elements) to F_3^* (2 elements) is onto, so some
units get sent to -1 and some to 1. So delta is not the trivial character.
It is a real, non-fake sign. That little fact is the whole engine of the
paper. Everything else is bookkeeping to get this sign somewhere it can be
detected.
So the coefficient computation hands us ONE LINE:
Q_x( delta ) -- a copy of k((x)) carrying the character delta.
The paper writes it "Q_x(delta)", read as "the field k((x)) with the
nontrivial norm action on it". This is the norm line. It lives on top of
the periodicity line, not on it.
--------------------------------------------------------------------------
11. STEP B, SECOND HALF: WHY THE NORM LINE IS NOT A PERIODICITY POWER
--------------------------------------------------------------------------
Now the single most beautiful step of the paper. Show that the norm line
Q_x(delta) can NEVER look like a periodicity power omega-bar^{ tensor b}.
That is the whole obstruction.
FIRST, A FACT ABOUT INVARIANTS. Whatever fixed field k((x)) has under the
action of the full unit group G_F, that fixed field is just k. The reasoning
is lovely and short: suppose a nonzero f were fixed but not constant. Divide
or shift so its valuation in x is positive. Write its valuation as d. Then
the subring k[[f]] makes k[[x]] finite free over k[[f]] of rank d, so the
field extension k((x))/k((f)) is finite. But a fixed element would force the
infinite group G_F into a FINITE automorphism group of a finite extension,
which is impossible: G_F has infinitely many principal-unit automorphisms.
So the fixed field is exactly k. Nonconstant functions of x are always
moved by some element of G_F.
SECOND, THE HASSE INVARIANT. There is a canonical, intrinsic section h of the
second power of the periodicity line,
h = a_1 times (dT)^2,
where a_1 is the coefficient of T^p in the multiplication-by-p series of the
formal group. This section is INVARIANT under G_F (because any change of
coordinate changes both the coefficient a_1 and the differential dT in a
cancelling way). Moreover, in the chosen coordinate, h has a SIMPLE ZERO at
the parameter x: its x-adic valuation is 1. So h is a genuine invariant
element with valuation 1.
THIRD, THE ARGUMENT. Suppose, for contradiction, that
Q_x(delta) ~= omega-bar^{ tensor b} for some integer b.
Then a basis e of the right-hand side, but multiplied by some nonzero f of
the form Q_x, would satisfy g(e) = delta(g) e. Since delta^2 = 1, squaring
gives that f^2 / h^b is INVARIANT. By the first step that invariant must
lie in k, and taking x-adic valuations (remembering: norm line f has
valuation v_x(f), h has valuation 1) gives exactly
2 v_x(f) = b.
So b must be EVEN. But if b is even, then omega-bar^{ tensor b} equals
omega-bar^{ tensor (b/2)} squared, which is the square of a line, and
h^{ tensor (b/2)} is an INVARIANT basis of that line, so the line (the right
side) is trivial. Yet the left side cannot be trivial: if it had a nonzero
invariant u up to delta, then u^2 would be invariant and lie in k, forcing u
algebraic over k, hence u in k^*, forcing delta trivial, contradicting the
fact standing in black and white that delta is NOT trivial.
CONTRADICTION. Therefore:
Q_x(delta) is NOT isomorphic to omega-bar^{ tensor b} for any b.
The norm line can never be a periodicity power. That is the LEMMA OF THE
GENERIC CHARACTER. This is the exact place where the paper is stronger
than a shallow attempt: a shallow attempt would only look at the residue
field F_9^* (the "constant" units); the paper needs the FULL unit group,
including all the principal units, and their action as a nontrivial
semilinear action over k((x)), not just a scalar action over k.
So the challenge is now absolutely clear:
if, after the smoke clears, the real V_d of the overlap CONTAINS our
bad line Q_x(delta), the detector's "must look nice" condition is
violated and finite assembly is impossible.
The rest of the paper is transport: getting the line Q_x(delta), which is
computed in World 1, all the way to World 2 and into a genuine V_d of the
real overlap.
--------------------------------------------------------------------------
12. THE TRANSPORT PROBLEM, AND WHY IT IS HARD
--------------------------------------------------------------------------
Getting a coefficient line into actual homotopy is famously nontrivial. There
are two gaps this paper must cross:
(i) PERFECTION vs ORDINARY. The beautiful geometric computation happens
over a COMPLETED PERFECTION (a huge p-adically thick ring). But
homotopy cooperations only see the ORDINARY marking algebra. Passing
from the completed world back to the ordinary one is called
"decompletion". If we lose delta in this step, all is lost.
(ii) SPECTRA DESCENT. We must then realize the coefficient line in an
actual spectral sequence converging to the homotopy of our test
object T of the real overlap, and KEEP it from being washed out.
The paper handles (i) with a "decompletion" theorem and (ii) with Morava
descent plus a "through every jet" comparison. Sections 13-14 sketch what
each does.
--------------------------------------------------------------------------
13. DECOMPLETION: FROM COMPLETED PERFECTIONS TO ORDINARY COCHAINS
--------------------------------------------------------------------------
The completed perfections are much bigger than the ordinary marking algebra
L. We need a theorem saying the two give the SAME cohomology. This is a
classic "the difference is invisible to the cohomology" result, but here it
must preserve the delta action.
The key trick is a distribution operator D. The Cartier dual of the height-2
formal group gives a ring k[[D]] acting on the root stages of L. Three
properties make it the perfect decompression tool:
- D has kernel exactly L inside the perfection R = L^{perf}. So
R / ker = R / L, and iterating exposes L itself.
- D commutes with the group action, so it can be used to strip away
the extra perfection layers WITHOUT killing delta.
- D has a nice relation with Frobenius: D(f^p)=(Df)^p, and
D^q on the q-power layer follows suit.
With D in hand, the paper does a "finite resolution" and a "finite stage"
argument (the coefficient spaces are compact, separable, metrizable, so
Baire and compactness give finite-dimensional cohomology at every finite
stage). This yields:
DECOMPLETION THEOREM. The natural comparison
cohomology of the ordinary marking algebra L
== cohomology of its infinite perfection R,
is an equivalence of cochain complexes, naturally in any compact
parameter space, and equivariantly for all group actions.
The payoff is that the four-line picture of World 1 holds in the ordinary
marking algebra L, not just in the otherworldly completed perfection. So the
norm line Q_x(delta) is an honest feature of the ordinary marking algebra.
A second operation, an averaging over the order-2 sign subgroup, removes the
"roots" of the perfection cleanly and leaves exactly the four lines:
k in degrees 0 and 1, Q_x(delta) in degrees 3 and 4.
This is the ORDINARY COEFFICIENT THEOREM: four lines where we can see them,
with delta visible on the upper two. Now we can attempt the trip to
homotopy.
--------------------------------------------------------------------------
14. MORAVA DESCENT AND THE JETS COMPARISON: GETTING TO REAL HOMOTOPY
--------------------------------------------------------------------------
Now (ii): putting the norm line into an actual spectral sequence of spectra.
FIRST, THE CONSTANTS. Let S_k be the sphere spectrum with its coefficients
enlarged to include the big field k: it is a finite free S-algebra (it is,
as an underlying S-module, just finitely many copies of S). Put
Z_k = L_{K(3)} S_k.
Then Z_k, as an underlying object, is a finite sum of copies of
L_{K(3)} S. This enlargement is purely bookkeeping: if the height-2 test
detects a bad line on Z_k, the SAME bad line appears on the original
L_{K(3)} S, because a finite sum's constituents are the constituents of
each summand. So we can replace the overlap by Z_k for the calculation.
SECOND, MORAVA DESCENT. E_3 (the height-3 E-theory with our k) is a
DESCENDABLE algebra over Z_k. Descendability is a fancy word for: E_3 is
a faithful probe, and the failure of Z_k to already be E_3 is controlled by
a finite bound. Concretely, Z_k is the totalization of the Amitsur
cosimplicial object
Z_k -> E_3 -> E_3 smash E_3 -> E_3 smash E_3 smash E_3 -> ...
where each smash is over Z_k and K(3)-local. This builds a descent
spectral sequence on any test object.
Now the crucial move: exactness of the detector T means the descent tower
survives the test, and MATHEW'S THEOREM says the descendability bound is
finite. So the tested descent spectral sequence
E_2^{s,t} = H^s ( pi_t T of the Amitsur object ) ==> pi_{t-s} T(Z_k)
is STRONGLY CONVERGENT, with a FINITE filtration on its actual abutment
(after inverting x). That is not a we-wish-it-converged; it is a genuine
finite filtration. This is what will allow us to say "the norm line is a
subquotient, hence a constituent, of an actual homotopy group".
There is also a small technical nicety: on mod-3 homotopy there is a
"diagonal" summand, an idempotent that picks out the piece where the two
copies of k (one from E_2, one from S_k) agree. Everything below happens
on that diagonal summand; the notation V_d^Delta means "diagonal part".
THIRD, THE JETS COMPARISON. The second page of that spectral sequence
sits in pi_* T(E_3 smash ... smash E_3), and each entry is a completed
inverse limit over the parameter x. The coefficient theorem gave us the
answer over the perfections; this step compares with the ordinary
inverse limit over quotients k[[x]]/(x^m).
The mechanism is honest deformation theory. The height-three formal law at
this point is a "split" object -- a marked connected part of height 2 and an
etale quotient of height 1. The paper shows that you can LIFT this split
deformation through every square-zero jet R[x]/(x^m) -> R[x]/(x^{m+1}),
uniquely, matching given data at each level. The tools are:
- Hasse derivatives / finite Taylor expansion (ordinary calculus on
power series, kept finite by taking only jets of the given order);
- square-zero lifting of p-divisible groups, with the Lubin-Tate /
Kodaira-Spencer isomorphism telling you exactly which ring lifts
correspond to which group lifts.
- the split -- connected plus etale -- makes the special fiber easy
to analyze, and the fixed order-2 action pins every lift uniquely.
The JETS COMPARISON THEOREM then says: after carrying the coefficient
isomorphism of Section 10 through every jet, and taking the inverse limit
over m, the second page of the tested descent spectral sequence is
literally
E_2^{s,2j} = H^s(P, L)[[x]] tensor omega-bar^{tensor j},
E_2^{s,2j+1} = 0.
In words: the four-line picture of the ordinary coefficient theorem
survives verbatim into the second page of the real spectral sequence, with
the semilinear action through x accounted for.
This is exactly the technical depth at which a reader can get lost, so
keep the shape in mind:
descent gives a real spectral sequence;
descendability makes it converge to real homotopy;
jets compare the coefficient line with real second-page entries;
the four-line picture and delta both survive the trip.
--------------------------------------------------------------------------
15. THE GENERIC HOMOTOPY AND THE FINAL CONTRADICTION
--------------------------------------------------------------------------
Now invert x (i.e. tensor the second page with k((x))). Recall the
second page has four rows: periodicity lines in cohomological degrees 0 and
1, and the norm-twisted lines in 3 and 4, plus the internal degree bookkeeping
from even periods. When we pass to the abutment of the actual spectral
sequence, differentials in between rows COULD arise. The paper shows none
survive. Here is why, and it is exactly the norm-line obstruction again.
- Differentials between even rows are zero by parity: cohomological
changes of an odd amount can't move line to line because the rows
only exist in even degrees (which is where the periodicity is).
- The only nonzero possibility is a differential from an even-degree
row into a norm-twisted row. But any such map is a map between
one-dimensional lines over k((x)) which, if nonzero, would be a
constant. Conjugating by the periodicity line, it would identify
Q_x(delta) WITH a periodicity power. Lemma 11 forbids that.
So ALL differentials vanish, on the first page where they matter, and the
spectral sequence collapses after the second page. This is the GENERIC
HOMOTOPY PROPOSITION of the paper, and it says that in the abutment (the
real homotopy, after inverting x), the norm line SURVIVES as a subquotient
of V_d for particular d. In fact, the exact statement includes
V_{-3}^Delta(Z_k): the diagonal summand in total degree -3 gets a short
exact sequence
0 -> Q_x(delta) -> V_{-3}^Delta(Z_k) -> (a periodicity power) -> 0,
so Q_x(delta) is a SUBrepresentation of a genuine V_d, hence a simple
constituent of the whole test of Z_k.
THE CONTRADICTION. Put the two halves together.
IF L_2 L_{K(3)} S were in Thick{L_0 S, L_1 S, L_2 S},
THEN L_{K(2)} L_{K(3)} S would be in Thick{1_2} (Sect 8);
THEN the detector would put every V_d(Z_k) in Class C, i.e. with only
periodicity constituents (Sect 9);
BUT the generic homotopy gives V_{-3}^Delta(Z_k) with the subrepresentation
Q_x(delta), which is NOT a periodicity power (Sect 15).
Contradiction. Therefore the overlap is NOT in Thick{ L_0 S, L_1 S, L_2 S }.
That is the theorem.
The shape of the argument is worth keeping in three lines:
reduce to one height-2 unit;
compute a bad line in continuous cohomology;
show the bad line survives into real homotopy.
Anything assembleable would have no bad line.
But the overlap has one. QED.
--------------------------------------------------------------------------
16. WHY PRIME THREE, HEIGHT THREE -- AND WHY NOT OTHER CASES
--------------------------------------------------------------------------
The theorem is stated at p=n=3 for a real reason; it is not a "let's pick a
small number" choice.
- The machinery needs the residue-norm sign to be NONTRIvIAL. That
requires the residue field of F to be strictly bigger than F_p, and at
p=3 the cheapest case is F_9: F_9^* -> F_3^* surjects, so some unit
has norm -1. At p=2 the residue field would be F_4, and the same
thing happens; but that case is forbidden elsewhere (see below).
- The machinery needs the norm character to act SEMILINEARLY over
k((x)) in a way that has a nontrivial principal-unit part. At p=3
this works out.
- The height has to be 3 to be the first height strictly above
"everything splits". Height 2 at prime 3 splits (Goerss-Henn-Mahowald);
the first case beyond that gives our famous "doesn't quite work".
- The paper is careful that at prime 2 (n=p=2) the failure of the
ORIGINAL wedge is not the same question: Beaudry-Goerss-Henn
describe extra Moore terms which the finite-building rules of this
paper CAN produce. So p=2 is not a counterexample to finite
assembly, even though the pretty splitting is false there.
- For primes p > n+1, there is a companion paper ("Filtered chromatic
splitting at generic primes") that gives an ACTUAL finite
filtration by lower local spheres; at those primes finite assembly
HOLDS. So p=3, n=3 sits exactly at the first edge case.
The companion rational obstruction at height 3 (again for p >= 5) shows
that a proposed STRONG wedge fails there, while finite assembly holds;
again, the two questions are genuinely different.
Put differently: the paper does not claim the general question is
everywhere false. It identifies the first p=3, n=3 case where even the
wild form fails, closing that gap carefully.
--------------------------------------------------------------------------
17. WHAT THIS PAPER DOES AND DOES NOT PROVE
--------------------------------------------------------------------------
Does: it proves L_2 L_{K(3)} S is not in Thick{L_0 S, L_1 S, L_2 S} at
p=3, hence finite assembly fails. That is a statement about closure under
finitely many cofibers -- no prescribed summand list, no bound on the
number of cofibers, no splitting assumption.
Does NOT: it does not decide the weaker "weak splitting" question, namely
whether the canonical map L_2 S -> L_2 L_{K(3)} S has a retraction. The
paper explicitly notes that its obstruction addresses only the finite-
construction (thick-subcategory) question. A weak splitting, even if it
exists, does not by itself put the object into Thick; and a finite assembly
does not by itself produce a weak splitting. The two questions are
genuinely separate, and the paper settles only the first. It also does not
attack larger p or higher n beyond a companion-filtered positive result
(p > n+1) and the p=2 correction.
So the honest summary of the situation after this paper:
at p=3, n=3: even the wild finite gluing cannot build the overlap;
at p > n+1: finite gluing DOES build it (companion paper);
at p=2: the strong wedge is false but the object remains buildable
by wild gluing, so no failure of finite assembly.
That is why the paper's title stresses one specific (p, n) and why the
result is regarded as clean and striking rather than sweeping.
--------------------------------------------------------------------------
18. WHY THIS IS INTERESTING
--------------------------------------------------------------------------
Two reasons worth keeping.
(1) It SEPARATES two questions that had always been mentioned near
each other but never really separated: "does the object split
nicely" and "can the object be built at all by finite means."
Before this paper, a failure of the strong form (as at p=2)
raised the hope that wild gluing still worked. This paper shows
that at p=3, n=3 wild gluing is ALSO impossible. That is a real
new boundary.
(2) It shows a HEIGHT-THEORETIC phenomenon: the wrongness of the
overlap is detected by a lower-height instrument (height 2) as a
failure of the overlap's coefficients to look like periods. The
"bad line" is not a small blemish; it is a nontrivial semilinear
character which, because the fixed field of the full unit group is
as small as possible, cannot be a periodicity power. That is the
mathematical content.
If you remember only one thing, remember this.
A periodic object always looks, at the coefficient level, like a
power of the fundamental 2-periodic line. Any object reachable by
finite gluing from the periodic unit shares that property. The
chromatic overlap at p=3, n=3 carries instead a nontrivial norm-line
character that is NOT any such power. So the overlap is not reachable
by finite gluing. That is the entire proof.
--------------------------------------------------------------------------
GLOSSARY (in the order you will meet it while reading)
--------------------------------------------------------------------------
Sphere spectrum S: the unit of stable homotopy theory; the atom from which
every spectrum is glued.
Stable homotopy category: the world where dimensions are not fixed and
suspension/desuspension is free; the setting for chromatic homotopy.
p-complete sphere S_p: the sphere with only p-power information kept. This
paper lives at p=3 throughout.
Height: a number attached to a formal group law; height 0 is additive
(rational), height 1 like the multiplicative law, height 2, 3, ...
genuinely higher-dimensional. The "colour" of the chromatic prism.
Formal group law: a power series F(X,Y) that behaves like a group addition
near zero. Its higher-order structure is the height.
Morava K-theory K(n): the sharp height-n microscope.
Johnson-Wilson / E-theory E(n): the blunt "everything up to height n"
microscope.
Localization L_E: idempotent functor replacing X by its closest E-local
approximation. Exact on triangles.
L_{K(n)}: exact-height-n localization; L_{K(n)}S is the height-n piece.
L_n: localization with respect to E(n); "heights at most n".
Overlap: L_{n-1} L_{K(n)} S, the part of the height-n color bent into
lower heights.
Cofiber / cone: what is left after gluing along a map.
Thick subcategory Thick{X_i}: objects obtainable by finitely many sums,
shifts, cofibers, and retracts. The formal "finite assembly".
Chromatic splitting conjecture: the conjecture that L_{n-1}L_{K(n)}S
decomposes cleanly into lower exact-height pieces.
Finite assembly question: can the overlap be built by arbitrarily wild
finite gluing? Strong form: no.
E_2 = Morava E-theory of height 2: the ring spectrum whose coefficients
are W(k)[[x]] and which carries the height-2 formal group.
x: the height-2 deformation parameter, generating W(k)[[x]]; inverting x
throws us into k((x)) where line-based arithmetic is easy.
Detector T: T(X) = L_{K(2)}(E_2 smash X)/p, an exact test functor. V_d(X)
is the degree-d coefficient line over k((x)).
Semilinear representation: a group action that moves both the vectors and
the scalars (through x). G_F, the units of the integers of an unramified
quadratic F/Q_3, acts this way.
Periodicity power omega^{tensor b}: the b-fold power of the fundamental
2-periodic line; the "periodic-looking" values.
Class C / Serre subcategory: classes of representations closed under
sub, quotient, extension.
Norm character delta: the residue field norm F_9^* -> F_3^* taken as a
character; the nontrivial sign that survives to homotopy.
Norm line Q_x(delta): k((x)) carrying the delta action; the bad line.
Hasse invariant h: the intrinsic invariant section a_1 (dT)^2 of omega^{2},
with valuation 1; the tool that rules out Q_x(delta) being periodic.
Decompletion: passage from the completed perfection back to the ordinary
marking algebra, an honest cohomological equivalence preserving the action.
Morava descent: expressing Z_k = L_{K(3)} S_k as (the totalization of) an
E_3-based Amitsur tower; descendability gives a strong bound and a
convergent spectral sequence.
Jets: the quotients k[[x]]/(x^m); the comparison through all m, via
square-zero deformation and Kodaira-Spencer, carries the coefficient
answers into real homotopy.
Generic homotopy: the abutment over k((x)) after the spectral sequence
collapses; supplies the short exact sequences that place Q_x(delta) inside
V_d(Z_k).
The result: L_2 L_{K(3)} S is NOT in Thick{L_0 S, L_1 S, L_2 S} at p=3,
n=3. Chromatic overlaps are not in general finitely assembleable from the
lower local spheres.
THE END.
--------------------------------------------------------------------------
APPENDIX A. CHAPTER MAP BACK TO THE ORIGINAL PAPER
--------------------------------------------------------------------------
If you want to read the original alongside this, here is which of my
sections corresponds to which .tex file inside
Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three:
this file original paper .tex file what it proves
---------------------------------------------------------------------
Sections 1-5 01-introduction (background) set the stage
Sections 6-8 01-introduction (statement) the main theorem
Section 9 02-detector Prop 2.3 detector
Sections 10-11 03-setup, 04-towers, 06-analytic
coefficient theorem,
Hasse invariant,
Lemma 10.1
Sections 12-13 07-decompletion ordinary four-line
coefficient theorem
Section 14 08-homotopy, 09-jets descent + jets
Section 15 10-obstruction generic homotopy and
the contradiction
Sections 16-18 01-introduction (discussion) scope and meaning
Roughly: original Sections 1-3 correspond to my 1-11; original 4-7 to my
10-13; original 8-9 to my 14; original 10 to my 15-18.
--------------------------------------------------------------------------
APPENDIX B. SEVEN QUESTIONS READERS ACTUALLY ASK
--------------------------------------------------------------------------
Q1. "Isn't this just saying the chromatic splitting conjecture is false?"
No. The chromatic splitting conjecture is a statement about a specific
DECOMPOSITION into summands. It was already known to fail in some cases.
This paper says even the much weaker "can the object be built by finitely
many wild gluings" fails at p=3, n=3 -- a strictly stronger obstruction
than a false decomposition.
Q2. "What exactly is a 'thick subcategory', in one sentence?"
The set of objects you can get from the building blocks by taking finitely
many finite sums, dimension shifts, mapping cones, and retracts. It is the
precise notion of "finite build".
Q3. "What is a 'constituent'?"
A smallest indecomposable piece of a module in the sense of sub-quotients
(like an atom of a representation). The theorem is about which constituents
appear.
Q4. "Why does the norm character make it fail?"
Two facts. (a) Any object built from the periodic unit has only periodicity
constituents. (b) The overlap carries an extra constituent Q_x(delta). The
lemma of Section 11 shows that Q_x(delta) is not any periodicity line,
because it has a nontrivial action that no single period can supply. That
is the contradiction.
Q5. "Why the full unit group O_F^*, and not just the residue units?"
Because the residue units alone might give a character indistinguishible
from a periodicity power on a smaller field. Using the full unit group,
including principal 3-adic units, forces the fixed field of the action on
k((x)) to be exactly k. That rules out the disguise.
Q6. "Where does the number 3 show up essentially?"
Three places, all tied to the prime p=3: residue field F_9 -> F_3 has a
surjective norm (nontrivial delta); the 3-adic extension F/Q_3 is
unramified quadratic; and the fact that at p=3 height 2 already splits
(Goerss-Henn-Mahowald), so height 3 is the first nontrivial case.
Q7. "What keeps the bad line from being swallowed by a spectral sequence
differential?"
The only potential differential would be a nonzero linear map between two
one-dimensional lines over k((x)), which would be an isomorphism and would
identify Q_x(delta) with a periodicity power. The lemma of Section 11
excludes exactly that, so all differentials vanish.
--------------------------------------------------------------------------
APPENDIX C. ONE-LINE SUMMARY OF EACH ORIGINAL SECTION
--------------------------------------------------------------------------
01-introduction: main theorem, history, relation to earlier chromatic
splitting results.
02-detector: T and V_d, Prop 2.3 (thick => all constituents
are periodicity powers).
03-setup: the p=3, height 3 arithmetic, groups P, H, J, G_F,
Lubin-Tate coordinates, K = k((t)), stage cochains.
04-towers: connected and etale marking towers, integral frames,
two-tower comparison, finite-etale descent.
05-finite-resolutions:
finite projective resolutions of F_p[[H]], compact
duality, dimension-8 resolution.
06-analytic: the geometric computation on the Fargues-Fontaine
curve, the two lines in degrees 0 and 3 of K_k, then
the four lines of the completed calculation.
07-decompletion: the distribution operator D, invariant differential,
finite-stage finiteness, decompletion from completed
perfections to ordinary L.
08-homotopy: spherical constants S_k, Morava descent, tested
descent spectral sequence, finite abutment filtration.
09-jets: lifting the split deformation through every x-jet,
comparison at all jets, J-equivariance, second page
H^s(P,L)[[x]] tensor omega-bar^{tensor j}.
10-obstruction: Hasse invariant, norm line not periodic,
generic homotopy, contradiction.
That is the whole paper.