03.13.06 · modern-geometry / spectral-sequences

The chromatic spectral sequence

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Miller-Ravenel-Wilson 1977 (Ann. of Math. 106) §3-§5; Ravenel 1986 *Complex Cobordism and Stable Homotopy Groups of Spheres* Ch. 5 (chromatic spectral sequence) and Ch. 6 (Morava stabiliser group, change of rings); Morava 1985 *Noetherian localisations of categories of cobordism comodules* (Ann. of Math. 121, 1-39); Devinatz-Hopkins 2004 *Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups* (Topology 43)

Intuition Beginner

White light passing through a prism splits into a spectrum of colours, each with its own frequency. The chromatic spectral sequence does the same thing to the stable homotopy of spheres: it splits an impossibly complicated object into coloured layers, where the "colour" is a notion of periodicity called height. The lowest layer is rational and simple. Each successive layer captures a more refined, faster-vibrating kind of pattern that the earlier layers could not see.

Why bother? The stable homotopy groups of spheres are among the hardest computations in mathematics, with no closed formula. Direct attack drowns in noise. The chromatic idea is to reorganise the noise by periodicity, so that within each layer the answer becomes repetitive and tractable. You compute one layer at a time, then reassemble.

The reassembly is the spectral sequence. Its first page lists the coloured layers side by side; its later pages record how the layers interact and correct one another. What it converges to is the input page of the deeper Adams-Novikov machine, which in turn computes the homotopy of spheres. So the chromatic sequence is a lens that resolves a hard object into a stack of periodic pictures.

Visual Beginner

Picture a vertical stack of horizontal strips, numbered from the bottom upward: strip zero, strip one, strip two, and so on. Strip $n$ holds the height- $n$ periodic patterns of stable homotopy. The bottom strip is rational and almost empty; each higher strip is richer and more intricate. Short connecting arrows run between adjacent strips, recording how a pattern in one layer is born from the layer just below it.

The picture captures the central pattern: homotopy is filtered by height, the layers are read off one at a time, and the connecting arrows assemble them into a single answer. The higher you climb, the finer the periodicity you are seeing, and the more the layer resembles the symmetries of a formal group of that height.

Worked example Beginner

Read off the bottom two layers of the chromatic stack and see what each one detects.

Step 1. The bottom layer, height zero, is what survives after you allow yourself to divide by every prime. This rationalises everything. The stable homotopy of spheres is finite in every positive degree, so after rationalising, only the degree-zero part remains: a single copy of the rational numbers. The height-zero layer is therefore one generator, sitting in degree zero, and nothing else.

Step 2. The next layer, height one, is the first place periodicity appears. It is governed by a single periodic operator, traditionally written with the symbol $v_{1}$ . Inverting $v_{1}$ exposes a repeating family of classes that recur at a fixed spacing as the degree grows.

Step 3. At a prime $p$ , that repeating family is detected in the homotopy of spheres as the image of a classical construction, the $J$ -homomorphism, studied in 03.08.11. Its sizes follow a denominator pattern controlled by which powers of $p$ divide certain numbers — the same arithmetic that governs Bernoulli denominators.

Step 4. So the first two layers already recover two known things: the rational homotopy of spheres (height zero) and the image of $J$ (height one). Each came out of a single periodicity operator.

What this tells us: the chromatic layers are not abstract bookkeeping. Layer zero is the rational answer, layer one is the image of $J$ , and each higher layer promises a deeper periodic family — the spectral sequence is the machine that stacks them into the full answer.

Check your understanding Beginner

Formal definition Intermediate+

Fix a prime $p$ and work in the abelian category of $B P_{*} B P$ -comodules, where $B P_{*} = Z_{(p)} [v_{1}, v_{2}, \dots]$ with $∣ v_{n} ∣ = 2 (p^{n} - 1)$ and $B P_{*} B P = B P_{*} [t_{1}, t_{2}, \dots]$ is the Hopf algebroid of cooperations from 03.13.05. The invariant prime ideals are $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ , with $I_{0} = (0)$ ; each $I_{n} \subset B P_{*}$ is a $B P_{*} B P$ -comodule. The element $v_{n}$ acts on the quotient $B P_{*} / I_{n}$ as a comodule self-map, and inverting it is exact on comodules (Hovey-Strickland; Landweber).

Definition (chromatic modules). Define comodules $N^{n}$ and $M^{n}$ recursively. Set $N^{0} = B P_{*}$ . Given $N^{n}$ , set $$ M^n = v_n^{-1} N^n, \qquad N^{n+1} = M^n / N^n, $$ so that there is a short exact sequence of $B P_{*} B P$ -comodules $$ 0 \longrightarrow N^n \longrightarrow v_n^{-1} N^n = M^n \longrightarrow N^{n+1} \longrightarrow 0. $$ Concretely $N^{n} = B P_{*} / I_{n}$ as a comodule (with $N^{0} = B P_{*}$ , $N^{1} = B P_{*} / (p) = B P_{*} / p^{\infty}$ after the first localisation step, and so on), and $M^{n} = v_{n}^{- 1} B P_{*} / I_{n}$ is its height- $n$ localisation. The module $M^{n}$ is $v_{n}$ -periodic: multiplication by $v_{n}$ is an isomorphism on it.

Definition (chromatic resolution). Splicing the short exact sequences gives a long exact sequence of comodules, the chromatic resolution of $B P_{*}$ : $$ 0 \longrightarrow BP_* \longrightarrow M^0 \longrightarrow M^1 \longrightarrow M^2 \longrightarrow \cdots, $$ where the map $M^{n} \to M^{n + 1}$ is the composite $M^{n} ↠ N^{n + 1} ↪ M^{n + 1}$ . This is a resolution of $B P_{*}$ by the $v_{n}$ -periodic comodules $M^{n}$ .

Definition (chromatic spectral sequence). Applying $Ext_{B P_{*} B P} (B P_{*}, -)$ to the chromatic resolution and assembling the resulting long exact sequences into an exact couple 03.13.01 produces the chromatic spectral sequence $$ E_1^{n, s} = \mathrm{Ext}^{s}{BPBP}(BP_, M^n) ;\Longrightarrow; \mathrm{Ext}^{n + s}{BPBP}(BP_, BP_*), $$ where the chromatic degree $n$ is the resolution index (the column / height) and $s$ is the homological degree in $Ext$ . The abutment $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ is the Adams-Novikov $E_{2}$ -page 03.12.38. The differential $d_{1} : E_{1}^{n, s} \to E_{1}^{n + 1, s}$ is induced by the chromatic resolution maps; the higher differentials $d_{r} : E_{r}^{n, s} \to E_{r}^{n + r, s - r + 1}$ record the connecting homomorphisms.

Counterexamples to common slips

The chromatic index $n$ is not the Adams-Novikov filtration $s$ . A class in $E_{1}^{n, s}$ sits in chromatic column $n$ (height $n$ ) and contributes to $Ext^{n + s}$ ; the total Adams-Novikov filtration is $n + s$ , not $n$ alone. The $α$ -family lives in $E_{1}^{1, 0}$ and lands in $Ext^{1}$ , so $n + s = 1$ with $n = 1$ , $s = 0$ .
The chromatic resolution is not a finite resolution and the modules $M^{n}$ are not finitely generated; $M^{0} = Q$ already, and $M^{1} = v_{1}^{- 1} B P_{*} / p^{\infty}$ is a divisible $p$ -torsion object. Convergence is in the sense of a half-plane exact couple, not a bounded one.
$v_{n}^{- 1} B P_{*} / I_{n}$ depends on the invariance of $I_{n}$ . The ideals $(p, v_{1}, \dots, v_{n - 1})$ are the only invariant prime ideals (Landweber-Morava); replacing them by a non-invariant sequence destroys the comodule structure and the construction fails.

Key theorem with proof Intermediate+

Theorem (chromatic spectral sequence; Miller-Ravenel-Wilson 1977). Let $p$ be a prime. The chromatic resolution $0 \to BP_ \to M^0 \to M^1 \to \cdots$ yields a spectral sequence* $$ E_1^{n, s} = \mathrm{Ext}^{s}{BPBP}(BP_, M^n) ;\Longrightarrow; \mathrm{Ext}^{n + s}{BPBP}(BP_, BP_*), $$ converging to the Adams-Novikov $E_{2}$ -page, with $d_{r}$ of bidegree $(r, 1 - r)$ in the $(n, s)$ indexing.

Proof. The chromatic resolution is an exact sequence of $B P_{*} B P$ -comodules. Break it into the short exact sequences $$ 0 \to N^n \to M^n \to N^{n+1} \to 0, \qquad N^0 = BP_*, \quad M^n = v_n^{-1} N^n. $$ Each short exact sequence of comodules induces a long exact sequence in $Ext_{B P_{*} B P} (B P_{*}, -)$ , since $Ext$ is the derived functor of comodule primitives and is a cohomological $δ$ -functor. Write $D^{n, s} = Ext_{B P_{*} B P}^{s} (B P_{*}, N^{n})$ and $E^{n, s} = Ext_{B P_{*} B P}^{s} (B P_{*}, M^{n})$ . The long exact sequences read $$ \cdots \to D^{n, s} \xrightarrow{i} D^{n-1, s} \xrightarrow{j} E^{n-1, s} \xrightarrow{k} D^{n, s+1} \to \cdots, $$ where $i$ comes from $N^{n} \to M^{n - 1}$ (the inclusion into the previous periodic module, after re-indexing $N^{n} = ker (M^{n} \to N^{n + 1})$ identified with the cokernel image), $j$ from $M^{n - 1} \to N^{n}$ , and $k$ is the connecting homomorphism. These assemble into a bigraded exact couple in the sense of 03.13.01.

The exact couple has derived couples in the standard way, producing a spectral sequence with $E_{1}^{n, s} = Ext_{B P_{*} B P}^{s} (B P_{*}, M^{n})$ and first differential $d_{1} = j \circ k : E_{1}^{n, s} \to E_{1}^{n + 1, s}$ , the map induced by the chromatic resolution map $M^{n} \to M^{n + 1}$ . The higher differentials $d_{r}$ are the iterated connecting homomorphisms, of bidegree $(r, 1 - r)$ .

Convergence. The resolution exhibits $B P_{*}$ as the totalisation of the cochain complex $M^{∙}$ in the derived category of comodules. The associated spectral sequence is the hyper- $Ext$ spectral sequence of this resolution, and it converges to $Ext_{B P_{*} B P}^{n + s} (B P_{*}, B P_{*})$ because the resolution computes $Ext (B P_{*}, B P_{*})$ as hyper- $Ext (B P_{*}, M^{∙})$ (Miller-Ravenel-Wilson §3.3). In each internal degree $t$ the modules $M^{n}$ vanish for $n$ exceeding the $v$ -adic complexity of that degree, so each diagonal of the abutment receives contributions from finitely many columns, and convergence is strong in each internal degree. The abutment is the Adams-Novikov $E_{2}$ -page by the identification of 03.12.38. $□$

Bridge. The chromatic spectral sequence builds toward the entire chromatic stratification of stable homotopy and appears again in 03.13.07 (Greek-letter elements), where the connecting homomorphisms $Ext^{0} (M^{n}) \to Ext^{n} (B P_{*})$ manufacture the $α$ , $β$ , $γ$ families column by column. The foundational reason it works is exactly that inverting $v_{n}$ on $B P_{*} / I_{n}$ is an exact operation on comodules, so the localisation tower $N^{n} \to M^{n} \to N^{n + 1}$ is a genuine resolution rather than an approximation; this is dual to the topological chromatic tower of Bousfield localisations $L_{n}$ , where the algebraic $M^{n}$ corresponds to the monochromatic $K (n)$ -local layer. The central insight is that periodicity height organises the otherwise structureless $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ into computable strata, and putting these together with the Morava change-of-rings theorem turns each column into the continuous cohomology of an explicit profinite group. This pattern recurs through chromatic homotopy theory: the same height filtration generalises from the algebraic $E_{2}$ -page here to the topological sphere in 03.12.38 (Adams-Novikov) and forward to 03.13.08 (the telescope conjecture), where the agreement between algebraic and topological layers is precisely what is at stake.

Exercises Intermediate+

Exercise 5 (medium, symbolic).

For $n = 1$ at an odd prime $p$ , the module $M^{1} = v_{1}^{- 1} B P_{*} / p^{\infty}$ . Describe $Ext_{B P_{*} B P}^{0} (B P_{*}, M^{1})$ and relate it to the $α$ -family.

Hint

$Ext^{0}$ is the comodule primitives. The primitives of $v_{1}^{- 1} B P_{*} / p^{\infty}$ are the $v_{1}$ -power classes $v_{1}^{t} / p^{j}$ with the appropriate invariance.

Answer

The comodule primitives are spanned by classes $x_{t, j} = v_{1}^{t} / p^{j}$ that are invariant under the coaction, which forces a congruence relating $t$ and $j$ (the $v_{1}$ -periodicity condition). Miller-Ravenel-Wilson show $Ext_{B P_{*} B P}^{0} (B P_{*}, M^{1}) ≅ Q / Z_{(p)} {v_{1}^{t}}$ controlled by the $p$ -adic valuation of $t$ — precisely $Z / p^{1 + ν_{p} (t)}$ for the class in internal degree $2 t (p - 1)$ . The connecting homomorphism $Ext^{0} (M^{1}) \to Ext^{1} (B P_{*})$ sends these to the $α$ -family $α_{t}$ , the $v_{1}$ -periodic generators detecting the image of $J$ 03.08.11. Rubric: full credit for the primitive description plus the $α_{t}$ identification.

Exercise 6 (hard, short-answer).

Explain why the chromatic spectral sequence converges in each internal degree even though every $M^{n}$ is infinitely generated and the resolution is infinite.

Hint

Fix an internal degree $t$ . Bound the chromatic length of $t$ using the degrees $∣ v_{n} ∣ = 2 (p^{n} - 1)$ .

Answer

Fix the internal degree $t$ . The module $M^{n} = v_{n}^{- 1} B P_{*} / I_{n}$ contributes in degree $t$ only through monomials whose $v_{n}$ -adic structure fits within $t$ ; since $∣ v_{n} ∣ = 2 (p^{n} - 1)$ grows without bound, for $n$ large enough $2 (p^{n} - 1) > t$ and $M^{n}$ has no nonzero classes in total degree $t$ in $Ext^{0}$ , and a finite-column bound propagates to all $s$ by the boundedness of the cobar filtration in fixed degree. Hence in each fixed internal degree only finitely many columns $n$ contribute, the exact couple is locally finite there, and the spectral sequence converges strongly degree by degree. Globally the abutment is the union over $t$ . Rubric: full credit for the degree-bound argument forcing finitely many contributing columns per internal degree.

Exercise 7 (hard, short-answer).

Using the Morava change-of-rings theorem, rewrite the $E_{1}$ -term $Ext_{B P_{*} B P} (B P_{*}, M^{n})$ in terms of the Morava stabiliser group, and state what this buys computationally.

Hint

$M^{n}$ is $v_{n}$ -periodic and $I_{n}$ -torsion; change of rings replaces $B P_{*} B P$ by the automorphisms of a height- $n$ formal group law.

Answer

The Morava change-of-rings theorem identifies $$ \mathrm{Ext}^{}{BPBP}(BP_, M^n) \cong H^{}_c\big(\mathbb{G}n; (E_n)* / I_n[v_n^{-1}]\big) \otimes (\text{a } \mathbb{Q}/\mathbb{Z} \text{ factor}), $$ the continuous cohomology of the Morava stabiliser group $G_{n} = Aut (Γ_{n})$ — the automorphism group of the height- $n$ Honda formal group law over $F_{p^{n}}$ , extended by the Galois group — acting on the Lubin-Tate ring $E_{n}^{*}$ 03.12.38. Computationally this replaces an intractable comodule $Ext$ by group cohomology of an explicit $p$ -adic Lie group, which for small $n$ is finite-dimensional and computable (e.g. $G_{1} = Z_{p}^{\times}$ gives the height-1 / image-of- $J$ answer). Rubric: full credit for the $H_{c}^{*} (G_{n}; -)$ identification and the computational reduction to group cohomology.

Lean formalization Intermediate+

Mathlib has neither $B P_{*} B P$ -comodules nor the localisation/quotient operations on them, so the chromatic resolution cannot be stated in Mathlib syntax. The intended formalisation would read schematically:

import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.Algebra.Homology.SpectralSequence
import Mathlib.Algebra.Homology.ExactCouple

variable (p : ℕ) [Fact p.Prime]

/-- The chromatic comodules N^n, M^n built by iterating v_n-localisation
    and quotient by the invariant ideals I_n = (p, v_1, …, v_{n-1}). -/
noncomputable def chromaticModule :
    ℕ → BPComodule p
  | 0 => BPstar p                          -- N^0 = BP_*
  | (n + 1) => cokernel (toLocalisation (chromaticModule n))

noncomputable def periodicModule (n : ℕ) : BPComodule p :=
  vInvLocalisation n (chromaticModule p n)  -- M^n = v_n^{-1} N^n

/-- The chromatic resolution as an exact cochain complex of comodules. -/
noncomputable def chromaticResolution : CochainComplex (BPComodule p) ℕ :=
  sorry  -- splice 0 → N^n → M^n → N^{n+1} → 0

/-- The chromatic spectral sequence: E_1^{n,s} = Ext^s(BP_*, M^n)
    converging to the Adams-Novikov E_2 = Ext(BP_*, BP_*). -/
noncomputable def chromaticSS :
    SpectralSequence ℤ × ℤ (BPComodule p) :=
  (chromaticResolution p).extExactCouple (BPstar p) |>.spectralSequence

theorem chromaticSS_converges (n s : ℤ) :
    (chromaticSS p).Converges
      (Ext (BPstar p) (BPstar p) (n + s)) :=
  sorry  -- Miller-Ravenel-Wilson 1977, §3.3

The proof gap is substantive: Mathlib lacks the Hopf algebroid $(B P_{*}, B P_{*} B P)$ , the comodule abelian category, the exactness of $v_{n}^{- 1} (-)$ , the invariant-ideal structure, and the change-of-rings theorem to the Morava stabiliser group. Each is a separate formalisation target listed in Mathlib gap analysis.

Advanced results Master

Theorem (Morava change of rings; Morava 1985, Miller-Ravenel-Wilson). Fix a prime $p$ and height $n \geq 1$ . There is an isomorphism $$ \mathrm{Ext}^{s}{BPBP}\big(BP_, v_n^{-1} BP_*/I_n\big) ;\cong; H^{s}_c\big(\mathbb{G}n;, (E_n)/I_n[v_n^{\pm 1}]\big), $$ where $G_{n} = S_{n} ⋊ Gal (F_{p^{n}} / F_{p})$ is the (extended) Morava stabiliser group, $S_{n} = Aut (Γ_{n})$ is the automorphism group of the height- $n$ Honda formal group law $Γ_{n}$ over $\overline{F}_{p}$ , and $H^_c$ is continuous cohomology.

The proof passes through the Hopf algebroid $(v_{n}^{- 1} B P_{*} / I_{n}, v_{n}^{- 1} B P_{*} B P / I_{n})$ , which after base change to the Lubin-Tate ring $E_{n}^{*}$ becomes the cohomology of the groupoid of height- $n$ formal group laws and their isomorphisms. The isomorphism classes are a single point (all height- $n$ formal group laws over $\overline{F}_{p}$ are isomorphic), and the automorphisms form $S_{n}$ , so the comodule $Ext$ becomes group cohomology. This is the algebraic shadow of the topological statement that the $K (n)$ -local sphere is the homotopy fixed points $E_{n}^{h G_{n}}$ (Devinatz-Hopkins).

Theorem (height-zero column). $Ext_{B P_{*} B P}^{s} (B P_{*}, M^{0}) = Q$ concentrated in $s = 0$ , internal degree $0$ . This is the rationalisation: $M^{0} = B P_{*} \otimes Q$ , the Hopf algebroid becomes the discrete groupoid over $Q$ , and the $Ext$ collapses to the rational homotopy of the sphere, a single $Q$ in degree zero.

Theorem (height-one column; the $α$ -family). At an odd prime $p$ , $$ \mathrm{Ext}^{0}{BPBP}(BP_, M^1) \cong \mathbb{Q}/\mathbb{Z}{(p)} \quad \text{in internal degrees } 2t(p-1), $$ with the class in degree $2 t (p - 1)$ of order $p^{1 + ν_{p} (t)}$ , where $ν_{p}$ is the $p$ -adic valuation. The connecting homomorphism $\delta: \mathrm{Ext}^0(M^1) \to \mathrm{Ext}^1(BP) $c a r r i es t h ese t o t h e$ \alpha $- f ami l y$ \alpha_{t} \in \mathrm{Ext}^1_{BP_BP}(BP_, BP_) $. U n d er t h e A d am s - N o v ik o v e d g e ma pt h ese d e t ec tt h e ima g eo f t h e$ J $- h o m o m or p hi s m [03.08.11], w h oseor d er s ma t c h t h e d e n o mina t or so f$ B_{t}/t $f or B er n o u l l in u mb er s$ B_t $. T h ec han g e - o f - r in g sr e a d in g i s$ H^*_c(\mathbb{G}_1; -) $w i t h$ \mathbb{G}_1 = \mathbb{Z}_p^\times $, w h oseco h o m o l o g y r eco v er se x a c tl y t hi s$ p$-adic valuation pattern.

Theorem (height-two column; the $β$ -family, Miller-Ravenel-Wilson). The $n = 2$ column $Ext_{B P_{*} B P}^{0} (B P_{*}, M^{2})$ is computed via $H^c(\mathbb{G}2; -) $, an d i t s ima g e u n d er t h e i t er a t e d co nn ec t in g h o m o m or p hi s m$ \mathrm{Ext}^0(M^2) \to \mathrm{Ext}^2(BP*) $i s t h e$ \beta $- f ami l y$ \beta{s/j} \in \mathrm{Ext}^2 $. * T h e$ v_2 $- p er i o d i c f ami l i es$ \beta_s $a r e t h e f i r s t g e n u in e l y n e w c h r o ma t i c p h e n o m e n o nb ey o n d t h e ima g eo f$ J $; t h e i r e x i s t e n ce an d t h e d i v i s ibi l i t y b oo k k ee p in g ($ \beta_{s/j} $in d e x e d b y$ v_2^s/(p^{i}, v_1^{j})$) is the headline computation of Miller-Ravenel-Wilson 1977. The detailed account of these families is taken up in 03.13.07.

Theorem (chromatic convergence, algebraic). The chromatic spectral sequence converges strongly to $\mathrm{Ext}{BPBP}(BP_, BP_) $in e a c hin t er na l d e g r ee, an d t h ec h r o ma t i c f i l t r a t i o ni t in d u ceso n t h e A d am s - N o v ik o v$ E_2 $- p a g e a g r ees, u n d er t h e A d am s - N o v ik o v s p ec t r a l se q u e n ce, w i t h t h e t o p o l o g i c a l c h r o ma t i c f i l t r a t i o n o f$ \pi_*^s $b y t h e B o u s f i e l d l oc a l i s a t i o n s$ L_n$.* The agreement of the algebraic and topological chromatic filtrations is the algebraic counterpart of Ravenel's chromatic convergence theorem; the precise comparison of the filtration steps is where the telescope conjecture 03.13.08 enters.

Synthesis. The chromatic spectral sequence is the foundational reason that the Adams-Novikov $E_{2}$ -page, otherwise an opaque comodule $Ext$ , decomposes into computable height- $n$ strata. The central insight is that the invariant ideals $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ and the periodicity operators $v_{n}$ stratify $B P_{*}$ by height, and the chromatic resolution $0 \to B P_{*} \to M^{0} \to M^{1} \to \dots$ realises that stratification as an exact resolution by $v_{n}$ -periodic comodules. Putting these together with the Morava change-of-rings theorem, each column collapses to continuous cohomology of the stabiliser group $G_{n}$ acting on Lubin-Tate space, which is exactly the structure that identifies the height- $n$ layer with the symmetries of a formal group of height $n$ . This is dual to the topological chromatic tower: the algebraic $M^{n}$ corresponds to the monochromatic $K (n)$ -local layer, and the bridge between the algebraic chromatic filtration here and the topological one in 03.12.38 is precisely the comparison whose failure or success is the content of the telescope conjecture. Putting these together, the Green Book's organising principle is visible in a single diagram — homotopy resolved into chromatic layers indexed by periodicity height, each governed by an explicit profinite symmetry group.

Full proof set Master

Proposition (the chromatic short exact sequences are exact on comodules). *For each $n$ , the sequence $0 \to N^{n} \to v_{n}^{- 1} N^{n} \to N^{n + 1} \to 0$ is a short exact sequence in the category of $B P_{*} B P$ -comodules.*

Proof. Exactness as $B P_{*}$ -modules is the definition of $N^{n + 1} = v_{n}^{- 1} N^{n} / N^{n}$ , so only the comodule (coaction-compatibility) statement requires argument. Localisation at $v_{n}$ is a filtered colimit of the maps $v_{n}^{k} : N^{n} \to N^{n}$ . Each such map is a comodule map because $v_{n} \in B P_{*}$ is a comodule primitive modulo $I_{n}$ : the right unit satisfies $η_{R} (v_{n}) \equiv v_{n} (mod I_{n})$ (Hazewinkel / Araki, used in 03.13.05), and $N^{n}$ is $I_{n}$ -torsion, so multiplication by $v_{n}$ commutes with the coaction on $N^{n}$ . A filtered colimit of comodule maps is a comodule map and the colimit $v_{n}^{- 1} N^{n}$ inherits a coaction; the localisation map $N^{n} \to v_{n}^{- 1} N^{n}$ is then a comodule map, and its cokernel $N^{n + 1}$ acquires the quotient coaction. Exactness of the coaction-compatible sequence follows because the forgetful functor to $B P_{*}$ -modules is exact and reflects exactness on the comodule category (the coaction is a map of comodules whose underlying module map is the localisation). $□$

Proposition (the $n = 0$ column is rational). $Ext_{B P_{*} B P}^{s, t} (B P_{*}, M^{0}) = Q$ if $(s, t) = (0, 0)$ and $0$ otherwise.

Proof. $M^{0} = v_{0}^{- 1} B P_{*} = p^{- 1} B P_{*} = B P_{*} \otimes Q = Q [v_{1}, v_{2}, \dots]$ . After tensoring with $Q$ , the Hopf algebroid $(B P_{*} \otimes Q, B P_{*} B P \otimes Q)$ is equivalent to the discrete unit Hopf algebroid: rationally, $B P_{*} \otimes Q = Q [m_{1}, m_{2}, \dots]$ on the logarithm coefficients $m_{i}$ , and the right unit acts so that $B P_{*} B P \otimes Q = (B P_{*} \otimes Q) [b_{1}, b_{2}, \dots]$ with the $b_{i}$ exterior-free generators on which the cobar differential is invertible. The cobar complex computing $Ext_{B P_{*} B P \otimes Q} (Q [v_{*}], Q [v_{*}])$ is therefore acyclic except in homological degree $0$ , where the primitives are the constants $Q$ in internal degree $0$ . Hence $Ext^{0, 0} = Q$ and all other groups vanish. This reproduces $π_{*}^{s} \otimes Q = Q$ in degree $0$ . $□$

Proposition (first chromatic differential raises the column by one). In the chromatic spectral sequence, $d_{1} : E_{1}^{n, s} \to E_{1}^{n + 1, s}$ is the map on $Ext_{B P_{*} B P}^{s} (B P_{*}, -)$ induced by the composite $M^{n} ↠ N^{n + 1} ↪ M^{n + 1}$ .

Proof. In the exact couple $(D, E, i, j, k)$ of the proof above, $D^{n, s} = Ext^{s} (B P_{*}, N^{n})$ , $E^{n, s} = Ext^{s} (B P_{*}, M^{n})$ , with $j : D^{n, s} \to E^{n, s}$ induced by $N^{n} ↪ M^{n}$ and $k : E^{n, s} \to D^{n + 1, s}$ the connecting map of $0 \to N^{n + 1} \to M^{n + 1} \to N^{n + 2} \to 0$ re-indexed so that $k$ lands in $Ext^{s} (B P_{*}, N^{n + 1})$ . The first differential of the spectral sequence of an exact couple is $d_{1} = j \circ k$ . Tracing: $k$ sends $E^{n, s} = Ext^{s} (M^{n})$ to $Ext^{s} (N^{n + 1})$ via the surjection $M^{n} ↠ N^{n + 1}$ , and $j$ embeds $Ext^{s} (N^{n + 1})$ into $Ext^{s} (M^{n + 1})$ via $N^{n + 1} ↪ M^{n + 1}$ . The composite is induced by $M^{n} \to N^{n + 1} \to M^{n + 1}$ , which is the chromatic resolution map, of bidegree $(1, 0)$ . $□$

Connections Master

The chromatic spectral sequence converges to the Adams-Novikov $E_{2}$ -page $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ of 03.12.38; that page is the input to the Adams-Novikov differentials computing the stable homotopy of spheres. The chromatic sequence is the standard tool for organising the otherwise intractable Adams-Novikov $E_{2}$ -term by height, so the two spectral sequences are run in series: chromatic first, Adams-Novikov second.
The entire construction is built on the Hopf algebroid $(B P_{*}, B P_{*} B P)$ of 03.13.05: the comodules $M^{n}$ , $N^{n}$ , the invariant ideals $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ , and the periodicity operators $v_{n}$ all live in the comodule category introduced there. The exactness of $v_{n}^{- 1} (-)$ and the primitivity of $v_{n}$ modulo $I_{n}$ are inherited directly from the structure of $B P_{*} B P$ .
The connecting homomorphisms $Ext^{0} (M^{n}) \to Ext^{n} (B P_{*})$ manufacture the Greek-letter families $α, β, γ$ of 03.13.07, the named periodic elements of the Adams-Novikov $E_{2}$ -page. The chromatic spectral sequence is the machine that produces and indexes these families, column by column.
The height-one column recovers the image of the $J$ -homomorphism of 03.08.11: the $α$ -family $α_{t}$ detects $Im J$ , and its orders follow the Bernoulli-denominator pattern. This is the lowest-height, most classical instance of the chromatic picture, and the bridge between 1960s computations and the chromatic programme.
The algebraic chromatic filtration constructed here is compared, via the Adams-Novikov spectral sequence, with the topological chromatic tower of Bousfield localisations $L_{n}$ ; the precise agreement of the finite (telescopic) and $K (n)$ -local layers is the content of the telescope conjecture 03.13.08, disproved at heights $n \geq 2$ in 2023.

Historical & philosophical context Master

The chromatic point of view emerged from the 1977 Annals paper of Haynes Miller, Douglas Ravenel, and Steve Wilson, Periodic phenomena in the Adams-Novikov spectral sequence ^{[Miller-Ravenel-Wilson 1977]}. Building on Jack Morava's unpublished work on the cohomology of formal-group automorphism groups and on the Adams-Novikov spectral sequence introduced by Novikov, they recognised that the periodicity operators $v_{n}$ in $B P_{*}$ organise stable homotopy into layers indexed by the height of a formal group law. The chromatic resolution and its spectral sequence gave the first systematic computation of $v_{2}$ -periodic families, the $β$ -family, beyond the $v_{1}$ -periodic image of $J$ that Adams had analysed in the 1960s.

Morava's contribution, circulated for years before its 1985 Annals publication Noetherian localisations of categories of cobordism comodules ^{[Morava 1985]}, was the change-of-rings theorem identifying each chromatic column with the continuous cohomology of the stabiliser group $G_{n}$ of a height- $n$ formal group law. This connected stable homotopy to the arithmetic of Lubin-Tate deformation theory and to the symmetries of formal groups, a link that became the organising principle of Ravenel's 1986 monograph Complex Cobordism and Stable Homotopy Groups of Spheres ^{[Ravenel 1986]} — the "Green Book" — whose fifth chapter is the chromatic spectral sequence and whose sixth develops the Morava theory. The chromatic programme reframed the homotopy groups of spheres not as a list of numbers but as a stratified object whose strata are governed by the moduli of formal groups, a perspective that drove the nilpotence and periodicity theorems of Devinatz, Hopkins, and Smith and continues to organise the field.

Bibliography Master

@article{MillerRavenelWilson1977,
  author  = {Miller, Haynes R. and Ravenel, Douglas C. and Wilson, W. Stephen},
  title   = {Periodic phenomena in the {Adams}-{Novikov} spectral sequence},
  journal = {Annals of Mathematics (2)},
  volume  = {106},
  number  = {3},
  pages   = {469--516},
  year    = {1977},
  doi     = {10.2307/1971064}
}

@book{Ravenel1986,
  author    = {Ravenel, Douglas C.},
  title     = {Complex Cobordism and Stable Homotopy Groups of Spheres},
  series    = {Pure and Applied Mathematics},
  volume    = {121},
  publisher = {Academic Press},
  year      = {1986},
  note      = {Second edition, AMS Chelsea, 2004; the ``Green Book''}
}

@article{Morava1985,
  author  = {Morava, Jack},
  title   = {Noetherian localisations of categories of cobordism comodules},
  journal = {Annals of Mathematics (2)},
  volume  = {121},
  number  = {1},
  pages   = {1--39},
  year    = {1985},
  doi     = {10.2307/1971192}
}

@book{HoveyStrickland1999,
  author    = {Hovey, Mark and Strickland, Neil P.},
  title     = {Morava {K}-theories and localisation},
  series    = {Memoirs of the American Mathematical Society},
  volume    = {139},
  number    = {666},
  publisher = {American Mathematical Society},
  year      = {1999}
}

@article{DevinatzHopkins2004,
  author  = {Devinatz, Ethan S. and Hopkins, Michael J.},
  title   = {Homotopy fixed point spectra for closed subgroups of the {Morava} stabilizer groups},
  journal = {Topology},
  volume  = {43},
  number  = {1},
  pages   = {1--47},
  year    = {2004},
  doi     = {10.1016/S0040-9383(03)00029-6}
}

Prerequisites

03.13.05 pending
03.12.38 pending

Tier anchors

beginner: Ravenel 1986 *Complex Cobordism and Stable Homotopy Groups of Spheres* (Academic Press) §1.4 'the chromatic point of view' — informal opening; Hopkins's Bourbaki survey of chromatic homotopy theory
intermediate: Miller-Ravenel-Wilson 1977 *Periodic phenomena in the Adams-Novikov spectral sequence* (Ann. of Math. 106, 469-516) §3-§4; Ravenel 1986 (the Green Book) §5.1-§5.2
master: Miller-Ravenel-Wilson 1977 (Ann. of Math. 106) §3-§5; Ravenel 1986 *Complex Cobordism and Stable Homotopy Groups of Spheres* Ch. 5 (chromatic spectral sequence) and Ch. 6 (Morava stabiliser group, change of rings); Morava 1985 *Noetherian localisations of categories of cobordism comodules* (Ann. of Math. 121, 1-39); Devinatz-Hopkins 2004 *Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups* (Topology 43)

References

Miller-Ravenel-Wilson — Periodic phenomena in the Adams-Novikov spectral sequence · Ann. of Math. (2) 106 (1977), 469-516 — the chromatic resolution, the chromatic spectral sequence, and the height-1 (alpha-family / image of J) computation
Ravenel — Complex Cobordism and Stable Homotopy Groups of Spheres · Pure and Applied Mathematics 121 (Academic Press 1986), Ch. 5 — the chromatic spectral sequence; Ch. 6 — the Morava change-of-rings theorem and the stabiliser group
Morava — Noetherian localisations of categories of cobordism comodules · Ann. of Math. (2) 121 (1985), 1-39 — the change-of-rings theorem identifying Ext over BP_*BP localised at height n with continuous cohomology of the Morava stabiliser group
Hovey-Strickland — Morava K-theories and localisation · Mem. Amer. Math. Soc. 139 (1999), no. 666 — invariant prime ideals, the Landweber filtration, and the algebraic structure underlying the chromatic resolution
Lurie — Chromatic Homotopy Theory (252x lecture notes) · Harvard course notes 2010, Lectures 20-24 — modern account of the chromatic resolution and the Morava stabiliser group action on Lubin-Tate space

Estimated time

beginner: 20m
intermediate: 50m
master: 85m