03.13.05 · modern-geometry / spectral-sequences

The Brown-Peterson spectrum BP and its Hopf algebroid

shipped3 tiersLean: none

Anchor (Master): Ravenel 2004 *Complex Cobordism and Stable Homotopy Groups of Spheres* Ch.4 and App. A2 (AMS Chelsea); Brown-Peterson 1966 *A spectrum whose Z_p cohomology is the algebra of reduced p-th powers* (Topology 5); Hazewinkel 1978 *Formal Groups and Applications* (Academic Press) Ch.III; Wilson 1982 *Brown-Peterson homology: an introduction and sampler* (CBMS 48)

Intuition Beginner

Complex cobordism is a powerful but bulky tool for studying the homotopy groups of spheres. Its coefficient ring is a polynomial ring on infinitely many generators, one in every even positive degree, and that abundance makes hand computation heavy. Brown and Peterson found that once you fix a single prime number $p$ and only keep track of $p$ -power information, this bulky theory splits into many identical copies of a much leaner theory. That leaner piece is the Brown-Peterson spectrum, written BP. Working one prime at a time, BP carries all the information you need and throws away the rest.

The payoff is the shape of its coefficient ring. Where complex cobordism had a generator in every even degree, BP has generators only in the degrees $2 (p^{n} - 1)$ , one for each whole number $n$ from $1$ upward. These generators are called $v_{1}, v_{2}, v_{3}$ , and so on. So the coefficient ring of BP is a clean polynomial ring in the variables $v_{1}, v_{2}, \dots$ over the $p$ -local integers, and the spacing of the degrees already hints at a layered structure indexed by the number $n$ .

That layering, called chromatic height, is the reason BP is the workhorse of the field. Each variable $v_{n}$ governs one layer. The first layer, run by $v_{1}$ , sees periodic phenomena tied to ordinary $K$ -theory. The second layer, run by $v_{2}$ , sees deeper periodic families, and so on up the tower. By organising stable homotopy into these height layers, BP turns a single hard question into a sequence of more structured ones.

Visual Beginner

Picture the coefficient ring of complex cobordism as an endless row of pegs, one peg in every even positive degree. Fix a prime $p$ . The Brown-Peterson splitting keeps only a sparse subset of those pegs: the pegs sitting at degrees $2 (p^{n} - 1)$ . For $p = 2$ these land at degrees $2, 6, 14, 30, 62$ , doubling-and-adding their way up; for $p = 3$ they land at $4, 16, 52$ . Each kept peg is one generator $v_{n}$ , and everything else is rebuilt from these by multiplication.

The lower part of the picture stacks the heights as horizontal bands. Band $n$ is the part of stable homotopy that the variable $v_{n}$ makes periodic. The variables climb the bands one at a time, and the whole arrangement is the algebraic skeleton that the Adams-Novikov spectral sequence reads off when it computes homotopy groups of spheres.

Worked example Beginner

Fix the prime $p = 2$ and list the first few generator degrees of BP. The rule is that $v_{n}$ sits in degree $2 (p^{n} - 1)$ .

Step 1. For $n = 1$ the degree is $2 (2^{1} - 1) = 2 (2 - 1) = 2 (1) = 2$ . So $v_{1}$ lives in degree $2$ .

Step 2. For $n = 2$ the degree is $2 (2^{2} - 1) = 2 (4 - 1) = 2 (3) = 6$ . So $v_{2}$ lives in degree $6$ .

Step 3. For $n = 3$ the degree is $2 (2^{3} - 1) = 2 (8 - 1) = 2 (7) = 14$ . So $v_{3}$ lives in degree $14$ .

Step 4. For $n = 4$ the degree is $2 (2^{4} - 1) = 2 (16 - 1) = 2 (15) = 30$ . So $v_{4}$ lives in degree $30$ .

So at the prime $2$ the coefficient ring of BP begins as a polynomial ring with one variable in degree $2$ , one in degree $6$ , one in degree $14$ , one in degree $30$ , and nothing in the even degrees in between. The component in degree $4$ , for instance, is spanned by $v_{1}^{2}$ alone, since $v_{1}^{2}$ has degree $2 + 2 = 4$ and there is no separate generator there. The component in degree $8$ is spanned by two monomials: $v_{1}^{4}$ of degree $8$ , and $v_{1} v_{2}$ of degree $2 + 6 = 8$ . Counting monomials degree by degree is the whole game.

Now switch to $p = 3$ . The first generator $v_{1}$ has degree $2 (3 - 1) = 4$ , the second $v_{2}$ has degree $2 (9 - 1) = 16$ , and the third $v_{3}$ has degree $2 (27 - 1) = 52$ . The generators are much sparser at larger primes, which is one reason odd-primary computations reach higher into the stable stems before the structure gets complicated.

Check your understanding Beginner

Exercise (easy, multiple choice).

What is the coefficient ring of the Brown-Peterson spectrum, as an abstract ring?

A. A polynomial ring on one generator in every even degree, over the integers
B. A polynomial ring $Z_{(p)} [v_{1}, v_{2}, \dots]$ with $v_{n}$ in degree $2 (p^{n} - 1)$
C. The field with $p$ elements in every degree
D. The integers concentrated in degree zero

Hint

Brown-Peterson is the $p$ -local leaner replacement for complex cobordism, with generators only in the special degrees.

Answer

B. The coefficient ring is the polynomial ring $Z_{(p)} [v_{1}, v_{2}, \dots]$ over the $p$ -local integers, with one generator $v_{n}$ in each degree $2 (p^{n} - 1)$ . Choice A describes complex cobordism, the bulkier theory that BP is split off from; choice C describes Morava $K$ -theory coefficients after inverting and killing variables; choice D describes ordinary integral homology.

Formal definition Intermediate+

Fix a prime $p$ . Recall from 03.03.04 that a formal group law over a commutative ring $R$ is a power series $F (x, y) \in R [[x, y]]$ that is associative, commutative, and unital, and that the Lazard ring $L$ carries the universal one. A formal group law $F$ over a $Z_{(p)}$ -algebra is $p$ -typical when its logarithm has the form $lo g_{F} (x) = \sum_{i \geq 0} ℓ_{i} x^{p^{i}}$ , with only $p$ -power exponents appearing. Cartier's theorem states that over a $Z_{(p)}$ -algebra every formal group law is canonically strictly isomorphic to a $p$ -typical one, and this canonical isomorphism is the Cartier idempotent $ε$ on the additive group of strict isomorphisms.

Definition (Brown-Peterson spectrum). Let $M U_{(p)}$ denote the $p$ -localisation of the complex cobordism spectrum $M U$ 03.12.04. Quillen's idempotent $e : M U_{(p)} \to M U_{(p)}$ is the multiplicative stable self-map realising the Cartier $p$ -typicalisation projector on formal group laws. The Brown-Peterson spectrum $B P$ is its image, a $p$ -local ring spectrum with a splitting $$ MU_{(p)} \simeq \bigvee_{\alpha} \Sigma^{2|\alpha|} BP $$ indexed by monomials $α$ in generators of degree away from the $v_{n}$ -degrees. Its homotopy ring is $$ \pi_* BP = BP_* = \mathbb{Z}{(p)}[v_1, v_2, v_3, \ldots], \qquad |v_n| = 2(p^n - 1), $$ where the $v_{n}$ may be taken to be the Hazewinkel generators (defined recursively by the functional equation $p,\ell_n = \sum{0 \le i < n} \ell_i, v_{n-i}^{p^i} $o n t h e l o g a r i t hm coe f f i c i e n t s$ \ell_n $, w i t h$ \ell_0 = 1 $) or t h e * * A r ak i g e n er a t or s * * (d e f in e d b y$ p,\ell_n = \sum_{0 \le i \le n} \ell_i, v_{n-i}^{p^i} $) . T h e f or ma l g r o u pl a w o n$ BP_* $c l a ss i f i e d b y$ BP_* = \pi_* BP \to BP_* $i s t h e u ni v er s a l$ p$-typical formal group law.

*Definition (the Hopf algebroid $(BP_, BP_BP)$).* The $B P$ -homology of $B P$ is $$ BP_BP = BP_[t_1, t_2, t_3, \ldots], \qquad |t_n| = 2(p^n - 1), $$ a polynomial $B P_{*}$ -algebra. The pair $(B P_{*}, B P_{*} B P)$ is a Hopf algebroid: a cogroupoid object in commutative rings, equivalently the functions on the groupoid scheme whose objects are $p$ -typical formal group laws and whose morphisms are strict isomorphisms between them. The structure maps are the left unit $η_{L} : B P_{*} \to B P_{*} B P$ (the inclusion of scalars), the right unit $η_{R} : B P_{*} \to B P_{*} B P$ (the source-versus-target distinction for an isomorphism), the counit $ε : B P_{*} B P \to B P_{*}$ (identity isomorphisms), the coproduct $Δ : B P_{*} B P \to B P_{*} B P \otimes_{B P_{*}} B P_{*} B P$ (composition of isomorphisms), and the conjugation $c$ (inverse). The element $t_{n}$ records the coefficient of $x^{p^{n}}$ in the strict isomorphism, and the formulas $η_{R} (v_{n})$ are determined by requiring that the universal $p$ -typical formal group law over the source be carried to that over the target.

The simplest right-unit formula, modulo decomposables and the invariant ideal below, reads $η_{R} (v_{n}) \equiv v_{n} (mod (p, v_{1}, \dots, v_{n - 1}))$ , so each $v_{n}$ is invariant modulo the lower ones.

Counterexamples to common slips

$B P$ is not all of $M U_{(p)}$ : it is one wedge summand. The other summands are suspensions of $B P$ , so no information is lost, but $B P_{*}$ is a proper polynomial subquotient of $M U_{(p) *} = Z_{(p)} [x_{1}, x_{2}, \dots]$ , having generators only in degrees $2 (p^{n} - 1)$ rather than in every even degree.
The $v_{n}$ are not canonical: Hazewinkel and Araki generators differ (they agree modulo $p$ and modulo decomposables but not on the nose). Statements that depend only on the ideals $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ are independent of the choice; statements about exact polynomial expressions are not.
$η_{L} \neq = η_{R}$ on $B P_{*} B P$ : this asymmetry is the whole point of a Hopf algebroid as opposed to a Hopf algebra. Over a Hopf algebra the two units coincide; here $η_{R} (v_{1}) = v_{1} + p t_{1}$ at the simplest level (Araki, $p$ odd, leading terms), so the two units genuinely differ.
$B P_{*} B P$ corepresents strict isomorphisms (leading coefficient $1$ ), not all isomorphisms. Dropping strictness would adjoin a unit and change the groupoid; the strict version is what feeds the Adams-Novikov cobar complex.

Key theorem with proof Intermediate+

Theorem (Quillen; Brown-Peterson). Fix a prime $p$ . There is a $p$ -local ring spectrum $B P$ and a splitting $M U_{(p)} ≃ ⋁_{α} Σ^{2∣ α ∣} B P$ such that $\pi_ BP = \mathbb{Z}{(p)}[v_1, v_2, \ldots] $w i t h$ |v_n| = 2(p^n - 1) $, an d t h e f or ma l g r o u pl a w i t c a r r i es i s t h e u ni v er s a l$ p $- t y p i c a l o n e . M or eo v er$ (BP, BP_BP) $w i t h$ BP_BP = BP_[t_1, t_2, \ldots] $i s a H o p f a l g e b r o i d cor e p r ese n t in g s t r i c t i so m or p hi s m so f$ p$-typical formal group laws, and $$ \mathrm{Ext}^{,}{BPBP}(BP_, BP_) \cong E_2\text{-term of the Adams-Novikov spectral sequence for } \pi_* S^0_{(p)}. $$

Proof. By Quillen's theorem 03.03.04, $π_{*} M U = L$ is the Lazard ring and $M U$ carries the universal formal group law $F^{M U}$ . Localising at $p$ , the ring $M U_{(p) *}$ classifies formal group laws over $Z_{(p)}$ -algebras. Cartier's structure theorem provides, over any such algebra, a natural strict isomorphism $φ_{F}$ from $F$ to a $p$ -typical formal group law $ε (F)$ ; the assignment $F \mapsto ε (F)$ is idempotent. By the universal property of $M U_{(p)}$ as the representing object, this idempotent natural transformation is induced by a ring endomorphism $e_{*}$ of $M U_{(p) *}$ , and by Quillen's identification of multiplicative operations on $M U_{(p)}$ with such transformations it is realised by an idempotent stable self-map $e$ of the ring spectrum $M U_{(p)}$ .

Since $M U_{(p)}$ is a wedge of $p$ -local spheres after a suitable cell structure, idempotents split: $B P := Im (e)$ exists as a $p$ -local summand, and $M U_{(p)} ≃ B P \lor Im (1 - e)$ . Iterating the analysis of $1 - e$ produces the full wedge decomposition into suspended copies of $B P$ . The homotopy ring $π_{*} B P$ is the image of $e_{*}$ , which is exactly the subring classifying $p$ -typical laws; Hazewinkel's functional equation (or Araki's) exhibits this image as $Z_{(p)} [v_{1}, v_{2}, \dots]$ with $∣ v_{n} ∣ = 2 (p^{n} - 1)$ , because a $p$ -typical formal group law over a torsion-free ring is determined by its logarithm coefficients $ℓ_{n}$ in degree $2 (p^{n} - 1)$ , and the $v_{n}$ are the integral generators these coefficients produce.

For the Hopf algebroid, $B P_{*} B P = π_{*} (B P \land B P)$ is computed from the $M U$ -analogue $M U_{*} M U = M U_{*} [b_{1}, b_{2}, \dots]$ by applying the same idempotent to both smash factors; the surviving generators are the $t_{n}$ in degrees $2 (p^{n} - 1)$ , giving $B P_{*} B P = B P_{*} [t_{1}, t_{2}, \dots]$ . The pair $(B P_{*}, B P_{*} B P)$ is the Hopf algebroid of the flat ring spectrum $B P$ in the sense of 03.12.38: the left and right units, counit, coproduct, and conjugation are the spectrum-level structure maps, and under the identification of $B P_{*}$ with $p$ -typical formal group laws they become source, target, identity, composition, and inverse of strict isomorphisms. Finally, by the general theory of the Adams-Novikov spectral sequence 03.12.38, its $E_{2}$ -term is the cohomology of the cobar complex $Ω^{∙} (B P_{*}, B P_{*} B P)$ , which is by definition $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ . $□$

Bridge. This theorem builds toward the entire chromatic program. The foundational reason BP works is exactly that $p$ -typicality is an idempotent condition on formal group laws, so the universal complex-orientable theory $M U$ splits at each prime into identical leaner pieces whose homotopy is a polynomial ring on the height generators $v_{n}$ . This is exactly the algebraic counterpart of the chromatic filtration: the variable $v_{n}$ is the height- $n$ periodicity operator, and the Hopf algebroid $(B P_{*}, B P_{*} B P)$ packages all strict isomorphisms of $p$ -typical formal group laws into a single corepresenting object. The central insight is that computing stable homotopy is, after this reduction, the homological algebra of comodules over $B P_{*} B P$ , and the bridge is the identification of $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ with the Adams-Novikov $E_{2}$ -term. Putting these together, the invariant prime ideals $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ stratify $Spec (B P_{*})$ by height, and this stratification generalises the height invariant of a single formal group law to a filtration of the whole moduli stack. The structure introduced here appears again in 03.13.06 (the chromatic spectral sequence), where the invariant ideals organise an exact couple, and in 03.13.07 (the Greek-letter families), where connecting maps in that couple manufacture periodic families in the stable stems.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the ideal $I_{n} = (p, v_{1}, \dots, v_{n - 1}) \subset B P_{*}$ is invariant, meaning $η_{R} (I_{n}) \subseteq I_{n} \cdot B P_{*} B P$ , using the congruence $η_{R} (v_{k}) \equiv v_{k} (mod I_{k})$ .

Hint

For $k < n$ you have $I_{k} \subseteq I_{n}$ . Apply the congruence to each generator of $I_{n}$ .

Answer

The generators of $I_{n}$ are $p, v_{1}, \dots, v_{n - 1}$ . The element $p$ is central and $η_{R} (p) = p \in I_{n}$ . For each $k$ with $1 \leq k \leq n - 1$ , the right-unit congruence gives $η_{R} (v_{k}) = v_{k} + (terms in I_{k} \cdot B P_{*} B P)$ . Since $k \leq n - 1$ we have $I_{k} = (p, v_{1}, \dots, v_{k - 1}) \subseteq I_{n}$ , and $v_{k} \in I_{n}$ as well. Hence $η_{R} (v_{k}) \in I_{n} \cdot B P_{*} B P$ for every generator, so $η_{R} (I_{n}) \subseteq I_{n} \cdot B P_{*} B P$ . This is precisely the statement that $I_{n}$ is an invariant ideal of the Hopf algebroid. Rubric: full credit for handling $p$ and the inductive containment $I_{k} \subseteq I_{n}$ .

Exercise 4 (medium, symbolic).

The quotient Hopf algebroid $(B P_{*} / I_{n}, B P_{*} B P / I_{n} B P_{*} B P)$ governs height-exactly- $n$ phenomena. Identify the coefficient ring $K (n)_{*}$ of Morava $K$ -theory obtained by inverting $v_{n}$ and killing all other generators, and state its degree-wise structure.

Hint

Set $p = 0$ , $v_{i} = 0$ for $i \neq = n$ , and invert $v_{n}$ .

Answer

$K (n)_{*} = F_{p} [v_{n}^{\pm 1}]$ with $∣ v_{n} ∣ = 2 (p^{n} - 1)$ . Starting from $B P_{*} = Z_{(p)} [v_{1}, v_{2}, \dots]$ , reduce mod $p$ to get $F_{p} [v_{1}, v_{2}, \dots]$ , kill every $v_{i}$ with $i \neq = n$ , and invert $v_{n}$ . The result is the graded field $F_{p} [v_{n}^{\pm 1}]$ , concentrated in degrees that are multiples of $2 (p^{n} - 1)$ , with every nonzero graded piece one-dimensional over $F_{p}$ . This is the coefficient ring of the $n$ -th Morava $K$ -theory $K (n)$ ; for $n = 1$ it is a summand of mod- $p$ $K$ -theory, and $K (0) = H Q$ , $K (\infty) = H F_{p}$ by convention. Rubric: full credit for $F_{p} [v_{n}^{\pm 1}]$ plus the degree.

Exercise 5 (medium, short-answer).

Explain why $(B P_{*}, B P_{*} B P)$ is a Hopf algebroid rather than a Hopf algebra, by contrasting the roles of $η_{L}$ and $η_{R}$ .

Hint

A Hopf algebra has a single unit; a Hopf algebroid distinguishes source and target.

Answer

A Hopf algebra $(A, Γ)$ has $Γ$ an $A$ -bialgebra with one unit map $A \to Γ$ ; it corepresents a group scheme acting on a point. A Hopf algebroid has two unit maps $η_{L}, η_{R} : A \to Γ$ , making $Γ$ an $A$ -bimodule with generally distinct left and right module structures; it corepresents a groupoid scheme with nontrivial object space. For $B P$ , the object space is the affine scheme of $p$ -typical formal group laws (coordinatised by $B P_{*} = Z_{(p)} [v_{n}]$ ) and the morphism space is strict isomorphisms (coordinatised by $B P_{*} B P$ ). The two units encode the source and target formal group laws of an isomorphism: $η_{L} (v_{n})$ is the source law's coefficient, $η_{R} (v_{n})$ the target's. Because a strict isomorphism genuinely changes the law (for example $η_{R} (v_{1}) \neq = η_{L} (v_{1}) = v_{1}$ ), the two units differ, so the structure is a true algebroid. Rubric: full credit for the source-target distinction and one explicit inequality.

Exercise 6 (medium, symbolic).

Compute the leading-term right-unit formula $η_{R} (v_{1})$ in $B P_{*} B P$ for an odd prime $p$ , using Araki generators, given that the universal isomorphism is $g (x) = x +_{F} t_{1} x^{p} + \dots$ and the relation $η_{R} (v_{1}) = v_{1} + p t_{1}$ modulo decomposables.

Hint

The relation is stated; report it and interpret why the $p t_{1}$ correction term appears.

Answer

$η_{R} (v_{1}) = v_{1} + p t_{1}$ (modulo decomposable corrections in higher $t_{i}$ and products). The interpretation: a strict isomorphism shifts the height- $1$ data $v_{1}$ by an amount proportional to the isomorphism's first coefficient $t_{1}$ , scaled by $p$ . Reducing modulo $p$ gives $η_{R} (v_{1}) \equiv v_{1} (mod p)$ , confirming that $v_{1}$ is invariant modulo $I_{1} = (p)$ ; the $p t_{1}$ term is exactly the failure of invariance integrally, and it is what makes $α$ -family elements in $Ext^{1}$ have the orders they do. Rubric: full credit for the formula plus the mod- $p$ invariance reading.

Exercise 7 (hard, short-answer).

Describe the cobar complex computing $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ and identify $Ext^{0}$ and the source of $Ext^{1}$ classes.

Hint

The cobar $n$ -cochains are $B P_{*} \otimes \overset{ˉ}{Γ}^{\otimes n}$ with $\overset{ˉ}{Γ}$ the augmentation coideal of $Γ = B P_{*} B P$ ; $Ext^{0}$ is the primitives.

Answer

Write $Γ = B P_{*} B P$ and $\overset{ˉ}{Γ} = ker (ε)$ the augmentation coideal, a free $B P_{*}$ -module on the $t_{n}$ and their products. The cobar complex has $C^{s} = \overset{ˉ}{Γ}^{\otimes_{B P_{*}} s}$ (with $C^{0} = B P_{*}$ ), and differential the alternating sum of the coproduct applied in each slot together with the two units at the ends. Its cohomology is $Ext_{B P_{*} B P}^{s, *} (B P_{*}, B P_{*})$ . In cohomological degree $0$ , $Ext^{0} = B P_{*}^{Γ}$ is the ring of primitives, namely the elements invariant under all strict isomorphisms; this is $Z_{(p)}$ concentrated in degree $0$ (only constants are invariant). In degree $1$ , $Ext^{1}$ is cocycles modulo coboundaries in $\overset{ˉ}{Γ}$ ; the $α$ -family lives here, arising as $v_{1}$ -divided classes $α_{i} = ⟨$ images of $v_{1}^{i} / p^{?} ⟩$ produced by the connecting map of the short exact sequence $0 \to B P_{*} p B P_{*} \to B P_{*} / p \to 0$ . These detect the image-of- $J$ elements in $π_{*}^{s}$ . Rubric: full credit for the cobar description plus identifying $Ext^{0} = Z_{(p)}$ and the $α$ -family as the $Ext^{1}$ source.

Exercise 8 (hard, short-answer).

Sketch why $π_{*} B P$ being concentrated in even degrees forces the Adams-Novikov $E_{2}$ -term to have a sparseness property, and contrast this with the classical Adams spectral sequence over the Steenrod algebra.

Hint

$B P_{*}$ even and $B P_{*} B P$ even means the cobar complex is evenly graded; compare with $Ext_{A}$ which has a $1$ -line in every degree.

Answer

Both $B P_{*} = Z_{(p)} [v_{n}]$ and $B P_{*} B P = B P_{*} [t_{n}]$ are concentrated in even internal degrees (each generator has even degree $2 (p^{n} - 1)$ ). Hence the cobar complex $\overset{ˉ}{Γ}^{\otimes s}$ is concentrated in even internal degree $t$ for all $s$ , so $Ext_{B P_{*} B P}^{s, t} (B P_{*}, B P_{*}) = 0$ whenever $t$ is odd. The Adams-Novikov $E_{2}$ -term therefore vanishes off the even- $t$ slots, which already forces many differentials to vanish for parity reasons and makes the spectral sequence converge faster in low stems than the classical Adams one. By contrast, the classical Adams $E_{2} = Ext_{A} (F_{p}, F_{p})$ over the Steenrod algebra has classes in a dense range of bidegrees — the $h_{i}$ on the $1$ -line, the Adams periodicity lines, and so on — with no such parity vanishing, because $A$ has generators in every positive degree. The price for Adams-Novikov sparseness is that the algebra $B P_{*} B P$ is larger and the integral (rather than mod- $p$ ) bookkeeping is heavier. Rubric: full credit for the even-degree vanishing argument plus the contrast with the dense Steenrod-algebra $Ext$ .

Advanced results Master

Theorem (Quillen idempotent and the splitting; Quillen 1969). Fix a prime $p$ . There is a multiplicative idempotent operation $e : M U_{(p)} \to M U_{(p)}$ realising the Cartier $p$ -typicalisation of formal group laws, and $B P = Im (e)$ satisfies $M U_{(p)} ≃ ⋁_{α} Σ^{2∣ α ∣} B P$ with $\pi_ BP = \mathbb{Z}_{(p)}[v_1, v_2, \ldots] $,$ |v_n| = 2(p^n - 1)$.*

The operation $e$ is built from the universal property: $M U_{(p) *}$ corepresents formal group laws over $Z_{(p)}$ -algebras, the Cartier idempotent is a natural endomorphism of that functor, and Quillen's calculus of $M U$ -operations realises it on the spectrum. Brown and Peterson's original 1966 construction ^{[Brown-Peterson 1966]} produced $B P$ directly, as the $p$ -local spectrum whose mod- $p$ cohomology is $A / A (Q_{0}, Q_{1}, \dots)$ , the quotient of the Steenrod algebra by the Milnor primitives; Quillen's later viewpoint identifies that spectrum with the summand above.

Theorem (Hazewinkel and Araki generators). The $p$ -typical part of the universal logarithm $lo g (x) = \sum_{i \geq 0} ℓ_{i} x^{p^{i}}$ over $BP_ \otimes \mathbb{Q} $d e t er min es tw o in t e g r a l g e n er a t in g sy s t e m s f or$ BP_* $. T h eH a z e w ink e l g e n er a t or ss a t i s f y$ p,\ell_n = \sum_{0 \le i < n} \ell_i, v_{n-i}^{p^i} $an d t h e A r ak i g e n er a t or ss a t i s f y$ p,\ell_n = \sum_{0 \le i \le n} \ell_i, v_{n-i}^{p^i} $(w i t h$ v_0 = p $in t h e A r ak i co n v e n t i o n) . B o t h g i v e$ BP_* = \mathbb{Z}_{(p)}[v_1, v_2, \ldots] $an d a g r ee m o d u l o$ (p, \text{decomposables})$.*

The two systems are related by a triangular change of variables over $Z_{(p)}$ . Hazewinkel's are slightly more convenient for the functional-equation lemma; Araki's make the formula $[p]_{F} (x) = \sum_{n}^{F} v_{n} x^{p^{n}}$ for the $p$ -series cleaner. Statements phrased in terms of the ideals $I_{n}$ are insensitive to the choice ^{[Hazewinkel 1978]}.

Theorem (invariant prime ideals; Landweber 1973; Morava). The invariant prime ideals of the Hopf algebroid $(BP_, BP_BP)$ are exactly $$ I_0 = (0), \quad I_n = (p, v_1, \ldots, v_{n-1}) \ (1 \le n < \infty), \quad I_\infty = (p, v_1, v_2, \ldots). $$ These form a totally ordered chain $I_{0} \subset I_{1} \subset \dots \subset I_{\infty}$ , and they are the only prime ideals of $BP_ $s t ab l e u n d er$ \eta_R$.*

Invariance means $η_{R} (I_{n}) \subseteq I_{n} \cdot B P_{*} B P$ , equivalently that $I_{n}$ is a sub-comodule. The Landweber filtration theorem ^{[Landweber 1973]} strengthens this: every finitely generated invariant $B P_{*} B P$ -comodule that is a $B P_{*}$ -module admits a finite filtration whose quotients are suspensions of $B P_{*} / I_{n}$ . The chain $I_{0} \subset I_{1} \subset \dots$ is the algebraic shadow of the chromatic filtration: $V (n) = Spec (B P_{*} / I_{n + 1})$ cuts out the height- $\geq n$ locus in the moduli of $p$ -typical formal group laws, and $B P_{*} / I_{n}$ is the height-exactly- $(\geq n)$ comodule whose $v_{n}$ -localisation governs the $n$ -th chromatic layer.

Theorem (Morava $K$ -theory quotients). For each $0 \leq n < \infty$ the graded field $K(n)_ = \mathbb{F}p[v_n^{\pm 1}] $a r i ses f r o m$ BP $b y$ K(n)_ = v_n^{-1} BP_*/(p, v_1, \ldots, v_{n-1}, v_{n+1}, v_{n+2}, \ldots) $. T h e M or a v a$ K $- t h eor i es d e t ec tt h e h e i g h t f i l t r a t i o n : a f ini t e$ p $- l oc a l s p ec t r u m$ X $ha s t y p e$ \ge n $e x a c tl y w h e n$ K(m)_* X = 0 $f or a l l$ m < n$.*

Each $K (n)$ is a field spectrum, and the functor $X \mapsto K (n)_{*} X$ measures the height- $n$ contribution. The variable $v_{n} \in B P_{*}$ is the periodicity generator of this layer; inverting it produces the periodic family bookkeeping carried out via the chromatic spectral sequence 03.13.06 and named in the Greek-letter elements 03.13.07.

Theorem (Adams-Novikov $E_{2}$ -term via the cobar complex). The cobar complex $\Omega^\bullet(BP_, BP_BP)$ of the Hopf algebroid computes $$ E_2^{s,t} = \mathrm{Ext}^{s,t}{BPBP}(BP_, BP_*) \Longrightarrow \pi_{t-s}(S^0_{(p)}), $$ the Adams-Novikov spectral sequence, and it vanishes for $t$ odd.

Because both $B P_{*}$ and $B P_{*} B P$ are even, the cobar complex is even, forcing the $E_{2}$ -page into even total internal degree and eliminating many differentials by parity ^{[Adams 1974]}. The $E_{2}$ -term reorganises by chromatic height through the chromatic spectral sequence, whose $E_{1}$ -page is built from $Ext$ of the localised quotients $M^{n} = v_{n}^{- 1} B P_{*} / I_{n}^{\infty}$ .

Synthesis. The Brown-Peterson spectrum is the foundational reason that stable homotopy at a prime is governed by the geometry of formal group laws. The central insight is that $p$ -typicality is idempotent, so $M U_{(p)}$ splits into copies of a spectrum whose homotopy $B P_{*} = Z_{(p)} [v_{n}]$ is a polynomial ring graded by chromatic height, and the Hopf algebroid $(B P_{*}, B P_{*} B P)$ corepresents the groupoid of $p$ -typical formal group laws and their strict isomorphisms. This is exactly the moduli-stack description: $Spec (B P_{*})$ is the scheme of laws, $Spec (B P_{*} B P)$ the scheme of isomorphisms, and $Ext_{B P_{*} B P} (B P_{*}, B P_{*})$ is the cohomology of the resulting stack, which is the Adams-Novikov $E_{2}$ -term. Putting these together, the invariant ideals $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ stratify the stack by height and the Landweber filtration theorem reduces every finitely generated invariant comodule to suspensions of the height strata $B P_{*} / I_{n}$ , so the bridge from algebra to homotopy is the recognition that each chromatic layer is a $v_{n}$ -periodic phenomenon. This generalises the single height invariant of one formal group law to a global filtration, and it is dual to the topological chromatic tower of Bousfield localisations: the algebraic filtration by $I_{n}$ on $B P_{*}$ is the comodule shadow of the topological filtration by Morava $K$ -theories on the sphere. The same data feeds forward into 03.13.06 and 03.13.07, where the height stratification becomes a computational engine for the stable stems.

Full proof set Master

Proposition (idempotents split in the $p$ -local stable category, giving $B P$ ). Let $e : M U_{(p)} \to M U_{(p)}$ be the Quillen idempotent. Then there is a spectrum $B P$ and maps $i : B P \to M U_{(p)}$ , $r : M U_{(p)} \to B P$ with $r i = id_{B P}$ and $i r = e$ .

Proof. The $p$ -local stable homotopy category is idempotent-complete: it has all countable coproducts and is triangulated, and in such a category every idempotent $e = e^{2}$ on an object $X$ splits via the mapping telescope of $X e X e \dots$ . Concretely, set $B P = hocolim (X e X e \dots)$ with $X = M U_{(p)}$ . The structure map $i : B P \to X$ and the canonical $r : X \to B P$ into the telescope satisfy $r i = id$ because the telescope of an idempotent retracts onto its image, and $i r = e$ because the composite reproduces one application of $e$ followed by the inclusion. Applying $π_{*}$ , which commutes with the homotopy colimit of this sequential diagram, gives $π_{*} B P = Im (e_{*} : π_{*} M U_{(p)} \to π_{*} M U_{(p)})$ . By Quillen's calculation this image is $Z_{(p)} [v_{1}, v_{2}, \dots]$ with $∣ v_{n} ∣ = 2 (p^{n} - 1)$ . The complementary idempotent $1 - e$ splits off $Im (1 - e)$ , and iterating its analysis decomposes $M U_{(p)}$ into the wedge of suspended copies of $B P$ indexed by the monomials in the discarded generators. $□$

Proposition (the $v_{n}$ are invariant modulo $I_{n}$ ). *In $B P_{*} B P$ one has $η_{R} (v_{n}) \equiv v_{n} (mod I_{n})$ , where $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ .*

Proof. Work over $B P_{*} \otimes Q$ , where the logarithm $lo g_{F} (x) = \sum_{i} ℓ_{i} x^{p^{i}}$ and the universal strict isomorphism $g (x) = \sum_{i} t_{i} x^{p^{i}}$ (with $t_{0} = 1$ ) are available. The right unit acts on the logarithm coefficients by $η_{R} (ℓ_{n}) = \sum_{i + j = n} ℓ_{i} t_{j}^{p^{i}}$ , the formula expressing that $g$ carries the source law's logarithm to the target's. Translating to the Araki $v$ -generators via $p ℓ_{n} = \sum_{0 \leq i \leq n} ℓ_{i} v_{n - i}^{p^{i}}$ and reducing modulo $(p, v_{1}, \dots, v_{n - 1})$ : every term of $η_{R} (v_{n}) - v_{n}$ involves either a factor of $p$ , or a $t_{j}$ with $j \geq 1$ multiplied by some $v_{k}$ with $k < n$ (because the lower logarithm coefficients $ℓ_{1}, \dots, ℓ_{n - 1}$ become divisible by the lower $v_{k}$ after clearing denominators), or a decomposable in the lower $v_{k}$ . Each such term lies in $I_{n} \cdot B P_{*} B P$ . Hence $η_{R} (v_{n}) - v_{n} \in I_{n} \cdot B P_{*} B P$ , which is the claimed congruence. $□$

Proposition (the chain $I_{n}$ is invariant and prime). Each $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ is a prime ideal of $BP_ $s t ab l e u n d er$ \eta_R$.*

Proof. Primality: $B P_{*} / I_{n} = F_{p} [v_{n}, v_{n + 1}, \dots]$ is a polynomial ring over a field, hence an integral domain, so $I_{n}$ is prime. Invariance: the generators are $p$ and $v_{1}, \dots, v_{n - 1}$ . For $p$ , $η_{R} (p) = p \in I_{n}$ . For $v_{k}$ with $k \leq n - 1$ , the previous proposition gives $η_{R} (v_{k}) \equiv v_{k} (mod I_{k})$ , and since $k \leq n - 1$ we have both $I_{k} \subseteq I_{n}$ and $v_{k} \in I_{n}$ , so $η_{R} (v_{k}) \in I_{n} \cdot B P_{*} B P$ . Therefore $η_{R} (I_{n}) \subseteq I_{n} \cdot B P_{*} B P$ , i.e. $I_{n}$ is a sub-comodule. That these are the only invariant primes is Landweber's and Morava's theorem: any invariant prime $P$ is generated by an initial segment of the $v_{n}$ together with $p$ , because invariance forces $P$ to be homogeneous and closed under the comodule structure, and the only such chains are the $I_{n}$ . $□$

Proposition (cobar $Ext^{0}$ is $Z_{(p)}$ ). $\mathrm{Ext}^0_{BP_BP}(BP_, BP_) = BP_*^{,\Gamma} $, t h e p r imi t i v es, e q u a l s$ \mathbb{Z}_{(p)} $co n ce n t r a t e d in d e g r ee$ 0$.*

Proof. By definition $Ext^{0}$ is the kernel of $d^{0} = η_{R} - η_{L} : B P_{*} \to B P_{*} B P$ , the equaliser of the two units, i.e. the elements $x \in B P_{*}$ with $η_{R} (x) = η_{L} (x)$ . An element $x$ is primitive exactly when it is invariant under every strict isomorphism. In positive degree the right unit moves every generator: $η_{R} (v_{n}) = v_{n} + (terms involving the t_{i})$ , and these correction terms are nonzero in $B P_{*} B P$ for $n \geq 1$ , so no positive-degree polynomial in the $v_{n}$ is fixed (the leading $t$ -dependent term cannot cancel). In degree $0$ , $B P_{*}$ is $Z_{(p)}$ , on which both units are the identity inclusion of scalars, so every scalar is primitive. Hence $Ext^{0} = Z_{(p)}$ in degree $0$ and vanishes otherwise. This matches $π_{0} (S_{(p)}^{0}) = Z_{(p)}$ , the $s = 0$ edge of the Adams-Novikov spectral sequence. $□$

*Proposition ( $B P_{*} B P$ is even, forcing $E_{2}$ -parity vanishing).* $\mathrm{Ext}^{s,t}{BPBP}(BP_, BP_) = 0 $f or a l l$ s $w h e n e v er$ t$ is odd.*

Proof. The cobar $s$ -cochains are $\overset{ˉ}{Γ}^{\otimes_{B P_{*}} s}$ with $\overset{ˉ}{Γ} = ker (ε : B P_{*} B P \to B P_{*})$ . As a $B P_{*}$ -module, $B P_{*} B P = B P_{*} [t_{1}, t_{2}, \dots]$ is free on monomials in the $t_{n}$ , each of even degree $2 (p^{n} - 1)$ ; the augmentation coideal $\overset{ˉ}{Γ}$ is spanned by the positive-degree such monomials, all of even degree. A tensor power over the even ring $B P_{*}$ of even-degree free modules is again even. Hence each cochain group $C^{s}$ is concentrated in even internal degree $t$ , the cobar differential preserves internal degree, and the cohomology in odd internal degree is computed from zero groups. Therefore $Ext^{s, t} = 0$ for odd $t$ . $□$

Connections Master

Formal group laws and $p$ -typicality 03.03.04. The entire construction of $B P$ rests on the formal-group-law dictionary of 03.03.04: the Lazard ring $L = π_{*} M U$ , Quillen's universal formal group law, and Cartier's theorem that over a $Z_{(p)}$ -algebra every law is canonically strictly isomorphic to a $p$ -typical one. The Cartier idempotent there becomes the Quillen idempotent here, and the height invariant of a single formal group law there becomes the chain of invariant ideals $I_{n}$ here. Without the $p$ -typical normal form, $B P_{*}$ would not be a polynomial ring on the $v_{n}$ .
Spectrum and $M U$ 03.12.04. $B P$ is a wedge summand of the $p$ -localisation of the complex cobordism spectrum $M U$ of 03.12.04. That unit supplies the ring spectrum $M U$ , its Thom-spectrum construction, and the identification of its coefficients with the Lazard ring; $B P$ inherits the ring-spectrum structure through the splitting. The Morava $K$ -theories $K (n)$ named in 03.12.04 are realised here as the quotient-and-localisation $v_{n}^{- 1} B P_{*} / (p, v_{i} : i \neq = n)$ .
Adams-Novikov spectral sequence 03.12.38. The Hopf algebroid $(B P_{*}, B P_{*} B P)$ is the $E_{2}$ -input object for the Adams-Novikov spectral sequence constructed in 03.12.38 as the Bousfield-Kan spectral sequence of the cobar resolution at the ring spectrum $R = B P$ . That unit builds the cobar complex and proves convergence to $π_{*} (S_{(p)}^{0})$ ; the present unit supplies the explicit algebra $B P_{*} B P = B P_{*} [t_{1}, t_{2}, \dots]$ whose $Ext$ that spectral sequence computes, together with the units, coproduct, and parity vanishing.
Chromatic spectral sequence 03.13.06. The invariant ideals $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ introduced here are the input to the chromatic spectral sequence of 03.13.06: iterating localisation $v_{n}^{- 1}$ and quotient by $I_{n}$ produces the chromatic resolution $B P_{*} \to M^{0} \to M^{1} \to \dots$ , an exact couple whose derived spectral sequence converges to the Adams-Novikov $E_{2}$ -term. The height stratification of $Spec (B P_{*})$ established here is precisely what that construction filters along.
Greek-letter elements 03.13.07. The periodic $α$ -, $β$ -, $γ$ -families of 03.13.07 are manufactured from the structure assembled here. The $α$ -family arises from the connecting map of $0 \to B P_{*} p B P_{*} \to B P_{*} / p \to 0$ together with $v_{1}$ -divisibility (height $1$ ); the $β$ -family from $v_{2}$ -periodicity over $B P_{*} / I_{2}$ (height $2$ ); and so on. The right-unit formulas and the invariance of the $v_{n}$ modulo $I_{n}$ proved here are exactly the divisibility bookkeeping those families require.

Historical & philosophical context Master

Brown and Peterson constructed their spectrum in 1966 in A spectrum whose $Z_{p}$ cohomology is the algebra of reduced $p$ -th powers (Topology 5) ^{[Brown-Peterson 1966]}, characterising it by its mod- $p$ cohomology as a module over the Steenrod algebra: the quotient $A / A (Q_{0}, Q_{1}, \dots)$ by the ideal generated by the Milnor primitives. Their motivation was to find the smallest $p$ -local spectrum carrying the essential complex-cobordism information, isolating a single copy of the relevant Steenrod-algebra quotient. The connection to formal group laws was not yet visible.

Quillen's 1969 note On the formal group laws of unoriented and complex cobordism theory (Bull. AMS 75) ^{[Quillen 1969]} transformed the subject. By identifying $π_{*} M U$ with the Lazard ring and exhibiting the universal formal group law on $M U$ , Quillen recast the Brown-Peterson splitting as the idempotent extracting the $p$ -typical part of that universal law, in the sense of Cartier's earlier work on formal groups. This is the moment complex cobordism became a geometric object: the spectrum of a moduli problem for formal group laws. Hazewinkel's 1978 treatise Formal Groups and Applications (Academic Press) ^{[Hazewinkel 1978]} supplied the explicit functional-equation generators $v_{n}$ that made $B P_{*}$ a usable polynomial ring.

Landweber's 1973 classification of invariant prime ideals (Illinois J. Math. 17) ^{[Landweber 1973]}, proved independently by Morava, identified the chain $I_{n} = (p, v_{1}, \dots, v_{n - 1})$ as the only invariant primes and established the filtration theorem reducing invariant comodules to the height strata $B P_{*} / I_{n}$ . Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres ^{[Ravenel 2004]} organised this material into Chapter 4 as the algebraic backbone of the chromatic program, where the Hopf algebroid $(B P_{*}, B P_{*} B P)$ and its invariant ideals became the standard computational platform for the Adams-Novikov spectral sequence.

Bibliography Master

@article{BrownPeterson1966,
  author  = {Brown, Edgar H. and Peterson, Franklin P.},
  title   = {A spectrum whose {$\mathbb{Z}_p$} cohomology is the algebra of reduced {$p$}-th powers},
  journal = {Topology},
  volume  = {5},
  year    = {1966},
  pages   = {149--154}
}

@article{Quillen1969,
  author  = {Quillen, Daniel},
  title   = {On the formal group laws of unoriented and complex cobordism theory},
  journal = {Bull. Amer. Math. Soc.},
  volume  = {75},
  year    = {1969},
  pages   = {1293--1298}
}

@book{Ravenel2004,
  author    = {Ravenel, Douglas C.},
  title     = {Complex Cobordism and Stable Homotopy Groups of Spheres},
  edition   = {2nd},
  series    = {AMS Chelsea Publishing},
  volume    = {347},
  publisher = {American Mathematical Society},
  year      = {2004}
}

@book{Hazewinkel1978,
  author    = {Hazewinkel, Michiel},
  title     = {Formal Groups and Applications},
  series    = {Pure and Applied Mathematics},
  volume    = {78},
  publisher = {Academic Press},
  year      = {1978}
}

@article{Landweber1973,
  author  = {Landweber, Peter S.},
  title   = {Annihilator ideals and primitive elements in complex bordism},
  journal = {Illinois J. Math.},
  volume  = {17},
  year    = {1973},
  pages   = {273--284}
}

@book{Wilson1982,
  author    = {Wilson, W. Stephen},
  title     = {Brown-Peterson Homology: An Introduction and Sampler},
  series    = {CBMS Regional Conference Series in Mathematics},
  volume    = {48},
  publisher = {American Mathematical Society},
  year      = {1982}
}

@book{Adams1974,
  author    = {Adams, J. Frank},
  title     = {Stable Homotopy and Generalised Homology},
  series    = {Chicago Lectures in Mathematics},
  publisher = {University of Chicago Press},
  year      = {1974}
}

Prerequisites

03.03.04
03.12.04
03.12.38

Tier anchors

beginner: Ravenel 2004 *Complex Cobordism and Stable Homotopy Groups of Spheres* (2nd ed., AMS Chelsea) Ch.1 informal overview; Hopkins's Berkeley lecture notes on complex-oriented cohomology, opening sections
intermediate: Ravenel 2004 *Complex Cobordism and Stable Homotopy Groups of Spheres* Ch.4 §4.1-4.3 (AMS Chelsea); Adams 1974 *Stable Homotopy and Generalised Homology* (Univ. Chicago Press) Pt. II; Quillen 1969 *On the formal group laws of unoriented and complex cobordism theory* (Bull. AMS 75)
master: Ravenel 2004 *Complex Cobordism and Stable Homotopy Groups of Spheres* Ch.4 and App. A2 (AMS Chelsea); Brown-Peterson 1966 *A spectrum whose Z_p cohomology is the algebra of reduced p-th powers* (Topology 5); Hazewinkel 1978 *Formal Groups and Applications* (Academic Press) Ch.III; Wilson 1982 *Brown-Peterson homology: an introduction and sampler* (CBMS 48)

References

Brown-Peterson — A spectrum whose Z_p cohomology is the algebra of reduced p-th powers · Topology 5 (1966), 149-154 — original construction of the BP spectrum as a p-local wedge summand realising the quotient of the Steenrod algebra by the Milnor primitives
Quillen — On the formal group laws of unoriented and complex cobordism theory · Bull. Amer. Math. Soc. 75 (1969), 1293-1298 — idempotent splitting of MU_(p), identification of pi_* MU with the Lazard ring, and the p-typical universal formal group law on BP
Ravenel — Complex Cobordism and Stable Homotopy Groups of Spheres · 2nd ed., AMS Chelsea 347 (2004); Ch.4 the structure of BP_*BP, the Hopf algebroid, Hazewinkel and Araki generators, invariant prime ideals; App. A2 Hopf-algebroid formalism
Hazewinkel — Formal Groups and Applications · Academic Press (1978); Ch.III the universal p-typical formal group law and the Hazewinkel generators v_n via the functional equation
Wilson — Brown-Peterson homology: an introduction and sampler · CBMS Regional Conf. Ser. Math. 48, Amer. Math. Soc. (1982) — survey of BP_*, BP_*BP, the Landweber filtration theorem, and the invariant ideals I_n
Landweber — Annihilator ideals and primitive elements in complex bordism · Illinois J. Math. 17 (1973), 273-284 — classification of invariant prime ideals of BP_*BP as the I_n = (p, v_1, ..., v_{n-1})
Adams — Stable Homotopy and Generalised Homology · Chicago Lectures in Mathematics (1974); Pt. II the Hopf-algebroid structure (A_*, A_*A) for a ring spectrum and the cobar Ext computing the Adams-Novikov E_2-term

Estimated time

beginner: 20m
intermediate: 55m
master: 95m