A finite Smith–Toda complex V(4) at the prime 1009

Proof. Write the logarithm as \log_F(x)=\sum_{i\geq0}l_i x^{p^i}, where l_0=1. The Hazewinkel recursion is

Applying the logarithm to the formal sum for [p]_F and comparing the coefficient of x^{p^n} shows that the coefficient of the indecomposable v_n in a_n is

Indeed the two terms involving l_n are pl_n and p^{p^n}l_n, and the coefficient of v_n in l_n is 1/p. All other terms involve lower v_i. The coefficients a_n are integral, so the grading and polynomial independence of the v_i give a_n-c_nv_n\in(v_1,\ldots,v_{n-1})R. Each c_n is a p-local unit.

The invariance and primitivity follow inductively. The element p is invariant. Once the lower coefficients vanish, the first surviving term of the p-series is c_m v_m x^{p^m}. A strict coordinate change has leading term x, so comparison of this first surviving term on the two sides of the coordinate-change identity fixes c_m v_m, and hence v_m. This proves both invariance of the next ideal and the asserted primitivity.

After all base generators have been killed, the formal group is additive, and composition of additive p-polynomials gives Equation (2.2). Modulo I_m, a correction to this coproduct must have a positive base coefficient, of weight at least u_m. For i<m its weight is already too large. For i=m the only possibility is a constant multiple of v_m with no positive-weight cooperation factor. Applying either counit excludes that term. This proves the stated range.□

Proof. Modulo I_4, the first two formal-sum coefficients are unit multiples of v_4,v_5. There is no decomposable correction involving v_4 to the second of these coefficients: u_5=pu_4+1 is not a multiple of u_4. The formal group law has no nonlinear term of total ordinary degree less than p^4, because a coefficient of such a term would have positive weight less than u_4. Thus, through exponent p^5, its p-series is the ordinary sum of its x^{p^4} and x^{p^5} terms. The nonlinear terms in adding these two inputs have exponent greater than p^5.

In the identity intertwining this p-series with a strict coordinate change, only x+t_1x^p from that change can affect these two exponents. On the side where the change is applied after the p-series, the coefficient of x^{p^5} acquires a term proportional to v_4^p t_1. On the other side it acquires a term proportional to v_4t_1^{p^4}. All nonlinear corrections to the strict formal-sum change enter too late after taking the p^4-th power. Consequently

where c,d\in\mathbb F_p and c\ne0. The nonzero coefficient follows also directly from the unit factors c_4,c_5 in Equation (2.3); their reductions cannot annihilate the first displayed contribution. Divide this coboundary by c and put e=d/c. Since t_1 is primitive modulo I_4, all its p-power iterates are primitive, and the resulting cohomology relation is Equation (2.4).□

Proof. Every reduced cooperation factor is a polynomial monomial containing at least one t_i, and hence has weight at least one. The coefficient has nonnegative weight. Adding these weights proves the claim.□

Proof. In each of the three degrees, any positive coefficient monomial consumes at least u_m weight. It would leave at most w-u_m<s-1 for at least s-1 reduced bar factors, which is impossible. A coordinate t_i with i\geq m+1 exceeds the entire weight budget. Thus these cochain groups have scalar coefficients and only t_1,\ldots,t_m in their bars. Their coproducts are additive by Lemma 2.1. This identifies the two groups adjacent to the desired cohomology group as well as the group itself, so both kernels and images agree. If w<u_m, the same exclusion of coefficients and coordinates works without a cochain-length restriction.□

Proof. The coefficient and cooperation gradings give the assertion on the cobar page. A differential preserves the sparse support only if q divides r-1. Subsequent pages are subquotients of earlier pages. Since r\geq2 on or after the cohomology page, the first allowed length is q+1.□

Proof. The first two inequalities are the rearrangement of w\geq s. Here

so the remaining inequalities in Equation (2.7) hold at the fixed prime. For a cochain length d\in\{s-1,s,s+1\}, a coefficient containing v_4 would leave at most k weight for d\geq qk>k reduced factors. It therefore cannot occur. A bar containing t_4 would leave at most k weight for the other d-1\geq qk-1>k bars, and likewise cannot occur. Higher generators exceed the weight bound. The remaining coproducts are additive by Lemma 2.1; equivalently, apply Lemma 2.4. This identifies both adjacent differentials and proves the cohomology assertion.□

Proof. Write every exponent in base p and assign a digit factor t_i^{p^j} the filtration 2i-1+M[j=0]. For a monomial, add these filtrations with the digit multiplicities, without carrying. Carrying p copies of one digit to the next digit lowers filtration: its decrease is

A nonprimitive coproduct summand of t_i^{p^j} replaces it by t_l^{p^j}\otimes t_{i-l}^{p^{j+l}} and decreases filtration by exactly one. These observations show that this is a multiplicative coalgebra filtration, and that its associated graded Hopf algebra is the tensor product of primitive truncated polynomial Hopf algebras

For one such factor the dual algebra is again a truncated polynomial algebra: factorial rescaling identifies its basis in exponents less than p with the ordinary polynomial basis. The free resolution alternating multiplication by its generator and its (p-1)st power computes its cohomology as one exterior generator in degree one and one polynomial generator in degree two. Their weights are respectively the weight of z_{i,j} and p times that weight. Tensoring these resolutions gives (3.1), including the stated auxiliary filtrations. The exterior square is zero since its cohomological degree is odd and p is odd.

The filtration-one part of the reduced coproduct gives the splitting formula for h. For the polynomial generator associated to a digit z, a filtration-one contribution would replace one of its p copies by a split pair and leave p-1 unsplit copies of z. In the associated-graded cohomology of the z factor, a nonzero multiplicity has the form pa+\epsilon, with a\geq0 and \epsilon\in\{0,1\}. The multiplicity p-1 has neither form, so that component has zero cohomology. Carries cannot contribute to this first differential by the strict inequality above. This proves d_1b=0.

In a fixed positive internal weight only finitely many digit factors occur, every monomial exponent is bounded, and a normalized bar has positive weight. Its length is consequently bounded as well. Thus all the filtrations under discussion are finite in each weight, and their spectral sequences converge to the indicated cobar cohomology. The construction restricts to P_r because its coproduct uses only the first r coordinates.□

Proof. Include the first case in the notation of the second by setting U=u_n and \rho=0, while retaining its indicated coefficient comodule. If the normalized cochain group in degree (s,w) is nonzero, Lemma 2.3 gives

The exact values

give Table 1. Each upper bound is obtained by substituting the displayed maxima in Equation (4.1) and taking the integer part. A zero upper bound on k excludes all k\geq1.

These bounds also justify the additive comparison in the adjacent cochain degrees. For either remaining product case, w<u_{n+1}, so no positive coefficient or coordinate t_i with i\geq n+1 can occur. The retained coproducts are additive modulo I_{n+1}, and the last assertion of Lemma 2.4 applies in every degree. For a remaining v_n target, a positive coefficient costs at least u_n, leaving at most k weight for d\geq s-1=qk>k reduced bars when d\in\{s-1,s,s+1\}. Such a coefficient cannot occur. A bar containing t_n or a higher coordinate leaves at most k for the other d-1\geq qk-1>k bars and cannot occur either. Hence the same three cochain degrees are additive. In all cases, both adjacent differentials are identified, so the desired cohomology is identified with that for P.

Every remaining weight in Table 1 is at most 2{,}041{,}216, whereas

A May letter covering position 3 or above has weight at least p^3. Thus only positions 0,1,2 can occur. We use the common exterior and polynomial alphabet of Equation (3.3), with the interval coverage conventions of Section 3. Allowing this entire alphabet only enlarges the upper bound if fewer coordinates were retained in the preceding comparison.

Suppose a monomial of the desired bidegree exists. Let m\leq6 be its number of exterior letters and L=(s-m)/2 its total polynomial exponent. Every polynomial letter has weight at least p, so

For all profiles under consideration, U\leq2p^2+4p+6 and \rho\leq8. Consequently

If k\geq3, the left side exceeds the last expression by at least

It follows that 1\leq k\leq2.

Write U=F_0+pF_1+p^2F_2. In a product profile the digits are

with missing e_i set to zero. For a v_n target they are initial strings of ones. In every case,

Let H_r count the exterior letters covering position r, and let n_B,n_C be the polynomial exponents of B,C. Adding weights gives the exact identity

We next extract its digits without an unproved no-carry assumption.

First, 0\leq H_0\leq3 and 1\leq F_0+k\leq8<p. Reduction modulo p therefore gives H_0=F_0+k. Also B,C each cost at least p^2, so

the last strict inequality has difference p^2-4p-8=1{,}014{,}037>0. Hence n_B+n_C\leq2.

Put \delta=(m+\rho-1)/2. It is an integer because s-m is even. For a target with \rho=0, that parity forces m odd, hence m\geq1; for a product profile \rho\geq2. Thus \delta\geq0. The bounds m\leq6, \rho\leq8 and the same parity give \delta\leq6. Now L=(p-1)k-\delta. Subtracting the position-zero equation from Equation (4.4), dividing by p, and rearranging gives

Since H_1\leq4, the left side lies in [-14,3]. Its only possible multiple of p=1009 is zero. We have therefore proved the three exact relations

Among the exterior letters covering position zero, only x,a fail to cover position two. Thus H_2\geq H_0-2. Equations (4.3) and (4.5) imply

As k\geq1, equality is forced: k=1 and F_0=F_1=F_2=f. All the intervening inequalities are equalities, giving

The bounds on H_0,H_2 leave only f=1,2.

If f=1, the exterior letters x,a must occur and c,b,z must be absent. Only y is optional, so H_1\leq2 and m\geq2. On the other hand, the last relation in Equation (4.5) gives H_1=2+\delta>2, since \delta=(m+\rho-1)/2>0. If f=2, the letters x,a,c must occur and b,z must be absent. Again only y is optional, so H_1\leq3 and m\geq3, whereas the same relation gives H_1=3+\delta>3. Both possibilities are impossible. There is no monomial on the May upper-bound page, and hence no cobar cohomology in any of the stated bidegrees.□

Proof. Proceed one attachment at a time. If the previously constructed spectrum is the localization of a finite spectrum Z, an attaching map belongs to \pi_d(Z_{(p)})=\pi_d(Z)\otimes\mathbb Z_{(p)} and can be written a/m with p\nmid m. After localization, multiplication by m on its sphere source is an equivalence. The localized cone of the integral attaching map a is therefore equivalent to the cone of a/m. Induction over the finitely many cells proves the assertion.□

Proof. Start with V(0), the cofiber of p:S_{(p)}\to S_{(p)}. Since multiplication by p is injective on R, its BP-homology is R/(p), with the canonical quotient coaction and bottom generator. Its cells are in dimensions zero and one, as in Equation (4.6).

We give first the product-extension argument, which applies to this initial complex and to every later complex once its asserted homology and cells have been obtained. The subcomplex

has a map to V(n) given by the two identity maps, which agree on the common bottom sphere. Every remaining cell corresponds to a pair of nonempty subsets of \{0,\ldots,n\}. If e_i counts occurrences of i in that pair, its dimension is qU+\rho, with precisely the parameters in the second part of Lemma 4.1. The obstruction to extending across the cell lies in \pi_{qU+\rho-1}V(n).

An Adams–Novikov term in that stem has qw-s=qU+\rho-1, or

Since 2\leq\rho\leq8<q, nonnegative filtration requires k\geq1. Every corresponding E_2 group is zero by Lemma 4.1. Strong convergence and finite filtration, from Proposition 2.5, give \pi_{qU+\rho-1}V(n)=0. Extending cell by cell therefore produces \mu_n, agreeing with both identity maps on U_n. This constructs the initial product on V(0) as well.

Suppose now that V(n-1), including its product, has been constructed, where 1\leq n\leq3. By Lemma 2.1, v_n is a primitive element of BP_*V(n-1)=R/I_n. It defines a filtration-zero E_2 class. Its possible positive differential targets are exactly Equation (2.6); the first part of Lemma 4.1 makes them all zero. There is no incoming differential to filtration zero. The edge assertion in Proposition 2.5 consequently realizes v_n by a homotopy map

Use the product to form

and define V(n) as its cofiber.

The induced map on BP-homology is multiplication by v_n: the coefficient pairing is R-bilinear, the homology is cyclic on the unit, and \widetilde v_n has the specified Hurewicz image. This multiplication is injective on the polynomial quotient R/I_n. The cofiber exact sequence therefore gives precisely R/I_{n+1}, with no additional shifted kernel summand. The bottom sphere map surjects onto this cyclic homology, so its kernel I_{n+1} identifies the coaction with the canonical quotient coaction, as well as fixing the generator in degree zero.

The mapping cone adds the old cells shifted by qu_n+1, giving exactly Equation (4.6). The product-extension argument already proved then constructs \mu_n. This completes the induction. All attachments are finite p-local cell attachments, and Lemma 4.2 gives finiteness in the sense of the stated realization problem. No associativity of any \mu_n has been asserted or used.□

Proof. The bottom two cells form the mod-p Moore spectrum. All other cells have dimension at least qu_1+1=q+1>1, so they do not change \pi_0. Thus p\eta_3=0 in \pi_0Y. Applying the left unit identity of its binary multiplication gives

□

Proof. If w<s, the degree-(s,w) cochain groups for both Y and P are zero, as is the May page, and all the conclusions are immediate. We may therefore suppose w\geq s. The normalized bound in Lemma 2.3 then gives (q-1)k\leq u_4-1. Consequently

where u_4<2p^3 and q/(q-1)<2. Thus no letter can reach position four. Also w<u_5. A coefficient containing v_4 leaves at most k weight for normalized bars, whereas the three lengths s-1,s,s+1 are all greater than k, because s-1=qk>k. Higher coefficient generators are excluded by w<u_5. The adjacent-chain comparison of Lemma 2.8 therefore identifies the required cohomology with that of P.

There are ten possible exterior intervals on positions 0,1,2,3. Since s is odd, m is odd and hence m\leq9. Every polynomial letter has weight at least p, so

But

It follows that k\leq p+2.

For k\geq2 we have L\geq2p-6. If any letter reaches position three, that letter has weight at least p^3, while all other polynomial letters have total weight at least (L-1)p. This is a valid lower bound whether the distinguished letter is exterior or polynomial. Thus

since the strict difference is p^2-9p-3>0 at p=1009. This contradiction excludes position three in that range.□

Proof. We prove exhaustion simultaneously for w=u_n+1, with n=4,5. Since u_n+1<p^n, the available exterior intervals are [a,b) with 0\leq a<b\leq n, and the polynomial intervals cover positions 1,\ldots,n-1. The target weight has base-p digits (2,1,\ldots,1). At position zero, H_0\leq n<p, so H_0=2 and there is no carry into position one. As s=2p-1 is odd, write

There are exactly two exterior intervals starting at zero; at most n(n-1)/2 others are available. Hence m\leq11, 2\leq D\leq6, and

The lower bound on D follows from m being odd and at least two.

Let c_r be the carry into position r, with c_1=c_n=0, and let B_r count polynomial intervals covering position r. The largest possible exterior coverage at a position is nine. Inductively, the quantity H_r+B_r+c_r is at most 9+(p-2)+1=p+8<2p; therefore every carry is zero or one. The digit equations are

If there were no carry, then B_r\leq1 for every r, and L\leq\sum B_r\leq4, contrary to L\geq p-6=1003.

Call a position high when c_{r+1}=1. A high position has B_r\geq p-9, and a nonhigh position has B_r\leq1. There cannot be two blocks of high positions. Indeed, choose high positions a<b in different blocks and a nonhigh position c between them. An interval covering both a and b also covers c, whence

Here 2p-19=1999 and p-2=1007. This contradicts (5.5). Thus the high positions form a single block [l,r] with 1\leq l\leq r\leq n-2. At the next position (5.6) reads B_{r+1}=-H_{r+1}, so

At the beginning of the block, B_l=p+1-H_l\leq L, giving H_l\geq D+1. If l<r, its end gives B_r=p-H_r\leq L, hence H_r\geq D.

A singleton block is impossible. Every exterior interval covering its position l must end at l+1 by (5.7). When l=1 there are only two such intervals. When l>1, at most one begins before l, since H_{l-1}\leq1, and at most one begins at l. Thus in either case H_l\leq2, contradicting H_l\geq D+1\geq3. We henceforth have l<r.

If l>1, at most two exterior intervals cover both l and r: they must end at r+1, and at most one starts before l, while at most one starts at l. At least one of the two intervals starting at zero misses both positions, since H_{l-1}\leq1. Consequently

This contradicts H_l+H_r\geq2D+1=m+2. Therefore l=1. There are now at most two exterior intervals covering both endpoints, namely [0,r+1) and [1,r+1). Equality in the same counting identity is forced: both are present, every exterior interval meets position 1 or position r, and

It follows that B_1=B_r=L. Every polynomial interval therefore covers both positions, and by (5.7) every one is exactly [1,r+1). No exterior or polynomial interval can occur after the block. The carry supplies the target digit at r+1; all further digits would be zero. Since the required remaining digits are all one, necessarily r=n-2. Thus

It remains to list a small, now forced set of exterior intervals. For n=4, the intervals [0,3) and [1,3) are mandatory. The other start-zero interval must be [0,2); [0,1) would meet neither endpoint. The only optional intervals are [1,2) and [2,3). The coverage requirements in (5.8) require both or neither. This is exactly (5.4).

For n=5, the mandatory intervals are [0,4) and [1,4). The remaining start-zero interval is [0,t) with t=2 or 3. The only optional intervals are [1,2),[1,3),[2,4),[3,4); [2,3) meets neither endpoint and is forbidden. Write their inclusion indicators as e_{12},e_{13},e_{24},e_{34}\in\{0,1\}, and let their sum be o. The three required coverage equations are

where o\in\{0,2,4\}. For t=2 the solutions are no optional intervals, the pair \{12,24\}, the pair \{13,34\}, and all four. For t=3 the only solution is the pair \{12,34\}. This proves the exhaustive Table 2, including its length-seven entry.

For completeness, the splitting calculation is also explicit. The length-three entry has, for example, a nonzero component on 01,04,12,14. For the three length-five entries in their displayed order, take the following four rows, ordering exterior intervals lexicographically:

The common polynomial factor is b_{3,0}^{p-3}. The minor using the first, second, and fourth rows is 3, which is nonzero in \mathbb F_{1009}. Thus these three images are linearly independent. Finally, (5.3) shows that acb has a nonzero splitting differential and that acybz is closed: every term in the latter differential repeats an exterior factor. This finishes the lemma.□

Proof. Use the filtration in Lemma 3.1 with the explicit choice M=3p. If the source cohomology is nonzero, any nonzero class has leading detector a nonzero scalar multiple of acybz B^{p-3}, by Lemma 5.2. Its filtration is

The cocycle [t_1^{p^4}] represents h_{1,4}, with degree, weight, and filtration shifts (1,p^4,1). The proposed leading product therefore has filtration

Every possible incoming May differential to this product has source cohomological degree q+1 and weight u_5+1. Table 2 is its complete initial-page source list. The length-three and length-five source filtrations are respectively

all three length-five sources have the latter filtration. The length-three and length-five entries disappear under d_1, since their displayed splitting maps are injective. None can be a splitting boundary for the product itself, because its polynomial factor is a power of B=b_{2,0}, whereas their polynomial factor is a power of b_{3,0}. The length-seven source has filtration

For M=3p the difference is

As all May differentials lower filtration, this last source cannot hit the proposed product on any later page. There are no other incoming sources.

This argument does not require B to represent a permanent class by itself. Choose an actual cocycle for the given nonzero source cohomology class and multiply that entire cocycle by [t_1^{p^4}]. The result is an actual cocycle. Its leading associated-graded cup product is the nonzero monomial acybz B^{p-3}h_{1,4}, since the new exterior letter is distinct from all the old ones. An actual cocycle has no outgoing May differential; its leading detector could disappear only by becoming a boundary, which the preceding incoming-source calculation excludes at every page. Its product cohomology class is therefore nonzero.□

Proof. Equation (5.2) and Proposition 5.3 give \sigma(d_{q+1}(v_4))=0. Lemma 5.1 identifies this specialization with the original target group. Hence d_{q+1}(v_4)=0.□

Proof. Lemma 5.1 allows only positions 0,1,2 and 2\leq k\leq p+2. Thus (3.4) applies. There are six exterior letters; m is odd, so write

Reducing the weight modulo p gives H_0\equiv1+k\pmod p, with 0\leq H_0\leq3. The only possibilities are k=2 or k=p+d with -1\leq d\leq2.

When k=2, H_0=3 forces all of x,a,c. The other exterior letters are an even subset of \{y,b,z\}. Put T=H_2+n_B+n_C. Substitution in (3.4) gives

For the four possible exterior subsets the value e=H_1-3-f is -2 or -1. Therefore n_C\equiv e\pmod p. Since 0\leq n_C\leq L\leq2p-3, the only possible lift is n_C=p+e: the lift e is negative, and 2p+e\geq2p-2>L. Equation (5.11) now gives T=p. All the remaining counts follow as displayed:

The negative counts exclude the second and fourth rows. The first and third are exactly the k=2 row of Table 3.

Now let k=p+d, -1\leq d\leq2. Then H_0=1+d, and the weight equation becomes

Since n_C\leq T and H_1\geq0, it follows that

Here 4+f\leq6<p-1, so T\leq2-d\leq3 and n_C\leq3. Rearranging (5.12) now gives

The right side lies between -3 and 9, strictly between -p and p. Both sides therefore vanish, proving the exact equations

For d\geq1, these require 3+f\leq H_1\leq\min(2f+1,4), impossible for each of f=0,1,2. For d=0, exactly one of x,a,c occurs, so H_1\leq3. Equation (5.13) forces f=1, n_C=0, and H_1=3. The only exterior triples are ayb and cyb. Their H_2 values are respectively 1 and 2, giving the two k=p monomials in the table.

For d=-1, only y,b,z can occur. If m=1, the choices y,b,z have (H_1,H_2) equal to (1,0),(1,1),(0,1). Equation (5.13) excludes z and gives respectively (n_C,n_B)=(0,3),(0,2) for the other two. If m=3, the exterior factor is ybz, with f=1, H_1=H_2=2; the same equations give (n_C,n_B)=(0,1). Since L=N_0-f, these are exactly the three k=p-1 monomials. Every nonnegative count and every exterior subset has now been considered.□

Proof. The subalgebra map includes a subset of the exterior and polynomial generators in (3.1); its initial-page map is therefore injective in every bidegree. Likewise, multiplication by the polynomial A^a is injective in every bidegree on that page. It is induced by a filtered cochain map: t_1 is primitive, and

is an actual cocycle, obtained by taking the integral divided reduced coproduct and then reducing modulo p. It has leading class A. For the second comparison shift the source cohomological degree, weight, and auxiliary filtration by 2a, pa, and ap(1+M).

Take the filtered mapping cone of either cochain map. The long exact sequence of associated-graded cohomology, together with injectivity in all bidegrees, identifies its initial cohomology page with the cokernel of the specified page map; there is no extra kernel from the next degree. That cokernel is zero in the stated target bidegree. Finite filtration therefore gives zero cone cohomology there. The ordinary cone long exact sequence gives the required surjection. Transport through the stated commuting comparisons preserves that surjectivity.□

Proof. The inclusion of the Frobenius subring makes P_r free of rank p^r, with basis t_1^{e_1}\cdots t_r^{e_r}, 0\leq e_i<p. It is therefore faithfully flat. Its kernel has coordinate algebra

All the mixed terms in its coproduct vanish, so each e_i is primitive. This proves the description of the kernel and the exactness of (6.2).

Here is the acyclicity calculation needed to derive invariants. Right multiplication by a kernel point is, by (6.1), simply t_i\mapsto t_i+e_i: every mixed term contains a positive p-power of a kernel coordinate. Hence multiplication gives an isomorphism of K_r-comodules

The first tensor factor has trivial K_r-coaction. The second is the regular comodule. In particular, O(G_r)^{K_r}=O(Q_r), with its regular quotient coaction.

A cofree comodule is injective: its cofree functor is right adjoint to the exact forgetful functor to vector spaces. Equation (6.4) shows that every cofree G_r-comodule is K_r-acyclic and that its K_r-invariants are cofree over Q_r. Every injective is a summand of a cofree comodule, so these assertions also hold for injectives. Finally, M^{G_r}=(M^{K_r})^{Q_r}. Resolving these two invariants functors successively, using the indicated cofree resolutions, gives (6.3). This argument derives K_r-invariants; it does not assert their exactness on arbitrary modules. The spectral sequence is first quadrant, so each fixed total cohomological degree has a finite diagonal. All coordinate weights are positive and each weight component is finite dimensional. These observations give the asserted convergence, also weight by weight.□

Proof. The dual algebra of a primitive truncated coordinate \mathbb F_p[e]/(e^p) is a truncated polynomial algebra after rescaling the dual basis by the invertible factorials j!, j<p. Its resolution alternates multiplication by its generator and by the (p-1)st power of that generator. Applying the augmentation gives one class in each cohomological degree. Its algebra of extensions is an exterior degree-one generator and a polynomial degree-two generator. The shift by two of the periodic resolution represents the latter, and its iterates are nonzero in every even degree. The former has square [e\mid e]=\tfrac12d[e^2], which is zero at this odd prime. Tensoring these resolutions gives (6.5), including the indicated weights.

For completeness, the naturality of the polynomial generators is Frobenius semilinear. A primitive coordinate T has degree-two cocycle

where the displayed coefficients are integers reduced modulo p. For a linear coordinate T=\sum_i\lambda_i e_i, the integral polynomial

has reduced coproduct \beta(\sum_i\lambda_i e_i)-\sum_i\lambda_i^p\beta(e_i). Thus a matrix (\lambda_{ij}) on degree-one coordinates induces (\lambda_{ij}^p) on these degree-two cohomology classes. This is a polynomial identity in the \lambda_{ij}, so it also applies when they are quotient coordinate functions.

If e is a kernel point, direct use of (6.1) gives

These coefficients belong to O(Q_r), so the action descends to the quotient. The degree-one and degree-two naturality just proved gives (6.6) and (6.7). The exterior matrix is triangular with diagonal entries one, so its top exterior power xac is fixed. The formulas also prove separate preservation of the two degrees.□

Proof. In (6.6) and (6.7), every nonidentity term has lower coefficient weight, its difference being the positive weight of its quotient coordinate. The cobar differential therefore preserves the increasing coefficient filtration. In the associated graded complex only the trivial coefficient action remains, giving (6.8). At fixed total weight the coefficient weights lie between zero and w, so the filtration is finite. The stated description of its first possible differential follows by taking the leading nonzero weight-lowering component of the cobar differential. Finally, expanding

proves the stated action on (6.9).□

Proof. The next exterior weights are p^2 and p^2+p; the third-coordinate weight, if present, is p^2+p+1. The next polynomial weight is p^2. All are outside the asserted range. Lemma 3.1 therefore gives exactly (6.13). To verify the table, first reduce its weight equation modulo p: its remainder is e+g\in\{0,1,2\}. For weights divisible by p it forces e=g=0, after which the degree equation fixes j and f. For the listed weights congruent to two it forces e=g=1; substitution into both equations gives the entries shown. The weight-two entries follow in the same way, since j=f=g=0 at that weight. All weights in the table are within the asserted range, because 3p+2<p(p-1) at p=1009.

The cocycles z=[\xi^p] and C=\beta(\xi) come from the primitive coordinate \xi. Their product zC^{p-2} is an actual cocycle with the indicated nonzero leading May monomial. The preceding cohomological degree at its weight has no May monomial in any filtration. It consequently cannot become a boundary on any May page. An actual cocycle has no outgoing May differential, proving its nonvanishing. The factor C^{p-2} is therefore nonzero as well.

For the other two classes, use the cobar convention d[u\mid v]=d[u]\mid v-u\mid d[v] with d[\eta]=[\xi\mid\xi^p]. The corrected cocycle

is closed: its two differentials are -[\xi\mid\xi\mid\xi^p] and [\xi\mid\xi\mid\xi^p], respectively. In the ordinary May filtration its first summand has degree four and the correction has degree three. Its leading detector is thus yb. The table excludes both an incoming and an outgoing May differential, so \upsilon=[U]\neq0. Similarly

is a closed representative with leading detector ybz. Its correction has smaller May filtration. The absence of a degree-two May monomial at weight 2p+2 excludes every possible incoming differential, and the explicit cocycle excludes outgoing differentials. Hence \upsilon z\neq0.□

Proof. We prove vanishing for G_3 by treating the first two rows of Table 4. Their kernel degrees are 2p+1 and 2p-1, so their coefficient differentials cannot mix.

For M=xac\operatorname{Sym}^{p-1}(A,B,E), we have \mathcal R=p(p-1) and quotient degree a=2p-4. After the weight exclusion of E, the finite module is xacA^{p-1}\langle1,D,\ldots,D^{p-1}\rangle. The sole candidate from Table 3 is D C^{p-2}, suppressing the invariant factor xacA^{p-1}. Its quotient weight is p(p-2) by (6.11). The translation of D gives, on the first possible coefficient page,

The target has quotient degree 2p-3 and weight p(p-1). Both source and target are nonzero actual quotient-cohomology classes by Lemma 6.4; they are present on this page because all earlier coefficient differentials vanish. Thus (6.15) removes the candidate.

We record the adjacent-degree checks. A differential on this page entering its source would have D-degree two, quotient degree 2p-5, and quotient weight p(p-3); this group is zero by Lemma 6.4. The target, having D-degree zero, has no outgoing coefficient differential. Its first-page incoming source can only have D-degree one, and (6.13) in degree 2p-4, weight p(p-2), gives only C^{p-2}. Thus no other source can cancel the nonzero map in (6.15). In the original total degree, the same equation, or the exhaustive candidate table, leaves no other coefficient-leading term in this vertical summand. Hence a change of representative cannot replace the removed class.

Now take the full second summand M=xac\operatorname{Sym}^{p-2}(A,B,E), with \mathcal R=p^2-p+1 and quotient degree 2p-2. Table 3 leaves only its D-degree-zero candidate, whose coefficient xacA^{p-2} is invariant. Its quotient detector yzC^{p-2} is already a splitting boundary, since db=yz and dC=0 on the May page. This also has a direct cobar realization: yz=d[\eta] and C is an actual cocycle, so yzC^{p-2} is exact. It is the only May monomial in this quotient bidegree, as follows by putting a=2p-2, w=p^2-p+1 in (6.13). It therefore contributes no actual quotient cohomology in (6.8). These are all initial candidates of total degree 4p-3 and weight u_4+2. The coefficient spectral sequences consequently vanish there, so all corresponding terms of (6.3) are zero. Its convergence proves the result.□

Proof. We treat the last two rows of Table 4 for G_2, using the finite module (6.9) in each case. For n=N_0, the residual weight is \mathcal R=3p+1 and the quotient degree is one. We show directly that every one-cocycle vanishes. The leading candidates could survive only if lower powers of D supplied corrections. The crossed-cocycle equation will rule these out: it requires unequal mixed coefficients where every available correction has equal ones.

Write \xi(g)=r, \eta(g)=u, \xi(h)=s, \eta(h)=v, so that

A normalized one-cocycle is a polynomial f(D;g) satisfying

Since D has weight p and 4p>3p+1, it must have the form

All these powers are in the finite module, since N_0\geq3. For a polynomial function \varphi write \delta\varphi(g,h)=\varphi(gh)-\varphi(g)-\varphi(h). Comparing the four coefficients in (6.16) gives

Homogeneity forces

Here the coefficients a_3,b_2,c_2,d_1,e_1,r_0,s_0,t_0 lie in \mathbb F_p. The second equation of (6.17), together with

forces c_2=3a_3 and b_2=-3a_3. The next equation has coefficient 2b_2 on r^pv and zero on us^p. On the other hand,

and \delta(\xi^{2p+1}) contains neither u nor v. Its left side must therefore have equal coefficients on r^pv and us^p. Since 2b_2=-6a_3 and 6\neq0 in \mathbb F_{1009}, we obtain a_3=0, and then b_2=c_2=0. No term in f_0 can alter this coefficient-of-D equation.

Now f_1 is primitive. Equation (6.18) first forces d_1=0, and the coefficient of r^{2p}s in \delta(\xi^{2p+1}) then forces e_1=0. Finally f_0 is primitive. In its reduced coproduct the monomial u^2s^{p-1} has coefficient t_0, so t_0=0. The monomial us^{2p} then has coefficient s_0, so s_0=0. The coefficient of r^{3p}s in \delta(\xi^{3p+1}) is one, giving r_0=0. Every normalized cocycle is thus zero, proving the required H^1 vanishing, including all lower-D corrections.

For n=N_0-1, the residual weight is \mathcal R=3p+2 and the desired quotient degree is three. The first coefficient differential comes from the term 2D\xi^p in

Using the actual class \upsilon of Lemma 6.4, it is

The coefficient differential first prepends z. Its product z\upsilon agrees with \upsilon z: both are actual cocycles with the same leading May monomial ybz, with coefficient +1 after two exterior interchanges, and the list in Equation (6.13) leaves no other initial May monomial or lower-filtration cohomology in this bidegree. Thus the product order in Equation (6.19) is justified directly. Both powers of D lie in the finite module, since n\geq3. Equation (6.11) gives quotient degree and weight (2,p+2) for the source and (3,2p+2) for the target. Both therefore have the required residual weight 3p+2. All earlier coefficient differentials vanish, so source and target are still their actual quotient-cohomology classes on this first possible page. They are both nonzero by Lemma 6.4.

A same-page incoming differential to the source would require D-degree three and quotient H^1 of weight two, which is zero. The only same-page incoming source for the target is D^2H^2(Q_2)_{p+2}, the one-dimensional span displayed in (6.19). A same-page outgoing differential from the target is proportional to \upsilon z^2=0. Indeed [\xi^p\mid\xi^p]=\tfrac12d[\xi^{2p}], so this last vanishing also follows directly in the cobar complex. The term \xi^{2p} in the expansion of D^2 lowers two D-levels and cannot cancel its first differential. Since 2\neq0 in \mathbb F_p, (6.19) removes the target.

There is no other degree-three coefficient-leading term. In fact, the possibilities for D-degrees 0,1,2,3 require quotient degree three at respective weights 3p+2,2p+2,p+2,2; by Lemma 6.4, only the second has a candidate, namely ybz. Larger D-degrees exceed the residual weight. Thus there is no replacement in a lower coefficient filtration. The initial candidate table excludes every other kernel degree or exterior summand at this total bidegree. Both possible terms of (6.3) consequently vanish, and the proposition follows.□

Proof. Apply Propositions 6.5 and 6.6, and the exhaustive list in Table 3.□

Proof. By q-sparsity, a positive differential out of (5,W) has target

All possible Adams–Novikov terms in the stem in (7.5) have

We treat both by writing

First we give explicit bounds permitting the additive calculation. The normalized-cobar inequality of Lemma 2.3 implies

At the specified prime,

Consequently j\leq1012 and

Thus no nonconstant coefficient of R/I_4 occurs, in any cobar degree at these weights. The only possible digit positions are 0,1,2, and the cooperation formulas are additive. Lemma 2.4 therefore identifies the relevant cobar calculation with the indicated range of P.

Use the six exterior letters x,a,c,y,b,z and the three polynomial letters A,B,C of Equation (3.4). If m is the number of exterior letters and L=(s-m)/2 the number of polynomial letters, then m\leq6 and

It follows that

Since p(p-1)-1=1{,}017{,}071 and 3\cdot1{,}017{,}071>W+1, this improves the bound to j\leq2.

For completeness, we enumerate this remaining finite calculation. Let H_i count the exterior letters covering position i, and let n_A,n_B,n_C be the polynomial exponents. The weight equation is

In (7.6), m is even and reduction modulo p gives H_0=1+\delta+j, whenever a solution exists. Put \kappa_2=H_2+n_B+n_C. All position contributions are nonnegative; (7.7) also gives w<3p^2, so \kappa_2\leq2 and n_C\leq2. Substituting L=3+(p-1)j-m/2 into (7.8) yields

For 0\leq j\leq2, its right side lies between -3 and 8. Its only multiple of p=1009 is zero. Hence

For j=2, either H_0=4, which is impossible, or H_0=3 forces c to occur whereas \kappa_2=0. For j=1,\delta=1, all of x,a,c occur and \kappa_2=1 excludes b,z. Thus H_1\leq3, but the last equation of (7.9), with even m\geq4, requires H_1\geq4. For j=1,\delta=0, one has H_0=2. If m=2, the required inequality H_1\geq3 is impossible. If m=4, one must have H_1=4 and n_C=0; the four letters covering position one are exactly a,c,y,b, which have H_2=2, contrary to \kappa_2=1. Finally m=6 would give H_0=3. All positive-j groups in (7.6) therefore have empty initial May pages.

It remains to examine j=0,\delta=1, where H_0=2 and m is 2 or 4. For m=2 both letters start at zero, and (7.9) permits only ac, followed by n_B=1,n_C=0,n_A=1. For m=4 choose two of x,a,c and two of y,b,z. The equations become

The following table lists all nine choices; a dash means that at least one exponent is negative.

Thus the complete initial-page list at (6,W+1) is

With exterior order x,a,c,y,b,z and da=xy, db=yz, dc=xb+az, their splitting differentials are

The first image has polynomial factor AB and cannot cancel either of the other images. The two remaining cycle directions are boundaries, since

The splitting cohomology of (7.10) is zero. The May spectral sequence, finite in each of these weights, proves all the claimed E_2 vanishings.

For orientation, the same equations also enumerate the source (5,W). Here H_0=1, m is odd, and L=(5-m)/2. They force H_2+n_B+n_C=2 and H_1+L-n_C=4. If m=1, then H_1+L\leq1+2<4; if m=5, then at least two of x,a,c occur, contrary to H_0=1. Thus m=3, which forces n_C=0 and H_1=3. The only monomials are aybB and cybA. We need no assertion that either one survives the May spectral sequence.

No topological Adams–Novikov differential can hit filtration five: its length would have to be at least q+1>5. The outgoing target groups have just been proved zero. Thus every actual class of E_2^{5,W}(Y) survives and is represented by a class \gamma\in\pi_{qW-5}Y. The vanishings for \delta=1 exhaust all filtrations in the stem q(W+1)-6; strong convergence proves (7.5).□

Proof. The cooperation t_1 is primitive. In weight one the normalized sphere cobar is

because \eta_R(v_1)-v_1=pt_1. Hence its degree-one cohomology is \mathbb Z/p on [t_1].

The integral polynomial

defines the two-cobar cocycle b_{1,0} by substituting the two primitive copies of t_1. Its differential is zero: the two-cocycle identity follows by expanding (x+y+z)^p, or by dividing the identity d^2(t_1^p)=0 in the torsion-free integral cobar. Its image in the additive specialization is nonzero in cohomology. At weight p the only coordinate available is t_1; the only one-cochain of that weight is t_1^p, whose differential is zero in characteristic p, whereas (7.11) is a nonzero two-cochain. Thus it is not a boundary.

The first positive topological differential length permitted by sparsity is q+1. From (2,p) an outgoing differential has target (qj+3,p+j) for j\geq1, violating w\geq s; no incoming differential can reach filtration two. The same argument applies to (1,1). Therefore these cocycles survive to the stated sphere classes.

For any positive stem i with a nonzero normalized-cobar term, either s=0 and i=qw\geq q, or s\geq1 and

This proves the lower sphere vanishing. In stem q-1 the same inequality permits only (s,w)=(1,1), so there is no extension ambiguity and \pi_{q-1}S^0_{(p)}=\mathbb Z/p\{\alpha\}. Finally, the sphere action and its cobar pairing in Proposition 2.6 identify the detector of multiplication by \beta with b_{1,0}=A.□

Proof. Let C be the cofiber of b_0:S^0\to U_0. The assumed integral homology groups and successive applications of connective Hurewicz show that C is (q-2)-connected and

Choose a generator S^{q-1}\to C. Its obstruction to lifting to U_0 lies in \pi_{q-2}S^0, which is zero by Lemma 7.2. A lift, together with b_0, gives S^0\vee S^{q-1}\to U_0. Its cofiber D is (q-1)-connected. The integral homology exact sequence gives

In particular this is a free rank-one module. Use Hurewicz to choose a generator of \pi_qD and take its boundary in \pi_{q-1}(S^0\vee S^{q-1}). Attaching the corresponding cell gives a map Q\to U_0 extending the two lower cells. Its component on S^{q-1} is multiplication by a unit times p, since its homology image is exactly p\mathbb Z_{(p)}. Its component on the bottom belongs to \pi_{q-1}S^0=\mathbb Z/p\{\alpha\}, so it is a\alpha for some a\in\mathbb F_p.

The map Q\to U_0 induces isomorphisms on integral homology in degrees zero and q-1. The universal coefficient sequence then gives isomorphisms on mod-p cohomology in degrees zero and q; the latter is the class arising from the torsion in degree q-1. The assumed nonzero reduced power therefore pulls back to a nonzero reduced power of the bottom class of Q. If a=0, the actual attaching map would give a wedge

The degree-zero class in this wedge is pulled back from S^0, so its positive reduced powers vanish by naturality. This is a contradiction. Hence a\ne0. The construction used an integral homology gap and relative Hurewicz in a finite range; no assertion that U_0 itself is finite is involved.□

Proof. Write d=pq-2=2m, where

Let \rho_{\mathbb C} be the complex regular representation of C_p. The cyclic extended power of the even sphere S^{2m} is the Thom spectrum of m\rho_{\mathbb C} over BC_p:

Applying this functor to a sphere map representing \beta, then collapsing BC_p in the extended power of S^0, gives a map

whose restriction to a fiber sphere S^{pd} is \beta^p. Only the symmetric multiplication of the sphere spectrum is used in constructing this map.

The group \mathbb F_p^\times acts by permutations of the cyclically labelled smash coordinates and by the corresponding automorphisms of C_p. This normalizer action is compatible with (7.12), on whose target it is trivial. Normalize the bottom degree by setting

The idempotent e_0 splits in spectra. Denote its image by U_0, with maps i_0:U_0\to U, r_0:U\to U_0 satisfying i_0r_0=e_0 and r_0i_0=1. The projected Thom fiber b_0:S^0\to U_0 induces an isomorphism on H_0(-;\mathbb Z_{(p)}): the complex orientation makes every normalizer permutation act as one on the bottom class. Moreover, normalizer invariance of \Phi gives a map

We describe the homology and the reduced power of this summand. The complex bundle is integrally oriented, so the integral Thom isomorphism identifies H_*(U;\mathbb Z_{(p)}) with H_*(BC_p;\mathbb Z_{(p)}). The periodic cyclic-group resolution, with maps g-1 and 1+g+\cdots+g^{p-1}, gives

These groups are finitely generated degreewise, as are their direct summands for U_0. The same resolution, or the circle bundle of Chern class p over infinite complex projective space, gives

An automorphism a\in\mathbb F_p^\times multiplies both s and t by a. Equivalently it acts by a^j on integral homology in degree 2j-1. Averaging consequently retains, through degree q, exactly the mod-p classes in degrees 0,q-1,q and the integral groups

Let u be the normalized Thom class. We claim

For a universal complex line bundle with first Chern class l and Thom class u_l, write \mathcal P^1u_l=c\,l^{p-1}u_l by the Thom isomorphism. Pullback along the zero section and the identity \mathcal P^1(l)=l^p imply c=1 in the polynomial cohomology of infinite complex projective space. Naturality gives the formula for every line bundle. The Cartan formula for a direct sum of lines therefore gives the sum of their (p-1)st Chern powers as the multiplier of \mathcal P^1 on its Thom class. The regular representation splits into the characters with first Chern classes it, i\in\mathbb F_p. Thus the multiplier here is

Both this class and u are normalizer-invariant. The reduced power commutes with the idempotent, so the same nonzero formula holds in U_0.

Apply Lemma 7.3 to (7.13) and (7.14). It supplies a three-cell spectrum

with a nonzero modulo p and u a p-local unit.

Restrict \Phi_0 to \Sigma^{pd}Q. Its bottom value is \beta^p; let its value on the other lower cell be \zeta. The top cell gives the relation

The action of p on Y is null by Corollary 4.4. Apply the unit to this relation and use a\ne0 modulo p to obtain \alpha\beta^p=0 in \pi_*Y.□

Proof. The two-cell coaction. We first compute the coefficient comodule needed for duality. The dual DJ is the cofiber of \alpha':S^{q-1}\to S^0, where \alpha'=\pm\alpha after fixing cell orientations, and its bottom inclusion is D\varepsilon. Put \mathcal F=\operatorname{fib}(S^0\xrightarrow{\eta_{BP}}BP). A chosen BP-nullhomotopy of \alpha' gives a lift \widetilde\alpha:S^{q-1}\to\mathcal F and a map H:DJ\to BP extending the unit, in the following map of cofiber sequences:

The BP-module BP\wedge DJ splits because BP_{q-1}=0. Let e_0 be its bottom generator and choose an upper generator z\in BP_qDJ whose top quotient is 1 and for which \rho_*z=0, where \rho=\mu_{BP}(BP\wedge H):BP\wedge DJ\to BP retracts the bottom inclusion. With \epsilon:\Gamma\to R the counit, the element g=(BP\wedge H)_*z satisfies

Here \eta_L is induced by the bottom map BP\wedge\eta_{BP}:BP\to BP\wedge BP. After smashing the cofiber diagram with BP, its connecting map \partial:\Gamma_q\to\pi_{q-1}(BP\wedge\mathcal F) has kernel \eta_L(R_q) and gives

Thus g represents the first Adams–Novikov detector of \alpha'. It is also the actual off-diagonal coaction: writing \psi(z)=1\otimes z+\omega\otimes e_0, naturality under H gives \Delta g=1\otimes g+\omega\otimes1. Apply \operatorname{id}\otimes\epsilon and use \epsilon(g)=0 to obtain \omega=g. By Lemma 7.2 this detector is \pm[t_1], so c\not\equiv0\pmod p.

For completeness, fix the left-comodule convention \psi(am)=\eta_L(a)\psi(m) with tensor product using \eta_R, so \eta_R(v_1)-\eta_L(v_1)=pt_1. Replacing z by z+av_1e_0 changes its off-diagonal term to g+a(\eta_L-\eta_R)(v_1)=g-pa\,t_1. Changing H by the composite DJ\to S^q\xrightarrow{av_1}BP instead changes g by \eta_R(av_1); renormalizing the upper generator to z-av_1e_0 subtracts \eta_L(av_1), giving g+pa\,t_1. Opposite conventions reverse these signs, and dual cell orientations change the detector by a sign; none changes its nonzero residue modulo p. The underlying BP-module splitting therefore need not split the comodule. Replacing z by c^{-1}z now makes the off-diagonal term exactly t_1. Its top quotient is then c^{-1} rather than 1; we no longer need the earlier top normalization. This change fixes e_0 and the bottom inclusion D\varepsilon, which are the data used in the following symmetric-power calculation.

The symmetric coefficient module. Duality changes each permutation into its inverse, and hence preserves the average. Consequently

Under this identification D\varepsilon_T is the pure bottom tensor e_0^{\otimes n}. All the cell degrees of DJ are even. If z_S is the tensor with upper generator in positions S and bottom generator elsewhere, define its normalized symmetric basis by

Every denominator is a p-local unit. Expanding the translation of each tensor gives the coefficient

on r_jt_1^{k-j} in the coaction of r_k. Thus we can denote this basis by 1,r,\ldots,r^{p-1}, with coaction (7.15). Each r^k has weight k because it is represented by tensors with k upper generators of degree q.

The module BP_*DT is free. Smashing its free BP-module splitting with Y gives

with the same translated basis and with D\varepsilon_T identified with constants. This argument uses a free factor, so it requires no flatness of R/I_4.□

Proof. Finite duality identifies

and carries f to (D\varepsilon_T\wedge1_Y)(\eta\beta). By Lemma 7.5, its induced E_2 class is represented by the constant-valued beta cocycle. We first show that this cocycle is a boundary in the translated coefficient module. Consider the integral polynomial

The displayed sum defines it before reduction modulo p; it uses only the allowed coefficient powers r,\ldots,r^{p-1}. The exact integral identity

follows by cancellation of the four pth powers. After reduction, the coefficient coaction (7.15) and primitivity of t_1 make the left side the cobar differential of F(r,t_1), with the common overall sign determined by the cobar convention. The right side is the constant cocycle (7.11). Thus the bottom image of \beta has zero image on E_2.

There is no possible higher-filtration homotopy value hidden by this boundary. The spectrum DT\wedge Y is connective and its coefficient weights are nonnegative. In stem pq-2, its q-sparse terms have

For j\geq1 the inequality w\geq s would give p-2\geq(q-1)j, which is impossible. Naturality places the image of \beta in filtration at least two, and the boundary just found raises it to filtration at least three. All later associated-graded groups are zero, so convergence with finite filtration makes the actual image zero.

The bottom beta image is therefore zero, and finite duality makes f itself nullhomotopic.□

Proof. Lemma 7.1 gives \alpha\gamma=0 in precisely the stem q(W+1)-6. Proposition 7.4 gives \alpha\beta^p=0 after applying the unit to Y. By (7.4), choose lifts

of \gamma and \eta\beta^p, respectively. Fix any parenthesization of the n-fold iteration of \mu. Smash one copy of \widetilde\gamma with p-2 copies of \widetilde b, collect the J factors, and apply that chosen iteration to the Y factors. This gives

Its top projection is \beta^{p(p-2)}\gamma. To verify this without an associativity assumption, for any sphere scalar a use the two unit homotopies to identify

The scalar \beta^p is even. In the chosen binary tree, every subtree without \gamma is a scalar times the unit, and the unique subtree containing \gamma is the product of its scalars times \gamma. This proves the asserted top value by induction on the tree.

Now project to the symmetric retract after those Y factors have already been multiplied:

For every permutation \sigma of the J factors, naturality of the symmetry and the zero-dimensional top sphere give an equality of stable maps

Consequently

No permutation of the multiplied Y factors occurs in this identity.

The actual null map f of Proposition 7.6 now gives the following homotopy-commutative diagram:

Commutativity is the left unit identity for \mu and the definition of f. Its top arrow is null, so composing the diagram with \Sigma^dL_T and using (7.21) proves \beta^{p(p-2)+1}\gamma=0.□

Proof. Let \theta\in E_2^{5,W}(Y). By Lemma 7.1, it is a permanent cycle represented by a homotopy class \gamma. The sphere class \beta is also permanent. The pagewise product A^K\theta therefore supports no nonzero outgoing differential; if either factor has zero image on a page, the product is already zero. By Proposition 7.7 its filtered homotopy product is zero. Product compatibility and finite convergence then imply that A^K\theta is zero at E_\infty.

If it is nonzero at E_2, it must consequently be hit by an incoming differential. Its cohomological filtration is

An incoming differential of length r has source filtration S-r\geq0, so r\leq S. This bounds the page of disappearance, rather than merely its eventual abutment. The exact comparisons are

Numerically,

Thus A^K\theta is zero before the page of the prospective d_{qp+1}(v_4). Multiplication by the further power A^{N-K} is defined on every page by the sphere permanent cycle \beta; it shows that A^N\theta is zero there as well. The surjection (7.2) accounts for every E_2 class of the target, proving the assertion.□

Proof of Theorem 1.1. The only possible positive differentials from v_4 have target (s,w)=(qk+1,u_4+k). Proposition 5.4 rules out k=1. Lemma 5.1 and Table 3 leave, for k\geq2, only k=2,p-1,p. Corollary 6.7 eliminates the first two target groups already at E_2, and Proposition 7.8 makes the last target zero before its possible differential. No differential can hit filtration zero. Hence v_4\in E_2^{0,u_4}(Y) is permanent.

The edge homomorphism of the convergent Adams–Novikov spectral sequence supplies a map a:S^{qu_4}\to Y whose BP-Hurewicz image is exactly v_4. Extend it by the existing binary multiplication to

and take its cofiber X. The right unit identity shows that the bottom generator is sent to v_4. Since BP_*Y=R/I_4 is cyclic as an R-module, this determines the entire induced map as multiplication by v_4. That element is a nonzerodivisor in R/I_4=\mathbb F_p[v_4,v_5,\ldots]. The homology exact sequence of the cofiber therefore gives the short exact sequence

The composite bottom map S^0\to Y\to X induces the quotient map R\to R/(I_4,v_4). It is a comodule map and is surjective, so it identifies the induced coaction with the canonical quotient coaction, and identifies its generator in degree zero.

Finally, X is the cofiber of a map between finite p-local spectra. It is finite p-local in the sense defined in the introduction; equivalently, the finite cell construction can be realized by a finite spectrum before localization by clearing prime-to-p denominators in its attaching maps, as in Lemma 4.2. More explicitly, the mapping cone adds to the cells of Y their translates by qu_4+1. Together with Equation (4.6), this gives a finite p-local CW model for X with one cell in each dimension

In particular, this model has 32 cells.□

Proof of Proposition 2.5. We first identify the ordinary totalization tower exactly. Let \mathcal P_0([n]) be the poset of nonempty subsets of \{0,\ldots,n\}. The functor

sends an inclusion to the corresponding order-preserving injection. It is homotopy-initial. To prove this, for m\leq n identify (u_n\downarrow[m]) with the nonempty face poset of the simplicial complex L_{n,m} whose vertices are (i,j), 0\leq i\leq n, 0\leq j\leq m, and whose simplices have strictly increasing first coordinates and weakly increasing second coordinates. Its nerve is the barycentric subdivision of L_{n,m}.

The complex L_{n,m} is contractible for n\geq m, by induction on n. The case n=m=0 is a point. For the inductive step start with the cone with vertex (n,m) on the entire subcomplex L_{n-1,m}. Adjoin the cones with vertices (n,j), 0\leq j<m, on their bases L_{n-1,j}. Two distinct vertices with first coordinate n cannot belong to a simplex. Each new cone therefore intersects the preceding union in exactly its base. That base is contractible by induction, since n-1\geq j. Attachment along this simplicial subcomplex is a homotopy pushout and preserves contractibility. This proves the claim.

Homotopy-initiality, applied to the cosimplicial unit resolution, now gives, naturally in n and Z,

This is the standard cubical description of the partial Amitsur totalizations; compare Mathew et al. [13, Propositions 2.11 and 2.14]. Indeed derived partial totalization is the homotopy limit over \Delta_{\leq n}: the restricted standard simplex is a Reedy-cofibrant resolution of the constant point diagram. Adjoining the empty vertex Z on the right of Equation (A.1) gives the (n+1)-fold cube formed by smashing copies of the unit S_{(p)}\to BP. If \mathcal F is its fiber, iterated fibers and exactness of smash identify its total fiber with \mathcal F^{\wedge(n+1)}\wedge Z. Thus

The comparisons commute with restriction from n+1 to n: on the fibers this is the map induced by the last factor \mathcal F\to S_{(p)}. There is no suspension in Equation (A.2).

The unit induces an isomorphism on \pi_0, and \pi_1BP=0, so \mathcal F has no homotopy in degrees at most zero. If Z is c-connective, \mathcal F^{\wedge s}\wedge Z is at least (c+s)-connective. Equation (A.2) therefore identifies the coaugmentation Z\to K(Z) as an equivalence, and identifies the relative totalization tower as

Its homotopy groups vanish in each fixed stem for sufficiently large s. The exact couple consequently has the asserted abutment, no inverse-limit ambiguity, and finite filtration in each stem.

For completeness, the successive layer is

where \overline{BP}=\operatorname{cofib}(S_{(p)}\to BP). This is also \operatorname{fib}(K_s(Z)\to K_{s-1}(Z)), with the usual normalization and the displayed s desuspensions.

The iterated extended terms compute the cobar complex. Indeed BP\wedge BP, as a BP-module, is a wedge of suspended free BP-modules with basis the polynomial monomials in the t_i. Iteration identifies their homotopy with the iterated tensors of \Gamma over R. There are no derived tensor corrections, whether or not BP_*Z is flat over R. Cofaces give the cobar differential; the codegeneracy splittings give its normalized form. The exact-couple grading gives the displayed differential shift. Finally the first map is Z\to BP\wedge Z. Its image is exactly the permanent filtration-zero part by the finite-filtration convergence just proved.□

Proof of Proposition 2.6. We construct maps

which induce the stated cobar pairing and recover the actual map f.

The box pairing. We use Batanin’s cosimplicial box product in the form of McClure and Smith [15, Definition 2.1, Proposition 2.3, and Remark 2.4]; they attribute its Kan-extension description to Cordier and Porter. We construct the pairing before any fibrant replacement and then transport the entire pairing. For cosimplicial spectra define A\mathbin\square B as the left Kan extension of (a,b)\mapsto A^a\wedge B^b along ordinal concatenation

Equivalently, in degree n it is the zigzag pushout of the terms A^a\wedge B^b with a+b=n, with neighboring terms identified by the maps

The same definition applies to cosimplicial based spaces. For the original unit resolutions, concatenating the two BP blocks and applying f once gives maps

They are natural in both ordinals: cofaces and codegeneracies within a block insert a unit or multiply adjacent factors within that block. Kan-extension adjunction therefore gives an actual cosimplicial map

On the degree-n summand a+b=n in the pushout description, this map multiplies the last factor of the first BP block with the first factor of the second block. This is the single overlapping slot. The two descriptions agree by applying the codegeneracy that identifies the two junction vertices of [a+b+1].

Deriving the pairing. We record the model-category check that permits us to derive this construction. The box product is a Quillen bifunctor for the Reedy model structures. Let h_a([n])=\Delta([a],[n])_+, and let \partial h_a be the subdiagram of noninjective maps. These are the representables and boundaries defining the generating Reedy cells. Indeed, the boundary of the covariant representable consists of maps factoring through a nonidentity inverse map out of [a], hence of noninjective maps by the unique surjection–injection factorization in \Delta. This representable boundary is distinct from the latching subobject in target degree n, which consists of maps factoring through a proper injection into [n]. Kan extension gives h_a\square h_b=h_{a+b+1}. In the relative box boundary

the remaining maps are exactly the monotone maps injective on each of the two blocks. Such a map is either globally injective or identifies only the two junction vertices. A map that is not surjective onto its target lies in that target degree’s latching subobject. The only remaining surjections are the junction identification onto [a+b] and the identity of [a+b+1]. Thus the junction-identifying and globally injective non-latching cells occur in degrees a+b and a+b+1, respectively. Attach the junction cell first; every further collapse of it is already in the boundary, since it collapses two vertices within one block. Attach the globally injective cell second: its junction codegeneracy lands in the first cell, and its other codegeneracies land in the boundary. This gives a two-cell Reedy filtration of the relative box boundary. For two generating Reedy cofibrations, each of these attachments is tensored with the ordinary spectral pushout-product of their underlying generating cofibrations. The spectral pushout-product axiom therefore makes both attachments cofibrations, and makes them trivial cofibrations if either input is trivial. This proves the Reedy pushout-product axiom, including the acyclic case, for generating cofibrations. Closure under pushouts, composition, and retracts proves the assertion for all cofibrations.

Choose Reedy-cofibrant replacements A_i\to T(Z_i), i=1,2, that are trivial fibrations, and trivial cofibrations A_i\to R_i with R_i Reedy fibrant. For the target choose a Reedy fibrant replacement T(Z_3)\to R_3 directly. Equation (A.5) gives A_1\square A_2\to R_3. By the preceding Quillen check, A_1\square A_2\to R_1\square R_2 is a trivial Reedy cofibration. The lifting axiom extends the map to

The space of such extensions is contractible up to the usual relative homotopy. This transports the original pairing; it places no condition on the literal BP slots of the fibrant replacements. All resulting tower and spectral-sequence pairings are transported along the equivalences with the original diagrams. In particular, when some Z_i coincide, their source and target replacements need not be identical.

The filtered diagonal. We use the simplex subdivision map given in McClure and Smith [15, Lemma 3.6], credited to Grayson in their Remark 3.7. We record its formula to verify the relative filtration. For t=(t_0,\ldots,t_n)\in\Delta^n, put

For a cut index k, x is supported in 0,\ldots,k and y in k,\ldots,n. The cut region is a copy of \Delta^k\times\Delta^{n-k}, with inverse t=(x+y)/2. Its boundary seams identify the last face of the first factor with the first face of the second factor, precisely as in the box pushout. Thus the cut defines

It is natural under all monotone maps: they aggregate consecutive intervals, and aggregation commutes with the cut. In particular this is a map of ordinary cosimplicial objects, including codegeneracies.

Define the relative simplex object

where the skeleton for s=0 is the basepoint. The simplex, its skeleta, and these quotients are Reedy cofibrant: their latching maps are the corresponding boundary inclusions or identities. Consequently

is a derived relative end and the actual fiber of the fibration \operatorname{Tot}R_i\to\operatorname{Tot}_{s-1}R_i. The degree-n box summands have dimensions k and n-k. If n<s+t, at least one is below its respective skeleton cutoff. Naturality applies the same observation to every face of a larger simplex, including faces with missing ambient vertices. Hence the cut descends to maps

Enriched evaluation, followed by m_R and precomposition with \delta_{s,t}, gives the actual relative pairing

It commutes with the tower inclusions in both inputs. By Equation (A.3), these are the exact relative fibers of the unit-resolution filtration, with filtration indices s,t,s+t.

The cobar and homotopy pairings. Put L_s=\operatorname{cofib}(G^{s+1}\to G^s). The relative pairing induces L_s(Z_1)\wedge L_t(Z_2)\to L_{s+t}(Z_3): raising either input filtration by one raises the output filtration by one. The boundary formula for the product of relative cells is \partial(a\cdot b)=\partial a\cdot b+(-1)^{|a|}a\cdot\partial b. It gives a pairing of the exact couples and the Leibniz rule on every derived page. On the layer of dimension s+t, the only contributing cut cell has front and back dimensions s,t. The inverse t=(x+y)/2 gives its usual oriented front-face/back-face cup map, with coefficient one. The normalized cobar identification therefore gives exactly the asserted E_2 cup pairing.

We finally verify the homotopy target without assuming that a replacement preserves units. For cofibrant Z_i, the diagram \Delta^\bullet_+\wedge Z_i is Reedy cofibrant and maps levelwise equivalently to the constant diagram Z_i. Its original all-unit map to T(Z_i) lifts to a map \alpha_i:\Delta^\bullet_+\wedge Z_i\to A_i along the trivial fibration A_i\to T(Z_i). The composite of \alpha_i with A_i\to R_i represents the coaugmentation. The box product of the \alpha_i, precomposed with \delta, lands in A_1\square A_2. Since m_R extends the original map on that object, the composite is exactly the target coaugmentation composed with f, transported through T(Z_3)\to R_3. On totalizations the pairing therefore satisfies

The coaugmentations e_i:Z_i\to K(Z_i) are the equivalences proved above. Thus the limit pairing is the actual homotopy map f, and the E_\infty pairing is its associated-graded pairing for the filtration by the images of \pi_*G^s\to\pi_*K.

For the final assertion, the coefficient pairing is R-bilinear and sends the two units to the unit. Every element of R/I is a scalar multiple of that unit. The pairing is consequently (r\bmod I,r'\bmod I)\mapsto rr'\bmod I. This uses neither flatness of the quotient nor associativity of the target map.□