THE CHROMATIC EXT GROUPS Ext 0 Γ(m+1)(BP, M 1 2 ) (DRAFT VERSION) June 12, 2001

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1 THE CHROMATIC EXT GROUPS Ext Γm+BP M DRAFT VERSION IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL June Contents Introduction Preliminaries Elementary calculations 6 d x k for k Some lemmas 8 6 d x k for k 6 7 Proof of Lemma 68 for m Proof of Lemma 68 for m = 8 9 Proof of Lemma 68 for m = Proof of Lemma 68 for m = Proof of the main theorem References 5 Introduction Let BP be the Brown-Peterson spectrum for a fixed prime p In [Rav86 65] the third author has introduced the spectrum T m which has BP -homology BP T m = BP [t t m ] and is homotopy equivalent to BP below dimension p m+ Then the Adams- Novikov E -term converging to the homotopy groups of T m E T m = Ext BP BP BP BP T m is isomorphic by [Rav86] Corollary 7 to where Ext Γm+ BP BP Γm + = BP BP / t t m = BP [t m+ t m+ ] In particular Γ = BP BP by definition To get the structure of this Ext group we can use the chromatic method introduced in [MRW77] Define the chromatic ule M s n by M s n = v n+sbp /p v v n v n v n+s The third author acknowledges support from NSF grant DMS-9856

2 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL and consider the chromatic spectral sequence converging to Ext Γm+ BP /I n with E st = Ext t Γm+M s n Shimomura calls this Ext group the general chromatic E -term In this paper we will determine the ule structure of Ext Γm+M The analogous result for m = was obtained long ago by Miller-Wilson in [MW76] and is as follows Theorem [Miller-Wilson] As a k -ule Ext ΓM is the direct sum of a the cyclic subules generated by x s k /vak for k and s Z pz where x = v x = v p vp v v x p k for k even and for k x k = x p k p /p vpk v pk p k + for k odd and a = a = p pak for k even and for k ak = pak + p for k odd; and b K /k generated by /v j for j Before this result was proved the naive conjecture about this group would have had the exponents ak being p k for all k It was clear that v spk Ext ΓBP M v pk but the existence of deeper elements such as x v a = vp vp v p v p vp v p p + v p +p came as a surprise as did the fact that the limiting value as k of ak/p k is p + p + /p + p instead of Our result Theorem 69 below has the same form as Theorem but with x k and ak replaced by x k and 67 and âk and 6 and with k replaced by a larger ring k defined in In order to avoid the excessive appearance of the index m we will use the following notation v i = v m+i t i = t m+i Kn = Kn [v n+ v n+m ] ĥ ij = h m+ij kn = kn [v n+ v n+m ] and ω = p m

3 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION In 7 we will define elements x k v BP /I and integers âk whenever the condition < p m + /p of Theorem is satisfied and compute the reduced right unit d on x k These will depend on m Using these computations we can obtain the structure of the target Ext group In particular for large m we have Theorem Assume that p = and m 6 or that p > and m 5 As a k -ule Ext Γm+ M is the direct sum of a the cyclic subules generated by x s k /vbak for k and s Z pz where x k = v for k = x p vp v p v p vp v pω p bv for k = x p 5 + vp+p5 v p5 W for k = 6 x p 8 ba6 vp v p b b6 X p v p ba6 v p b b6+p ω p 6 x 6 for k = 9 x p k for k 8 x p k + vbak bak v b bk b bk x p k x k x p k 5 for k ; see Lemma 5 and Proposition 59 for definitions of W and X and and âk = p k for k p + p k for k 5 p + p + p k for 6 k 8 p â6 + â5 for k = 9 p k 9 â9 â5 + âk for k ; b K / k generated by /v j for j This is a part of our main result Theorem 69 In fact we have determined the structure of Ext Γm+ M for p = and m or p > and m In JAMI conference held at Johns Hopkins University in March Shimomura reported that he extended our result and obtained the structure of Ext Γm+M n for m n n For n = this result coincides to our Theorem Recently he also determined the structure of higher Ext groups [Shi] In the above case the assymptotic behavor of the exponents is given by âk lim k p k = p + p + p p a slightly larger value than for the case m = In addition there is a new form of periodicity in our statement with no precedent in Theorem namely for all k 9 we have x k x p k = vpk +p k +p k v pm+k p k p k p k x p k x k x p k 5

4 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL and âk = p k + p k + p k + âk For lower m the period is 6 instead of A similar result for the chromatic ule M is obtained in [NR] or [Shi] in which the period is instead of Preliminaries For a Γm + -coule M consider the cobar complex C Γm+ M } where C n Γm+ M = Γm + BP BP Γm + BP M with n factors of Γm + Then Ext Γm+ BP M is the cohomology of this chain complex We will abbreviate Ext Γm+ BP M to Ext Γm+ M as usual By the change-of-ring isomorphism [Rav86 Theorem 6] we have Ext Γm+ M n = Ext BP BP M n BP T m = Ext Σn Kn Kn T m This object is already known by [Rav86] Corollary 656: Theorem If n < p m + /p then Ext Γm+ M n = Kn E ĥij : i n j Z/n In particular we have Ext Γm+ M = K E ĥij : i j Z/ From this information we can get the structure of Ext Γm+ M using the Bockstein spectral sequence Lemma cf [MRW77] Remark Assume that there exists a k -subule B t of Ext t Γm+ M for each t < N such that the following sequence is exact: Ext Γm+ M B v B δ Ext Γm+ M /v /v /v B N v B N δ Ext N Γm+ M where δ is the restriction of the coboundary map δ : Ext t Γm+ M Ext t+ Γm+ M Then the inclusion map i t : B t Ext t Γm+ M is an isomorphism between k -ules for each t < N Proof Because Ext t Γm+ M is a v -torsion ule we can filter B t and Ext t Γm+ M as } P t j = x B t : v j x = and Q t j = x Ext t } Γm+ M : v j x = Assume that the inclusion i k is an isomorphism for k t the t = case is obvious and consider the following commutative ladder diagram:

5 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 5 B t δ Ext t Γm+ M /v Pt j Pt j Ext t+ = i t i t i t M δ Ext t Γm+ M /v v Qt j Qt j Ext t Γm+ Using the Five Lemma we can show that P t j is isomorphic to Q t j j by induction on j v δ Γm+ M δ Ext t+ Γm+ M In and 67 we will define elements x k of v BP /I which is congruent to v spk ulo v and integers âk in and 6 such that each x s k /vl is a cycle of Ext Γm+ M for all l âk Then the structure of B is expressed as follows: Lemma As a k -ule B = k x s k v bak : k s > and p s } K / k is isomorphic to Ext Γm+ M if the set } x s k δ : k s > and p s v bak Ext Γm+ M is linearly independent where δ is a coboundary map in Lemma Proof We will show that the following sequence is exact Ext Γm+ M B v B δ Ext Γm+ M /v The only part of this that is not obvious is that Ker δ Im v To show this separate the Z/p-basis of B into two parts } x s k A = : k s > and p s and v bak } x s } B = k v l : k s > p s and l < âk v j : j > Then it is obvious that δx λ Ext Γm+ M for xλ A and that δy µ = Ext Γm+ M for yµ B Thus for any element z = λ a λx λ + µ b µy µ of B a λ b µ Z/p we have δz = λ a λδx λ The condition implies that all a λ are zero when δz = and so v µ b µy µ /v = z This completes the proof

6 6 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL Elementary calculations In this section we will introduce elements ŵ and ŵ 5 which we need to define our x k First we recall the right unit on v i Lemma Assume that p and m In Γm + /p v the right unit map η R : Â Γ on the element v i is given by η v i = v i for i η v = v + v t p vpω t η v = v + v t p + v t p vpω t v p ω t and η v 5 v 5 + v + v t p + v t p v pω t v p ω t v p+ v pp add v t 7 when p m = Moreover when p and m we have η v 6 v 6 + v 5 t p5 + v + v t p vpω t 5 v p ω t p v p+ t v p ω t v Define elements ŵ i for i 5 by ŵ = v v and ŵ 5 = v v 5 v ŵ p + vp+ v p p ŵ Then we have the following lemma Lemma For any prime p we have dŵ = t p + v v dŵ 5 t p + v v +v p+ v p t vpω p t v vωp t v p ω v t for m t v p ω t v p v p v t p + ω p vp v t p t p t for m vp p vp+ v pω v p p add vv 7 t when p m = v p+ Proof dŵ is easily computed using Lemma For ŵ 5 we find that d v 5 v + v t p + v t p vpω t v p ω t v p+ add v t 7 for p m = v pp t p v p+ We can read off the fact that dv = for m and d v v from Lemma We have

7 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 7 d v ŵ p v dv p+ v pp ŵ v p+ v pp t p + vp v p p t vpω p p t vpω + v v t p v p ω t v p+ Summing these congruences and multiplying v gives dŵ 5 d x k for k 5 For all p and m we can construct x k k 5 and compute differentials on these one at a time Lemma Define integers âk as in Theorem and bk for k 5 by âk = p k if k p + p k if k 5 and bk = if k p k if k 5 Define elements as in Theorem x k for k 5 by x = v x = x p vp ŵp vp v pω p x and x k = x p k for k Lemma Assume that p and m In the ule M Γm + we have d x k v bak t pk+ t pk+ v bak v b bk v pk+ ω v pk p pω In particular these equivalences hold ulo v +bak for k for k 5 Proof We obtain d x k for k from η R v Lemma For d x we have d x p vp p5 t d v p ŵp v p p5 t + vp d v p v pω p x = v p vpω p t p v p vpω p vpω t t p Notice that p + pω < minp ω pω + p } = p ω for all p and m Summing congruences in we have d x v p +p v p + vp +pω v pω p t v p ω v pω+p v p ω

8 8 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL Noticing that the inequality p + pω > p + p holds only if m we find that d x v p +p v p p v p +p v p t v p +p v p +pω for m = for m By assumption we may consider only for the case that m and set â = p + p The formulas for k 5 are obvious To define x k for higher k we will prepare some lemmas in the next section The definitions of x k k 6 and computations of the chromatic differential d on x k are separated into sections 7 according to the value of m The results are stated in section 6 5 Some lemmas Here we will prove some lemmas for later use In the rest of this paper we will treat Ext Γm+M whenever the condition in Theorem 5 < p m + /p is satisfied This is equivalent to m for p m for p = Lemma 5 There is an element W such that dw + vp v p vp v p v p p5 d x 5 v pω p v p ω p +p ω t v epm where p p if m = e p m = p p if p m = p + p p otherwise Proof We find the following congruences:

9 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 9 dŵ p 5 + vp dv ba v p v p5 ω x v p v p v p v p5 ω ω p vp v p p7 t + ω p p vp v p p5 t v p p5 vp+p v pω p v p p5 v p+p v p5 v p p +v p+p v p v p5 ω d v ba v p ω p +p x v p ω p d v p5 pba5 v p v p xp 5 v p5 v p p v p d v pba v p ω p p v p xp v p ω p p vpω p v p dv p+p ba v pω p v p p5 x v p+p v pω p v p ω p +p ω t t p7 v p p5 t p5 v pω p v p ω p p v pω p v pω p v p+p p Summing these congruences gives the desired formula The integer e p m is the minimum exponent of v in these indeterminacies Lemma 5 There is an element Y such that dy v pωp + t v p vp v p p d x + v pω ω vp t v e pm where e p m = p p for p m = p + p otherwise Proof We find the following congruences:

10 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL dŵ t p + v v p t vpω t d v p x t p dv v p ω p ŵ p 5 v v p ω p v p v p p +v p+p v p dv ba v p p ω p x v v p ω p t p5 + vp v p dv p ω v p ω x = v p ω v p ω v t p vpω t dv +p ba v p ω v p x v +p v pω+p d v p v p v p x v v p v p t p5 + vp ω p p v p ω v pω p v p ω t p ω p vp v p v p v p p vp+p v pω p v p p t p v +p+p v p v p dv p+p v p ω p p v p p x v +p+p v p ω p p The sum of the right sides is v pω t +v p + v p ω t p t p5 p5 t p t v p p +vp+p + v p ω+p t p v p p t p v +pω p p v +pω p v +pω p v p+p vpω v p ω v p ω Multiplying v p ω+ gives the desired formula The integer e p m is the minimium exponent of v in these indeterminacies ŵ 6 = Now we define ŵ 6 by v 6 v pba v 5 x p + v ba v b b v x for m above expression + v p v p ωp + x p Y p v pωp + x p Y for m = Lemma 5 In the ule v BP /I Γm + we have dŵ 6 t p v vpω 5 t v ω vp t v p ω t v t Proof We find that 55 d v 6 v 5 t p5 + v + v t p dv ba v b b v x v vp 5 t v p ω v v pω p t v p ω t

11 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION Notice that η R v 5 = v 5 for m and so we have 56 d v pba v 5 x p = v pba dv 5 x p + ηv 5d x p = v pba v 5 d x p v 5 t p5 v pω p Multiplying the sum of 55 and 56 by v we obtain the desired formula for m For m = we know that η R v 5 = η R v = v 5 + v t p t vp and so we have d v p v 5 x p v p v t p v p t vp x p + v 5 + v t p v t p t vp x p + vp v 5 t p5 vp t v p p5 t 57 v p x p t p + xp t v 5 t p5 v We also find that 58 dv p v p ωp + x p Y p v p x p t p d v pωp + x p Y xp t v Multiplying the sum of and 58 by v gives the desired formula for m = Using this ŵ 6 we define X by X = ŵ p 6 v ba v p v p5 ω 5 x v ba v p +p v p6 ω x +v pω p v ω pp pωp + v p6 ω Y Then we have Proposition 59 For m dx ω p vp v p Proof We find the following congruences

12 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL 5 dŵ p 6 v p dv ba v p v p5 ω 5 x v p v p5 ω 5 v p5 ω 5 v p v p6 ω ω p vp d v ba v p +p v p6 ω x v ba v p +p v p6 ω v ba dv pω p v ω pp pωp + v p6 ω Y v pω p v ω pp v p6 ω t v p v pω p v p + +pω vp v pω p v p ω p p v pω t Summing these congruences gives the desired formula In case that m = we have the following analogy instead Proposition 5 For m = there is an element X such that d X ω p vp vp vpωp p p 6 v p6 ω Proof Instead of the last congruence of 5 we find that v p p dv pω p v ω pp pωp + v p6 ω Y v p p v ω pp pωp + v p6 ω This gives dx ω p vp Then define X by v pωp + vpω vω pp pωp + v p6 ω t v p + v pω p v p p This gives the desired formula X = X v p vpωp p p 6 v p6 ω X Define integers a i 6 and b i 6 for i = by and the element M by a 6 = p p + p + a 6 = p + p 5 + pω b 6 = p + p b 6 = p + p + pωp p + M = x p 5 + vp+p5 v p5 W + v a6 p v b6 Y v a6+pω p v b6 pωp + v p ω Y

13 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION Proposition 5 For m we have dm v a 6 v b 6 xp 5 d x 5 +v a 6+pω v b 6 pωp + v p ω v a 6 v a 6+pω p v b 6 pωp + v b 6 v p ω xp d x t d x xp v epm where ep m = p + p 5 + min pω p p e p m pω p + e p m Proof We find that d x p 5 vp+p5 dv p+p5 v p5 W v p5 v p+p5 v p5 dv a 6 p v b 6 Y v p +pω p + va6 v b6 vp p5 d x 5 v a 6 p v p ω p p +p ω v a6 p v b6+pωp + v a6 v b6 t t vp p d x d v a6+pω p v b6 pωp + v p ω Y v a 6+pω p v b 6 v p ω t +v a 6+pω v b 6 pωp + v p ω v a 6+pω p v b 6 pωp + v p ω v p+p5 +e pm + v a6+pω p v b6 v p ω v a 6 p +e pm vp p d x t Summing these congruences gives the desired formula 6 d x k for k 6 Define integers c i and d i for i = by t v a 6+pω p +e pm 6 c = a 6 + â5 c = a 6 + â d = b 6 + b5 d = b 6 + b and integers li for i = by

14 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL 6 l = l = if a 6 a 6 if a 6 > a 6; if c c if c > c Then we define âk and bk for k 6 they were defined for k 5 in by 6 p k 6 a l 6 for all m and 6 k 8 p âk = â6 + c l for all m and k = 9 p k 9 â9 â5 + âk for m 5 and k p k 9 â9 â + âk 6 for m and k ; bk = p k 6 b l 6 for m and 6 k 8 p b6 + p ω p 6 + d l for m and k = 9 p k 6 b 6 p ω p for m = and 6 k 7 p b 6 for m = and k = 8 p b6 + p ω p + b for m = and k = 9 p k 9 b9 b5 + bk for m 5 and k p k 9 b9 b + bk 6 for m and k ; Define ŷ i for i by ŷ = v p ba6 v p b b6 X p v p ba6 v p b b6+p ω p 6 x 6 ŷ = v p ba6+a 6 v p b b6+p ω p 6 +b 6 p ω p X v ba7 v b b7 x 7 ŷ = v p ba6 v p b b6 X ŷ = v p ba6 v p b b6+p ω p x 6 For m 5 define x k v BP for k 6 by M pk 6 for 6 k 8 x p 8 + ŷ for k = 9 unless p m = 5 6 x k = x p 8 + ŷ + ŷ for k = 9 and p m = 5 x p k + vbak bak v b bk b bk x p k x k x p k 5 for k For m = define x k for k 6 by 65 x k = M pk 6 for 6 k 8 x p 8 + ŷ + ŷ for k = 9 x p k vbak bak 6 v b bk b bk 6 x p k 6 x k 6 x p k 7 for k

15 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 5 For m = define x k for k 6 by M v ba6 v b6 p ω p X for k = 6 x p 6 for k = 7 66 x k = M p for k = 8 x p 8 + ŷ + ŷ for k = 9 x p k vbak bak 6 v b bk b bk 6 x p k 6 x k 6 x p k 7 for k For m = define x k for k 6 by M v ba6 v b b6 p ω p X + v p p p 6 vpωp v p6 ω for k = 6 x p 6 for k = 7 67 x k = M p for k = 8 x p 8 + ŷ for k = 9 x p k vbak bak 6 v b bk b bk 6 x p k 6 x k 6 x p k 7 for k Then we have Lemma 68 Assume that p and m satisfy 5 Modulo the differential on x k k 6 is expressed as v bak v b bk t pk v bak v b bk t pk vbak v pk 6 b 6 p ω p k v ba8 v b b8 t p8 vba8 v p b 6 v bak v b bk t pk+ v bak v b bk v +bak for m and 6 k 8 t pk+ for m = and 6 k 7 for m = and k = 8 for m = and 6 k 7 for m = and k = 8 v bak bak v b bk b bk x p k d x k for m 5 and k 9 v bak bak 6 v b bk b bk 6 x p k 6 d x k 6 for m and k 9 We will prove this lemma in Sections and Then our main result is Theorem 69 As a k -ule Ext Γm+ M is the direct sum of a the cyclic subules generated by x s k /vbak for k and s Z pz where x k and âk are elements defined in section and this section b K / k generated by /v j for j 7 Proof of Lemma 68 for m 5 Notice that the exponent ep m in Proposition 5 is pω p p p + p 5 + min p + p p pω + p = p + p5 + p + p p which is always larger than a 6 + â5 + p + p + = p + p 5 + p 5 + p + p + p +

16 6 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL for all primes p By Proposition 5 we have d x 6 v a6 v b6 xp 5 d x 5 v a6 v b6 if p m = 5 v a 6 v b 6 xp 5 d x 5 otherwise v a6+ba5+p +p+ In particular we have d x 6 v ba6 v b b6 v ba6+ba5 in both cases For d x 9 we find that 7 d x p 8 vp ba6 v p b b6 t p9 Assume that p m 5 For dŷ we find that v p ba6+ba5 7 d v p ba6 v p b b6 X p v p ba6 v p b b6 t p9 ω p 6 vp d v p ba6 v p b b6+p ω p 6 x 6 v p ba6 v p b b6+p ω p 6 xp 5 d x 5 v p ba6+p 6 v ba9+p +p+ Notice that the second formula fails if p m = 5 This gives 7 dŷ v p ba6 v p b b6 t p9 + ba6 vp v p b b6+p ω p 6 x p v ba9+p +p+ Summing 7 and 7 we obtain d x 9 v ba9 ba5 v b b9 b b5 x p 5 d x 5 5 d x 5 v ba9+p +p+ unless p m = 5 We will see that a similar congruence holds even for p m = 5 after an appropriate change of x 9 Define integers nk by p + p + for k p + p for k nk = p + p for k p + p for k Instead of 6 we may define integers âk for k 9 inductively on k by pâk + p 5 + p for k âk = pâk + p for k pâk for k and

17 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 7 This suggests that we may compute d x k ulo v bak+nk inductively on k Assume that the congruence 7 d x k v pk ba9 ba5 v pk b b9 b b5 x p k 5 d x k 5 holds ulo v bak +nk For k it follows by direct calculations Moreover this gives d x k whenever k or since x k = x p k In other cases we denote x k x p k by ẑ k Notice that ẑ k is related with ẑ k by Then dẑ k is dẑ k = v bak bak v b bk b bk ẑ k = v bak bak v b bk b bk x p k ẑk d = v bak bak v b bk b bk d x p k ẑk x p k ẑ k + η R x p k d ẑ k Notice that ẑ k divided by âk âk 8 so that we can ignore d x p k because âk + nk âk âk âk âk 8 Thus we have 75 d ẑ k v bak bak v b bk b bk x p k dẑ k On the other hand by assumption 7 we have = nk + âk 8 < âk d x p k ba9 ba5 vpk 9 v pk 9 b b9 b b5 x p p k 5 d x p k 5 v bak bak v b bk b bk 76 x p k d xp k 5 v bak+nk v bak+nk Summing 75 and 76 we obtain as desired d x k v bak bak v b bk b bk x p k d x k v bak+nk Now we consider the p m = 5 case instead of the second one in 7: We have the following congruence 77 d v p ba6 v p b b6+p ω p 6 x 6 v p ba6 v p b b6+p ω p 6 xp 5 d x 5 +v p ba6+a 6 v p b b6+p ω p 6 +b 6 v ba9+p +p+

18 8 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL Define ŷ by ŷ = v p ba6+a 6 v p b b6+p ω p 6 +b 6 p ω p X v ba7 v b b7 x 7 We see that the differential on X v ba7 v b b7 x 7 is expressed as This gives dx v ba7 v b b7 x 7 v p ω p v p 78 dŷ v p ba6+a 6 v p b b6+p ω p 6 +b 6 v p ba6+a 6+p Summing 77 and 78 we have the same one as the second of 7 8 Proof of Lemma 68 for m = Notice that the exponent ep m in Proposition 5 is p 5 p p p + p 5 + min p + p p p 5 + p = p + p5 + p 5 + p which is always greater than or equal to a 6 + â + p + = p + p 5 + p 5 + p + p + p + for all primes p By Proposition 5 we have d x 6 v a 6 v b 6 va 6 v b 6 v a 6+ba+p+ In particular we have d x 6 v ba6 for any prime p For d x 9 we have 8 d x p 8 vp ba6 Instead of 7 we have v b b6 v p b b6 t p9 xp v a 6 d x v p a 6 8 dŷ v p ba6 v p b b6 t p9 + vba9 ba v b b9 b b v ba9+p+ Moreover as same as 78 we have 8 dŷ v ba9 ba v b b9 b b Summing 8 8 and 8 we obtain xp v ba9 ba+p d x

19 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 9 d x 9 v ba9 ba v b b9 b b x p d x v ba9+p+ for any prime p Define integers nk by 8 nk = p + for k 6 p + p for k 6 p + p for k 5 6 p p + for k 6 p p + p for k 6 p p + p for k 6 Instead of 6 we may define integers âk for k 9 inductively on k by pâk + p 5 p + p + p for k 6 âk = pâk + p for k 6 pâk for k This suggests that we may compute d x k ulo v bak+nk inductively on k Assume that the congruences 85 d x k v pk ba9 ba v pk b b9 b b x p k 7 d x k 7 holds ulo v bak +nk For k 6 case it follows by direct calculations Moreover this gives d x k whenever k since x k = x p k In other cases we denote x k x p k by ẑ k Notice that ẑ k is related with ẑ k 6 by ẑ k = v bak bak 6 v b bk b bk 6 x p k 6ẑk 6 Then dẑ k is expressed as d ẑ k = v bak bak 6 v b bk b bk 6 d = v bak bak 6 v b bk b bk 6 d x p k 6ẑk 6 x p k 6 ẑ k 6 + η R x p k 6 Notice that ẑ k 6 is divided by v bak 6 bak so that we can ignore d because } d ẑ k 6 x p k 6 Thus we have âk + nk âk âk 6 âk 6 âk = nk + âk < âk 6 86 d ẑ k v bak bak 6 v b bk b bk 6 x p k 6 dẑ k 6 v bak+nk

20 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL On the other hand by assumption 85 we have d x p k vbak bak 6 v b bk b bk 6 x p p k 7 d x p k 7 v bak bak 6 v b bk b bk 6 87 x p k 6 d xp k 7 v bak+nk Summing 86 and 87 we obtain as desired d x k v bak bak 6 v b bk b bk 6 x p k 6 d x k 6 v bak+nk 9 Proof of Lemma 68 for m = Notice that the exponent ep m in Proposition 5 is p p p p + p 5 + min gp p + p = p + p5 + p + p which is always greater than or equal to a 6 + â + p + = p 6 + p 5 + p + p + p + p + for all primes p By Proposition 5 and 68 we have dm v a 6 v b 6 va 6 v b 6 xp d x 9 d v a 6 v b 6 p ω p X v a6 v b6 p ω p ω p vp v ba6+p This gives d x 6 v a 6 v b 6 + va 6 v b 6 x p v a 6+ba+p+ In particular we have d x 6 v ba6 d x v a 6 v b b6 ω p vb6 p v a6+ba+p+ v b 6 p ω p v ba6+ba Using the first congruence in 9 we find that d x 8 v p ba6 v p ba6+ba For d x 9 we have 9 d x p 8 vp ba6 dŷ v p ba6 v p b b6 t p8 b 6 vp v p b b6 t p9 b 6 vp v p b b6 t p9 + vba9 ba v b b9 b b v p ba6+ba xp v ba9+p+ d x

21 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION Moreover we find that 9 dŷ v p ba6 v p b 6 Summing 9 and 9 we have ω p vp d x 9 v ba9 ba v b b9 b b x p d x for all primes p Define integers nk by 9 nk = p + for k 6 p + p for k 6 p + p for k 5 6 p for k 6 p for k 6 p 5 for k 6 v p ba6+p v ba9+p+ Instead of 6 we may define integers âk for k 9 inductively on k by pâk + p + p for k 6 âk = pâk + p for k 6 pâk for k This suggests that we may compute dx k ulo v bak+nk inductively on k We can prove this proposition in the same fashion as in case that m = Proof of Lemma 68 for m = Notice that the exponent ep m in Proposition 5 is p p p p + p 5 + min p p p p + fp which is larger than a 6 + â + = p + p 5 + p + p + p + only if p > We define N by N = M v a 6 v b 6 p ω p X By Proposition 5 we have This gives dm v a 6 v b 6 xp d x d v a 6 v b 6 p ω p X v a6 v b6 p ω p ω p vp vp v a 6+ba+ p p 6 vpωp v p6 ω v a 6+p p

22 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL dn v a 6 v b 6 x p d x v a6 v b6 p ω p p p 6 vp vpωp v a6+ba+ Then it is easy to see that v p6 ω d x 6 v a 6 v b 6 x p for p > In particular we have d x v p ω p v a 6+ba+ d x 6 v ba6 v b b6 v ba6+ba for p > Using the first congruence in we find that For d x 9 we have d x 8 v p ba6 v p b 6 v p ba6+p d x p 8 vp ba6 v p b 6 dŷ v ba9 ba v b b9 b b x p v p ba6+p d x v p ω p v ba9+ Summing we obtain d x 9 v ba9 ba v b b9 b b x p d x for p > Define integers nk by nk = for k 6 p for k 6 p for k 5 6 p for k 6 p for k 6 p 5 for k 6 v p ba6+ba+ Instead of 6 we may define integers âk for k 9 inductively on k by pâk + p + p for k 6 âk = pâk + p for k 6 pâk for k This suggests that we may compute dx k ulo v bak+nk inductively on k We can prove this proposition in the same fashion as in case that m =

23 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION Proof of the main theorem Here we will prove Theorem 69 First we consider the case that m 5 Theorem included By Lemma it suffices to show that the set δ x s k/v bak in linearly independent over } : k s > and p s k /v = Z/p[v v m v ] Ext Γm+ M It follows from that this group is the free K -ule on the nine classes represented by } t t p t p t t p t p t t p t p In Γm + /I we have η R v = v + v t p vpω t η R v 5 = v 5 + v + v t p vpω t v p ω t and η R v 6 = v 6 + v 5 t p5 + v + v t p vpω so in Ext Γm+M we have t 5 v p ω t v p ω t and t p = v pω t t p = v p ω t + v v pω t v t p = v p ω t + v v p ω t + v pω t 5 v v 5 t p5 This means that for m we can replace the K -basis of Ext of with } t p t p t p5 t p8 so its basis over k /v is } v t t p vt t p vt vt vt t p5 vt vt vt vt t p8 : t Now define integers ĉk for k by for k 8 ĉ k = p p k + ĉk for k 9 For k > write k = k + k with 5 k 8 Then the above is equivalent to ĉk = p p k p k p

24 IPPEI ICHIGI HIROFUMI NAKAI AND DOUGLAS C RAVENEL Then Lemmas and 68 imply that ulo that d x k ±v bak v b bk v bck v +ak x s k δ v bak t p+k for k t p+k for k for k > and k for k > and k for k > and k t p8 for k > and k This and the multiplicative property of the right unit imply = ±sv b bk v s pk +bck t p+k for k t p+k for k for k > and k for k > and k for k > and k t p8 for k > and k By Lemma it suffices to show that these elements with k and s > not divisible by p are linearly independent over k The ones for k are clearly independent of those for k > so it suffices to consider the exponents of v above for k > ie to show that the set } v s pk +bck : k > s > p s is linearly independent over k /v For a fixed value of k the exponents appearing in are congruent to ĉk ulo p k but not congruent since p s to p k + ĉk ulo p k+ Now implies that for k > ĉk p k p pk p p k+ so s p k + ĉk sp k p pk p p k+ Hence our condition is that the exponents associated with k are congruent to p pk p ulo p k but not ulo p k+ and these conditions are mutually exclusive for differing k Proof for m The argument is the same subject to the following changes The integers ĉk are defined by ĉk = for k 8 p p k 6 + ĉk 6 for k 9 For k > write k = k + 6k with k 8 Then the above is equivalent to

25 THE CHROMATIC EXT GROUPS Ext Γm+ BP M DRAFT VERSION 5 Then gets replaced by x s k δ = ±sv b bk v bak ĉk = p p k v s pk +bck p 6k p 6 t p+k for k t p+k for k 5 and t pk for 6 k 8 and Notice that there is another term for m = and 6 k 8 We can ignore it because it is linear independent with the above coboundary We can argue for linear independence as before References [MRW77] H R Miller D C Ravenel and W S Wilson Periodic phenomena in the Adams- Novikov spectral sequence Annals of Mathematics 6: [MW76] H R Miller and W S Wilson On Novikov s Ext ulo an invariant prime ideal Topology 5: 976 [NR] H Nakai and D C Ravenel The structure of the general chromatic E -term Ext Γm+ m preprint [Rav] D C Ravenel The microstable Adams-Novikov spectral sequence Contemp Math 65:9 9 [Rav86] D C Ravenel Complex Cobordism and Stable Homotopy Groups of Spheres Academic Press New York 986 [Shi] KShimomura The homotopy groups π L T n V To appear [Shi] KShimomura The homotopy groups π L nt m V n To appear Kochi University Japan Himeji Institute of Technology Japan and University of Rochester Rochester NY 67