FINITE RESOLUTIONS DIGEST NORTHWESTERN TAF SEMINAR FEBRUARY 21, Introduction

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1 FINITE RESOLUTIONS DIGEST NORTHWESTERN TAF SEMINAR FEBRUARY, 04 Remark 0.. My main reference is A Resolution of the Kpq-Local Sphere at the Pme 3 by Goerss, Henn, Mahowald and Rezk, and Finite Resolutions of the Kpnq-local sphere by Henn.. Introduction Let E n be Morava E-theory and G n be the Morava Stabilzier group. Recall that G n acts on E n. Devinatz and Hopkins showed that for any subgroup H G n, we can form the fixed point spectrum En hh. Further, E hgn n L Kpnq S 0. Let s first look at the case n. In this case, the Morava Stabilizer group is just G Z p, and The action of Z p E Z p ru s. is just multiplication on u. Throughout, F G n will be a maximal finite subgroup of G n. Now # Z C Z if p ; p Z p C p if p, So, F # C if p ; C p if p, Let Z p rrg n ss be the completed group ng and for a finite subgroup H Z p rrg n {Hss Z p rrg n ss b ZprHs Z p. Note that if p H, this is a projective Z p rrg n ss-modules.

2 Now, in the case n, there is an exact sequence 0 Ñ Z p rrg {F ss l e ÝÝÑ Z p rrg {F ss Ñ Z p Ñ 0, where e is the canonical generator of Z p rrg {F ss and l p is a topological generator for G {F. If p is odd, this is a projective resolution of G. In either case, these resolutions have topological realization L Kpq S 0 E h G Ñ E hf ψ l id ÝÝÝÑ E hf. If p is odd, E hf is the Adams summand for K Z p and if p is even, it is KO Z. In this talk, I want to make precise this idea of realizing a finite algebraic resolution to a topological one to higher n s.. Morava Stabilizer Group If Γ is the Honda formal group law over F p n with p-sees rpspxq x pn, then S n AutpΓq. I want to highlight a couple features of S n. Let p n -root of unity ζ. Then rq p rζs : Q p s n and Z p rζs Q p rζs is the ng of integers in this field extension. Let W W pf p nq Z p rζs. Let φ be the Fröbenius on W, that is φpζq ζ p. Let for a P W. Then O n W S { ps n p, Sa φpaqsq S n O n. We can also think of the automorphisms of the pair pγ, F p nq, which is the big Morava Stabilizer group G n S n GalpF p n { F p q, where Gal has the natural action on S n O n. Recall, F G n is the maximal finite subgroup of G n. In W, there is a copy of C p n generated by ζ. Then

3 FINITE RESOLUTIONS DIGEST NORTHWESTERN TAF SEMINAR FEBRUARY, 04 3 Lemma.. If p n, then F C p n GalpF p n { F p q. If p n, F contains a cyclic group of order divisible by p, or a quaternion group. 3. Morava E-theory Let E n be the spectrum representing deformations of a formal group law of height n over complete local ngs. E n is a complex oented ng spectrum with pe n q Wrru,..., u ssru n s for u, and whose formal group law F En px, yq reduces to Γ over F p n. This is a kind of coveng space so that the group G n acts on pe n q. Recall that pe n q X π L Kpnq pe n ^ Xq. For finite spectra, this definition coincides with the usual one as E n ^ X is already Kpnq-local, since E n is Kpnq-local. Definition 3. (Morava modules). A Morava module is a complete pe n q -module M with a compatible action of G n. Examples are pe n q X for X such that Kpnq X is in even degrees. Let H be a closed subgroup of G n. Then the abelian group of continuous maps Hom c pg n {H, pe n q q has the structure of a Morava module and pe n q E hh n Hom c pg n {H, pe n q q as Morava modules. In particular, there is an isomorphism of Morava modules, pe n q E n Hom c pg n, pe n q q 4. Finite Resolutions Definition 4.. A spectrum I is E n -injective if the map S 0 ^ I Ñ L Kpnq pe n ^ Iq induced by the unit of E n is split.

4 4 A sequence X Ñ Y Ñ Z is E n -exact if for every E n -injective I, is an exact sequence of abelian groups. rx, Is Ð ry, Is Ð rz, Is An E n -resolution is a sequence of E n -injectives I s which is E n -exact. Ñ X Ñ I 0 Ñ I Ñ I Ñ... From such resolutions, once can construct E n -Adams Novikov spectral sequences which converge to the homotopy groups of L Kpnq S 0. Henn attbutes the following folklore theorem to Morava. Theorem 4. (Morava). If p n and p n, then L Kpnq S 0 admits an E n -resolution of length n. Proof sketch. The key idea is that if p n and p n, the cohomological dimension of G n is n. So there is a resolution 0 Ñ P n Ñ... Ñ P 0 Ñ Z p Ñ 0 where P i are a finitely generated projective Z p rrg n ss-modules. Now consider 0 Ñ pe n q Ñ Hom c Z pp 0, pe n q q Ñ... Ñ Hom c Z pp n, pe n q q Ñ 0 Now, since P i is a projective Z p rrg n ss-module, it is a direct summand of à r i Z p rrg n ss for some r i. So, it corresponds to an idempotent e i. As Morava modules, we have an isomorphism à à Hom c Z p Z p rrg n ss, pe n q q Hom c pg n, pe n q q pe n q E n. r i r i So e i induces a map pe n q j E n e ÝÑ i pen q j E n. j

5 FINITE RESOLUTIONS DIGEST NORTHWESTERN TAF SEMINAR FEBRUARY, 04 5 Finally, the Hurewicz homomorphism () re n, E n s Ñ Hom penq En ppe n q E n, pe n q E n q is an isomorphism, so this gives a map e i : E n Ñ j j E n and the spectrums X i telescopepe i q. which is the colimit of the iterations of e i. We get isomorphisms of Morava modules pe n q X i Hom c pp i, pe n q q Using () we can realize the maps in the resolution topologically. One needs to show that these are E n -injective and that the resolution is E n -exact. This approach is very abstract and does not directly produce a resolution with which one can compute. So we need to get our hands dirty. We need to produce algebraic resolutions in which we understand the P i s, and identify the spectrum X i corresponding to the idempotents e i. For this, it is useful to start with half the sphere. 5. Half the Sphere Recall that S n O n. The natural action of S n on O n gives a ρ : S n Ñ GL n pw pf p nqq. Composing ρ with the determinant gives a map det ρ : S n Ñ Z p C p Z p. Projecting onto Z p, we get a map, whose kernel is denoted S n Ñ S n Ñ S n Ñ Z p Ñ 0. The group G n is just G n S n GalpF p n { F p q. There always is a fiber sequence L Kpnq S n E h Gn n Ñ E h G n n Ñ E h G n n.

6 6 For this reason, we call E h G n n half the sphere. Corollary 5.. If p n and p n, then E h G n n admits an E n -resolution of length n. Proof. Since Z p has cohomological dimension, G n has cohomological dimension n. 6. The case when n 6.. The case when n, p 3. Recall that F C p GalpF p { F p q. Theorem 6. (Henn). Assume n, p 3, then there is an E -resolution where This can be extended to a resolution of L Kpq S 0. Ñ E h G Ñ E hf Ñ X Ñ X Ñ E hf Ñ X Σ pp q E hf _ Σ pp pq E hf. To prove this theorem, we need an algebraic resolution. Let λ p be the Z p rf s-module with underlying module Z p -module W Z p rζs and action of ζ C p W and Fröbenius acts as usual on W. Let g paq g p a for a P W and g P C p λ p Ò G F Z prrg ss b ZprF s λ p. Theorem 6. (Ravenel/Henn). There is an projective resolution of the tvial Z p rrg ss-module Z p 0 Ñ Z p rrg {F ss Ñ λ p Ò G F Ñ λ p Ò G F Ñ Z prrg {F ss Ñ Z p Ñ 0. Then this can be upgraded to a topological resolution as before, where we apply Hom c p, pe q q and prove that and prove that pe q E hf Hom c pg {F, pe q q Hom c Z p pz p rrg {F ss, pe q q pe q pσ pp q E hf q ` pe q pσ pp pq E hf q Hom c Z p pλ p Ò G F, pe q q. 6.. Constructing Algebraic resolutions. Nakayama The assumptions in the lemma are technical and will be satisfied in our applications.

7 FINITE RESOLUTIONS DIGEST NORTHWESTERN TAF SEMINAR FEBRUARY, 04 7 Lemma 6.3 (Nakayama). Let G be a finitely generated p-profinite group and f : M Ñ N a morphism of complete Z p rrgss-modules. If is surjective, then f is surjective. Further, if F p b ZprrGss f : F p b ZprrGss M Ñ F p b ZprrGss N TorpF p, fq : Tor ZprrGss q pf p, Mq Ñ Tor ZprrGss pf p, Nq is isomorphism for q 0 and surjective for q, then f is an isomorphism. q Poincaré Duality Subgroup Lemma 6.4. Let S be the p-sylow subgroup of G. For n, p 3, S group of dimension 3. As F -modules, we have isomorphisms is a poincaré duality H 0 ps ; F p q H 3 ps ; F p q F p F p b ZprrS ss Z prrg {F ss and H ps ; F p q H ps ; F q F p b Zp λ p F p b ZprrS ss λ p Ò G F. We will produce a diagram Z p rrg {F ss N 3 λ p Ò G F N λ p Ò G F N Z p rrg {F ss Z p Define N to be the kernel of the augmentation, so that 0 Ñ N Ñ Z p rrg {F ss ɛ ÝÑ Z p Ñ 0. Note that, as S -modules, Z p rrg {F ss Z p rrs ss.

8 8 Since Z p rrs ss has no higher S -homology, this induces an exact sequence 0 H ps, F pq H 0 ps, N {pq H 0 ps, F prrg {F ssq H 0 ps, F pq 0 F p b ZprrS ss λ p Ò G F F p b ZprrS ss N By Nakayama, this isomorphism lifts to a surjective map 0 Ñ N Ñ λ p Ò G F Ñ N Ñ 0, with kernel N. Similarly, we can construct a surjective map 0 Ñ N 3 Ñ λ p Ò G F Ñ N Ñ 0, with kernel N 3. Finally, we get an exact sequence 0 H 3 ps, F pq H 0 ps, N 3{pq H 0 ps, λ p Ò G F {pq H 0 ps, N {pq 0 F p b ZprrS ss Z prrg {F ss F p b ZprrS ss N 3 Using Nakayama again, this time we get an isomorphism Splice these all together to get the resolution. Z p rrg {F ss ÝÑ N The case when n, p 3 - Difficulties F contains 3-torsion. So Z 3 rrg {F ss is not projective. Lifting to a topological resolution and identifying the relevant spectra is harder. S is not a Poincaré duality group. H ps q is more complicated. Identifying N 3 is hard. The result will not consist of E -injectives. Obtaining a Adams-Novikov style spectral sequence is harder. We need to prove that some Toda brackets are zero and then build a tower of spectra. Let F be a maximal finite subgroup of G (F has order 4). Let D be the -torsion in G. Let χ be some representation of D induced by the sign representation for Z 3 on a quotient D{Q C. Theorem 7.. There is an exact sequence of G -modules, 0 Ñ Z 3 rrg {F ss Ñ χ Ò G D Ñ χ ÒG D Ñ Z 3rrG {F ss Ñ Z 3 Ñ 0.

9 with FINITE RESOLUTIONS DIGEST NORTHWESTERN TAF SEMINAR FEBRUARY, 04 9 This can be realized in the homotopy category by pe q Σ 8 E hd Hom c Z 3 pχ Ò G D, pe q q. Ñ E h G Ñ E hf Ñ Σ 8 E hd Ñ Σ 40 E hd Ñ Σ 48 E hf Ñ, where the composite of successive maps is null-homotopic and all the Toda brackets are zero. This can be extended to a similar resolution of L Kpq S 0. Corollary 7.. There is a tower of fibrations in the Kpnq-local category L Kpq S 0 X 3 X X E hf Σ 44 E hf Σ 45 E hf _ Σ 37 E hd Σ 6 E hd _ Σ 38 E hd Σ 7 E hd _ Σ E hf This gives se to a spectral sequence converging to π L Kpq S 0. The corresponding tower for half the sphere is E h G X X E hf Σ 45 E hf Σ 38 E hd Σ 7 E hd