THE HOMOTOPY OF THE K(2)-LOCAL MOORE SPECTRUM AT THE PRIME 3 REVISITED HANS-WERNER HENN, NASKO KARAMANOV AND MARK MAHOWALD Abstract.In this paper we use the approach introduced in [5] in order to* * analyze the homotopy groups of LK(2)V (0), the mod-3 Moore spectrum V (0) localized * *with respect to Morava K-theory K(2). These homotopy groups have already been calculated* * by Shimomura [12]. The results are very complicated so that an independent verificati* *on via an alternative approach is of interest. In fact, we end up with a result which is more * *precise and also differs in some of its details from that of [12]. An additional bonus of our approa* *ch is that it breaks up the result into smaller and more digestible chunks which are related to * *the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Ev* *en more, the Adams-Novikov differentials for LK(2)V (0) can be read off from those fo* *r TMF. 1. Introduction Let K(2) be the second Morava K-theory for the prime 3. For suitable spectra * *F , e.g. if F is a finite spectrum, the homotopy groups of the Bousfield localization LK(2)F can* * be calculated via the Adams-Novikov spectral sequence. By [3] this spectral sequence can be i* *dentified with the descent spectral sequence Es,t2= Hs(G2, (E2)tF ) =) sst-s(LK(2)F ) for the action of the (extended) Morava stabilizer group G2 on E2^ F where the * *action is via the Goerss-Hopkins-Miller action on the Lubin-Tate spectrum E2 (see [5] for a s* *ummary of the necessary background material). Here we just recall that the homotopy group* *s of E2 are non-canonically isomorphic to WF9[[u1]][u 1] where WF9denotes the ring of Witt * *vectors of F9, where u1 is of degree 0 and u is of degree -2. We also recall that G2 is a prof* *inite group and its action on the profinite module (E2)*F is continuous; group cohomology is, t* *hroughout this paper, taken in the continuous sense. The cohomological dimension of G2 is well-known to be infinite and therefore * *a finite pro- jective resolution of the trivial profinite G2-module Z3 cannot exist. However* *, in [5] a finite resolution of the trivial module Z3 was constructed in terms of permutation mod* *ules. More precisely, the group G2 is isomorphic to the product G12x Z3 of a central subgr* *oup (isomorphic to) Z3 and a group G12which is the kernel of a homomorphism G12! Z3, also calle* *d the reduced norm. One of the main technical achievements of [5] was the construction of a * *permutation resolution of the trivial module Z3 for the group G12. This resolution is self-* *dual in a suitable sense (cf. section 3.4) and has the form (1) 0 ! C3 ! C2 ! C1 ! C0 ! Z3 ! 0 with C0 = C3 = Z3[[G12=G24]] and C1 = C2 = Z3[[G12]] Z3[SD16]O. Here G24 is a* * certain subgroup of G12of order 24, isomorphic to the semidirect product Z=3 o Q8 of th* *e cyclic group of order 3 with a non-trivial action of the quaternion group Q8, and SD16is ano* *ther subgroup, isomorphic to the semidihedral group of order 16 (see section 2.2). Furthermore* *, O is a suitable one-dimensional representation of SD16, defined over Z3, and if S is a profinit* *e G12-set we denote the corresponding profinite permutation module by Z3[[S]]. ___________ Date: October 1, 2008. The authors would like to thank the Mittag-Leffler Institute, Northwestern Un* *iversity, Universit'e Louis Pas- teur at Strasbourg and the Ruhr-Universit"at Bochum for providing them with the* * opportunity to work together. 1 2 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald For any Z3[[G12]]-module M the resolution (1) gives rise to a first quadrant * *cohomological spectral sequence (2) Es,t1= ExttZ3[[G12]](Cs, M) =) Hs+t(G12, M) refered to in the sequel as the algebraic spectral sequence. By Shapiro's Lemma* * we have ( (3) E0,t1= E3,t1~=Ht(G24, M), E1,t1= E2,t1~=Hom Z3[SD16](O, M)t = 0 0 t > 0 . The bulk of our work is the calculation of this spectral sequence if M = (E2)*(* *V (0)). In this case the E1-term is well understood and can be interpreted in terms of modular forms* * in characteristic 3. In fact, it is determined by the following result which we include for the c* *onvenience of the reader and in which v1 denotes the well-known G2-invariant class u1u-2 2 M4. Fo* *r the definition of the other classes figuring in this result the reader is referred to section * *5.1. Theorem 1.1. Let M = (E2)*(V (0)). a) There are elements fi 2 H2(G24, M12), ff 2 H1(G24, M4) and eff2 H1(G24, M1* *2), an invertible G24-invariant element 2 M24, and an isomorphism of graded algebras H*(G24, M) ~=F3[[v61 -1]][ 1, v1, fi, ff, eff]=(ff2, eff2, v1ff, v1ef* *f, ffeff+ v1fi) . b) The ring of SD16-invariants of M is given by the subalgebra MSD16 = F3[[u4* *1]][v1, u 8] and Hom Z3[SD16](O, M) is a free MSD16-module of rank 1 with generator !2u4, i.* *e. HomZ3[SD16](O, M) ~=!2u4F3[[u41]][v1, u 8] . Remark_We note that v61 -1 is a G24-invariant class in the maximal ideal of M0 * *and hence a formal power series in v61 -1 converges in M and is also invariant. Similarly* * with u41. Of course, the name for is chosen to emphasize the close relation with the theor* *y of modular forms. For example we note that MG24 is isomorphic to the completion of M3 := F* *3[ 1, v1] with respect to the ideal generated by v61 -1, and M3 is isomorphic to the ring* * of modular forms in characteristic 3 (cf. [2] and [1]). Similarly, MSD16 is isomorphic to * *the completion of F3[v1, u 8] with respect to the ideal generated by u41= v41u8. The larger algeb* *ra F3[v1, u 4] is isomorphic to the ring M3(2) of modular forms of level 2 (in characteristic 3) * *(cf. [1]). The relation with modular forms could be made tight if in [5] we had worked with a * *version of E2 which uses a deformation of the formal group of a supersingular curve rather th* *an that of the Honda formal group. As (E2)*(V (0)) is a graded module, the spectral sequence is trigraded. The d* *ifferentials in this spectral sequence are v1-linear and continuous. Therefore d1 is completel* *y described by continuity and the following formulae in which we identify the E1-term via Theo* *rem 1.1. Theorem 1.2. There are elements _ 2,0,16k+8_ 3,0,24k k 2 E0,0,24k1,b2k+12 E1,0,16k+81,b2k+12 E1 k,2 E1 for each k 2 Z satisfying _ __ k k, b2k+1 !2u-4(2k+1),b2k+1 !2u-4(2k+1), k k (where the congruences are modulo the ideal (v61 -1) resp. (v41u8) and in the c* *ase of 0 we even have equality 0 = 0 = 1) such that 8 ><(-1)m+1b2.(3m+1)+1 k = 2m + 1 n-2 d1( k) = > (-1)m+1mv4.31 b2.3n(3m-1)+1k = 2m.3n, m 6 0 (3) : 0 k = 0 8 n _ ><(-1)nv6.31+2b3n+1(6m+1) k = 3n+1(3m + 1) n+2_ d1(b2k+1)= > (-1)nv10.31 b3n(18m+11) k = 3n(9m + 8) : 0 else The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited3 8 __ >>>(-1)m+1v21 2m_ 2k + 1 = 6m + 1 _ < (-1)m+n v4.3n13n(6m+5) 2k + 1 = 3n(18m + 17) d1(b2k+1)= > m+n+1 4.3n_ n >>:(-1) v1 3n(6m+1)2k + 1 = 3 (18m + 5) 0 else. It turns out that the d2-differential of this spectral sequence is determined* * by the following principles: it is non-trivial if and only if v1-linearity and sparseness of the* * resulting E2-term permit it, and in this case it is determined up to sign by these two properties* *. The remaining d3-differential turns out to be trivial. More precisely we have the following r* *esult. Proposition 1.3. a) The differential d2 : E0,1,*2! E2,0,*2is determined by 8 n _ ><(-1)m+n+1v6.31+1b3n+1(6m+1)k = 2.3n(3m + 1) _ d2( kff) = >(-1)m+n v10.3n+1+11b3n+1(18m+11)k = 2.3n(9m + 8) :0 else ( m 11_ d2( keff) = (-1) v1 b18m+11k = 6m + 5 0 else. b) The d3-differential is trivial. Remark_1_on_notation_Of course, the elements kff and keffare only names for e* *lements in the E2-term which are represented in the E1-term as products, but which are no long* *er products in the E2-term. Similar abuse of notation will be used in Theorem 1.4, Proposition* * 1.5, Theorem 1.6 and in section 6 and 8. Next we use that the element fi of Theorem 1.1 lifts to an element with the s* *ame name in H2(G12, M12) resp. in H2(G2, M12). In fact this latter element detects the imag* *e of fi1 2 ss10(S0) in ss10(LK(2)V (0)). The previous results yield the following E1 -term as a mod* *ule over F3[fi, v1]. Theorem 1.4. As an F3[fi, v1]-module the E1 -term of the algebraic spectral seq* *uence (2) for M = (E2)*=(3) is isomorphic to a direct sum of cyclic modules generated by the * *following elements and with the following annihilator ideals: a) For E0,*,*1we have the following generators with respective annihilator id* *eals 1 = 0 (fiv21) m fi m 6= 0 (v21) ff (v1) 2m+1ff (v1) 2.3n(3m-1)ff m 6 0 mod (3)(v1) 2meff (v1) 2m+1eff m 6 2 mod (3)(v1) 2.3n(3m+1)fffi (v1) 2.3n(3m-1)fffim 0 mod (3)(v1) 2m+1efffi m 2 mod (3) (v1) . b) For E1,*,*1we have the following generators with respective annihilator id* *eals b1 (fi) n b2.3n(3m-1)+1m 6 0 mod (3)(v4.31-2, fi) . c) For E2,*,*1we have the following generators with respective annihilator id* *eals _ 6.3n+1 b3n+1(6m+1)_ (v1 n, fi) b3n(6m+5) m 1 mod (3) (v10.31+1, fi) . 4 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald d) For E3,*,*1we have the following generators with respective annihilator id* *eals __ __2m (v21)n __3n(6m 1) (v4.31, fiv21) __mff (v1) meff (v1) . To get at H*(G12, (E2)*=(3)) we still need to know the extensions between the* * filtration quotients. They are given by the following result. Proposition 1.5. The F3[fi, v1]-module generators of the E1 -term of Theorem 1.* *4 can be lifted to elements (with the same name) in H*(G12; (E2)*=(3)) such that the relations * *defining the annihilator ideals of Theorem 1.4 continue to hold with the following exceptions v1ff = b1 v1 2.3n(9m+2)ff= (-1)m+1b2.3n+1(9m+2)+1 v1 2.3n(9m+5)ff= (-1)m+1b2.3n+1(9m+5)+1 v1 6m+1eff = (-1)m b2(9m+2)+1 v1 6m+3eff = (-1)m+1b2(9m+5)+1 _ __ fib3n+1(6m+1)_= __3n(6m+1)eff fib3n+1(18m+11)=_ __3n(18m+11)eff fib18m+11 = 6m+4ff . Apart from the last group of fi-extensions (which are simple consequences of * *the calculation of H*(G2, (E2)*=(3, u1)), cf. [4]) one can summarize the result by saying that* * nontrivial v1- extensions exist only between E0,1,*1and E1,0,*+41and there is such an extensio* *n whenever sparseness permits it, and then the corresponding relation is unique up to sign* *. Unfortunately this is not clear a priori, but needs proof and the proof gives the exact value* * of the sign. In contrast determining the sign for the fi-relations would require an extra effor* *t. The main results can now be stated as follows. Theorem 1.6. As an F3[fi, v1]-module H*(G12, (E2)*=(3)) is isomorphic to the di* *rect sum of the cyclic modules generated by the following elements and with the following a* *nnihilator ideals 1 = 0 (fiv21) m fi m 6= 0 (v21) ff (fiv1) 2m+1ff (v1) n+1-1 2.3n(3m-1)ff m 6 0 mod (3)(v4.31 , fiv1) 2meff (v1) 2m+1eff m 6 2 mod (3)(v31, fiv1) 2.3n(3m+1)fffi (v1) 2.3n(3m-1)fffim 0 mod (3)(v1) 2m+1efffi m 2 mod (3) (v1) _ 6.3n+1 b3n+1(6m+1)_ (v1 n, fiv1) b3n(6m+5) m 1 mod (3) (v10.31+1, fiv1) __ __2m (v21)n 3n(6m 1) (v4.31, fiv21) __ __2m+1ff (v1) __2mff m 6 2 mod (3)(v1) __2meff (v1) 3n(6m+5)eff m 6 1 mod (3)(v1) . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited5 We emphasize that even though this result is involved the mechanism which pro* *duces is quite transparent. The passage to the cohomology of G2 results now from the dec* *omposition G2 ~=Z3x G12and the fact that the central factor Z3 acts trivially on (E2)*=(3). Theorem 1.7. There is a class i 2 H1(G2, (E2)0=(3)) and an isomorphism of grade* *d algebras H*(G12; (E2)*=(3)) Z3 Z3(i) ~=H*(G2, (E2)*=(3)) . Remark_We warn the reader that there is something subtle about this K"unneth ty* *pe isomor- phism. In fact, the class ff of Theorem 1.1 is defined via the Greek letter for* *malism in H1(G24, -) as the Bockstein of the class v1 with respect to the obvious short exact sequen* *ce 0 ! (E2)*=(3) -3!(E2)*=(9) -! (E2)*=(3) ! 0 . The same formalism allows to define classes ff(F ) 2 H1(F, (E2)4=(3)) for any c* *losed subgroup F of G2 and these classes are well compatible with respect to restrictions amon* *g different sub- groups. However, with respect to the isomorphism of Theorem 1.7 the class ff(G2* *) corresponds to ff(G12) - v1i (cf. Corollary 7.2). We will insist on the notation ff(G2) and* * ff(G12) in order to avoid possible confusion when we deal with H*(G2, -). Fortunately similar notat* *ion is unneces- sary for the classes effand fi (cf. Corollary 7.2). The difference between ff(G* *2) and ff(G12) turns out to be important for studying the differentials in the Adams-Novikov spectra* *l sequence for ss*(LK(2)V (0)). In fact, these differentials can be derived from those of the Adams-Novikov s* *pectral sequence for ss*(LK(2)V (1)) which have been determined in [4]. Remark_2_on_notation_In the following theorem we give the E1 -term of the Adams* *-Novikov spectral sequence for ss*(LK(2)V (0)) as a subquotient of its E2-term which its* *elf has been de- scribed in Theorem 1.6 and Theorem 1.7 as a module over F3[fi, v1] (i) with * *generators represented in the E1-term of the algebraic spectral sequence (2) for M = (E2)** *V (0). As be- fore, generators of E1 which are represented by products in this E1-term are no* *t necessarily products in E1 . In order to distinguish between module multiplication and the* * name of a generator we write fi and v1 as right hand factors in such a product if they ar* *e only part of the name of a generator, e.g. in the case of 6m+1fiv1._We have_also renamed (for r* *easons which will be explained below) generators involving k by 48 k-2. Theorem 1.8. As a module over F3[fi, v1] (i) the E1 -term of the Adams-Novik* *ov spectral sequence for ss*(LK(2)V (0)) is the quotient of the direct sum of cyclic F3[fi,* * v1] (i)-modules with the following generators and annihilator ideals 1 = 0 (fiv21, fi3v1, fi6) 3mfi, m 6= 0 (v21, fi2v1, fi5) 6m+1fiv1 (v1, fi2) 6m+4fiv1 (v1, fi3) ff(G12) (fiv1, fi3) 2m+1ff(G12) m 6 2 mod (3) (v1, fi3) n+1-1 2.3n(3m-1)ff(G12)m 6 0 mod (3), n 1(v4.31 , fiv1, fi3) 2(3m-1)ff(G12) m 6 0 mod (3) (v111, fiv1, fi4) 6meff (v1, fi5) b2(9m+2)+1 (v21, fi) 6m+3eff (v31, fiv1, fi5) 2.3n(3m+1)ff(G12)fin 1 (v1, fi2) 2.3n(3m-1)ff(G12)fim 0 mod (3), n 1(v1, fi2) 2(3m-1)ff(G12)fim 0 mod (3) (v1, fi3) 6 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald _ n b3n+1(6m+1)v1_ n 0 (v6.31,nfi) b3n(6m+5)v1_ m 1 mod (3), n 6= 1 (v10.31, fi) b3(6m+5)_ m 1 mod (3) (v311, fiv1, fi5) 48_3n(6m+1)-3_ n 1 (v21, fi2v1, fi5) 48_3n(6m+5)-3_ m 6 1 mod (3), n 1 (v21, fi3v1, fi5) 48_3n(6m+5)-3_ m 1 mod (3), n 1 (v21, fi2v1, fi5) 48_(6m+1)-3v1_ (v1, fi2)n 48_3n(6m_1)-2v1 n 1 (v4.31-1, fiv1, fi3) 48_(6m+1)-2v21_ (v21, fi) 48 (6m+5)-2 (v41, fiv21, fi3v1, fi5) __ 48_2m-1ff(G12)_ m 6 0 mod (3) (v1, fi3) 48_3n(6m+1)-3ff(G12)n_ 0 (v1, fi2) 48_3n(6m+5)-3ff(G12)m_6 1 mod (3), n 1(v1, fi3) 48_3n(6m+5)-3ff(G12)m_ 1 mod (3), n 1(v1, fi2) 48_6mfef_ (v1, fi4) 48 6m+3eff m 6 1 mod (3) (v1, fi4) modulo the following relations (in which module generators are put into bracket* *s in order to distinguish between module multiplications and generators.) fi3[ kff(G12)] = fi2i[ kfiv1] k = 2(3m - 1) m 6 0 mod (3) fi2[ kff(G12)fi]= fi2i[ kfiv1] k = 2(3m - 1) m 0 mod (3) fi4[ kfi]_ = fi4i[ keff]_ k = 6m + 3 fi2[ 48_kff(G12)]=fi2i[_48 kv1]_k = 6m + 1 fi2[ 48_kff(G12)]=fi2v1i[_48_k]_k = 6m + 3 fi2[ 48_kff(G12)]=fi2v1i[_48_k] k = 3n(6m + 5) - 3m 6 1 mod (3), n 1 fi4[ 48 k] = fi3i[ 48 keff]k = 6m . We remark that some but not all of the relations figuring in this result coul* *d have been avoided by choosing different generators, e.g. if we had chosen, for k = 2(3m -* * 1) and m 0 mod (3), kff(G2)fi as a generator instead of kff(G12)fi. Furthermore we remark that this description of the E1 -term as an F3[fi, v1] * * (i)-module does not lift to ss*(LK(2)V (0)). In fact, it is not hard to see that there are* * exotic relations like v1 ff = fi2effwhich hold in ss*(EhG242^ V (0)), in particular v1[ ff(G12)] 6= 0* * in ss*(LK(2)V (0)). As stated above this result is obtained without much trouble from the calcula* *tion of the Adams-Novikov E2-term given in Theorem 1.6 and Theorem 1.7 by using knowledge o* *f the Adams-Novikov differentials for LK(2)V (1). However, even if the rigorous proof* * proceeds this way, we feel that the final result can be better appreciated from the following* * point of view. In [5] the algebraic resolution (1) for G12resp. the companion resolution for G* *2 (obtained by tensoring with a minimal1resolution for Z3) was "realized" by resolutions of th* *e homotopy fixed point spectrum EhG22resp. of LK(2)S0 via homotopy fixed point spectra with resp* *ect to the corresponding finite subgroups of G2. In particular there is a resolution (4) * ! LK(2)S0 ! X0 ! X1 ! X2 ! X3 ! X4 ! * with X0 = EhG242, X1 = EhG242_ 8EhSD162, X2 = 8EhSD162_ 40EhSD162, X3 = 48E* *hG242_ 40EhSD162and X4 = 48EhG242. We note that the 48-fold suspension appearing in * *the definition of X3 and X4 is the reason for the (abusive) change of notation from k in Theo* *rem 1.6 to 48 k-2 in Theorem 1.8. Furthermore, the spectrum EhG242can be identified with * *the K(2)- localization of the spectrum T MF of topological modular forms and EhSD162with * *"half" of the K(2)-localization of the spectrum T MF0(2) of topological modular forms of * *level 2 (cf. [1]). These spectra are of considerable independent interest and their Adams-No* *vikov spectral The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited7 sequences and their homotopy is well understood (cf. the appendix or [1], [5]).* * The Adams- Novikov differentials for LK(2)V (0) can be completely understood by those for * *EhG242^V (0) (cf. the remarks following Lemma 8.1 and Lemma 8.4 for more precise statements). The* * complicated final result described in Theorem 1.8 can thus be deduced, just as in the case * *of Theorem 1.6, from more basic structures by an essentially simple though elaborate mechanism. We believe that our results have the following advantages over those by Shimo* *mura [12]. In our approach the final result relates well to modular forms and the homotopy of* * the spectrum T MF of topological modular forms; in particular the approach helps to understa* *nd how the complicated structure of ss*(LK(2)V (0)) is built from that of the comparativel* *y simple homotopy of T MF . This is also reflected in our notation which is very different from t* *hat in [11] where classical chromatic notation is used. Furthermore we determine E1 as a module o* *ver F3[fi, v1]. In contrast Theorem 2.8. in [12] gives a direct sum decomposition as an F3[v1]* *-module (of E1 and not as claimed in [12] of ss*(LK(2)V (1))) and this decomposition only p* *artially reflects the F3[fi, v1]-module structure. In fact, many non-trivial fi-multiplications a* *re_not recorded in [12], for example those on the classes 2.3n(9m+2)ff(G12), 2.3n(9m+8)ff(G12), * *b3(18m+11), ... , nor are the additional fi-relations of Theorem 1.8. There are related discrepancie* *s on the height_ of fi-torsion; for example, in [12] all elements in the same bidegree as the el* *ements 48 6m appear_to be already killed by fi4. Finally the classes v9m+22,=v31of [12] whic* *h correspond to v1 48 (6m+1)-2in our notation and which support a non-trivial Adams-Novikov d9-* *differential (cf. Lemma 8.4) seem to be permanent cycles in [12]. The paper is organized as follows. In section 2 we recall background material* * on the stabilizer groups, we introduce important elements of G2 and we recall the definition of i* *ts subgroups SD16 and G24as well as that of an important torsionfree subgroup K of finite index i* *n G12. In section 3 we study the maps in the permutation resolution (1). In fact, in [5] the maps* * C3 ! C2 and C2 ! C1 of the permutation resolution (1) were not described explicitly so that* * the resolution was not ready yet to be used for detailed calculations. The subgroup K plays a * *crucial role in finding an approximation to the map C2 ! C1. We also show that the map C3 ! C2 * *is in a suitable sense dual to the map C1 ! C0. In section 4 we study the action of the* * stabilizer group on (E2)*=(3) and we derive formulae for the action of the elements of G2 introd* *uced in section 2. In section 5 we comment on Theorem 1.1 and we verify Theorem 1.2 (cf. Propos* *ition 5.7, Proposition 5.10 and Proposition 5.12). Most of the new results of these sectio* *ns, in particular the formulae for the action of the stabilizer group, the approximation of the m* *ap C2 ! C1 and the evaluation of the induced map E1,0,*1= Ext0Z3[[G12]](C1, (E2)*=(3)) ! Ext0Z3[[G12]](C2, (E2)*=(3)) =* * E2,0,*1 are taken from the second author's thesis [8]. The evaluation of this map is by* * far the hardest calculation in our approach. In section 6 we prove Proposition 1.3 and Proposit* *ion 1.5. In a short section 7 we discuss the subtleties of the K"unneth isomorphism of Theore* *m 1.7 and section 8 contains the discussion of the differentials in the Adams-Novikov spectral se* *quence and proves Theorem 1.8. For the convenience of the reader we have collected the descriptio* *n of the related Adams-Novikov spectral sequences for EhG242^ V (0), for EhG242^ V (1) and for L* *K(2)V (1) in an appendix. 2. Background on the Morava Stabilizer Group In the sequel we will recall some of the basic properties of the Morava stabi* *lizer groups Sn resp. Gn. The reader is refered to [10] for more details (see also [7] and [5] * *for a summary of what will be important in this paper). 2.1. Generalities. We recall that the Morava stabilizer group Sn is the group o* *f automorphisms of the p-typicalnformal group law n over the field Fq (with q = pn) whose [p]-* *series is given by [p] n(x) = xp . Because n is already defined over Fp the Galois group Gal(F* *q=Fp) of the 8 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald finite field extension Fp Fq acts naturally on Aut( n) = Sn and Gn can be ide* *ntified with the semidirect product Sn o Gal(Fq=Fp). The group Sn is also equal to the group of units in the endomorphism ring of * * n, and this endomorphism ring can be identified with the maximal order On of the division a* *lgebra Dn over Qp of dimension n2 and Hasse invariant 1_n. In more concrete terms, On can* * be described as follows: let W := WFq denote the Witt vectors over Fq. Then On is the non-co* *mmutative ring extension of W generated by an element S which satisfies Sn = p and Sw = w* *oeS, where w 2 W and woeis the image of w with respect to the lift of the Frobenius automo* *rphism of Fq. The element S generates a two sided maximal ideal m in On with quotient On=m ca* *nonically isomorphic to Fq. Inverting p in On yields the division algebra Dn, and On is i* *ts maximal order. Reduction modulo m induces an epimorphism Oxn-! Fxq. Its kernel will be denot* *ed by Sn and is also called the strict Morava stabilizer group. The group Sn is equipped* * with a canonical filtration by subgroups FiSn, i = k_n, k = 1, 2, . .,.defined by FiSn := {g 2 Sn|g 1 mod (Sin)} . The intersection of all these subgroups contains only the element 1 and Sn is c* *omplete with respect to this filtration, i.e. we have Sn = limiSn=FiSn. Furthermore, we ha* *ve canonical isomorphisms FiSn=Fi+_1nSn ~=Fq induced by x = 1 + aSin7! ~a. Here a is an element in On, i.e. x 2 FiSn and ~ais the residue class of a in On* *=m ~=Fq. The associated graded object grSn with griSn := FiSn=Fi+_1nSn, i = 1_n, 2_n, * *. .b.ecomes a graded Lie algebra with Lie bracket [~a,~b] induced by the commutator [x, y] :=* * xyx-1y-1 in Sn. Furthermore, if we define a function ' from the positive real numbers to it* *self by '(i) := min{i+1, pi} then the p-th power map on Sn induces maps P : griSn -! gr'(i)Sn w* *hich define on grSn the structure of a mixed Lie algebra in the sense of Lazard (cf. Chap. * *II.1. of [9]). If we identify the filtration quotients with Fq as above then the Lie bracket and * *the map P are explicitly given as follows (cf. Lemma 3.1.4 in [7]). Lemma 2.1. Let ~a2 griSn, ~b2 grjSn. Then a) ni pnj [~a,~b] = ~a~bp- ~b~a2 gri+jSn b) 8 ><~a1+pni+...+p(p-1)nii < (p - 1)-1 P ~a= >~a+ ~a1+pni+...+p(p-1)nii = (p - 1)-1 :~a i > (p - 1)-1. The right action of Sn on On determines a group homomorphism Sn ! GLn(W). The resulting determinant homomorphism Sn ! Wx extends to a homomorphism Gn ! Wx o Gal(Fpn=Fp) which factors through ZxpxGal(Fpn=Fp). By choosing a fixed isomorphism between * *the quotient of Zxpby its maximal finite subgroup with Zp we get the "reduced determinant" h* *omomorphism Gn ! Zp . We denote its kernel by G1nand the intersection of G1nwith Sn resp. S1nby Sn re* *sp. S1n. The center of Gn is equal to the center of Sn and can be identified with Zxp(if we * *identify Sn with Oxn) and the composite Zxp! Gn ! Zxp sends z to zn. Thus if p does not divide n we get an isomorphism Gn ~=Zpx G1n. The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited9 2.2. Important subgroups in the case n = 2 and p = 3. From now on we assume n =* * 2 and p = 3. Let ! be a primitive eighth root of unity in Wx := WxF9. Then (5) a = -1_2(1 + !S) is an element of S12of order 3. (This element was denoted s in [4] and [5].) We* * can and will in the sequel choose ! such that we have the following relation in W=(3) ~=F9 (6) !2 + ! - 1 0 . Next we let t := !2. Then we have ta = a2t. Furthermore, if OE 2 Gal(F9=F3) i* *s the Frobenius automorphism then the elements _ := !OE and t generate a subgroup Q8 which norm* *alizes the subgroup generated by a so that _, t and a generate a subgroup G24of S12of orde* *r 24 which is isomorphic to a semidirect product Z=3 o Q8. The elements ! and OE generate a subgroup SD16of S12of order 16, isomorphic t* *o the semidi- hedral group of order 16. Finally there is a torsionfree subgroup K of S12which has already played an i* *mportant role in [7]._It is defined as follows: Lemma 2.1 implies that an element 1 + xS in S2 o* *f order 3 satisfies x 6= 0 and __x+ __x1+3+9= 0, i.e. __x4= -1 where __xis the class of x in gr1_2S* *12~=F9. There are no such elements x such that __x2 F3. Hence, if we define K to be the kernel of th* *e homomorphism S12! gr1_2S12~=F9 ! F9=F3 then K is torsion free, and we have a split short exact sequence (7) 1 ! K ! S12! Z=3 ! 1 . K inherits a complete filtration from S2 by setting Fk_2K = Fk_2S2 \ K and it i* *s easy to check that the associated graded is given as 8 > 0 even (8) grk_2K = >F3 k = 1 :F9 k > 1 odd where T r denotes the trace from F9 to F3. The following elements will play an important role in our later calculations. (9) b := [a, !], c := [a, b], d := [b, c] . In the next lemma we record approximations to these elements which we will us* *e repeatedly. Lemma 2.2. a)a 1 + !S + S2 mod (S3) b)b 1 - S - !S2 mod (S3) c)c 1 - !2S2 - !S3 mod (S4) d)d 1 + !2S3 mod (S4) Proof.a) The approximation for a is immediate from its definition. b) By explicit calculation in O2 we find b = 1_4(1 + !S)!(1 - !S)!-1 = 1_4(1 + !S)(1 - !-1S) = 1_4(1 + 3!2 + (! + * *!3)S) and then we use that our choice of ! yields !3 + ! = -1 and !2 - 1 = -! in F9. c) Similarly we get c = 1_64(1 + !S)(1 + 3!2 + (! + !3)S)(1 - !S)(1 - 3!2 - (! + !3)S) = -1_8(1 + * *6!2 - 3!S) . d) Finally the formula for d can be obtained from Lemma 2.1. 10 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald The information in the following proposition will be important for a closer i* *nspection of the permutation resolution (1). Proposition 2.3. a)H*(K; F3) is a Poincar'e duality algebra of dimension 3. b)H2(K; F3) ~=H1(K; F3) ~=(F3)2. c)H1(K; Z3) ~=Z=9 Z=3 where Z=9 resp. Z=3 is generated by b resp. by c. d)H2(K; Z3) = 0. e)H0(K; Z3) ~=H3(K, Z3) ~=Z3. Proof.Parts (a) and (b) have already been shown in Proposition 4.4. of [7]. ______ ______ For part (c) we note that H1(K; Z3) ~=K=[K, K]where [K, K]is the closure of t* *he commu- tator subgroup [K, K] of K. By Lemma 2.1 the commutator map griK x grjK ! gri+jK, (x, y) 7! [x, y] is surjective when i = 3_2and j = k_2with k even, and also when i = 1_2and j = * *l_2with l > 1 odd. Thus F2K [K, K]. If i = 1_2and j = 1 then the image of the commutator map_is_* *the kernel of the trace map. Together with part (b) of Lemma 2.1 this shows that K=[K, K]~* *=Z=9 Z=3 and it is easy to check that b and c generate Z=9 resp. Z=3. The remaining parts (d) and (e) now follow from a simple Bockstein calculatio* *n. 3.The maps in the permutation resolution 3.1. Generalities. Let G be a profinite p-group. We say that G is finitely gene* *rated if H1(G, Zp) is finitely generated over Zp. The kernel of the augmentation Zp[[G]] ! Zp is d* *enoted by IG, or simply by I. We say that a Zp[[G]]-module M is I-complete if the filtration * *by the submod- ules InM, n 0, is complete. As in [5] we use a Nakayama type lemma to show th* *at certain homomorphisms are surjective. Its proof is the same as that of Lemma 4.3 of [5]. Lemma 3.1. Let G be a finitely generated profinite p-group and f : M ! N a homo* *morphism of IG-complete Zp[[G]]-modules. Suppose that H0(f) : H0(G, M) ! H0(G, N) is su* *rjective. Then f is surjective. In [5] we used the analogous Lemma 4.3 for G = S12in order to show that certa* *in Z3[[S12]]- linear maps are surjective. Here we use Lemma 3.1 for G = K together with the a* *ction of the element a on H0(K, -) resulting from the exact sequence (7) in order to show th* *at the same maps are surjective. The advantage of working with K will become clear when we * *will discuss the kernel of the map C1 ! C0 (see the remark after Proposition 3.4 below). We * *begin the construction of the permutation resolution exactly as in [5]. 3.2. The homomorphism @1. Let C0 = Z3[[G12]] Z3[G24]Z3 and e0 = e 1 2 C0 if * *e is the unit in G12. Let @0 : C0 ! Z3 be the standard augmentation and let N0 be the ke* *rnel of @0 so that we have a short exact sequence (10) 0 ! N0 ! C0 ! Z3 ! 0 . Proposition 3.2. a)As Z3[[K]]-module N0 is generated by the elements f1 := (e - !)e0,_f2_:= * *(e - b)e0 and f3 := (e - c)e0. If we denote the class of fi in H0(K, N0) by fithen * *we have an isomorphism __ __ __ H0(K; N0) ~=Z3{f1} Z=9{f2} Z=3{f3} . b)The action of a on H0(K, N0) is given by : __ __ __ __ __ __ __ __ __ a*f1= f1+ f2, a*f2= f2+ f3, a*f3= f3- 3f2 . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *11 c)H1(K; N0) = 0. Proof.a) We consider the long exact sequence in H*(K, -) associated to the shor* *t exact sequence (10). As C0 is free K-module we have H1(K; C0) = 0. Furthermore, H0(K; C0) ~=Z2* *3, generated by the classes of e0 and !e0 so that the end of this long exact sequence has th* *e following form 0 ! H1(K; Z3) ~=Z=9 Z=3 ! H0(K; N0) ! H0(K; C0) ~=Z23! H0(K; Z3) ~=Z3 ! 0 . Now (a) follows from Proposition 2.3 and the identification of H1(K, Z3) with I* *K=(IK)2 which sends x 2 K to the class of x - e in IK=(IK)2. b) By definition we have ae0 = e0 and thus a!e0 = a!a-1!-1!ae0 = [a, !]!e0 = b!e0 . Consequently (11) a(e - !)e0 = (e - a!)e0 = (e - b!)e0 = b(e - !)e0+ (e - b)e0 and we obtain the first formula by passing to K-coinvariants. Similarly, ab = cba and ae0 = e0 imply (12) a(e - b)e0 = (e - ab)e0 = (e - cba)e0 = (e - cb)e0 = c(e - b)e0+ (e - c* *)e0 and by passing to K-coinvariants we get the second formula. __ __ The third formula can now be deduced from the fact that a3*f1= f1. c) This follows from the long exact sequence in H0(K, -) associated to the ex* *act sequence (10) by using that Hi(K, C0) = 0 for i = 1, 2 and H2(K, Z3) = 0. Let C1 = Z3[[G12]] Z3[SD16]O where O is the non trivial character of SD16def* *ined over Z3 on which ! and OE both act by multiplication by -1. Let e1 be the generator of C1 * *given by e 1 where e is as before the unit in G12. Corollary 3.3. There is a Z3[[G12]]-linear epimorphism @1 : C1 ! N0 given by e1* * 7! (e - !)e0. Proof.The elements !2, OE! and !-1OE all belong to G24 and hence they act trivi* *ally on e0. Therefore we have !(e - !)e0 = (! - !2)e0 = -(e - !)e0 and OE(e - !)e0 = (OE - OE!)e0 = (!(!-1OE) - e)e0 = -(e - !)e0 . This implies that there is a well defined homomorphism C1 ! N0 which sends e1 t* *o (e - !)e0. To see that this homomorphism is surjective we note that C1 is free as Z3[[K]]-* *module of rank 3 with generators e1, ae1 and a2e1. Then we use Lemma 3.1 and Proposition 3.2. 3.3. The homomorphism @2. Now we turn towards the construction of the second ho* *momor- phism in our permutation resolution. This is substantially more intricate; in [* *5] its existence was established but no explicit formula was given. Let N1 be the kernel of @1 so that we have a short exact sequence (13) 0 ! N1 ! C1 ! N0 ! 0 . Proposition 3.4. a)H0(K; N1) ~= Z23. The inclusion of N1 into C1 induces an injection H0(K,* * N1) ! H0(K, C1) and identifies H0(K, N1) with the submodule generated by the cl* *asses _gi, i = 1, 2, of g1 = 3(a - e)2e1 and g2 = 9(a - e)e1. b)The action of a on H0(K, N1) is determined by a*_g1= -2_g1- _g2,a*_g2= 3_g1+ _g2. 12 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald c)H0(S12, N1) ~=Z=3 and if n1 is any element of N1 which agrees in H0(K, C1* *) with _g1 then its class in H0(S12, N1) is non-trivial. d)The elements ! and OE both act on H0(S12, N1) by multiplication by -1. Proof.a) We observe that C1 is a free K-module of rank 3 generated by e1, ae1 a* *nd ae2. Then (a) follows from Proposition 3.2 by using the long exact sequence in H0(K, -) a* *ssociated to the short exact sequence (13) . b) The formulae for the action of a already hold in H0(K, C1). c) This follows from part (b) by passing to coinvariants with respect to the * *action of a. d) This has already been observed in Lemma 4.6. of [5]. Remark_We remark that working with S12-coinvariants only (as in [5]) does not g* *ive us a good hold on a generator of N1. The reason is that the map H0(S12, N1) ~=Z=3 ! H0(S1* *2, C1) ~=Z3 is necessarily trivial and therefore such a generator cannot be easily associated * *with an element in C1. Working with K-coinvariants gives us a starting point, namely the element g* *1 2 C1, from which_we can try to construct an element n1 of N1 whose class in H0(K, C1) agre* *es with that of g1and thus projects to a generator of H0(S12, N1). A first step in the directio* *n of finding such a generator n1 is taken in Lemma 3.6 below. Corollary 3.5. Let C2 = Z3[[G12]] Z3[SD16]O, let n1 2 N1 be any element which * *projects non-trivially to the coinvariants H0(S12; N1) and let X n01:= 1_16 O(g-1)g(n1) . g2SD16 Then there is a Z3[[G12]]-linear epimorphism @2 : C2 ! N2 given by e2 7! n01. Proof.By construction the group SD16acts on n01via the mod-3 reduction of the c* *haracter O and thus there is a homomorphism @2 as claimed. Surjectivity of @2 follows from* * Lemma 3.1. Lemma 3.6. a)Let l1 = (a - b)e1, l2 = (a - c)l1 and l3 = 3cl2+ (e - c)2l2. Then @1(l1) = (e - b)e0, @1(l2) = (e - c)e0 , @1(l3) = (e - c3)e0 . b)There exist elements x, y 2 IK such that e - c3 = x(e - b) + y(e - c). c)If x, y 2 IK satisfy e - c3 = x(e - b) + y(e - c) then n1 := l3- xl1- yl2 belongs to N1 and projects non-trivially to H0(S12, N1). Proof.We start with the following two observations: oProposition 3.2 implies that @1((a - e)2e1) (e - c)e0 mod (IK)N0. oIn Z3[[K]] we have the relation 3c(e - c) + (e - c)3 = e - c3. a) By equations (11) and (12) of the proof of Proposition 3.2 we see that @1(l1) = (e - b)e0 and @1(l2) = (e - c)e0 . The result for @1(l3) is now obvious. b) By Proposition 2.3.c we know that IK is generated by e - b and e - c. Furt* *hermore c3 belongs to F2K, hence it is trivial in H1(K, Z3) by Proposition 2.3.c . Therefo* *re e - c3 belongs to IK2 and we get the existence of x, y 2 IK as required in (b). c) By (a) and (b) n1 belongs to N1. Furthermore it is clear that n1 and 3(a -* * e)2e1 agree in H0(K, C1), hence n1 projects non-trivially to H0(S12, N1). The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *13 The question becomes now how we can determine x and y. In fact, we do not hav* *e explicit formulae for x and y. However, in section 5.3 we will give approximations for t* *hem which are sufficient for our homological calculations. 3.4. The homomorphism @3. In [5] it was shown (by using that K is a Poincar'e d* *uality group) that the kernel of @2 can be identified with Z3[[G12=G24]]. However, the* * identification and thus the construction of @3 was not explicit. The following result shows t* *hat @3 can be replaced by the dual of @1, at least up to isomorphism. If G is a profinite group and M a continuous left Zp[[G]]-module then we defi* *ne its dual M* by Hom Zp[[G]](M, Zp[[G]]). This becomes a left Zp[[G]]-module via (g.')(m) * *= '(m)g-1 if g 2 G, ' 2 Hom Zp[[G]](M, Zp[[G]]) and m 2 M. We observe that for a finite subg* *roup H there is a canonical Zp[[G]]-linear isomorphism i X j X (14) Zp[[G=H]] ! Zp[[G=H]]* ~=Zp[[G]]H , g 7! g* : eg7! eg( h)g-1 $ ( * * h)g-1 . h2H h2H Proposition 3.7. a)There is an exact complex of Z3[[G12]]-modules * @* @* _ffl 0 -! C*0@1-!C*1-2!C*2-3!C*3-! Z3 -! 0 in which @*iis the dual of @i for i = 1, 2, 3. b)There is an isomorphism of complexes of Z3[[G12]]-modules * @* @* _ffl 0 -! C*0 @1-! C*1 -!2 C*2 -!3 C*3 -! Z3 - ! 0 # f3 # f2 # f1 # f0 #= 0 -! C3 @3-! C2 -@2! C1 -@1! C0 -ffl!Z3 - ! 0 . 1]] such that fiinduces the identity on TorZ3[[S20(F3, Ci) for i = 2, 3 if we* * identify C*iwith C3-ivia the isomorphism of (14). c)The homomorphism @*1: C*0! C*1is given by e*07! (e + a + a2)e*1. Proof.a) Each Ciis free as a Z3[[K]]-module and therefore the complex 0 -! C3 @3-!C2 @2-!C1 @1-!C0 -ffl!Z3 -! 0 is a free Z3[[K]]-resolution of Z3. Because K is of finite index in G12the coin* *duced module of Z3[[K]] is isomorphic to Z3[[G12]]. Therefore there are natural isomorphisms C*i= Hom Z3[[G12]](Ci, Z3[[G12]]) ~=HomZ3[[K]](Ci, Z3[[K]]) and the n-th cohomology of the complex HomZ3[[K]](Ci; Z3[[K]]) is Hn(K; Z3[[K]]* *). Because K is a Poincar'e duality group this is zero except when n = 3 and then it is isomorp* *hic to Z3. Finally, one sees as in Proposition 5 of [13] that the Z3[[G12]]-module structure on Hn(* *G12; Z3[[G12]]) ~= Hn(K; Z3[[K]]) is trivial. b) The augmentation Z3[[G12]] ! Z3 induces an isomorphism Hom Z3[[G12]](Z3, Z3) ~=HomZ3[[G12]](Z3[[G12]], Z3) . Thus the right hand square is commutative up to a unit in Z3 if we choose for f* *0 the isomorphism given in (14), and we can modify f0 by a unit so that it commutes on the nose. * *Then f0 induces an isomorphism Kerffl ~=Ker_ffland * @* _ C2 @2-!C1 @1-!Kerffl resp. C*1@2-!C*2-3!Kerffl is the beginning of a resolution of Kerffl resp. Ker_fflby projective Z3[[G12]* *]-modules and the isomorphism induced by f0 lifts to1a chain map fo between the projective resolu* *tions. By Lemma 4.5 of [5] we have TorZ3[[S2]]i(F3, Kerffl) ~=F3 if i = 0, 1 and this imp* *lies that the maps 1]] f1 : C1 ! C*2and f2 : C2 ! C*1induce isomorphisms on TorZ3[[S20(F3, -) and henc* *e they are 14 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald themselves isomorphisms by Lemma 4.3 of [5]. Finally, f3 is trivially an isomor* *phism because f2 and f1 are isomorphisms. The isomorphism of chain complexes (considered as automomorphism via (14)) in* *duces an automorphism of spectral sequences 1]] Z3[[S1]] TorZ3[[S2j(F3, Ci) ) Tori+j (2F3, Z3) which converges towards the identity, and this easily implies the remaining par* *t of (b). c) By (14) we have for each eg2 G12 X -1 -1 -2 X -1 * * 2 (e + a + a2)*(e*1)(eg) = eg O(h )h (e + a + a ) = eg O(h )h (e + * *a + a ) h2SD16 h2SD16 and X X * * 2 @*1(e*0)(eg) = e*0(eg(e - !)) = eg(e - !) h = eg(e - !) h (e + * *a + a ) . h2G24 h2Q8 P P Then we conclude via the identity (e - !) h2Q8h = h2SD16O(h-1)h. 4.On the action of the stabilizer group In this section we will produce formulae for the action of the elements a, b,* * c and d of the stabilizer group G2 on F9[[u1]][u 1], at least modulo suitable powers of th* *e invariant ideal generated by u1. It turns out that it is sufficient to have a formula for the a* *ction of a and b on u modulo (u61), and for the action of c and d on u modulo (u101). 4.1. Generalities. We recall (cf. [10]) that BP* ~=Z(p)[v1, v2, . .].where the * *Araki generators visatisfy the following equation (in BP* Q) X pi (15) p~k = ~ivk-i. 0 i k Here the ~i 2 BP* Q are the coefficients of the logarithm of the universal p-* *typical formal group law F on BP*, X i logF(x) = ~ixp i 0 (with ~0 = 1), and thus the [p]-series of F is given by X F i [p]F(x) = vixp . i 0 The homomorphism 8 i < uiu1-pn i < n BP* ! (En)* = WFpn[[u1, . .,.un-1]][u 1], vi7! : u1-p i = n 0 i > n defines a p-typical formal group law Fn over (En)*. Then the formal group law G* *n over (En )0 defined by Gn(x, y) = u-1Fn(ux, uy) is a universal deformation of n and is p-t* *ypical with p-series n-1 pn (16) [p]Gn(x) = px +Gn u1xp+Gn . .+.Gnun-1xp +Gn x . Next we recall how one can get at the action of an element g 2 Sn on (En)0. F* *or a given g we choose a lift eg2 (En)0[[x]] of g and let eGbe the formal group law defined * *by Ge(x, y) = eg-1Gn(eg(x), eg(y)) . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *15 Then there is a unique ring homomorphism g* : (En)0 ! (En)0 and a unique *-isom* *orphism from g*Gn to eGsuch that the composition hg : g*Gn ! eGeg!Gn is an isomorphism of p-typical formal group laws and can therefore be written (* *cf. Appendix 2 of [10]) as X i hg(x) = Gnti(g)xp i 0 for unique continuous functions (17) ti: Sn ! (En)0 . We note that (18) ti(g) gimod (3, u1) P p2 if g = igiSi2 Sn with gi = gi. Then we have (19) g*(u) = t0(g)u and the equation (20) hg([p]g*Gn(x)) = [p]Gn(hg(x)) can be used to recursively find better and better approximations for t0(g) as w* *ell as for the action of g on the deformation parameters u1, . .,.un-1. 4.2. A formula modulo (3) for the formal group law G2. From now on we restrict * *attention to the case n = 2 and p = 3. Lemma 4.1. The logarithm and exponential of the formal group law G2 satisfy u4 logG2(x)= x - u1_24x3+ _1__1-381_3- _1_72x9 mod (x27) 2 12u3 i u4 55u4j expG2(x)= x + u1_24x3+ 3u1_242x5+ ___1_243x7- _1__1-381_3-m_1_72+o___1_24* *4x9d(x11) . v v4 Proof.From (15) we get ~1 = -v1_24and ~2 = _1__1-382_3- _1_72. To obtain the re* *sult for logG2we use the classifying homomorphism for F2 and that logG2(x) = u-1logF2(ux). Corollary 4.2. The formal group law G2 satisfies x +G2 y x + y - u1(xy2 + x2y) + u21(xy4 + x4y) -u31(xy6 + x6y) - u31(x3y4 + x4y3) -(x3y6 + x6y3) + u41(x4y5 + x5y4) mod (3, (x, y)11) . Proof.This follows directly from x +G2 y = expG2(logG2(x) + logG2(y)). 4.3. Formulae for the action modulo (3). To simplify notation we will denote in* * the re- mainder of this section the mod-3 reduction of the value of the function tiof (* *17) on an element g again by ti(g), or even by tiif g is clear from the context. Proposition 4.3. Let g 2 Sn and let u1 be Araki's u1. Then the following equati* *ons hold a)g*(u1) = t20u1 b)t0+ t60t1u31= t90+ t31u1 c)t1- t80t1u41 t91+ t32u1- t180t31u21- t90t61u31 mod (u71) d)If g 1 mod (S2) then t2 t92+ t33u1 mod (u21). 16 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Remark_If we want to know g*(u1) modulo (u71) then (a) shows that it is enough * *to know t0 modulo (u61) and this can be calculated from (b) if we know t0 modulo (u1) and * *t1 modulo (u31). Furthermore, t1 can be calculated modulo (u31) from (c) if we know t0, t1 and t* *2 modulo (u1). In the same manner we can even calculate g*(u1) modulo (u81). Similarly, if we want to know g*(u1) modulo (u111) then (a) shows that it is * *enough to know t0 modulo (u101) and this can be calculated from (b) if we know t0 modulo (u1) * *and t1 modulo (u71). Furthermore, t1 can be calculated modulo (u71) from (c) if we know t0 mo* *dulo (u31), t1 modulo (u1) and t2 modulo (u21). Finally (d) can be used to calculate t2 modulo* * (u21) if we know t2 and t3 modulo (u1). Proof.In this proof we abbreviate G2 simply by G. We consider equation (20) hg([3]g*G)(x) = [3]G(hg(x)) over (E2)0=(3)[[x]] and compare coefficients of x3kfor k = 1, 2, 3, 4. By (16) * *we have 3 hg([3]g*G)(x) t0 g*(u1)x3+g*Gx9 +G t1 g*(u1)x3+g*Gx9 +Gt2 g*(u1)x3+g*Gx9 9+G t3 g*(u1)x3+g*Gx9 27mod (x82) [3]G(hg(x)) u1(t0x +G t1x3+G t2x9+ t3x27)3 +G(t0x +G t1x3+G t2x9)9 mod (x82) . a) For the coefficient of x3 we obviously get g*(u1)t0 = u1t30which proves (a* *). b) The coefficient of x9 in hg([3]g*G)(x) is equal to t0 + g*(u1)3t1 which by* * (a) is equal to t0+ u31t60t1. The coefficient of x9 in [3]G(hg(x)) is equal to the same coeffic* *ient in u1(t0x +G t1x3)3+G (t0x)9 which is clearly equal to u1t31+ t90and hence we get (b). c) The coefficient of x27in hg([3]g*G)(x) is equal to the coefficient of x27in 3 9 3 9 3 3 9 t0 g*(u1)x +g*Gx +G t1 g*(u1)x +g*Gx +G t2 g*(u1)x and the latter coefficient is equal to t1+ t2g*(u1)9+ c where c is the coefficient of x27in 3 9 3 9 t0 g*(u1)x +g*Gx +G t1g*(u1) x . Next we observe that Corollary 4.2 yields g*(u1)x3+g*Gx9 g*(u1)x3+ x9- g*(u1)3x15 -g*(u1)2x21+ g*(u1)6x21- g*(u1)9x27 mod (x28) . Applying Corollary 4.2 once more and using (a) and calculating modulo (u71) we * *obtain c -u1t20t1g*(u1)3 = -u41t80t1 mod (u71) and hence modulo (u71) the coefficient of x27in hg([3]g*G)(x) is equal to t1- u41t80t1 . On the other hand the coefficient of x27in [3]G(hg(x)) is equal to the same coe* *fficient in u1(t0x +G t1x3+G t2x9)3+G (t0x +G t1x3)9 and this coefficient is equal to u1t32+ t91+ d where d is the coefficient of x27in u1(t0x +G t1x3)3+G t90x9 . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *17 Next we observe that Corollary 4.2 yields (t0x +G t1x3)3 t30x3+ t31x9- u31t60t31x15 -u31t30t61x21+ u61t120t31x21- u91t180t31x27 mod (x2* *8) . Applying Corollary 4.2 once more and using (a) and calculating modulo (u71) we * *obtain u1(t0x +G t1x3)3+G t90x9 u1 t30x3+ t31x9- u31t60t31x15- u31t30t61x21) + t90x9 -u31t30x3+ t31x9- u31t60t31x15- u31t30t61x21)2t90x9 -u21t30x3+ t31x9- u31t60t31x15- u31t30t61x21)t180x18 +u61t30x3+ t31x9- u31t60t31x15- u31t30t61x21)4t90x9mo* *d(x28) and hence d -u31(t61- 2u31t90t31)t90- u21t180t31+ 4u61t180t31 mod (u71) . Therefore the coefficient of x27in [3]G(hg(x)) is equal to t91+ u1t32- u21t180t31- u31t90t61+ 2u61t180t31+ 4u61t180t31 mod * *(u71) and (c) follows. d) The coefficient of x81in hg([3]g*G)(x) is equal to the same coefficient in 3 9 3 9 3 3 9 9 3 27 t0 g*(u1)x +g*Gx +G t1 g*(u1)x +g*Gx +G t2 g*(u1)x +g*Gx +G t3(g*(u1)x ) which modulo (u31) is equal to the same coefficient in 3 9 27 81 t0 g*(u1)x +g*Gx +G t1x +G t2x and by Corollary 4.2 this is easily seen to be equal to t2 modulo (u21). On the other hand the coefficient of x81in [3]G(hg(x)) is equal to the same c* *oefficient in u1(t0x +G t1x3+G t2x9+ t3x27)3+G (t0x +G t1x3+ t2x9)9 and this coefficient is equal to u1t33+ t92+ e where e is the coefficient of x81in the series u1(t0x +G t1x3+G t2x9)3+G (t0x +G t1x3)9 . Now g 1 mod (S2) implies t1 0 mod (u1) and thus modulo (u21) we find that e* * is also the coefficient of x81in u1(t0x +G t2x9)3+G t90x9 and by Corollary 4.2 even of the coefficient of x81in u1(t0x +G t2x9)3 . Now Lemma 4.4 below shows that the coefficient of x27in t0x +G t2x9 is trivial * *modulo (u1) (if not, either the coefficient of x18y or of x9y2 in x +G y would have to be nontr* *ivial modulo (u1)), hence e is trivial modulo (u21) and the proof of (d) is complete. Lemma 4.4. X x +G2 y x + y + P8i+1(x, y) mod (3, u1) i 1 where P8i+1is a homogeneous polynomial of degree 8i + 1 without terms x8i+1and * *y8i+1. Proof.It is enough to show this for the graded formal group law F2 over (E2)*. * *This group law is a homogeneous series of degree -2 if x and y are given degree -2 and thus, i* *f we write X x +G2 y x + y + Pj(x, y) mod (3, u1) j 1 with homogeneous polynomials in x and y of degree -2j then the coefficients in * *Pj have to be in (E2)2j-2. Furthermore, this group law has its coefficients in the subring ge* *nerated by u1u-2 and u-8. However, (E2)*=(3, u1) ~=F9[[u 1]] and thus 2j - 2 has to be a multipl* *e of 16. 18 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Corollary 4.5. The following equations hold in (E2)0=(3). a)Let g = 1 + g1S + g2S2 mod (S3). Then we have t1 g1+ g32u1- g31u21- g61u31 mod (u41) t0 1 + g31u1- g1u31+ (g2- g32)u41+ g31u51+ (g21+mg61)u61od(u71) . b)Let g = 1 + g2S2 + g3S3 mod (S4). Then we have t2 g2+ g33u1 mod (u21) t1 g32u1+ g3u41+ (g32- g2)u51 mod (u71) t0 1 + (g2- g32)u41- g3u71+ (g2- g32)u81mod(u101) . Proof.a) From Proposition 4.3.c we obtain t1 t91+ t32u1- t180t31u21- t90t61u31 mod (u41) and by using (18) we immediately get the formula for t1 modulo (u41). Then Prop* *osition 4.3.b and (18) yield t0+ t60(g1+ g32u1- g31u21- g61u31)u31 1 + (g31+ g2u31)u1 mod (u71) from which we easily get the formula for t0 modulo (u71). The formula for g*(u1* *) follows now from Proposition 4.3.a . b) From Proposition 4.3.d and (18) we immediately obtain the formula for t2. * *Substituting the value for t2 into the formula of Proposition 4.3.c and using (18) yields t1- t80t1u41 (g32+ g3u31)u1- t180t31u21 mod (u71) . Substituting the values of t0 modulo (u71) and t1 modulo (u41) from (a) into th* *is yields t1- g32u51 (g32+ g3u31)u1- g2u51 mod (u71) from which we get the value of t1 modulo (u71). Next we substitute this value o* *f t1 together with the value of t0 modulo (u71) of (a) into the formula of Proposition 4.3.b and o* *btain 4 4 6 3 4 3 5 3 4 10 t0+ 1 + (g2- g2)u1 g2u1+ g3u1+ (g2 - g2)u1 u1 1 + g2u1 mod (u1 ) from which we easily get the formula for t0 modulo (u101). The following calculation will be used repeatedly in later sections. The resu* *lt is only given to the precision needed later. Lemma 4.6. Let g = 1 + g1S + g2S2 mod (S3) and let k be an integer. Then we have 8 < 1 + g31u1+ (k0- 1)g1u31+ (k0g41+ g2- g32)u41+ g31u51mod(3,ku61)= 3k0* *+ 1 t0(g)k : 1 - g31u1+ g61u21+ (k0+ 1)g1u31+ (g41- k0g41+ g32- g2)u41+ (k0g71- g31- g31g2+ g31g32)u51mod(3,ku61)=* * 3k0+ 2 . Proof.The result follows easily from Corollary 4.5 and from k X5 `k' j 6 t0(g)k = (1 + (t0(g) - 1) ) (t0(g) - 1) mod (3, u1) j=1 3 k Q k P P by using that j i jii mod (p) if k = ikipiand j = ijipiare the p-adic e* *xpansions of k and j respectively. Using Lemma 2.2 we finally get the following information on the action of a, * *b, c and d on (E2)*=(3). We use 1 and !2 as a basis of F9 considered as an F3-vector space (r* *ather then 1 and !). The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *19 Corollary 4.7. The action of the elements a, b, c and d on (E2)*=(3) satisfy th* *e formulae a*u1 u1- (1 + !2)u21- !2u31+ (1 - !2)u41- u51- (1 + !2)u61mod(u71) b*u1 u1+ u21+ u31- u41+ (1 + !2)u51+ (1 - !2)u61 mod (u71) c*u1 u1- !2u51+ (-1 + !2)u81- (1 + !2)u91 mod (u111) d*u1 u1+ !2u81 mod (u111) a*u 1 + (1 + !2)u1+ (-1 + !2)u31+ (1 + !2)u51u mod (u61) b*u 1 - u1+ u31- !2u41- u51u mod (u61) c*u 1 + !2u41+ (1 - !2)u71+ !2u81u mod (u101) d*u 1 - !2u71u mod (u101) . 5.The E2-term of the algebraic spectral sequence 5.1. The E1-term. We begin by giving some background on Theorem 1.1, or equival* *ently, on the E1-term of the spectral sequence (2) Es,t,*1= ExttZ3[[G12]](Cs, (E2)*=(3)) =) Hs+t(G12, (E2)*=(3)) . We note that for s = 1, 2 the module Cs is projective as Z3[[G12]]-module and t* *hus we have a Shapiro isomorphism ( 1 Hom Z [SD (]O, (E2)*=(3))t* * = 0 (21) Es,t1= ExttZ3[[G12]](O "G2SD16, (E2)*=(3)) ~= 3 16 0 t >* * 0 . The action of SD16on (E2)* is known (cf. the proof of Lemma 22 of [6] for an ex* *plicit reference) to be given by (22) !*u1 = !2u1 and !*u = !u and the Frobenius OE acts Z3-linearly by extending the action of Frobenius on W* * via (23) OE*u1 = u1 and OE*u = u . This implies immediately that (E2)*=(3)SD16is isomorphic to F3[[u41]][v1, u 8] * *as a graded alge- bra and that there is an isomorphism of (E2)*=(3)SD16-modules (24) Ext0Z3[SD16](O, (E2)*=(3)) ~=!2u4F3[[u41]][v1, u 8]. For s = 0, 3 we have a Shapiro isomorphism Es,t1= ExttZ3[[G12]](Z3[[G12=G24]], (E2)*=(3)) ~=Ht(G24, (E2)*=(3))* * . Let G12 be the subgroup of G24 generated by the elements a and t. The calculat* *ion of the cohomology algebra H*(G12, (E2)*=(3)) was deduced from that of H*(G12, (E2)*) i* *n section 1.3 of [4]. In precisely the same way one deduces the calculation of H*(G24, (E2)*=* *(3)) from that of H*(G12, (E2)*) which was given in section 3 of [5]. In particular there are * *classes 2 H0(G24, (E2)24=(3)),ff 2 H1(G24, (E2)4=(3)) eff2 H1(G24, (E2)12=(3)),fi 2 H2(G24, (E2)12=(3)) and an isomorphism of algebras (25) H*(G24, (E2)*=(3)) ~=F3[[v61 -1]][v1, 1, fi, ff, eff]=(ff2, eff2, v1f* *f, v1eff, ffeff+ v1fi) . In the sequel we need some control over the elements occuring in this isomorphi* *sm (cf. section 1.3 of [4]). First we recall that ff is defined as ffi0(v1) where ffi0 is the B* *ockstein with respect to the short exact sequence of continuous Z3[[G2]]-modules (26) 0 ! (E2)*=(3) 3!(E2)*=(9) ! (E2)*=(3) ! 0 . Similarly, v2 := u-8 determines an invariant in H0(G2, (E2)16=(3, u1)) and effi* *s defined as ffi1(v2) where ffi1 is the Bockstein with respect to the short exact sequence of continu* *ous Z3[[G2]]-modules (27) 0 ! 4(E2)*=(3) v1!(E2)*=(3) ! (E2)*=(3, u1) ! 0 . 20 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Next fi is defined to be the mod 3-reduction of ffi0ffi1(v2). These elements ar* *e thus defined as elements in H*(G2, (E2)*=(3)). We denote their restriction to H*(G24, (E2)*=(3)* *) by the same name. The relation between (which lifts to an invariant of the same name in H0(G2* *4, (E2)24)) and the classes u and u1 is more subtle. Here we record the following result. Proposition 5.1. (1 - !2u21+ u41)!2u-12 mod (3, u61). Proof.By (3.11) of [5] the integral lift of is defined as 2 = _______!_________ 4 4 x(a*x)(a*(a*x)) and by the proof of Lemma 3.1 of [5] we know x u mod (3, u1), hence !2u-1* *2mod (3, u1). Because is invariant with respect to G24, in particular with respect to Q8, w* *e get from (22) and (23) that is of the form (1 + ~2!2u21+ ~4u41)!2u-12 mod (3, u61) with ~i2 F3 F9 . Because is also invariant with respect to the action of a we get from (19) = a*( ) t0(a)-12(1 + ~2!2a*(u1)2+ ~4a*(u1)4)!2u-12 mod (3, u61) . The right hand side of this equation can be evaluated modulo (u61) by using Cor* *ollary 4.5.a and Corollary 4.7. By looking at the coefficients of u31and u51in the right han* *d side we obtain ~2 = -1 and ~4 = 1. 5.2. The d1-differential. First of all we note that all differentials are v1-li* *near. Lemma 5.2. Let k 6 0 mod (3). Then the differential d1 : E0,01! E1,01satisfies ae m+1 2 4 -12k 8 d1( k) (-1)(-1!)(1m++u1)u1m!2umod2-(u1)12kk=62m + 1 1umod(u1) k = 2m. Proof.By Corollary 3.3 the differential is induced by the homomorphism C1 ! C0 * *which sends e1 to (e - !)e0. Furthermore Proposition 5.1 and (22) give k k 1 - k!2u21+ ku41- 2u41!2ku-12k mod (u61) !*( k) 1 + k!2u21+ ku41- k2u41!2k-12ku-12kmod (u61) and the result follows easily. (Note that by (24) the congruence for d1( 2m+1) * *improves to a congruence modulo (u81) rather than only modulo (u61).) Proposition 5.3. For each integer k 6= 0 there exists an element k 2 E0,0,24k1* *such that a) k k mod (u61) b)the differential d1 : E0,01! E1,01satisfies 8 m+1 2 -12k 4 < (-1) ! u n mod (u1) n k = 2m + 1 d1( k) : (-1)m+1m!2u4.31-2u-12k mod (u4.31+2)k = 2.3nm, m 6 0 mod (3) 0 k = 0 . Proof.For k = 0, k odd or k = 2m with m 6 0 mod (3) we define k to be equal t* *o k. The formula for d1 is then satisfied by the previous result. If k = 2.3nm with n 0 and m 6 0 mod (3) we recursively define n-2)3k-2.3n+1 (28) 3k: = 3k- mv3(4.31 . The previous proposition gives n+1 m 2 4 -12(3k-2.3n+1) 6 d1( 3k-2.3 ) (-1) ! (1 + u1)u mod (u1) . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *21 and by induction on n we have m+1 2 4.3n-2-12k 3 m 2 4.3n+1-6-12.3k 4.3n+1+6 d1( k)3 (-1) m! u1 u (-1) m! u1 u mod (u1 ) . Therefore v1-linearity of d1 yields n-2) d1( 3k) = d1( 3k) - mv3(4.31 d1( 3k-2.3n+1) n+1-2 4.3n+1+2 (-1)m+1!2mu4.31 u12.3k mod (u1 ) and the induction step is established. Corollary 5.4. There is an isomorphism of F3[v1]-modules E0,02~=F3[v1]. Remark_ By (25) the elements k form a topological basis of the continuous grad* *ed F3[v1]- module E0,01(in fact, this has been implicitly used in the last corollary) and * *by (24) a topological basis of the continuous graded F3[v1]-module E1,01can be given by any family of* * elements b2k+1, k 2 Z, such that b2k+1 !2u-8k-4mod (u41). By Proposition 5.3 we know that the* *re are such elements b2.(3m+1)+1for m 2 Z and b2.3n(3m-1)+1for n 0, m 6 0 mod (3), such * *that the first formula of Theorem 1.2 holds. Because the E1-term is torsion free as a F3[v1]-m* *odule and the d1-differential is F3[v1]-linear it is clear that those b2k+1's are in the kern* *el of the differential d1 : E1,01! E2,01. To complete this family to a topological basis we need to ch* *oose elements b2k+1for k = 3n+1(3m + 1) with n 0, m 2 Z, for k = 3n(9m + 8) with n 0, m 2* * Z, and for k = 0. Thus we are lead to concentrate on the differential on !2u-4(2k+1)for su* *ch k. The crucial step is given by Proposition 5.6 below whose proof is quite elaborate and will * *be postponed to section 5.3. The proof of Proposition 5.5 will be given in section 6. Proposition 5.5. There exists an element b1 2 E1,01such that b1 !2u-4 mod (u4* *1) and v1ff = b1 in H*(G12; (E2)*=(3)). In particular, d1(b1) = 0. Proposition 5.6. Let k be an integer such that 8k+4 is not divisible by 3. Then* * the differential d1 : E1,01! E2,01satisfies d1(!2u8k+4) -(k0+ k02)!2u121u8k+4mod (u161) 8k + 4 = 3k0+ 1 d1(!2u8k+4) (k0- k02)!2u81u8k+4mod (u121) 8k + 4 = 3k0+ 2 . Proof.This will be proved in section 5.3. Proposition 5.7. For each integer k there exists an element b2k+12 E1,0,8(2k+1)* *1such that a)b2k+1 !2u-4(2k+1)mod8(u41) < (-1)m+1b2(3m+1)+1n k = 2m + 1, m 2 Z b)d1( k) = : (-1)m+1mv4.31-2b2.3n(3m-1)+1k = 2m.3n, m 6 0 mod (3) 0 k = 0 c)the differential d1 : E1,01! E2,01satisfies 8 n n+1 < (-1)n!2u6.31+2u-4(2k+1)n mod (u2.31n +6)k = 3n+1(3m + 1),m 2 Z d1(b2k+1) : (-1)n!2u10.31+2u-4(2k+1) mod (u10.31+6)k = 3n(9m + 8), m 2 Z 0 otherwise. Proof.For k = 3m+1 with m 2 Z we define b2k+1to be (-1)m+1d1( 2m+1). For k = 3n* *(3m-1) with n n 0 and m 6 0 mod (3) we note that Propositionn5.3 shows that d1( 2m.3n* *) is divisible by v4.31-2and we define b2k+1to be (-1)m+1mv-(4.31-2)d1( 2m.3n). For k = 0 we t* *ake the element given in Proposition 5.5. With these definitions (b) holds as well as t* *he last case of (c). For k = 3n+1(3m + 1) resp. k = 3n(9m + 8), with n 0 and m 2 Z, we define el* *ements b2k+1 by induction on n such that (a) and (c) are satisfied. In fact, for n = 0 we de* *fine b2k+1: = !2u-4(2k+1) and then Proposition 5.6 gives d1(b18m+7) !2u81u-4(18m+7) mod (u121), d1(b18m+17) !2u121u-4(18m+17)mod * *(u121) . 22 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Now suppose that b2k+1has already been defined for k = 3n+1(3m+1) resp. k = 3n(* *9m+8) with with n N and m 2 Z so that (a) and (c) are satisfied. Then we observe that by* * Proposition 5.2 the elements 6 2 4 -4(6k+3) * * 8 b32k+1+ b2(3k+1)+1= b32k+1+ (-1)k+1d1( 2k+1) ! + ! (1 + u1) u mod * *(u1) are divisible by v41and thus we can define 3 (29) b6k+1: = v-41b2k+1+ b2(3k+1)+1. Then it is clear that b6k+1 !2u-4(6k+1)mod (u41). Furthermore, d1d1( 2k+1) =* * 0 and because d1 commutes with taking third powers and is F3[v1]-linear we see that b* *oth (a) and (c) are satisfied for k = 3n+1(3m + 1) resp. k = 3n(9m + 8) with n N + 1 and m 2 * *Z and thus the induction step is complete. Corollary 5.8. There is an isomorphism of F3[v1]-modules Y n E1,02~= F3[v1]=(v4.31-2){b2.3n(3m-1)+1} x F3[v1]{b1} . n 0,m2Z\3Z Proof.Because the elements k and bk form a topological basis of the graded con* *tinuous F3[v1]- modules E0,01and E1,01this follows immediately from Proposition 5.7. Remark_By inspection one sees that the infinite product is finite in each bideg* *ree and therefore it can also be identified with the direct sum. To evaluate the homomorphism d1 : E2,01! E3,01we need the following result. Lemma 5.9. Let k be any integer. Then k (e*+ a*+ (a2)*)(uk) (k - k2)!2u21+ (k 3 + k - k2)u41ukmod (3, u51) . Proof.By (19) we have a*(uk) = ukt0(a)k and (a2)*(uk) = ukt0(a)ka*(t0(a)k). Cor* *ollary 4.5 gives k t0(a)k 1 + (1 + !2)u1+ (-1 + !2)u31 1 + k(1 + !2)u1+ k(-1 + !2)u31 - k2!2u21- k2(1 + !2)(-1 + !2)u41+ k3(1 + !2)3u31- k4u41 1 + k(1 + !2)u1- k2!2u21 +( k3- k)(1 - !2)u31- ( k2+ k4)u41 mod (3, u51) and by Corollary 4.7 we get a*(t0(a)k) 1 + k(1 + !2) u1- (1 + !2)u21- !2u31+ (1 - !2)u41 - k2!2 u21+ (1 + !2)u31 +( k3- k)(1 - !2)u31- ( k2+ k4)u41 1 + k(1 + !2)u1+ (k - k2)!2u21 +( k3+ k2)(1 - !2)u31- (k + k2 + k4)u41 mod (3, u51) . k Finally an easy calculation (which only uses that 2 -k(k - 1) mod (3) and k3* * k mod (3)) gives t0(a)ka*(t0(a)k) 1 - k(1 + !2)u1+ (k2 - k)!2u21 +(- k3+ k)(1 - !2)u31 +( k4+ k2 + k k3+ k - k2)u41mod(3, u51) and the result clearly follows. _ 2,0,4(2k+1) Proposition 5.10. For each integer k there exists an element b2k+12 E1 s* *uch that _ a)b2k+1 !2u-4(2k+1)moda(u41)e nv6.3n+2_bn+1 k = 3n+1(3m + 1) b)d1(b2k+1) = (-1)n 110.3n3+2_(6m+1) n (-1) v1 b3n(18m+11)k = 3 (9m + 8) The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *23 c)the differential d1 : E2,01! E3,01satisfies 8 2 -4(2k+1) 4 >>-u1u mod (u1) n 2k + 1 = 6m + 1 _ < !2u4.3nu-4(2k+1) mod (u4.3 +2)2k + 1 = 3n(18m + 17) d1(b2k+1) > 214.3n-4(2k+1) 14.3n+2 n >:-! u1 u mod (u1 ) 2k + 1 = 3 (18m + 5) 0 otherwise. Proof.For 2k+1 = 3n+1(3m+1)nresp. 2k+1n= 3n(9m+8) Proposition 5.7 shows thatnd1* *(b2k+1)_ is divisiblenby_v6.31+2resp. v10.31+2and can_thus be written as_(-1)nv6.31+2b3n* *+1(6m+1)resp. (-1)nv10.31+2b3n(18m+11)for unique elements b3n+1(6m+1)resp. b3n(18m+11)which * *satisfy (a) and (b). _ So we still need to define b2k+1if 2k + 1 can be written as 2k + 1_= 6m + 1 w* *ith m 2 Z or 2k +1 = 3n(18m+11 6) with n 0 and m 2 Z. In those cases we define b2k+1: = !2* *u-4(2k+1) and note that -4(2k + 1) 2 mod (3) if 2k + 1 = 6m + 1 and that -4(2k + 1) 7* * mod (9) if 2k + 1 = 18m + 5 resp. -4(2k + 1) 4 mod (9) if 2k + 1 = 18m + 17. Then (c)* * holds by Lemma 5.9 and Proposition 3.7.c, at least if we pretend that the differential i* *s induced by the map @*1: C*0! C*1after identification of C3 with C*0and of C2 with C*1via the i* *somorphisms given by (14). In reality the differential is induced by @*1only up to the aut* *omorphisms of Ei,01, i = 2, 3, induced by the isomorphisms fiof Proposition 3.7 and the isomo* *rphisms of (14). * *1]] However, by Proposition 3.7 these automorphisms induce the identity on TorZ3[[S* *20(F3, Ci) for i = 2, 3. Then Corollary 4.5 shows that they induce automorphisms of Ei,01as c* *ontinuous graded F3[v1]-modules which map !2u-4(2k+1)to itself modulo (u41) respectively * *!2ku-12kto itself modulo (u21) and part (c) follows. Corollary 5.11. There is an isomorphism of F3[v1]-modules Y n+1 _ Y n _ E2,02~= F3[v1]=(v2.31 +2){b3n+1(6m+1)} x F3[v1]=(v10.31+2){b3n(18m+11)* *} . n 0 n 0 m2Z m2Z Remark_By inspection one sees again that the infinite product is finite in each* * bidegree and therefore it can also be identified with the direct sum. __ 3,0 Proposition 5.12. For each integer k there exists an element k 2 E1 such that __ a) k k mod (u21) b)The differential d1 : E2,01! E3,01is given by 8 __ >>(-1)m+1v21 2m_ 2k + 1 = 6m + 1 _ < (-1)m+n v4.3n 3n(6m+5) 2k + 1 = 3n(18m + 17) d1(b2k+1) > m+n+114.3n_ n >:(-1) v1 3n(6m+1)2k + 1 = 3 (18m + 5) 0 otherwise. _ Proof.Proposition 5.10 shows that d1(b2k+1) is divisible by the appropriate pow* *er of v1. The sign isnthen determined_by comparing the coefficients of the "leadingnterm"nu21* *u-4(2k+1)resp. !2u4.31u-42k+1in d1(b2k+1) on one hand and in v21 2m resp. v4.31 3 (6m+3 2)on t* *he other hand. Corollary 5.13. There is an isomorphism of F3[v1]-modules Y __ Y n __ __ E3,02~= F3[v1]=(v21){ 2m} x F3[v1]=(v4.31){ 3n(6m+1), 3n(6m+5)} . m2Z n 0,m2Z Remark_By inspection one sees once again that the infinite product is finite in* * each bidegree and it can therefore also be identified with the direct sum. 24 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald 5.3. The proof of Proposition 5.6. By Corollary 3.5 the differential E1,01! E2,* *01is induced by the homomorphism C2 ! C1, e2 7! n01where X n01:= 1_16 O(g-1)g(n1) , g2SD16 and by Lemma 3.6 we can take for n1 any element of the form n1 = `e1 with (30) ` := 3c(a - c)(a - b) + (e - c)2(a - c)(a - b) - x(a - b) - y(a - c)(a* * - b) and x, y 2 IK satisfying e - c3 = x(e - b) + y(e - c). The next result gives ap* *proximations for x and y which are sufficient for our homological calculations. Lemma 5.14. Let ex = b-1d-1(e - d) - b-1d-1(e - b)b-1c-1(e - c) ey = b-1d-1(e - b)b-1c-1(e - b) . Then there exists z 2 IF5_2S12+ IK.IF2S12such that the following identity holds* * in IK e - c3 = ex(e - b) + ey(e - c) + z . Proof.From Lemma 2.1 and Lemma 2.2 we deduce c3 [b-1, d-1] mod F5_2S12. Thus by using the elementary formulae -1 -1 -1 -1 (31) 1 - [X, Y ] = XY (1 - Y )(1 - X ) - (1 - X )(1 - Y ) (32) 1 - XY = X(1 - Y ) + (1 - X) which hold in any associative algebra we obtain 1 (33) e - c3 b-1d-1 (e - d)(e - b) - (e - b)(e - d) mod IF5_2S2 . Using Lemma 2.1 and Lemma 2.2 again we get d = [b, c] [b-1, c-1] mod F2S12 and hence we obtain from (31) 1 e - d b-1c-1 (e - c)(e - b) - (e - b)(e - c) mod IF2S2 . Substituting this into (33) gives the result. We will thus be interested in analyzing the action of (34) 3c(a - c)(a - b) + (e - c)2(a - c)(a - b) - ex(a - b) - ey(a - c)(a * *- b) as well as in the influence of the "error term" z on the elements !2u-4k+2. Thi* *s analysis will be simplified by the following result in which I denotes the ideal IS12. Lemma 5.15. Let r 1 be an integer. Then we have the following inclusions of l* *eft ideals a)IFrS12 I3r-1(er- b) + I2.(3r-1-1)(er- c) + 3I r I2.3r-1+ 3I b)IFr_2S12 I3 -1(e - b) + I3 -2(e - c) + 3I I3 + 3I. Proof.We note that for every integer r 0 we have an isomorphism IFr_2S12~=limq>rI(Fr_2S12=Fq_2S12) and it will therefore be enough to show the corresponding statements for the co* *rresponding ideals in the finite quotient groups FrS12=FqS12. Next we remark that for every* * finite p-group G the ideal IG is a free Zp-module with basis e - g, g 2 G - {e}. Therefore it is* * enough to show that r r-1 e - g 2 I3 -1(e - b) + I2.(3 -1)(e - c) + 3I for everyg 2 FrS12=FqS* *12 resp. r r e - g 2 I3 -1(e - b) + I3 -2(e - c) + 3I for everyg 2 Fr+1_2S12=FqS1* *2. The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *25 (By abuse of notation we do not distinguish between g 2 S12and its image in the* * quotients S12=FqS12.) Furthermore by (32) it is enough to show this for a system of multi* *plicative generators of FrS12=FqS12resp. Fr+1_2S12=FqS12. By Lemma 2.1 the element c3q-1forms a basi* *s of the one dimensional F3-vector space FqS12=Fq+1_2S12, and d3q-1and b3qform a basis of th* *e two dimensional F3-vector space Fq+1_2S12=Fq+1S12and therefore it is enough to consider those e* *lements. We have c = [a, b] and d = [b, c], and thus (31) shows first e - c 2 I2 and then e - d 2 I2(e - b) + I(e - c) I3 . Furthermore e - g3 (e - g)3 mod (3) and hence, modulo (3), we obtain for any * *integer r 1 e - c3r-1r (1 - c)3r-1-1(er- c) I2.(3r-1-1)(er- c) e - b3 (e - b)3 -1(e - b) I3 -1(e - b) e - d3r-1 (e - d)3r-1-1(e - d) I3r-3(e - d) I3r-1(e - b) + I3r-2(e * *- c) and (a) and (b) follow. Remark_The previous lemma can in principle also be used to get better explicit * *approximations of the elements x and y of Lemma 5.14, at least modulo (3). For this one has to* * express the element c3[b-1, d-1]-1 in F5_2S12=FqS12as explicit product of the elements b3r-* *1, d3r-2and c3r-1 for q r 3 and then use (32) and the formulae in the proof of the previous l* *emma. We will now give a qualitative description of the action of powers of I on (E* *2)*=(3). The following lemma is an immediate consequence of Lemma 4.6 and of the formula g*(* *ul1uk) = t0(g)k+2lul1uk (cf. (19) and Lemma 4.3.a). Lemma 5.16. a)IS12sends the F9[[u1]]-submodule of F9[[u1]]uk generated by ul1uk to the * *submodule gen- erated by ul+11uk. b)If k+2l 1 mod(3) then IS12sends ul1uk to the additive subgroup of ukF9[* *[u1]] generated by ul+11uk and the ideal generated by ul+31uk. c)If k + 2l 0 mod (3) then IS12sends the F9[[u31]]-submodule generated by* * ul1uk to the F9[[u31]]-submodule generated by ul+31uk. Lemma 5.17. a)Let k + 2l = 3m + 1 and r 1 be an integer. Then (IS2)r sends ul1uk to a* *n element of the form (fful+3(r-1)+11+ fiul+3r1)uk modulo (ul+3r+11) for suitable ff, * *fi 2 F9. b)Let k + 2l = 3m + 2 and r 2 be an integer. Then (IS2)r sends ul1uk to a* *n element of the form (flul+3(r-2)+21+ ffiul+3(r-2)+41)uk modulo (ul+3(r-2)+51) for su* *itable fl, ffi 2 F9. Proof.a) This follows by an easy induction on r by using the previous lemma. b) If k + 2l = 3m + 2, the previous lemmma shows that (IS2)2 sends ul1uk to (* *~ul+21+ ~u41)uk modulo (ul+51) for suitable ~, ~ 2 F9. Now the result follows again by an easy * *induction on r 2 by using once more the previous lemma. The following immediate corollary tells us that for the evaluation0of the dif* *ferential we should concentrate0on the coefficients of u81and u101in the case of u3k +2resp. of u10* *1and u121in the case of u3k +1. It also gives us more flexibility for approximating `. Corollary 5.18. Let k0 be an integer and # 2 (3, I4). Then there exist ff, fi, * *fl, ffi 2 F9 which only depend on # modulo (3, I5) such that we have the following congruences. a)#*(u3k0+1) (ffu101+ fiu121)u3k0+1 mod (3, u131) for suitable ff, fi 2 * *F9. 26 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald b)#*(u3k0+2) (flu81+ ffiu101)u3k0+2 mod (3, u111) for a suitable fl, f* *fi 2 F9. The next lemma gives a simplified approximation to ` and hence to d1. Proposition 5.19. Let x0= (e - d) - (e - b)(e - c) and y0= (e - b)(e - b) and define i * * j e`0:= 1_ X O(g-1)g 3c(a - c)(a - b) + (e - c)2(a - c)(a - b) - x0(a - b) - y* *0(a - c)(a - b) . 16g2SD16 Then the differential d1 : E1,01! E2,01satisfies d1(!2u-4(2k+1)) (e`0)*(!2u-4(2k+1))mod(u161)-4(2k + 1) = 3k0+ 1 d1(!2u-4(2k+1)) (e`0)*(!2u-4(2k+1))mod(u121)-4(2k + 1) = 3k0+ 2 . Proof.First we note that Lemma 5.15 shows that the element z of Lemma 5.14 belo* *ngs to (I8(e - b) + I7(e - c)) + I.(I8(e - b) + I4.(e - c)) and therefore does not con* *tribute to the calculation of d1 modulo the specified precision. Furthermore e`0belongs to (3,* * I4). Now the last corollary shows that we have equality modulo (u131) resp. (u111) if we rep* *lace exand eyfrom Lemma 5.14 by x0 and y0, and then the following lemma implies equality even mod* *ulo (u161) resp. (u121). Lemma 5.20. Let k and l 0 be integers and ~ 2 F9. Then ae 1_ X O(g-1)g(~uluk) = 1_2(~ - ~3)ul1ukk + 2l 4 mod (8) 16g2SD16 1 0 else. Proof.By (22) and (23) we have P -1 l k P 7 j j l k P 7 j+1 j l k g2SD16O(g )g(~u1u )= Pj=0(-1) (! )*(~u1u ) + j=0(-1)P (! OE)*(~u1u ) = 7j=0(-1)j!j(k+2l)~ul1uk - 7j=0(-1)j!j(k+2l)~3ul* *1uk . P 7 Furthermore j=0(-1)j!j(k+2l)= 0 unless k + 2l 4 mod (8) in which case it is* * equal to 8. The result follows. The previous result tells us how to get the "leading term" in the differentia* *lPonce we know the coefficients ff, fi, fl and ffi of Corollary 5.18 in the case of # = e`:= * * 4i=1e`iwith e`1:= 3c(a - c)(a - b) + (e - c)2(a - c)(a - b) e`2:= -(e - d)(a - b) e`3:= (e - b)(e - c)(a - b) e`4:= -(e - b)(e - b)(a - c)(a - b) , and in fact ff and ffi will not even matter. The coefficients fi and fl of Coro* *llary 5.18 for the action of each e`iare given in the following result. Lemma 5.21. Let k be an integer not divisible by 3. For k = 3k0+ 1 there are* * elements ffi,k, fii,k2 F9, i = 1, 2, 3, 4, and for k = 3k0+ 2 there are elements fli,k, * *ffii,k2 F9, i = 1, 2, 3, 4, such that we have the following congruences ae 10 12 k 13 0 (e`i)*uk (ffi,ku1(+ffii,ku1l)umod(3,8u1 )k1=03kk+ 1 11 0 i,ku1+mffii,ku1o)ud(3,ku1=)3k + 2 . Furthermore we have fi1,k= 0 fl1,k= 0 fi2,k= -(1 + !2) fl2,k= -(1 + !2) fi3,k= (k0- 1)!2 fl3,k= (k0+ 1)!2 fi4,k= - k02+ k0- 1 + (k0+ 1)!2 fl4,k= 1 + k0- k02- k0!2. The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *27 Proof.The first part of the lemma is clear by Corollary 5.18. Furthermore, the* * case of fii,k and fli,kfollows immediately from Lemma 5.17 because by Lemma 5.15 we have e`12* * I6 + 3I. The actual evaluation of the other fii,kand fli,kuses Lemma 4.6 and is a l* *engthy but straightforward calculation whose verification we leave to the reader. Here we * *just note that Lemma 5.16 guarantees that for this calculation it is enough to know the action* * of I on ul1uk modulo (ul+61). Proof of Proposition 5.6. This is an immediate consequence of the previous lemm* *a, of Proposi- tion 5.19 and Lemma 5.20. 6.Higher differentials and extensions in the algebraic spectral sequence In this section we will give a proof of Proposition 5.5 and we will determine* * the extensions and higher differentials in the algebraic spectral sequence (2) in the case of * *M* = (E2)*=(3). This spectral sequence allows for non-trivial d2- and d3-differentials. Their e* *valuation will be reduced to studying the long exact sequence in ExtZ3[[G12]](-, M*) associated t* *o the short exact sequence (10). In fact, we have the following more general lemma. Lemma 6.1. Let R be a ring and n > 0 be an integer and let M be a left R-module* *. Suppose that 0 -! Cn+1 @n+1-!Cn @n-!. .@.1-!C0 @0-!L -! 0 is an exact complex of left R-modules such that Ci is projective for each 0 < i* * < n + 1. a)Then there is a first quadrant cohomological spectral sequence E*,*r, r * * 1, converging to Ext*R(L, M) with Es,t1= ExttR(Cs, M) ) Exts+tR(L, M) in which Es,t1= 0 for 0 < s < n + 1 and t > 0, and Es,t1= 0 for t 0 and* * s > n + 1. b)The higher differentials in this spectral sequence can be described as fo* *llows. Let N0 be the kernel of @0 and let j : N0 ! C0 denote the resulting inclusion. The* *n there are isomorphisms which are natural in M 8 1,0 > Et+1,0= Et+1,0 1 t n : E2n+1,t-nt+1 n+1 t > n such that the homomorphism ExttR(C0, M) ! ExttR(N0, M) induced by j ident* *ifies with d0,tt+1: E0,tt! Et+1,0tif 1 t n and with d0,tn+1: E0,tn+1! En+1,t-nn+* *1if t > n. (Note that by (a) these are the only potentially non-trivial differentials in this s* *pectral sequence.) Proof.a) The spectral sequence can be obtained as the spectral sequence of an e* *xact couple. In fact, if Niis the kernel of @ithen we have short exact sequences 0 ! Ni! Ci!* * Ni-1! 0 for 0 i n (with N-1 : = L) and the long exact sequences in Ext*R(-, M) comb* *ine to give an exact couple from which the spectral sequence is derived. Projectivity of th* *e modules Cifor 1 i n gives the vanishing results. b) For the first statement we note that N0 admits a projective resolution Qo * *which is obtained from splicing the exact complex 0 -! Cn+1 @n+1-!Cn @n-!. .@.2-!C1 @1-!N0 -! 0 with a projective resolution of Cn+1. The second statement is easily seen by in* *spection of the higher differentials in an exact couple. We leave the details to the reader. 28 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Remark_The higher differentials can therefore be evaluated if we know projectiv* *e resolutions Qo of N0 and Po of C0 as well as a chain map OE : Qo ! Po covering j. These dat* *a can also be assembled in a double complex Too with To0= Po, To1= Qo, vertical differenti* *als ffiP and ffiQ and "horizontal differentials" (-1)nOEn : Qn ! Pn. The lemma implies that * *the filtration of the spectral sequence of this double complex agrees (up to reindexing) with tha* *t of the spectral sequence of the lemma. Hence extension problems in the spectral sequence of the* * lemma can also be studied by using the double complex. We apply this lemma and the remark to the algebraic spectral sequence associa* *ted to the case of the exact complex (1). We will make use of explicit projective resoluti* *ons Qo of N0 and Po of C0 and a suitable chain map OE : Qo ! Po covering j. The essential step i* *s given in the following elementary result whose proof is left to the reader. Lemma 6.2. Let __Obe the Z3[Q8]-module whose underlying Z3-module is Z3 and on * *which t acts by multiplication by -1 and _ by the identity. Then the trivial Z3[G24]-mo* *dule Z3 admits a projective resolution Po0of period 4 of the following form a2-a-!1 "G24e+a+a2 G24a2-a__G24e+a+a2_ G24a2-a G24 Q8 -! 1 "Q8 -! O"Q8 -! O "Q8 -! 1 "Q8 -! Z3 . In the sequel we work with the induced projective resolution Po := Z3[[G12]] * * Z3[G24]Po0of C0 = Z3[[G12]] Z3[G24]Z3 and the resolution Qo of N0 which is obtained from sp* *licing the exact complex obtained from (1) 0 ! C3 ! C2 ! C1 ! N0 ! 0 with a projective resolution of C3 = C0, i.e. by splicing it with Po. The next * *result records all we need to know about the chain map OE. Lemma 6.3. There is a chain map OE : Qo ! Po covering the homomorphism j such t* *hat 1 OE0 : Q0 = C1 ! 1 "G2Q8= P0 sends e1 to (e - !)ee0. (Here the generator ee02 P0 is given by e 1 2 P0 and * *we continue to denote the generator of C1 introduced in section 3.2, and also given by e 1, * *by e1.) Proof.In fact, as in the proof of Corollary 3.3 we see that SD16 acts on (e - !* *)ee0via the character O. Hence OE0 is well defined and it is clear that OE0 covers j. Proof of Proposition 5.5. From its definition it is clear that ff 2 H1(G24, M4* *) is a perma- nent cycle in the algebraic spectral sequence for M* = (E2)*=(3) (the restricti* *on of the class with the same name in H1(G12, M4)), i.e. there are cochains c 2 Hom Z3[[G12]](* *P1, M4) and d 2 Hom Z3[[G12]](Q0, M4) such that c + d is a cocycle in the total complex of * *the double complex Hom Z3[[G12]](Too, M4) and such that c represents ff 2 Ext1Z3[[G12]](C0, M4) = * *H1(G24, M4). Fur- thermore, the cocycle c can be obtained as the mod-3 reduction of a cocycle rep* *resenting ffi0(v1) (cf. the discussion in section 5.1), i.e. we can take c = 1_3(a*2- a*)v1 and th* *is is known to be of the form !u-2 mod (u1) (cf. the proof of Lemma 1 in [4]). Then v1c is a cocy* *cle represent- ing v1ff. However, v1ff is trivial in Ext1Z3[[G12]](C0, M4) by Theorem 1.1, and* * hence there exists h 2 Hom Z3[[G12]](P0, M4) such that v1c = ffiP(h) = a2*h - a*h . Because v1c is equal to !u1u-4 mod (u21), h must have the form fflu-4 mod (u1) * *for some unit ffl 2 F9. Corollary 4.7 shows that (a*2- a*)(u-4) = -(1 + !2)u1u-4 mod (u1) and hence ffl = !2 by (6). In the double complex HomZ3[[G12]](Too, M4) the coch* *ain v1c is therefore cohomologous to -OE*0(h) -(e - !)*(!2u-4) -!2u-4 + !-2u-4 !2u-4 mod (u1) and hence v1(c+d) is cohomologous to !2u-4+v1d mod (u1) and this implies the pr* *oposition. Now we turn towards the calculation of higher differentials and extensions. The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *29 Proposition 6.4. a)The differential d2 : E0,12! E2,02in the algebraic spectral sequence (2) * *for M = (E2)*=(3) is given by 8 m+n+1 6.3n+1_ n < (-1) v1 n+b3n+1(6m+1)1_ k = 2.3 (3m + 1) d2( kff)= : (-1)m+n v10.31 +1b3n+1(18m+11)k = 2.3n(9m + 8) 0 otherwise ae m 11_ d2( keff)= (-1)0 v1 b18m+11 ko=t6mh+e5rwise. b)In H*(G12; (E2)*=(3)) we have the following relations v1 2.3n(9m+2)ff= (-1)m+1b2.3n+1(9m+2)+1 v1 2.3n(9m+5)ff= (-1)m+1b2.3n+1(9m+5)+1 v1 2k+1ff = 0 v1 6m+1eff = (-1)m b2.(9m+2)+1 v1 6m+3eff = (-1)m+1b2.(9m+5)+1 v1 2keff = 0 . Remark_ We repeat that after the fact one can check that the d2-differential is* *, up to sign, determined by v1-linearity and the principle that it is non-trivial whenever it* * has a chance to be so. In the proof of the proposition we make repeated use of the following lemma. Lemma 6.5. Let s > 0, x 2 Hs(G24, M*) and c 2 Hom Z3[[G12]](Ps, M*) be a repres* *enting cocycle. Suppose that vk1x = 0 2 Hs(G24, M*) and suppose that h 2 Hom Z3[[G12]* *](Ps-1, M*) satisfies ffiP(h) = vk1c. a)Then there are elements d, d0 2 Hom Z3[[G12]](Qs-1, M*) and d002 Hom Z3[[* *G12]](Qs, M*) such that OE*s-1(h) = d0+ vk1d and ffiQ(d0) = vk1d00. b)If d, d0and d00are as in (a) then j*(x) = (-1)s[d00] 2 Exts(N0, M*) . c)If d, d0and d00are as in (a) and d00= 0 then vk1x0= (-1)s[d0] 2 H*(G12, M*) for any x0in H*(G12, M*) which restricts to x. In particular, if d0= 0, t* *hen vk1x0= 0. Proof.We can write OE*s-1(h) 2 Hom Z3[[G12]](Qs-1, M*) M* as utf for some t 2* * Z and f a power series in u1. Then we write f = f0+uk1f1 with f0 a polynomial in u1 of de* *gree less than k and f1 a power series in u1. If we put d0= utf0 and d = u2k+tf1 then the first * *part of (a) holds. Next we use that the double complex Hom Z3[[G12]](Too, M*) is a double complex * *of torsionfree F3[[v1]]-modules and F3[[v1]]-linear differentials. Then we get ffiQ(d0) + vk1ffiQ(d) = ffiQ(d0+ vk1d) = ffiQ(OE*s-1(h)) = OE*s(ffiP(h)) = * *OE*s(vk1c) 0 mod (vk1) and vk1j*(c)= vk1(-1)sOE*s(c) = (-1)sOE*s(vk1c) = (-1)sOE*s(ffiP(h)) = (-1)sffiQOE*s-1(h) = (-1)sffiQ(d0+ vk1d) = (-1)svk1d00+ (-1)svk* *1ffiQ(d) . and thus the second part of (a) and (b) follow. If d00= 0 then the equations vk1c = ffiP(h), (-1)s-1vk1d + (-1)s-1d0= (-1)s-1OE*s-1(h) show that vk1(c+(-1)s-1d) and (-1)sd0are cohomologous in the double complex. Fu* *rthermore, d0is a cycle by assumption and hence vk1(c + (-1)s-1d) is a cycle. Because the * *double complex is v1-torsion free, c + (-1)s-1d is a cycle as well and (c) follows. 30 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Proof of Proposition 6.4. We begin with the case of kff. As in the proof of Pr* *oposition 5.5 let c 2 Hom Z3[[G12]](P0, M4) be a cocycle representing ff 2 Ext1Z3[[G12]](C0, * *M4). Then kff is represented by the cocycle kc. As k is G24-invariant we have kv1c = ffiP( kh) where h is as in the proof of Proposition 5.5 above. In particular h !2u-4 an* *d hence OE*0( kh) (e - !)*(!2k+2u-12k-4) (1 - (-1)-3k-1)!2k+2u-12k-4mod (u41)* * . (We note that the congruence is modulo (u41) by (24)!) In particular, if k is o* *dd we see that OE*0( kh) = v41z for some z 2 Hom Z3[[G12]](Q0, M*) and if k is even, k = 2.3n(3m 1), we get OE*0( kh) = (-1)m b3k+1+ v41z for some z 2 Hom Z3[[G12]](Q0, M*). Then Lemma 6.5 and Theorem 1.2 give easily * *the differentials and extensions on all elements kff for k 6= 0. In the case of keffthe definition of eff(cf. section 5.1) shows that effcan * *be represented by a cocycle ec2 Hom Z3[[G12]](P1, M12) such that v1ec= ffiP(v2). Because k is G24-* *invariant we have ffiP( kv2) = kffiP(v2). Furthermore, kv2 !2ku-12k-8mod (u21) and thus OE*0( kv2) (e - !)*(!2ku-12k-8) (1 - (-1)-3k-2)!2ku-12k-8mod (u41) . (Again we note that the congruence is modulo (u41) by (24)!) In particular, if * *k is even we deduce that OE*0( kv2) = v41z for some z 2 Hom Z3[[G12]](Q0, M*) and for k = 6m + j with j 2 {1, 3, 5} we have OE*0( kv2) = (-1)m+1!2j-2b18m+3j+2+ v41z for some z 2 Hom Z3[[G12]](Q0, M*) and as before Lemma 6.5 and Theorem 1.2 give* * easily the differentials and extensions on all elements keff. The following result together with Proposition 5.5 and Proposition 6.4 finish* *es off the proof of Proposition 1.3 and Proposition 1.5. Proposition 6.6. a)For i 3 the differentials di in the algebraic spectral sequence (2) for* * the mod-3 Moore spectrum are all trivial. b)For each integer k 2 Z and l > 0 we have v21 kfil = v1 kfilff = v1 kfilef* *f= 0 in H*(G12; (E2)*=(3)). Proof.The differential is linear with respect to the natural ExtZ3[[G12]](Z3, F* *3) = H*(G12, F3)- module structure on its target and source. Hence it is enough show that the cl* *asses kfiff, kfieffand kfi are d3-cycles and to prove (b) in the case l = 1. In both cases* * we use Lemma 6.5 once again. We begin with the case of kfi. Let c1 be a cocycle in Hom Z3[[G12]](P2, M12)* * representing fi. As v21fi = 0 in H2(G24, M20) there exists h1 2 Hom Z3[[G12]](P1, M20) such that v21c1 = ffiP(h1) = (e + a + a2)*h1 . Next we use that Q1 = C2 = Z3[[G12]] Z3[SD16]O. Hence we have (cf. (24)) Hom G12(Q1, M*) ~=!2u4F3[[u41]][v1, u 8] and by degree reasons we need to have OE*1(h1) = v31z for some z 2 Hom Z3[[G12]* *](Q1, M8), and then Lemma 6.5 shows that fi is not only the restriction of a permanent cycle (* *which we knew anyway), but also that v21fi = 0 in H2(G12, M20). Furthermore, as k is G24-inv* *ariant we have The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *31 ffiP( kh1) = kv21c1 and in order to apply Lemma 6.5 in the case of kfi we nee* *d to understand OE*1( kh1). For this we recall that Q1 = C2 is a free Z3[[S12]]-module generate* *d by e2. Hence we can write OE1(ee1) = xe2for (a unique) x 2 Z3[[S12]] and thus OE*1( kh1) = x*( * *kh1). Furthermore, because !2u-12 mod (u21) we have for any g 2 S12 g*( kh1) = g*( k)g*(h1) t0(g)12k kg*(h1) kg*(h1) mod (u21) , thus OE*1( kh1) = x*( kh1) kx*(h1) = kOE*1(h1) 0 mod (u21) and then Lemma 6.5 shows that kfi is a permanent cycle and in H2(G12; M) we ha* *ve v21 kfi = 0. In the case of kfiff we choose a representing cocycle c2 2 Hom Z3[[G12]](P3,* * M16) for fiff. Because v1fiff = 0 in H1(G24, M20) there exists h2 2 Hom Z3[[G12]](P2, M20) suc* *h that v1c2 = ffiP(h2). Then OE*2(h2) 2 Hom Z3[[G12]](Q2, M20) = Hom Z3[[G12]](P0, M20) ~=(M20)Q8 is divisible by v1 by degree reasons (cf. (22) or Remark 3.12.3 of [5]) and the* *n Lemma 6.5 shows that kfiff is a permanent cycle and in H2(G12, M*) we have v1 kfiff = 0. The c* *ase of v1fieffis completely analogous. Proposition 6.7. If x 2 H1(G12, M*) is represented in E1,0,*1then fix = 0 in H3* *(G12, M*). Proof.It is enough to show fib1 = 0 and fib2.3n(3m-1)+1= 0 whenever m 6 0 mod * * (3). By Proposition 5.5 we have b1 = v1ff and thus fib1 = v1fiff = 0 by the previous pr* *oposition. Similarly, by Proposition 6.4 we have b2.3n+1(3m-1)+1= v1 2.3n(3m-1)ff and b2(* *3m-1)+1= v1 2m-1effand using the previous proposition once more shows fib2.3n(3m-1)+1= * *0. The following result finishes off the proof of Proposition 1.5. Proposition 6.8. The following relations hold in H*(G12; M*) _ __ fib3n+1(6m+1)_= __3n(6m+1)eff fib3n+1(18m+11)=_ __3n(18m+11)eff fib18m+11 = 6m+4ff . Proof.First we observe that eff2 H*(G24, (E2)*=(3)) is non-divisible by v1 and * *this implies that the mod-u1 reduction homomorphism H*(G24, (E2)*=(3)) ! H*(G24, (E2)*=(3, u1)) m* *ust send effto !2u-4ff (cf. Theorem A.3). Likewise, this map must send 2k+1to !2v3k+1* *2u-4, and it clearly sends ff to ff. Thus the proposition follows from Theorem A.3.c and * *naturality. 7.Passing from G12to G2 Theorem 1.7 is a simple instance of a K"unneth isomorphism: in fact, if we ha* *ve an isomor- phism of profinite 3-groups G = F xZ3, and if Z3 acts trivially on a 3-profinit* *e module M, then the exterior product in cohomology induces an isomorphism (35) H*(F, M) Z3 Z3(i) ~=H*(F, M) Z3H*(Z3; Z3) ! H*(G2, M) . In particular this holds if G = G2, F = G12and M = (E2)*=(3) or M = (E2)*=(3, u* *1). We will need to know how the Bockstein homomorphisms ffi1 and ffi0 associated to the ex* *act sequences (26) and (27) behave with respect to these isomorphisms. The proof of the following lemma is a straightforward exercise with the doubl* *e complex ob- tained from tensoring a projective resolution of the trivial Z3[[F ]]-module Z3* * with the projective resolution of the trivial Z3[[Z3]]-module Z3 given by 0 ! Z3[[T ]] -T!Z3[[T ]] ! Z3 ! 0 and is left to the reader. (Here we have identified Z3[[T ]] with Z3[[Z3]] via* * the continuous isomorphism which sends T to t - e if t is a topological generator of Z3.) 32 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Lemma 7.1. Let G = F x Z3 and H be a closed subgroup of G, 0 ! M1 ! M2 ! M3 ! 0* * be a short exact sequence of continuous G-modules and ffiH be the associated Bockste* *in in H*(H, -). If Z3 acts trivially on M1 and M3, and if we identify H*(G, Mi) with H*(F, Mi) * *H*-1(F, Mi)i for i = 1, 3 via (35), M3 with H0(Z3, M3) and M1 with H1(Z3, M1), then ffiG(x)= ffiF(x)+ (-1)nHn(F, ffiZ3)(x)i x 2 Hn(F, M3) ffiG(yi)= ffiF(y)i y 2 Hn-1(F, M3) . Corollary 7.2. Let M = (E2)4k=(3) resp. M = (E2)4k=(3, u1) and identify H*(G2* *, M) with H*(G12, M) H*-1(G12, M)i via (35). a)For x 2 Hn(G12, (E2)4k=(3, u1)) we have ffi1G2(x) = ffi1G12(x). b)For x 2 Hn(G12, (E2)4k=(3)) we have ffi0G2(x) = ffi0G12(x) + (-1)nkxi. In particular, if we define ff(G2) := ffi0G2(v1)eff(G2) := ffi0G2(v2)fi(G2) := ffi0G2ffi1G2(v* *2) ff(G12) := ffi0G12(v1)eff(G12) :=fffi0G12(v2)i(G12) := ffi0G12ffi* *1G12(v2) then ff(G2) = ff(G12) - v1i, eff(G2) = eff(G12), fi(G2) = fi(G12) . Proof.The central factor Z3 of G2 is generated by the element t := 1 + 3 2 Zx3.* * In this case t acts trivially on (E2)0 and on u via t*(u) = 4u. Therefore t acts trivially * *on M and via multiplication by (1 + 3)-2k = 1 - 6k on (E2)4k=(9). Hence ffiZ3 is given by mu* *ltiplication by -2k k mod (3) and the result follows. 8.The Adams-Novikov spectral sequence for LK(2)V (0) As before we use the E1-term of the algebraic spectral sequence (2) for M = (* *E2)*V (0) to represent elements in the E2-term of the Adams-Novikov spectral sequence for ss* **(LK(2)V (0)). However, unlike in the introduction, we do not always insist on writing element* *s in terms of the F3[fi, v1] (i)-module generators of Theorem 1.4. This allows for simplif* *ied statements of Lemma 8.1, Corollary 8.2, Lemma 8.4 and Corollary 8.5 below. The E2-term satisfies Es,t2= 0 unless t 0 mod (4), hence its differentials* * dr are trivial if r 6 1 mod (4). Furthermore, the differentials are linear with respect to F3[fi* *, v1] (i), and the existence of the resolution (4) of [5] gives further restrictions on the behavi* *our of the Adams- Novikov differentials. In fact, they have to preserve the filtration on its E2-* *term given by the algebraic resolution for G2, and modulo this filtration the differentials are e* *asily determined by the differentials in the Adams-Novikov spectral sequence for EhG242^ V (0) (cf.* * Theorem A.1). However, to settle the ambiguities coming from potential contributions of small* *er filtration terms we need to fall back on knowledge of the differentials in ss*(LK(2)V (1)) (cf. * *Theorem A.4). The following Lemma records some immediate consequences of the knowledge of t* *he d5- differential in the Adams-Novikov spectral sequence for ss*(LK(2)V (1)). Lemma 8.1. The following identities hold in H*(G2, (E2)*=(3)) ~=E*,*2~=E*,*5of* * the Adams- Novikov spectral sequence for ss*(LK(2)V (0)). a)Let k 6 0 mod (3). Then there are constants fflk 2 { 1} such that d5( kefffi)= fflk k-1fi4v1 d5( kfi2)_ = fflk k-1ff(G2)fi4_ d5( 48_keff)=_ fflk 48_k-1fi3v1_ d5( 48 kfi)= fflk 48 k-1ff(G2)fi3 . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *33 b)Let k 0 mod (3). Then d5( kefffi) = 0 d5( kfi2)_ = 0 d5( 48_keff)=_ 0 d5( 48 kfi2)= 0 . Remark_By identifying vector space generators in the appropriate bidegrees it i* *s easy to see that there are unique elements ~i, ~i, i2 F3 such that __ __ d5( kefffi)= ~1 k-1fi4v1+ ~1 48 k-2ff(G12)fi2 + 1 48_k-2fi2v1i d5( kfi2)_ = ~2 k-1ff(G12)fi4_+ ~2 k-1fi4v1i + 2 48 k-2ff(G12)fi2i d5( 48_keff)=_ ~3 48_k-1fi3v1_ __ d5( 48 kfi)= ~4 48 k-1ff(G12)fi3 + ~4 48 k-1fi3v1i . Naturality and the geometric boundary theorem (cf. Theorem 2.3.4 of [10]) appli* *ed to the reso- lution (4) allow to determine the values of the ~i, i.e. to show the lemma modu* *lo elements of lower filtration. The Lemma confirms these values and also determines ~i and i* *; formally ~i and ican be deduced by simply replacing in the differentials for EhG242^ V (0)* * the elements ff by ff(G2). Proof.We start with kefffi. This is in the kernel of v1-multiplication and mus* *t therefore (after 4-fold suspension) be in the image of the Bockstein homomorphism ffi1G2in H*(G2* *, -) and ffi1G12 in H*(G12, -) associated to the short exact sequence (27). Similarly with kfiv* *1. By Theorem A.3 and by degree reasons we must therefore have ffi1G12((!2u-4)3k+2fi) = 4 kefffi, ffi1G12((!2u-4)3k+2ff(G12)) = * *4 kfiv1 and, by Corollary 7.2, we even get (36) ffi1G2((!2u-4)3k+2fi) = 4 kefffi, ffi1G2((!2u-4)3k+2ff(G12)) = 4 k* *fiv1 . and by fi-linearity ffi1((!2u-4)3k+2fffi3) = 4 kfi4v1. Then the geometric bou* *ndary theorem and Theorem A.4.b show that 6kefffi and 6k+3efffi are permanent cycles and th* *e value of the differential in the other cases is as stated (with a suitable constant fflk). __ The case of 48 kefffi can be treated similarly but in this case it would als* *o suffice to use the strategy described in the remark above. The sign is clearly the same as in the * *previous case. The remaining two cases are deduced from what has already been established by* * using the Bockstein ffi0G2in H*(G2, -) associated to the short exact sequence (26) and th* *e geometric boundary theorem. By Corollary 7.2 we have __ __ ffi0G2( kefffi) = kfi2,ffi0G2( 48 keff)_=_ 48 kfi __ ffi0G2( kfi4v1) = kff(G2)fi4,ffi0G2( 48 kfi3v1) = 48 kff(G2)fi3 . In fact, by the Corollary we only need to determine ffi0G12and this is straight* *forward in the case of __ __ 48 kfi3v1 and 48 kefffi3. In the other two cases it is straightforward modulo* * terms of lower filtration and by degree reasons there are no error terms of lower filtration. Corollary 8.2. The d5-differential in the Adams-Novikov spectral sequence for L* *K(2)V (0) is linear with respect to F3[fi, v1] (i) and is trivial on all F3[fi, v1] (i* *)-module generators of H*(G12, (E2)*=(3)) of Theorem 1.6 except the following: 34 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald d5( m fi) = m-1ff(G2)fi3 m 6 0 mod (3) d5( 2meff) = 2m-1fi3v1 m 6 0 mod (3) d5( 6m+1eff) = 6mfi3v1 d5( 6m+5efffi) = 6m+4fi4v1 _ __ d5(b3n+1(6m+1))_ = 48_3n(6m+1)-3fi2v1_ n 0 d5(b3n(6m+5)) = 48 3n-1(6m+5)-3fi2v1 m 1 mod (3), n 2 __ __ d5( 48_2m)_ = 48_2m-1ff(G2)fi2_ m 6 0 mod (3) d5( 48_3n(6m+1)-2)_= 48_3n(6m+1)-3ff(G2)fi2n_ 0 d5( 48_3n(6m+5)-2)_= 48_3n(6m+5)-3ff(G2)fi2n_ 1 d5( 48_2mfef)_ = 48_2m-1fi3v1_ m 6 0 mod (3) d5( 48 3n(6m+5)-2eff)= 48 3n(6m+5)-3fi3v1, m 6 1 mod (3), n 1 . Proof.Linearity with respect to F3[fi, v1] (i) is clear. The rest is an imme* *diate consequence of the previous lemma together with sparseness, Proposition 1.5, Theorem 1.6, a* *nd the fact that fi-multiplication is injective above cohomological degree 3. Corollary 8.3. E6 is the quotient of the direct sum of cyclic F3[fi, v1] (i)* *-modules with the following generators and annihilating ideals 1 = 0 (fiv21, fi3v1) m fi 0 6= m 0 mod (3) (v21, fi2v1) 6m+1fiv1 (v1, fi2) 6m+4fiv1 (v1, fi3) m fiv1 m 2 mod (3) (v1) ff(G12) (fiv1, fi3) 2m+1ff(G12) m 6 2 mod (3) (v1, fi3) 2m+1ff(G12) m 2 mod (3) (v1) n+1-1 2.3n(3m-1)ff(G12)m 6 0 mod (3), n 1(v4.31 , fiv1, fi3) 2(3m-1)ff(G12) m 6 0 mod (3) (v111, fiv1, fi4) 6meff (v1) b2(9m+2)+1 (v21, fi) 6m+3eff (v31, fiv1) 2.3n(3m+1)ff(G12)fin 1 (v1, fi2) 2(3m+1)ff(G12)fim 0 mod (3) (v1) 2.3n(3m-1)ff(G12)fin 1 (v1, fi2) 2(3m-1)ff(G12)fim 0 mod (3) (v1, fi3) _ n b3n+1(6m+1)v1_ n 0 (v6.31,nfi) b3n(6m+5)v1_ m 1 mod (3), n 2 (v10.31,nfi) b3n(6m+5) m 1 mod (3), n = 0, 1(v10.31+1, fiv1) __ 48_3n(6m+1)-3_ n 1 (v21, fi2v1) 48_3n(6m+5)-3_ m 6 1 mod (3), n 1 (v21, fi3v1) 48_3n(6m+5)-3_ m 1 mod (3), n 1 (v21, fi2v1) 48_(6m+1)-3v1_ (v1, fi2) 48 (6m+5)-3v1 (v1) __ 4.3n-1 48_3n(6m_1)-2v1 n 1 (v1 , fiv1, fi3) 48_(6m+1)-2v1_ (v31, fiv1) 48 (6m+5)-2 (v41, fiv21, fi3v1) The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *35 __ 48_2m-1ff(G12)_ m 6 0 mod (3) (v1, fi3) 48_6m+5ff(G12)_ (v1) 48_3n(6m+1)-3ff(G12)n_ 0 (v1, fi2) 48_3n(6m+5)-3ff(G12)m_6 1 mod (3), n (v11, fi3) 48_3n(6m+5)-3ff(G12)m_ 1 mod (3), n (1v1, fi2) 48_6mfef_ (v1) 48 (6m+5)-2eff m 6 1 mod (3) (v1) modulo the following relations in which module generators are put into paranthe* *sis (in order to distinguish between module multiplications and generators) fi3( kff(G12)) = fi2i( kfiv1), k = 2(3m - 1) m 6 0 mod (3) fi2( kff(G12)fi)=_fi2i( kfiv1),_k = 2(3m - 1) m 0 mod (3) fi2( 48_kff(G12))=fi2v1i(_48_k),k_= 2m - 1 m 6 0 mod (3) fi2( 48 kff(G12)= fi2v1i( 48 k) k = 3n(6m + 5) - 3m 6 1 mod (3), n 1 . Proof.In principle this is a straightforward consequence of the previous coroll* *ary and Theorem 1.6. Complications arise because some of the time integers are distinguished by* * their residue class modulo (2), some of the time by their residue class modulo (3) and some o* *f the time the distinction is more involved. __ The most complicated case is perhaps that of the classes 48 m for m even. Th* *ere one uses that an even integer can be uniquely written in the form k - 3 with k odd and a* *n odd integer k can be uniquely written either as 3n(6m 1) with n 1. This together with t* *he previous __ corollary and Theorem 1.6 leads to the result for the first block of generators* * involving 48 m . The other blocks can be treated similarly. The following Lemma records immediate consequences of the knowledge of the d9* *-differential in the Adams-Novikov spectral sequence for ss*(LK(2)V (1)). Lemma 8.4. The following identities hold in the E9-term of the Adams-Novikov * *spectral sequence for ss*(LK(2)V (0)). a)Let k 2 mod (3). Then there are constants ffl0k2 { 1} such that d9( kff(G2)fi2) = ffl0k k-2fi7 d9( kfi2v1)_ = ffl0k k-2efffi6_ d9( 48_kff(G2)fi)=_ ffl0k 48_k-2fi6_ d9( 48 kfiv1) = ffl0k 48 k-2efffi5 . b)Let k 0, 1 mod (3). Then d9( kff(G2)fi2) = 0 d9( kfi2v1)_ = 0 d9( 48_kff(G2)fi)=_0 d9( 48 kfiv1) = 0 . Remark_By identifying vector space generators in the appropriate bidegrees it i* *s easy to see that there are unique elements ~i, ~i, i2 F3 such that __ d9( kff(G12)fi2)= ~5 k-2fi7 + ~5 k-2efffi6i_+ 5 48 k-3fi5i_ d9( kfi2v1)_ = ~6 k-2efffi6_+ ~6 48_k-3fi5_+ 6 48 k-3efffi4i d9( 48_kff(G12)fi)=_~7 48_k-2fi6_+ ~7 48 k-2efffi5i d9( 48 kfiv1) = ~8 48 k-2efffi5 . As before naturality and the geometric boundary theorem applied to the resoluti* *on (4) allow to show the lemma modulo elements of lower filtration, i.e. to determine the va* *lues of the ~i. Again the Lemma confirms these values and also determines ~i and i, and formal* *ly ~i and i can be deduced by simply replacing in the differentials for EhG242^ V (0) th* *e elements ff by ff(G2). 36 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald Proof.The proof resembles that of Lemma 8.1. We start with the class kfi2v1. F* *rom (36) and fi-linearity of ffi1G2we get ffi1G2((!2u-4)3k+2ff(G2)fi) = 4 kfi2v1, ffi1G2((!2u-4)3k-4fi6) = 4 k* *-2efffi6 and as before the geometric boundary theorem and Theorem A.4.c yield the value * *of the differ- ential (with a suitable constant ffl0k). __ The case of 48 kfiv1 can be treated similarly but we can also use the strate* *gy described in the remark above. Again the sign is clearly the same as in the previous case. The remaining two cases can once again be deduced by using the Bockstein ffi0* *G2in H*(G2, -) associated to the short exact sequence (26). As in the proof of Lemma 8.1 we ob* *tain __ __ ffi0G2( m fi2v1) = m ff(G2)fi2,ffi0G2(_48 m fiv1) = _48_m ff(G2)fi ffi0G2( m efffi6) = m fi7,ffi0G2( 48 m efffi5) = 48 m fi6 and the geometric boundary theorem gives the result. Corollary 8.5. The d9-differential in the Adams-Novikov spectral sequence for L* *K(2)V (0) is linear with respect to F3[fi, v1] (i) and is trivial on all F3[fi, v1] (i* *)-module generators of E*,*6given in Corollary 8.3 except the following: d9( m fiv1) = m-2feffi5 m 2 mod (3) d9( 6m+5ff(G12)) = ( 6m+3fi5 + 6m+3efffi4i) d9(_2(3m+1)ff(G12)fi)= ( 6mfi6_+ 6mefffi5i)_ d9(b18m+11)_ = ( 48_6mfi4 + 48 6mfeffi3i) d9( 48_6m+2v1)_ = 48_6mfeffi4_ d9( 48_6m+5v1)_ = _48 6m+3efffi4 m 6 1 mod (3) d9( 48_6m+5v1)_ = b3(6m+5)fi5_ __ m 1 mod (3) d9( 48 6m+5ff(G12))= ( 48 6m+3fi5 + 48 6m+3efffi4i). Proof.Linearity with respect to F3[fi, v1] (i) is clear. Then we note that the* * fi-torsion classes in the E6-term are in too low a cohomological degree to interact via d9 and hen* *ce d9 is trivial on them. The rest is an immediate consequence of Proposition 1.5, Corollary 8.3 an* *d the previous Lemma, and the fact that fi-multipication is injective in the relevant bidegree* *s. This finally allows us to calculate the homotopy of ss*(LK(2)V (0)). Proof of Theorem 1.8. By using the last corollary it is straightforward to veri* *fy that the E10- term has the structure described in Theorem 1.8. Then we see that Es,*10= 0 for* * s > 11 and hence there is no room for higher differentials and we get E10= E1 . Appendix A. The Adams-Novikov spectral sequences converging towards ss*(EhG242^ V (0)), ss*(EhG242^ V (1)) and ss*(LK(2)V (1)) The behaviour of the spectral sequence for ss*(EhG242^ V (0)) can be deduced * *from that for ss*(EhG242). We record this in the following result. Theorem A.1. a)The differentials in the Adams-Novikov spectral sequence Es,t2~=F3[[v61 -1]][v1, 1, fi, ff, eff]=(ff2, eff2, v1ff, v1eff, ffeff+ v1* *fi) =) sst-s(EhG242^ V (0)) are linear with respect to F3[ 3, v1, fi, ff, eff]. The only nontrivial* * differentials are d5 and d9. They are (redundantly) determined by d5( ) = fffi2, d5( eff) = efffffi2 = v1fi3 and d9( 2ff) = fi5, d9( 2v1) = efffi4 . The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *37 b)We have an inclusion of subalgebras E0,*1~=F3[[v61 -1]][v1, v1 , 3] . In positive filtration Es,t1has an F3-vector space basis given by the 16 * *elements which are represented on E2 by ff, fiff, ff, fi ff, effff = fiv1, fieffff = fi* *2v1, effff = fi v1, fi effff = fi2 v1, fij, j = 1, 2, 3, 4, fikeff, k = 0, 1, 2, 3 and their * *multiples by powers of 3. Proof.This is a consequence of the behaviour of the spectral sequence convergin* *g to ss*(EhG242) (cf. [4], [5]). First one observes that every element in the transfer from ss*(* *E2) is a permanent cycle. Together with the fact that v1 is a permant cycle as well, this shows th* *at the only classes on the 0-line which can carry non-trivial differentials are the classes kvffl1* *with k 2 Z and ffl = 0 or k 6= 0 and ffl = 1. Furthermore we recall that the elements ff and fi are pe* *rmanent cycles in the Adams-Novikov spectral sequence converging to ss*(EhG242) and effin that fo* *r V (0) detecting homotopy classes fow which we use the same names. Now the spectral sequence for ss*(EhG242) shows d5( ) = fffi2 and from the f* *act that the spectral sequence for ss*(EhG242^ V (0)) is a module over that for ss*(EhG242) * *we deduce d5( eff) = ffefffi2 = v1fi3, d5( kv1) = 0, k 1, 2 mod (3) , and d5 happens to be a derivation. In particular we obtain an isomorphism of al* *gebras E0,*6~=F3[[v61 -1]][ 3, v1, v1, 2v1] , and in positive filtration Es,t6has an F3-vector space basis given by the eleme* *nts fik, k > 0, fikv1, k = 1, 2, fikff, k = 0, 1, fikeff, k 0, fik v1, k = 1, 2, fik ff, k = * *0, 1, fik 2v1, k > 0 and fik 2ff, k 0 and their multiples by 3. Next the Adams-Novikov spectral sequence for ss*(EhG242) implies d9(fil 3k+2f* *f) = fil+5 3k, hence the module structure of the spectral sequence for ss*(EhG242^V (0)) with * *respect to that of ss*(EhG242) gives d9(fil+1 3k+2v1) = d9(fil 3k+2effff) = fil+5 3keffand thus d* *9(fil 3k+2v1) = fil+4 3keff. Furthermore sparseness shows that d9 is trivial on all other cla* *sses of positive cohomological degree. In the resulting E10-term we have Es,*2= 0 for s > 8 and * *thus there is no more room for higher differentials. Hence we get E10= E1 and the structure o* *f E1 agrees with the stated result. The following result has already been proved in [4] for the subgroup G12inste* *ad of G24. In fact, part (a) is an immediate consequence of the formulae (22) and (23) and th* *e fact that the action of G24on (E2)*=(3, u1) factors through an action of the quotient G24=Z=3* * = Q8 SD16. The proof of part (b) and (c) is analogous to the case of G12. We leave the det* *ails to the reader. Theorem A.2. (cf. Theorem 9 of [4]) a)The E2-term of the Adams-Novikov spectral sequence converging to ss*(EhG2* *42^ V (1)) is given by E*,*2~=H*(G24, (E2)*=(3, u1)) ~=F3[!2u 4, fi, ff]=(ff2) . b)The only non-trivial differentials in this Adams-Novikov spectral sequenc* *e are d5 and d9. They are determined by linearity with respect to F3[!2u 4, fi, ff] and th* *e formulae ae d5((!2u 4)k) = 0 (!2u 4)k-3fffik2k 0, 1,32, 4, 5mod,(9)6,m7,o8d(9) and ae d9((!2u 4)kff) = 0 (!2u 4)k-3fi5kk 0,61,,2,73,,4,m5od(9)8mod(9) . 38 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald c)There are elements in ss8k(EhG242^ V (1)) represented by (!2u-4)k, k = 0,* * 1, 2, and in ss8k+3(EhG242^ V (1)) represented by (!2u-4)kff, k = 0, 1, 2, 3, 4, 5, su* *ch that there is an isomorphism of modules over F9[ 3, fi] i ss*(EhG242^ V (1))F~=9[ 3] F9[fi]=(fi5){1, !2u-4, (!2u-4)2} j F9[fi]=(fi2){ff, . .,.(!2u-4)5ff} . Next we turn towards the algebraic spectral sequence (2) in the case of M = (* *E2)*=(3, u1) and the Adams-Novikov spectral sequence converging towards ss*(LK(2)V (1)). Thi* *s has already been discussed in [4] and [11]. Here we merely translate those results into a f* *orm suitable for our discussion. As before v2 is defined to be u-8. Theorem A.3. (cf. section 4 of [4]) a)As a module over F3[v21, fi, ff]=(ff)2 the E1-term of the algebraic spect* *ral sequence (2) for (E2)*=(3, u1) is given as follows (with fi and ff acting trivially on* * Es,*,*1if s = 1, 2, and the elements es, s = 0, 1, 2, 3, serving as module generators in trid* *egree (s, 0, 0). ) 8 >!2u4F3[v21]es s = 1, 2 :0 s > 3 . b)The differentials in this spectral sequence are F3[v21, fi, ff]-linear. T* *he only non-trivial differential in this spectral sequence is d1 and is determined by d0,01(!2u-12e0) = !2u-12e1, d1,01= 0, d2,01= 0 . c)The following fi-extensions hold in H*(G12, (E2)*=(3, u1)) fi!2u4vk2e2 = vk2ffe3 . d)H*(G12; (E2)*=(3, u1)) is a free module over F3[v21, fi] on generators e0, ffe0, !2u-4ffe0, !2u-4fie0, !2u-4e2, e3, !2u-4e3, !2u-4ffe3 . e)There is an isomorphism of F3[v21, fi] (i)-modules (even of algebras) H*(G2, (E2)*=(3, u1)) ~=H*(G12, (E2)*=(3, u1)) (i) . Proof.a) The action of G12on (E2)*=(3, u1) is trivial on its Sylow-subgroup S12* *, and on the quotient group G12=S2 ~=SD16the action is given by the formulae in (22) and (23* *). With this information the calculation of the E1-term is straightforward and is left to th* *e reader. b) As module over F3[v21, fi, ff] the E1-term is generated by es and !2u-12es* * for s = 0, 3 and !2u-4es for s = 1, 2. The map of algebraic spectral sequences (2) induced b* *y the canonical homomorphism_(E2)*=(3) ! (E2)*=(3, u1)_of Z3[[G12]]-modules sends 2k to v3k2e0* *, b2k+1 to !2u-4vk2e1, b2k+1to !2u-4vk2e2 and 2kto v3k2e3. This and Theorem 1.2 determin* *e the d1- differential and the E2-page. The abutment of the spectral sequence is known by* * Corollary 19 of [4] and comparing the E2-term with the abutment shows that the spectral sequ* *ence collapses at its E2-term. c) From the same corollary we know that H*(S2, (E2)*=(3, u1)) is free as modu* *le over F9[fi], hence H*(G2, (E2)*=(3, u1)) and then also H*(G12, (E2)*=(3, u1)) are free as mo* *dules over F3[fi]. This requires nontrivial fi-multiplications on E2,02and by degree reasons these* * multiplications must be as claimed. d) This is an immediate consequence of (a), (b) and (c). e) This is an easy consequence of the isomorphism G2 ~=G12xZ3 and the fact th* *at the central factor Z3 acts trivially on (E2)*=(3, u1). The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited* *39 Next we note that as before the existence of the resolution (4) of [5] puts a* *dditional restrictions on the Adams-Novikov differentials for LK(2)V (1). In fact, again by naturality* * and the geometric boundary theorem, these differentials can be easily read off from those for EhG* *242^V (1), at least modulo the filtration on H*(G2, (E2)*=(3, u1)) determined by the algebraic reso* *lution (2). It turns out that the potential terms of lower filtration are always trivial altho* *ugh showing this requires a non-trivial effort which has essentially been carried out in [4]. Theorem A.4. (cf. section 3 of [4]) a)The only non-trivial differentials in the Adams-Novikov spectral sequence* * converging to ss*(LK(2)V (1)) are d5 and d9. They are both determined by the fact that * *they are linear with respect to F3[v29, fi] (i) and by the following formulae in which* * we identify vk+32e3 with 48vk2e3 etc. . b)The differential d5 is given by d5(vk2ffe0) = 0 d5(vk2!2u-4ffe0) = 0 d5(vk2!2u-4e2) = 0 d5( 48vk2!2u-4ffe3)= 0 for all k, and ae d5(vk2e0) = 0 vk-2 2 -4 2 k 0, 1, 5 mod (9) 2 ! u fffi e0 k 2, 3, 4, 6, 7,m8od(9) ae d5(vk2!2u-4fie0)= 0 vk-1 3 k 0, 4, 5 mod (9) 2 fffi e0 k 1, 2, 3, 6, 7,m8od(9) ae d5( 48vk2e3) = 0 48vk-2 2 -4 2k 0, 1, 5 mod (9) 2 ! u fffi e3k 2, 3, 4, 6,m7,o8d(9) ae d5( 48vk2!2u-4e3)= 0 48vk-1 2 k 0, 4, 5 mod (9) 2 fffi e3 k 1, 2, 3, 6, 7,m8od(9) . c)The differential d9 is given by d9(vk2e0) = 0 k 0, 1, 5 mod (9) d9(vk2!2u-4fie0) = 0 k 0, 4, 5 mod (9) d9( 48vk2e3) = 0 k 0, 1, 5 mod (9) d9( 48vk2!2u-4fie3)=0 k 0, 4, 5 mod (9) ( d9(vk2ffe0) = 0 k-3 k 0, 1, 2, 5, 6, 7 mod (9) v2 fi5e0 k 3, 4, 8 mod (9) ( d9(vk2!2u-4ffe0) = 0 k-3 k 0, 1, 2, 4, 5, 6 mod (9) v2 !2u-4fi5e0 k 3, 7, 8 mod (9) ( d9(vk+22!2u-4e2) = 0 k-5 k 0, 1, 2, 5, 6, 7 mod (9) 48v2 fi4e3 k 3, 4, 8 mod (9) ( d9( 48vk2!2u-4ffe3)= 0 k-3 k 0, 1, 2, 4, 5, 6 mod (9) 48v2 fi5!2u-4e3 k 3, 7, 8 mod (9) . 40 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald d)As a module over P = F3[v29, fi] (i) there is an isomorphism ss*(LK(2)V (1))P~==(fi5){vk2e0}k=0,1,5 P=(fi3){vk2ffe0}k=0,1,2,5,6,7 P=(fi4){vk2!2u-4fie0}k=0,4,5 P=(fi2){vk2!2u-4ffe0}k=0,1,2,4,* *5,6 P=(fi4){ 48vk2e3}k=0,1,5 P=(fi3){vk+22!2u-4e2}k=0,1,2,5,6,7 P=(fi5){ 48vk2!2u-4e3}k=0,4,5 P=(fi2){ 48vk2!2u-4ffe3}k=0,1,* *2,4,5,6. Proof.This is an immediate reformulation of the main theorem of [4]. We just ha* *ve to use the following dictionary which translates between the F9[v21, fi]-module generators* * used in Corollary 19 of [4] and those of H*(S12, (E2)*=(3, u1)) of Theorem A.31. 1_ 1_ 1_ By degree reasons the generators 1, ff, v22ff, v22fi, ffa35and v22fiffa35of [* *5] must correspond, up to a unit in F9, to e0, ffe0, !2u-4ffe0,1!2u-4fie0, !2u-4v22e2 and 48!2u-4f* *fe3 of Theorem _ A.3. The generators fia35 and v22fia35 are not determined (not even up to a un* *it) by their bidegree, but if one takes into account that they are in the kernel of the rest* *riction map to H*(G12; (E2)*=(3, u1)) then they are also determined up to a unit and they must* * therefore agree, up to a unit, with the elements 48e3 and 48!2u-4e3. References 1.Mark Behrens, A modular description of the K(2)-local sphere at the prime 3,* * Topology 45 (2006), no. 2, 343-402. 2.P. Deligne, Courbes elliptiques: Formulaire (d'apr`es J. 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Institut de Recherche Math'ematique Avanc'ee, C.N.R.S. - Universit'e Louis Pa* *steur, F-67084 Stras- bourg, France Ruhr-Universit"at Bochum, Fakult"at f"ur Mathematik, D-44780, Germany Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A. ___________ 1For the translation we pass from H*(G12; (E2)*=(3, u1)) to H*(S12; (E2)*=(3,* * u1)) in Theorem A.3 and from H*(S2; (E2)*=(3, u1)) to H*(S12; (E2)*=(3, u1)) in [4]. This passage is straigh* *tforward and left to the reader.