THE ADAMS-NOVIKOV SPECTRAL SEQUENCE FOR L K(n) (X), WHEN X IS FINITE DANIEL G. DAVIS

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1 THE ADAMS-NOVIKOV SPECTRAL SEQUENCE FOR L K(n) (X), WHEN X IS FINITE DANIEL G. DAVIS 1. Summary If X i a finite pectrum, there i an Adam-Novikov pectral equence Hc (S n; π t (E n X)) W (F p n) Zp π (L K(n) X). It i well-known that taking Gal(F p n/f p )-fixed point of thi pectral equence yield the pectral equence Hc (S n Gal(F p n/f p ); π t (E n X)) π (L K(n) X). Part of thi lat aertion i the tatement that Hc (S n ; π t (E n X)) Gal = H c (S n Gal(F p n/f p ); π t (E n X)). In thi note, we give a proof of thi iomorphim. 2. The detail Let E n be the Lubin-Tate pectrum with E n = W (F p n)[[u 1,..., u n 1 ]][u ±1 ], where the degree of u i 2 and the complete power erie ring over the Witt vector i in degree zero. Let G n = S n Gal(F p n/f p ), where S n i the nth Morava tabilizer group. Definition 2.1. Let K n, (X) = lim I π (E n M I X), for an arbitrary pectrum X. We recall Propoition 7.4 of [4]: If K n, (X) i finitely generated over E n,then there i a pectral equence (2.1) E, 2 = Hc (S n ; K n, (X)) W (F p n) Zp π (L K(n) X). Remark 2.2. The econd in the upercript of the continuou cohomology above ha the following meaning: if N i a graded G n -module, Hc t (S n ; N )=Hc (S n ; N t ). Spectral equence (2.1), contructed by Hopkin and Ravenel (ee [3, 2.2], [5, pg. 241], [7, pg. 46], and the proof in [4, pg. 116]), i contructed by taking an invere limit over {I} of Adam-Novikov pectral equence of the form E, 2 = W (F p n) Zp Ext, BP (BP BP,BP (L n X M I )) W (F p n) Zp π (L n X M I ), and applying the Morava change of ring iomorphim W (F p n) Zp Ext, BP (BP BP,BP (L n X M I )) = Hc (S n ; π t (E n M I X)). Date: September 16,

2 Now aume that X i a finite pectrum, o that K n, (X) =π (E n X) ia finitely generated E n -module. Notice that Gal act on the abutment of (2.1) by acting only on the Witt vector. There i alo a canonical action of Gal on the E 2 term: given σ Gal, the automorphim σ :Map c (Sn k,π (E n M I X)) Map c (Sn k,π (E n M I X)) i defined by (σ h)( 1,..., k )=σ h(σ 1 1,..., σ 1 k ). Thi induce the action of Gal on Hc (S n ; K n, (X)) = lim I Hc (S n ; π (E n M I X)). We recall the following definition from [1, Lem. 5.4] and [2, proof of Lem. 3.22]. Definition 2.3. Let N be a W (F p n)-module and a Gal-module. If Gal act by Z p - module automorphim and if σ(cm) =(σc)(σm), whenever σ Gal, c W (F p n), and m N, thenn i called a twited Gal-W (F p n) module. We will make ue of the following, a verion of [1, Lem. 5.4]. Lemma 2.4. If N i a twited Gal-W (F p n) module, then the incluion N Gal N extend to a Gal-equivariant iomorphim W (F p n) Zp N Gal N of W (F p n)-module. Lemma 2.5. If N i a twited Gal-W (F p n) module with the dicrete topology, then, for any j 0, Map c (Sn j,n) i a twited Gal-W (F pn) module. Proof. It i clear that Map c (Sn,N)iaW j (F p n)-module, and the Gal-action i a above. We verify that the module tructure i twited: (σ ch)( 1,..., j )=σ ((ch)(σ 1 1,..., σ 1 j )) = (σc)(σ h(σ 1 1,..., σ 1 j )) =(σc)((σ h)( 1,..., j )) = ((σc)(σ h))( 1,..., j ). Finally, the following how that Gal act by Z p -homomorphim, uing the fact that Gal act trivially on Z p : for any p-adic integer c, (σ (ch))( 1,..., j )=c(σ h(σ 1 1,..., σ 1 j )) = (c(σ h))( 1,..., j ). Lemma 2.6. If N i a finite twited Gal-W (F p n) module with the dicrete topology, then H (Gal; N) =0, > 0. Proof. Since S n o G n, the continuou projection G n G n /S n = Gal make N a dicrete G n -module. Thu, the hypothee imply that N i a dicrete twited G n -W (F p n) module [2, Def. 3.21]. Recall [op. cit., pg. 23] that given a dicrete twited G n -W (F p n) module N and a cloed ubgroup H of G n, there i a cochain complex DH (N) withh (DH ( )) equivalent to Hc (H; ) on the category of dicrete twited G n-w (F p n) module. The cochain are given by D j H (N) =Mapl c(sn j+1,n) H, 2

3 where the action on h Map l c (Sk n,n), by (g, σ) S n Gal = G n,igivenby ((g, σ)h)( 1,..., k )=σ h(σ 1 (g 1 1 ),σ 1 2,..., σ 1 k ), and the differential δ :Map l c (Sj+1 n,n) H Map l c (Sj+2 n,n) H i defined by δh( 0, 1,..., j+1 )= ( j i=0 ( 1)i h( 0,..., i i+1,..., j+1 ) ) +( 1) j+1 1 j+1 h( 0,..., j ). Again, we point out that in the above, σ 1 i refer to the action of Gal on S n, intead of being the product of two element in G n. When H = Gal, the Gal-action pecified above i identical to the action defined previouly. Alo, we need to verify that the Gal-action on the cochain complex Map c (Sn +1,N) i by map of cochain complexe. We compute: (σ δ(h))( 0,..., j+1 )=σ (δ(h)(σ 1 0,..., σ 1 j+1 )) = ( j i=0 ( 1)i σ h(σ 1 0,..., (σ 1 i )(σ 1 i+1 ),..., σ 1 j+1 ) ) +( 1) j+1 σ ((σ 1 j+1 ) 1 h(σ 1 0,..., σ 1 j )). Recall that for S n, σ = σσ 1, where the right-hand ide i a product of element in G n,andfor(, σ), (,σ ) S n Gal, (, σ)(,σ )=((σ ),σσ ). Therefore, Alo, (σ 1 i )(σ 1 i+1 )=σ 1 i σσ 1 i+1 σ = σ 1 ( i i+1 ). σ ((σ 1 j+1 ) 1 h(σ 1 0,..., σ 1 j )) = (σ(σ 1 j+1 ) 1 ) h(σ 1 0,..., σ 1 j ), where σ(σ 1 j+1 ) 1 i a product of element in G n.thu, Alo, σ(σ 1 j+1 ) 1 = σ(σ 1 j+1 σ) 1 =(1,σ)(σ 1 1 j+1σ, 1) =(σ (σ 1 1 j+1 σ),σ)=( 1 j+1,σ). (δ(σh))( 0,..., j+1 )= ( j i=0 ( 1)i (σh)( 0,..., i i+1,..., j+1 ) ) + ( 1) j+1 1 j+1 (σh)( 0,..., j ) = ( j i=0 ( 1)i σ h(σ 1 0,..., σ 1 ( i i+1 ),..., σ 1 j+1 ) ) + ( 1) j+1 1 j+1 (σ h(σ 1 0,..., σ 1 j )). Since 1 j+1 σ =( 1 j+1, 1)(1,σ)=( 1 j+1,σ), we ee that σ δ(h) =δ(σh), o that σ i indeed a map of cochain complexe. Since Map c (Sn,N)iatwitedGal-W j (F p n) module, there i a Gal-equivariant iomorphim W (F p n) Zp Map c (Sn j,n)gal Map c (Sn j,n) of W (F p n)-module. A in [1, proof of Lem. 5.15], ince Gal act by map of cochain complexe and the above iomorphim i induced by the incluion, there i actually an iomorphim W (F p n) Zp Map c (Sn +1,N) Gal Map c (Sn +1,N) 3

4 of cochain complexe. Since W (F p n) = n i=1 Z p i a free Z p -module, there i an iomorphim n i=1 H (Gal; N) = n i=1 H (Map c (Sn +1,N) Gal ) = H (Map c (Sn +1,N)) = H ({e}; N). Becaue H ({e}; N) = 0, whenever >0, the concluion of the theorem follow. Theorem 2.7. If X i a finite pectrum, then for all and t, Hc (S n; π t (E n M I X)) Gal = H c (G n ; π t (E n M I X)). Proof. Conider the canonical retriction map Hc (G n ; π t (E n M I X)) Hc (S n ; π t (E n M I X)) Gal, and the Lyndon-Hochchild-Serre pectral equence E p,q 2 = H p (Gal; Hc q (S n ; π t (E n M I X))) Hc p+q (G n ; π t (E n M I X)). It i enough to prove that E p,q 2 =0forallp 1,q 0, ince thi implie that Hc q (G n ; π t (E n M I X)) = H 0 (Gal; Hc q (S n ; π t (E n M I X))) = Hc q (S n ; π t (E n M I X)) Gal. A a finite G n -module, π t (E n M I X) ialoane n,0 -module with the E n,0 - module tructure map G n -equivariant (ince G n act on E n by ring automorphim). Then W (F p n) E n,0 implie that π t (E n M I X) iatwitedgal-w (F p n) module. By the preceding reult, we only have to prove that H = Hc q (S n; π t (E n M I X)) i a finite twited Gal-W (F p n) module with the dicrete topology. Since S n i a compact p-adic analytic group, by [8, Prop ], H i a finite dicrete abelian group. By [6, Remark 7.2.2], Gal act on H by automorphim, and it i eay to check that thee automorphim are Z p -homomorphim. Since each Map c (Sn j,π t(e n M I X)) i a W (F p n)-module, the map in the cochain complex of continuou cochain are W (F p n)-homomorphim, o that H i a W (F p n)-module. Alo, it i eay to check that H i twited, ince the continuou cochain are. Thu, H i a finite twited Gal-W (F p n) module and a dicrete abelian group. Corollary 2.8. If X i a finite pectrum, then, for all and t, Hc (S n; K n,t (X)) Gal = H c (G n ; π t (E n X)). Proof. We have: Hc (S n ; K n,t (X)) Gal = lim Gal lim I Hc (S n ; π t (E n M I X)) = lim I Hc (G n; π t (E n M I X)). Therefore, a tated in [7, pg. 46], after taking Gal-fixed point, the pectral equence in (2.1) become (2.2) Hc (G n; π (E n X)) π (L K(n) X). 4

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