Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p^n

GALOIS MODULE STR UCTURE OF GALOIS COHOMOLOGY F OR EMBEDDABLE CYCLIC EXTENSION S OF DEGREE p n NICOLE LEMIRE, J ´ AN MIN ´ A ˇ C, ANDREW SCHUL TZ, AND JOHN SW ALLOW Abstract. Let p > 2 b e prime, a nd let n, m ∈ N b e given. F or cyclic extensions E /F of degr ee p n that con tain a primitive p th ro ot o f unity , we show th at the asso c iated F p [Gal( E /F )]-mo dules H m ( G E , µ p ) hav e a sparse decomp osition. When E / F is addition- ally a sube xtension of a cy c lic, degr ee p n +1 extension E ′ /F , we give a mor e re fined F p [Gal( E / F )]-decomp osition of H m ( G E , µ p ). 1. Introduction Absolute Galois groups capture a great deal of the arithmetic and algebraic prop erties of their underlying fields , tho ug h they are notori- ously in t r a ctable to compute. F or a giv en field E , one m ust often b e satisfied with studying in v arian t s attac hed to the corresp onding abso- lute Galois group G E , and in this resp ect the G alois cohomology g r oups H i ( G E , A ) fo r v arious G E -mo dules A are frequen t sub jects of in ves ti- gation. Of particular in terest are the gr o ups H m ( G E , µ p ) for a fixed prime p , where µ p represen ts the group of p th ro ots of unit y in G E . When E is itself a Galois exte nsion of a field F , the action of Gal( E /F ) on E × induces a natural a ction on H m ( G E , µ p ). Com bined with the F p -action on these cohomology groups, this naturally leads one to study these G alois cohomology groups a s F p [Gal( E /F )]-mo dules. In particular, one exp ects that this G alois mo dule structure will pro vide insigh t in to the correspo nding absolute Galois group G E . This program has been carried out in sev eral cases where G a l( E /F ) ≃ Z /p n Z and E con tains a primitiv e p th ro o t of unit y ξ p . In particular, the case n = m = 1 w a s resolv ed in [MS2], m = 1 a nd n ≥ 1 (without the restriction ξ p ∈ E ) in [MSS1], and m ≥ 1 and n = 1 in [LMS2]. As Date : July 7, 2 018. 1 2 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW desired, these computed mo dule structures ha ve already led to some in teresting results on the structure of absolute Galois gro ups: auto- matic realization results in [MS3, MSS2], a g eneralization of Schre ier’s Theorem in [LLMS2], a connection with D em u ˇ skin groups in [LLMS1], an interpretation of cohomological dimens ion in [L MS ], a nd a c harac- terization of certain gro ups whic h cannot a pp ear as absolute Galois groups in [BLMS]. The g o al o f t his pap er is t o b egin the in v estigat io n of a unified un- derstanding of the structures already computed b y determining some imp ortant results in the case m ≥ 1 and n ≥ 1. W e shall fo cus on the case p > 2 in this pap er. In m uc h the same w a y that this problem is the unification of the problems considered in [LMS2] and [MSS1 ], so to o will the metho dolo gy in o ur solution b e a combination of their in- dividual strategies. Indeed, careful refinemen ts o f the argumen ts from [LMS2], together with the appropriate mo dule-theoretic results, will already b e enough to give us the f ollo wing Theorem 1.1. L et p > 2 b e a given prime. If Ga l( E /F ) ≃ Z /p n Z and ξ p ∈ E , then the F p [Gal( E /F )] -mo dule H m ( G E , µ p ) is a dir e ct s um of inde c omp osable summands which ar e either of dimension p n or of dimension at most 2 p n − 1 . Since there a re p n isomorphism classes of indecompo sable F p [ G ]- mo dules — one for eac h cyclic submo dule of F p -dimension i , 1 ≤ i ≤ p n — this result shows that the decomp o sition of H m ( G E , µ p ) is relativ ely sparse. A more refined decomp o sition is av ailable, how ever, if we imp ose an additional assumption on the extension E /F . When Gal( E /F ) ≃ Z /p n Z a nd ξ p ∈ E , we say that E /F is an emb e ddable extension if E / F is an inte rmediate extension in a larger Galois extension E ′ /F so that Gal( E ′ /F ) / / ≃   Gal( E /F ) ≃   Z /p n +1 Z / / Z /p n Z , where the horizontal arrows a r e the natural pro jections. In the case of em b eddable extensions, w e can then use results from [MSS1] — particularly the prop erties o f so-called “exceptional” ele- men ts of E (see Prop osition 2.10) — to giv e the fo llo wing result. In GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 3 the statemen t of the result, w e use E j to denote the intermediate field of degree p j o ver F within the extension E /F . Theorem 1.2. L et p > 2 b e a given prime. I f E /F is an emb e ddabl e extension and a n is an exc eptional element, then as an F p [Gal( E /F )] - mo dule we h a ve H m ( G E , µ p ) ≃ X 0 ⊕ X 1 ⊕ · · · ⊕ X n − 1 ⊕ Y 0 ⊕ Y 1 ⊕ · · · ⊕ ⊕ Y n , wher e • for e ach 0 ≤ i ≤ n , b oth Y i and X i ar e dir e ct sums of inde c om- p osable mo dules of dimension p i , with Y i ⊆ res E /E i ( H m ( G E i , µ p )) and X i ⊆ ( a n ) ∪ res E /E i ( H m − 1 ( G E i , µ p )) ; and • for e ach i ≥ 0 , res E /F  cor E i /F ( H m ( G E i , µ p ))  = ( Y i ⊕ · · · ⊕ Y n ) G . Though t he strategies for em b eddable extensions cannot b e trans- lated directly into a decomp osition of the Galois mo dule structure of H m ( G E , µ p ) when E /F is not embeddable, this is nonetheless an im- p ortant step tow a r ds resolving the more general case. As an indication of this, w e note that for a non-embeddable extension E /F , an y prop er sub extension is embeddable. F or “b ot t o m-up” inductiv e a rgumen ts (i.e., those whic h rely on studying sub extensions whic h share a com- mon base field), then, the em b eddable case is of critical imp ortance. These kinds of argumen ts we re a lr eady used to great effect in resolving the case m = 1 , n > 1 in [MSS1], so it is lik ely that a resolution of the general ( no n-em b eddable) case for higher cohomo lo gy will also include this strategy . Section 2 outlines the basic ingredien ts necessary for the pro ofs of the main theorems, recalling imp ortant facts ab out Galois cohomolog y , mo dule theory and field theory . Section 3 then giv es a description of a submo dule Γ( m, n ) ⊆ H m − 1 ( G E n − 1 , µ p ) whic h is critical fo r our inductiv e approac h. Building on these results, Section 4 describ es the ma jor tec hnical results needed to pro vide a pro of of Theorem 1.2 in Section 5. Remark 1.3. Though the pr o of of The or em 1.2 r elies o n working in an em b e dda ble extension, the o ther machinery we develop holds for extensions E /F with Gal( E /F ) ≃ Z /p n Z and ξ p ∈ E ( p > 2 a prime) without insis ting on emb e ddability. The case p = 2 r equires sp ecial treatmen t and is a w or k in pro gress. 4 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW 2. Pre liminar y Re sul ts 2.1. Reduced Milnor K -theory. Though we ’v e phrased our results in the language o f Galois cohomology , the driving force in these pro ofs is the connection betw een these coho mology groups and reduced Mil- nor K - t heory that was first describ ed b y the so-called Blo c h-K a to conjecture, the conten t of whic h is stated in the fo llowing ( recen tly pro ven ) theorem. W e shall denote b y k m E the reduced Milnor K - groups K m E /pK m E . Theorem 2.1. F or a fi eld E c on taining a primitive p th r o ot of unity ξ p , the natur al map k m E → H m ( G E , µ p ) is an isomorphism. Mor e over, if G = G al( E /F ) for some sub extension F , then the isom orphism is G -e quivaria n t on the two G -mo dules . The pr o cess o f proving the Blo c h-Kat o conjecture b egan with Merkur- jev and Suslin [MS1], who v erified the case m = 2 for all primes p . The case where p = 2 and m is arbitrary w as resolv ed b y V o ev o d- sky [V1]. Recen tly Rost a nd V o ev o dsky together with W eibel’s patch pro ved the Blo c h- Kato conjecture for a ll p and m . F or details, see [R1, R2 , V2 , HW, W1, W2, W3]. In what follows, w e will emplo y The- orem 2.1 without men tion to iden tify Galois cohomology and reduced Milnor K-theory . The strategy w e emplo y will require generalizations to k -theory of some w ell-known results from field theory , namely Hilb ert’s Theorem 90 and Kummer theory . In this new setting, b oth of these results deal with extensions E /F that are degree p and Galois. In the results that follo w, N E /F denotes the map induced on K - theory by the field norm, and ι E /F denotes the map induced on K -theory b y inclusion. The results b elow can b e deduced fr o m t he pap ers cited ab o ve in the pro of of the Blo c h-Ka to conjecture, and they are in fact imp ortan t parts of the pro o f. An exp o sition of the precise results leading to these Prop ositions is con tained in Section 2 of [LMSS]. Prop osition 2.2 (Hilbert 90 for K -theory) . I f Gal( E /F ) = h σ i ≃ Z /p Z , then the se quenc e (2.3) K m E σ − 1 − − − → K m E N E /F − − − → K m F is exact. GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 5 Prop osition 2.4 (Kummer Theory) . Continuing with the assumptions of Pr op osition 2.2, supp o s e that ξ p ∈ E and that E = F ( p √ a ) for a ∈ F × . Then the se quenc e (2.5) k m − 1 E N E /F − − − → k m − 1 F { a }·− − − − → k m F ι E /F − − → k m E is exact. Finally , w e need a result whic h allows one to easily compute the norm of a sp ecial class of sym b ols. Prop osition 2.6 (Pro jection F orm ula, [FV, Chap. 9 , Thm. 3.8 ]) . L et E /F b e a Galois extensio n of fields, and l e t e ∈ E × and γ ∈ K m − 1 F b e given. Then (2.7) N E /F  { e } · ι E /F ( γ )  = { N E /F ( e ) } · γ . 2.2. Field Results. Since our fo cus is on embeddable extensions, there are sev eral simple G a lois-theoretic consequences whic h will b e useful to r ecord. Recall that w e denote b y E i the in termediate field of degree p i o ver F within E /F . Hence w e will in terc hangeably refer to E as E n and F a s E 0 . W e will write G i for the quotien t group Gal( E i /F ) ≃ Z /p i Z and H i for the subgroup Gal( E /E i ) ≃ Z /p n − i Z . F o r conv enience, w e will carry t his notation o ve r to abbreviate relev an t inclusion and norm maps: ι i j will denote ι E i /E j and N i j will denote N E i /E j , b oth for fields and their K - t heory . Since w e ha v e a ssumed ξ p ∈ E w e m ust hav e ξ p ∈ E i for ev ery 0 ≤ i ≤ n , and so it fo llo ws b y Kummer Theory that for ev ery 0 ≤ i ≤ n − 1 w e may find elem en ts a i ∈ E i so that E i +1 = E i ( p √ a i ) . In fact, it w as sho wn in [MSS1, Prop. 1] that these a i can b e selected to satisfy the follo wing norm compatibility criteria: (2.8) N i j a i = a j for an y j ≤ i ≤ n − 1 . It is also sho wn tha t for 0 ≤ i ≤ n − 1, the p t h p ow er class of eac h of these a i is fixed by its resp ectiv e Galois group: (2.9) τ ( a i ) ∈ a i E × p i for ev ery τ ∈ G i . In [MSS1], exceptional elemen t s for the extension E / F are defined as a kind of “minimal” extension of the ab o ve equations to i = n . The definition there is expressed in terms of classes of elemen ts in E × /E × p ; 6 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW w e no w presen t an equiv alent fo rm ulation for elemen ts in E × . F or a general field E /F with Ga l( E /F ) ≃ Z /p n Z a nd containing ξ p , a n exceptional elemen t a n ∈ E × for the extension E /F is an elemen t with N E /F ( a n ) ∈ E × p \ F × p , and suc h that a σ − 1 n ∈ E × i ( E /F ) E × p , where i ( E /F ) = min N E /F ( a ) ∈ E × p \ F × p  i : a σ − 1 ∈ E × i E × p  . (Here, E −∞ is tak en t o b e { 1 } .) Hence the p ossible v alues for i ( E /F ) are from the set {−∞ , 0 , 1 , · · · , n } ; in [MSS1, Theorem 3], w e show that in fa ct i ( E /F ) ≤ n − 1. One can sho w that exceptional elemen ts exist under the h yp othesis p > 2 ([MSS1, Prop. 2]), and that t he cyclic F p [Gal( E /F )]-submo dule generated by an ex ceptional elemen t has F p -dimension p i ( E /F ) +1 ([MSS1 , Prop. 7]). F or an exceptional elemen t a n , this fact, together with the condition N E /F ( a n ) ∈ E × p \ F × p , ensures that N n j ( a t n ) = a j e p j for some t ∈ Z \ p Z and e j ∈ E × j ([MSS1, Lemma 8]) . (One migh t nat ura lly think to select a n so that N n j ( a n ) = a j in analog y to (2.8 ), but our w eak er condition is chosen to accoun t for the set { a i } b eing non-canonical.) Our em b eddability h yp othesis is equiv alent to t he conditio n that i ( E /F ) = −∞ , so that a σ − 1 n ∈ E × p in this case. This condition will b ecome imp o rtan t in the final stages o f our pro of of Theorem 1.2, but isn’t neces sary for other results w e describ e. F or o ur purp oses, the relev ant prop erties of exceptional elemen ts are outlined in the following Prop osition 2.10. F or e ach 1 ≤ i < n , the eleme nt a i is exc eptional for the extension E i /F . Any exc eptiona l ele ment a n of an emb e dda b le extension E /F satisfies N n n − 1 ( a n ) = a t n − 1 e p for som e t ∈ Z \ p Z a n d e ∈ E × n − 1 , an d furthermor e a σ − 1 n ∈ E × p . Pr o of. The first statemen t follo ws directly from the definition of ex- ceptionalit y and conditions (2.8) a nd ( 2 .9). The final statement follow s from [MSS1, Thm. 3] together with the result of Alb ert [A] whic h shows E n /E 0 is embeddable if and only if ξ p ∈ N n 0 ( E × n ). Finally , t o show that an exceptional elemen t a n of an em b eddable extension E /F satisfies N n n − 1 ( a n ) = a t n − 1 e p as ab o v e, one applies [MSS1, Lemma 8].  GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 7 As a final remark, w e p oint o ut that as op erators on E × /E × p , w e ha ve the iden tity p n − 1 X i =0 σ i ≡ ( σ − 1) p n − 1 . Hence w e hav e that N n 0 ( e ) ≡ e ( σ − 1) p n − 1 mo d E × p . More generally , w e ha v e that the norm op erator N i j is given b y the action of ( σ p j − 1) p i − j − 1 ≡ ( σ − 1) p i − p j . F or the F p [ G i ]-mo dules k m E i , w e hav e the related iden t ity ι i j ◦ N i j ≡ ( σ − 1) p i − p j . W e will mak e frequen t use of these identities throughout the remainder of the pap er. 2.3. Mo dule Theory. Finally , w e remind the reader of the esse n tial facts ab out F p [ G ]-mo dules; with t he exception of E × /E × p , we shall write our F p [ G ]-mo dules additiv ely . Muc h of the theory of F p [ G ]-mo dules comes from the fact that F p [ G ] is a discrete v aluation ring with maximal ideal generated b y σ − 1 , where for the rest o f the pap er w e use σ to denote a generator of G . F or instance, this tells us that the cyclic submodule generated b y an elemen t w is isomorphic to the indecomp osable F p [ G ]-mo dule A ℓ ( w ) := F p [ G ] / ( σ − 1) ℓ ( w ) , where ℓ ( w ) — the so-called length of w — is defined a s the minim um v alue of ℓ so that ( σ − 1) ℓ w = 0. In turn this implies that the F p - dimension of the cyclic submo dule generated b y w is ℓ ( w ). F rom this it is not difficult to see that ℓ ( w + v ) ≤ max { ℓ ( w ) , ℓ ( v ) } , with equality whenev er ℓ ( w ) 6 = ℓ ( v ). W e can also pro v e the following Lemma 2.11 (Exclusion Lemma) . S upp o s e that { U α } α ∈A ar e F p [ G ] - submo dules of a fixe d F p [ G ] -mo dule W . Then the submo dules { U α } ar e F p [ G ] -indep endent if and only if the F p -subsp ac es { U G α } ar e indep en- dent. Equivalently, X α ∈A U G α = M α ∈A U G α ⇐ ⇒ X α ∈A U α = M α ∈A U α . Pr o of. W e pro ve the result by induction. Notice that it suffices to pro ve the result for a finite collection of submo dules, since a dep en- dence amongst an infinite collection o f submo dules is defined to b e a dep endence amongst some finite sub collection. 8 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW In the case of the t wo mo dules U and V , the F p [ G ]-indep endence of U and V implies the F p -indep endence of U G and V G . S o supp ose that U G and V G are F p -indep enden t, and w e sho w tha t U ∩ V = { 0 } . Supp ose that x ∈ U ∩ V , and note that ( σ − 1) ℓ ( x ) − 1 x ∈ U G ∩ V G . Since U G ∩ V G = { 0 } by assumption, we hav e ( σ − 1) ℓ ( x ) − 1 x = 0. Since ( σ − 1) ℓ ( w ) − 1 w 6 = 0 whenev er w 6 = 0, w e conclude that x = 0. T o prov e the result for a collection of m submo dules U 1 , · · · , U m , notice that b y induction the F p -indep endence o f { U G i } m − 1 i =1 implies V = P i 1, and so the ab o ve recip e pro duces a nontrivial decomposition. This fact can in turn be used to show that the decomposition o f a n F p [ G ]- mo dule W is essen tially unique, in the sense tha t it determines the indecomp o sable t yp es which app ear in an F p [ G ]-decomp osition of W , together with their multiplic it ies. This is r ecorded in the following Corollary 2.14. F or an F p [ G ] -mo dule W , supp ose W = ⊕ α ∈A W α wher e e ach W α is inde c omp osable. Then A is a disj oint union of subsets A 1 , A 2 , · · · , A p n wher e • |A i | is the c o dimension o f V i +1 within V i , and • for e ach α ∈ A i ther e is an F p [ G ] -isomorphism W α ≃ A i . 3. The Submodule Γ( m, n ) As Corollary 2.14 suggests, the driving force in determining a de- comp osition of k m E in v o lv es understanding the submo dule ( k m E n ) G ⊆ k m E n , particularly the filtr a tion V p n ⊆ V p n − 1 ⊆ · · · ⊆ V 2 ⊆ V 1 = k m E G n , where V i := im(( σ − 1) i − 1 ) ∩ ( k m E n ) G . In the case that n = 1, the authors of [LMS2] w ere able to control this filtration b y carefully studying the in terplay b et w een k er( ι E /F ) and im( N E /F ). In par t icular, they sho w ed that elemen ts in k er ( N E /F ) had particularly nice mo dule- theoretic prop erties, and that “ most” other elemen ts came from the submo dule ker( ι E /F ◦ N E /F ). The challenge w as then to construct 10 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW a submo dule X ⊆ k m E that w as sufficien tly “small” and so that N E /F ( X ) = k er( ι E /F ). This submodule X could then be used to con- trol the mo dule-theoretic prop erties of other elemen ts in k m E , th us forcing the resulting strat ified decomp osition. Our approac h will tak e this same basic strategy , though we will fo cus m uc h of o ur a tten tion on the sub extension E n /E n − 1 and its asso ciated inclusion and norm maps. W e will start b y giving an in-depth study of the mo dule structure of k er( ι n n − 1 ). W e p oin t out that the results of this section do not use the em b eddabilit y of E /F ; instead, w e only use the fact that Gal( E /F ) ≃ Z /p n Z a nd that ξ p ∈ E , where p > 2 is a prime. Exact Sequence (2.5) tells us that k er( k m E n − 1 ι n n − 1 − − − → k m E n ) = { a n − 1 } · k m − 1 E n − 1 , where a n − 1 has E n − 1 ( p √ a n − 1 ) = E n and satisfies conditions (2.8) and (2.9). F urthermore, w e kno w that (3.1) k er( k m − 1 E n − 1 { a n − 1 }·− − − − − − → k m E n − 1 ) = N n n − 1 ( k m − 1 E n ) . Hence to understand ke r ( ι n n − 1 ), w e m ust find a complemen t Γ( m, n ) to N n n − 1 ( k m − 1 E n ) in k m − 1 E n − 1 . The main result of this section is finding this complemen t , as recorded in the follo wing Prop osition 3.2. T her e ex ists a submo dule Γ( m, n ) ⊆ k m − 1 E n − 1 such that (1) Γ( m, n ) = ⊕ n − 1 i =0 Z i wher e e ach Z i ⊆ ι n − 1 i ( k m − 1 E i ) is a dir e ct sum o f fr e e F p [ G i ] -mo dules, and Z G n − 1 i ⊆ ι n − 1 0 ( N i 0 ( k m − 1 E i )) ; (2) Γ( m, n ) ⊕ N n n − 1 ( k m − 1 E n ) = k m − 1 E n − 1 ; (3) { a n − 1 } · k m − 1 E n − 1 = { a n − 1 } · Γ( m, n ) ; and (4) as F p [ G n − 1 ] -mo dules, Γ( m, n ) ≃ { a n − 1 } · Γ( m, n ) under the m a p γ 7→ { a n − 1 } · γ . A few remarks are in order. First, the uniqueness of an F p [ G n − 1 ]- decomp osition of k m − 1 E n − 1 implies that all compleme n ts to N n n − 1 ( k m − 1 E ) are isomorphic as F p [ G n − 1 ]-mo dules, pro vided that a complemen t to N n n − 1 ( k m − 1 E n ) exists. F urthermore, pro p erties (3) and (4) follow di- rectly from prop erty (2): the former b ecause of Equation (3.1), and the latter b ecause a n − 1 has trivial G n − 1 -action (condition (2.9) ). Hence the con tent of this theorem is in sho wing that N n n − 1 ( k m − 1 E n ) is a summand of k m − 1 E n − 1 , and that this latter mo dule is appropriately stratified. GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 11 T o pro ve this result, w e shall use induction. F or o ur base cases, supp ose that either n = 1 or m = 1. In either case N n n − 1 ( k m − 1 E n ) is a submo dule of the trivial F p [ G n − 1 ]-mo dule k m − 1 E n − 1 , a nd hence has a complemen t whic h is also trivial as an F p [ G n − 1 ]-mo dule. Suc h a complemen t obv iously satisfies condition 1. Supp ose then that m, n > 1, and assume b y induction the existence of a submo dule Γ( m − 1 , n ) ⊆ k m − 2 E n − 1 whic h satisfies the conclusions of Prop osition 3.2. Lemma 3.3. F or γ ∈ ι n − 1 0 ( k m − 1 E 0 ) with ι n n − 1 ( γ ) = 0 , ther e ex i s ts α ∈ k m − 1 E n − 1 so that ι n − 1 0 ( N n − 1 0 ( α )) = γ and ι n n − 1 ( α ) = 0 . Pr o of. By Exact Sequence (2 .5 ) w e ha v e γ = { a n − 1 } · g for some g ∈ k m − 2 E n − 1 . Prop osition 3.2 sa ys that w e may tak e g ∈ Γ( m − 1 , n ) G n − 1 ⊆ ι n − 1 0 ( k m − 2 E 0 ) ⊆ ι n − 1 n − 2 ( k m − 2 E n − 2 ) , sa y g = ι n − 1 n − 2 ( ˆ g ). W e no w compute N n − 1 n − 2 ( γ ) in t w o w ays. On the one hand, since γ ∈ im( ι n − 1 0 ) w e ha ve N n − 1 n − 2 ( γ ) = 0. On the other ha nd, since γ = { a n − 1 } · ι n − 1 n − 2 ( ˆ g ) , the Pro jection F ormula (2.7) giv es 0 = N n − 1 n − 2 ( γ ) = N n − 1 n − 2 ( { a n − 1 } · g ) = { a n − 2 } · ˆ g . By Exact Sequence (2 .5 ) w e conclude that ˆ g ∈ N n − 1 n − 2 ( k m − 2 E n − 1 ), and therefore ι n − 1 n − 2 ( ˆ g ) is in the image of ( σ − 1 ) p n − 1 − p n − 2 . This show s g lies in the fixed part of a submo dule of k m − 2 E n − 1 of length at least p n − 1 − p n − 2 + 1 > p n − 2 . Since b y induction Γ( m − 1 , n ) is a direct sum of free F p [ G i ]-submo dules for 0 ≤ i ≤ n − 1, g = ( σ − 1) p n − 1 − 1 ( α ′ ) for some α ′ ∈ k m − 2 E n − 1 . Letting α = { a n − 1 } · α ′ w e ha v e ι n n − 1 ( α ) = 0 and ι n − 1 0 ( N n − 1 0 ( α )) = ( σ − 1) p n − 1 − 1 ( α ) = ( σ − 1 ) p n − 1 − 1 ( { a n − 1 } · α ′ )) = { a n − 1 } · ( σ − 1) p n − 1 − 1 ( α ′ ) = { a n − 1 } · g = γ , as desired.  Lemma 3.4. Ther e exis ts a mo dule d e c omp osition k m − 1 E n − 1 = X 0 ⊕ · · · ⊕ X n − 2 ⊕ Y 0 ⊕ · · · ⊕ Y n − 1 satisfying the c on d itions of The or em 1.2, and with the pr op erties • X i ⊆ { a n − 1 } · k m − 2 E n − 1 for e ach i , and • Y n − 1 = K ⊕ N ⊕ ˆ Y n − 1 , wher e e ach of thes e submo d ules is fr e e over F p [ G n − 1 ] , an d so that (1) K ⊆ k er ι n n − 1 and 12 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW (2) N ⊆ N n n − 1 ( k m − 1 E n ) . Pr o of. W e shall let our decomp osition come f r om an arbitrary decom- p osition X 0 ⊕ · · · ⊕ X n − 2 ⊕ Y 0 ⊕ · · · ⊕ Y n − 1 of k m − 1 E n − 1 pro vided b y induction, sub ject to a few conditions on X and Y w e are free to imp ose. First, Prop o sition 2.10 giv es that a n − 1 is an exceptional elemen t for the extension E n − 1 /F , and so Theorem 1.2 tells us that the decompo sition can b e c hosen so that X i ⊆ { a n − 1 } · ι n − 1 i ( k m − 2 E i ) ⊆ { a n − 1 } · k m − 2 E n − 1 . Second, Corollary 2.12 giv es us a great deal of freedom in c ho osing the submo dule Y n − 1 . Sp ecifically , since ι n − 1 0 ◦ N n − 1 0 is give n b y t he ac- tion of ( σ − 1) p n − 1 − 1 , w e ma y c ho ose any F p -basis I of ι n − 1 0 ( N n − 1 0 ( k m − 1 E n − 1 )) and — for ev ery x ∈ I — an elemen t α x ∈ k m − 1 E n − 1 so that ι n − 1 0 ( N n − 1 0 ( α x )) = x . Then Corolla ry 2.1 2 say s that Y n − 1 can b e ta ken to be ⊕ x ∈I h α x i F p [ G n − 1 ] . W e c ho ose our basis I as the disjoint union of I K , I N and ˆ I , where (1) I K is a basis for k er ι n n − 1 ∩ ι n − 1 0 ( N n − 1 0 ( k m − 1 E n − 1 )); (2) I N is a basis for a complemen t to k er ι n n − 1 ∩ ι n − 1 0 ( N n 0 ( k m − 1 E n )) in ι n − 1 0 ( N n 0 ( k m − 1 E n )); (3) and ˆ I is a basis for a complemen t to hI K , I N i F p in ι n − 1 0 ( N n − 1 0 ( k m − 1 E n − 1 )) . By Lemma 3.3, for ev ery x ∈ I K there exists α x so tha t ι n − 1 0 ( N n − 1 0 ( α x )) = x and α x ∈ ker ι n n − 1 . Hence we define K := ⊕ x ∈I K h α x i F p [ G n − 1 ] ⊆ ker ι n n − 1 . F or eac h x ∈ I N , there exists β ∈ k m − 1 E n so that ι n − 1 0 ( N n 0 ( β )) = x , and therefore ι n − 1 0 ( N n − 1 0 ( N n n − 1 ( β ))) = x . Hence w e define N := ⊕ x ∈I N h N n n − 1 ( β ) i F p [ G n − 1 ] ⊆ N n n − 1 ( k m − 1 E n ) . F or eac h x ∈ ˆ I we c ho ose arbitrary α x ∈ k m − 1 E n − 1 to satisfy ι n − 1 0 ( N n − 1 0 ( α x )) = x , and w e let ˆ Y n − 1 := ⊕ x ∈ ˆ I h α x i F p [ G n − 1 ] .  GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 13 W e will show that the submo dule Γ( m, n ) of Prop o sition 3.2 is Y 0 ⊕ · · · ⊕ Y n − 2 ⊕ ˆ Y n − 1 . W e pro ceed by determining a complemen t for N n n − 1 ( k m − 1 E n ) in k m − 1 E n − 1 , b eginning with a calculation of k er ( ι n n − 1 ). Lemma 3.5. Using the notation fr om L emm a 3.4, k er  k m − 1 E n − 1 ι n n − 1 / / k m − 1 E n  = X 0 ⊕ · · · ⊕ X n − 2 ⊕ K . Pr o of. Since X i ⊆ { a n − 1 } · k m − 2 E n − 1 , Exact Sequence (2.5) giv es X i ⊆ k er ι n n − 1 . Lemma 3.4 also giv es K ⊆ k er ι n n − 1 . W e complete the pro o f b y sho wing that k er( ι n n − 1 ) ∩  Y 0 ⊕ · · · ⊕ Y n − 2 ⊕ ˆ Y n − 1 ⊕ N  = { 0 } . (3.6) T o do this w e show that the fixed submo dule of the direct sum ab ov e has trivial in tersection with k er ι n n − 1 (after whic h w e can app eal to the Exclusion Lemma (2.1 1)). Since N , ˆ Y n − 1 ⊆ Y n − 1 , Theorem 1.2 gives k er ι n n − 1 ∩  Y 0 ⊕ · · · ⊕ Y n − 2 ⊕ ˆ Y n − 1 ⊕ N  G n − 1 ⊆ ker ι n n − 1 ∩ ι n − 1 0 ( k m − 1 E 0 ) . Lemma 3.3, on the other hand, sho ws that k er ι n n − 1 ∩ im ι n − 1 0 ⊆ ker ι n n − 1 ∩ ι n − 1 0 ( N n − 1 0 ( k m − 1 E n − 1 )) = hI K i = K G n − 1 . Since the fixed par t s of eac h of the mo dules Y i (0 ≤ i ≤ n − 2) , ˆ Y n − 1 and N are F p -indep enden t fro m the fixed part of K , the Exclusion Lemma (2.11) implies tha t Equation (3.6) is true.  Lemma 3.7. Using the notation fr om L emm a 3.4, im  k m − 1 E n N n n − 1 / / k m − 1 E n − 1  = X 0 ⊕ · · · ⊕ X n − 2 ⊕ K ⊕ N . Pr o of. Let a n b e an exceptional elemen t o f E /F , and c ho ose t so that N n n − 1 ( a t n ) ∈ a n − 1 E × p n − 1 . An elemen t γ ∈ k er ι n n − 1 tak es the form γ = { a n − 1 } · g b y Exact Sequence (2.5), and so the Pro jection F orm ula (2.7) giv es N n n − 1 ( { a t n } · ι n n − 1 ( g )) = { a n − 1 } · g . Hence b y Lemma 3.5, k er ι n n − 1 = X 0 ⊕ · · · ⊕ X n − 2 ⊕ K ⊆ N n n − 1 ( k m − 1 E n ) . Of course N is constructed so that N ⊆ N n n − 1 ( k m − 1 E n ), and so we ha ve X 0 ⊕ · · · ⊕ X n − 2 ⊕ K ⊕ N ⊆ N n n − 1 ( k m − 1 E n ) . 14 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW F or the opp osite con tainmen t, it is enough to sho w N n n − 1 ( k m − 1 E n ) ∩  Y 0 ⊕ · · · ⊕ Y n − 2 ⊕ ˆ Y n − 1  = { 0 } . By the Exclusion Lemma (2.11), this is equiv alen t to sho wing the as- so ciated fixed submo dules are F p -indep enden t. W e will v erify this by sho wing ( N n n − 1 ( k m − 1 E n )) G n − 1 ⊆ X 0 ⊕ · · · ⊕ X n − 2 ⊕ K ⊕ N . Let γ b e an elemen t in ( N n n − 1 ( k m − 1 E n )) G n − 1 , sa y γ = N n n − 1 ( α ) for some α ∈ k m − 1 E n . If ι n n − 1 ( γ ) = 0 then γ ∈ ke r ι n n − 1 = X 0 ⊕ · · · ⊕ X n − 2 ⊕ K b y L emma 3.5, and w e are done. Otherwise γ / ∈ ke r ι n n − 1 , a nd so ι n n − 1 ( γ ) = ι n n − 1 ( N n n − 1 ( α )) 6 = 0. Since ι n n − 1 ◦ N n n − 1 is represe nted b y the p olynomial σ p n − 1 + · · · + σ p n − 1 ( p − 1) ≡ ( σ − 1 ) p n − p n − 1 , this implies that ℓ ( α ) > p n − p n − 1 ≥ 2 p n − 1 . The decomp o sition of k m − 1 E n pro vided b y Theorem 1.1 implies ι n n − 1 ( γ ) is the fixed part of a submodule of dimension p n ; i.e., ι n n − 1 ( γ ) = ι n 0 ( N n 0 ( β )) for some β ∈ k m − 1 E n . If w e let δ = ι n − 1 0 ( N n 0 ( β )), then w e hav e ι n n − 1 ( γ − δ ) = 0. Hence w e ha ve γ − δ ∈ k er ( ι n n − 1 ), from which it follows that γ ∈ ι n − 1 0 ( N n 0 ( k m − 1 E n )) + k er ι n n − 1 . Recall, ho w eve r , that N G n − 1 = hI N i F p w as c hosen as a complemen t to ke r ι n n − 1 ∩ ι n − 1 0 ( N n 0 ( k m − 1 E n )) ⊆ h I K i F p in ι n − 1 0 ( N n 0 ( k m − 1 E n )). Hence w e ha ve h I K , I N i F p ⊇ ι n − 1 0 ( N n 0 ( k m − 1 E n )), a nd so γ ∈ h I K , I N i F p + k er ι n n − 1 ⊆ X 0 ⊕ · · · ⊕ X n − 2 ⊕ K ⊕ N .  Pr o of of Pr op osition 3.2. F or each 0 ≤ i < n − 1 define Z i := Y i , and define Z n − 1 := ˆ Y n − 1 . W e define Γ( m, n ) := Z 0 ⊕ · · · ⊕ Z n − 1 . The previous lemmas sho w that Γ( m, n ) satisfie s (1) and ( 2 ), a nd w e ha ve already v erified that prop erties (3) and (4) f ollo w from (2).  W e record the follo wing corollary , since it will b e useful la ter. Corollary 3.8. If g ∈ Γ( m, n ) G n − 1 and N n − 1 n − 2 ( { a n − 1 } · g ) = 0 , then fo r some α ∈ Γ( m, n ) we ha v e g = ι n − 1 0 ( N n − 1 0 ( α )) . GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 15 Pr o of. Since Γ( m, n ) G n − 1 ⊆ ι n − 1 0 ( k m − 1 E 0 ), it follows that g = ι n − 1 n − 2 ( ˆ g ) for some ˆ g ∈ k m − 1 E n − 2 . By the Pro jection F ormula (2.7) w e therefore ha ve 0 = N n − 1 n − 2 ( { a n − 1 } · g ) = { a n − 2 } · ˆ g , and so Exact Sequence (2.5) implies ˆ g = N n − 1 n − 2 ( α ′ ) for some α ′ ∈ k m − 1 E n − 1 . Hence we ha ve g = ι n − 1 n − 2 ( ˆ g ) = ι n − 1 n − 2  N n − 1 n − 2 ( α ′ )  = ( σ − 1) p n − 1 − p n − 2 α ′ ∈ im ( σ − 1) p n − 1 − p n − 2 . Since Γ( m, n ) is a direct sum of cyclic submo dules of dimensions p i for 0 ≤ i ≤ n − 1, w e must hav e g ∈ im( σ − 1) p n − 1 − 1 . Hence g ∈ Z G n − 1 n − 1 .  4. Fixed Elements and Norms The k ey result of this section is Prop osition 4.4. This result uses Hilb ert 90-like r esults a nd f a cts ab out abstract F p [ G ]-mo dules to prov e that elemen ts in k er( N n n − 1 ) ha ve “ nice” mo dule-theoretic prop erties. Again, w e will not assume in this section that the g iven extension E /F is em b eddable — w e will just use the facts that Gal( E /F ) ≃ Z /p n Z , that ξ p ∈ E and that p > 2 is prime. In our setting w e need to b e careful ab out t he p ossible difference in length b et we en the F p [ G i ]-submo dule g enerated b y an elemen t γ ∈ k m E i and the F p [ G n ]-submo dule of k m E n generated b y ι n i ( γ ). T o w ards this end, w e giv e results for determining when an elemen t lies in the submo dule im( ι n i ) and — when it do es — for controlling the F p [ G i ]- lengths o f represen ta tiv es from k m E i for this elemen t. W e also establish notation to distinguish these po ten tially differen t notions of length: for an eleme n t γ ∈ k m E i , w e write ℓ G i ( γ ) to denote the length of the F p [ G i ]-submo dule generated b y γ . In the same wa y , the F p -dimension of the F p [ H i ]-submo dule generated b y γ ∈ k m E n is denoted ℓ H i ( γ ). Since H i = h σ p i i , we not e that ℓ H i ( γ ) = min { ℓ : ( σ p i − 1) ℓ γ = 0 } = min { ℓ : ( σ − 1) p i ℓ γ = 0 } . Lemma 4.1. If N n n − 1 ( γ ) = 0 and γ ∈ ( k m E n ) H n − 1 , then ther e exists ˆ γ ∈ k m E n − 1 such that ι n n − 1 ( ˆ γ ) = γ and ℓ G n − 1 ( ˆ γ ) = ℓ G ( γ ) . A dditional ly, if ℓ G ( γ ) ≤ p n − 1 − p n − 2 we may insist N n − 1 n − 2 ( ˆ γ ) = 0 . Pr o of. [LMS2, Lemma 3] shows that the sequence k m E n − 1 ι n n − 1 / / ( k m E n ) H n − 1 N n n − 1 / / { a n − 1 } · k m − 1 E n − 1 16 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW is exact, so f or N n n − 1 ( γ ) = 0 we ma y conclude γ = ι n n − 1 ( ˆ γ ) for some ˆ γ ∈ k m E n − 1 . Notice also that when n = 1 the length condition is trivial, so we may assume that n ≥ 2 . W e no w argue that ˆ γ may b e t a k en so t ha t ℓ G n − 1 ( ˆ γ ) = ℓ G ( γ ). W e cannot hav e ℓ G n − 1 ( ˆ γ ) < ℓ G ( γ ), since if ( σ − 1) x ˆ γ = 0 ∈ k m E n − 1 then ( σ − 1) x γ = ( σ − 1) x ι n n − 1 ( ˆ γ ) = ι n n − 1 (( σ − 1) x ˆ γ ) = 0 . So supp ose t ha t ℓ := ℓ G n − 1 ( ˆ γ ) > ℓ G ( γ ). Our g oal is to use Corollary 3.8 to adjust ˆ γ by an elemen t { a n − 1 } · α ∈ k m E n − 1 in o rder to pro duce an elemen t of smaller length whose image under inclusion is γ . F or this w e study f := ( σ − 1) ℓ − 1 ˆ γ . First, b y induction w e kno w k m E n − 1 = X 0 ⊕ · · · ⊕ X n − 2 ⊕ Y 0 ⊕ · · · ⊕ Y n − 1 , where b y Theorem 1.2 w e hav e X i ⊆ { a n − 1 } · ι n − 1 i ( k m − 1 E i ) ⊆ k er ι n n − 1 . Hence we ma y take ˆ γ ∈ Y 0 ⊕ · · · ⊕ Y n − 1 . Since f := ( σ − 1) ℓ − 1 ˆ γ ∈ ( Y 0 ⊕ · · · ⊕ Y n − 1 ) G n − 1 w e hav e f ∈ ι n − 1 0 ( k m E 0 ). Since n ≥ 2, w e therefore conclude (4.2) N n − 1 n − 2 ( f ) = 0 . On the other hand, since ℓ > ℓ ( γ ) we kno w f ∈ k er ( ι n n − 1 ). Exact Sequence (2.5) and Proposition 3 .2 then imply f ∈ { a n − 1 } · Γ( n, m ) G n − 1 , sa y f = { a n − 1 } · ι n − 1 0 ( g ). Recalling Equation (4 .2), the Pro jection F orm ula (2.7) giv es 0 = N n − 1 n − 2 ( f ) = N n − 1 n − 2  { a n − 1 } · ι n − 1 0 ( g )  = { a n − 2 } · ι n − 2 0 ( g ) . This allows us to apply Corollary 3.8, and w e conclude that ι n − 1 0 ( g ) = ι n − 1 0 ( N n − 1 0 ( α )) fo r some α ∈ Γ( m, n ). Since ℓ G n − 1 ( { a n − 1 } · α ) = p n − 1 b y Prop osition 3.2(4) and ι n n − 1 ( { a n − 1 } · α ) = 0, w e see that ˆ γ − ( σ − 1) p n − 1 − ℓ G n − 1 (ˆ γ ) ( { a n − 1 } · α ) has G n − 1 -length smaller than ℓ G n − 1 ( ˆ γ ) and has image γ under ι n n − 1 . W e iterate this pro cess un til w e hav e constructed an elemen t ˆ γ so that ι n n − 1 ( ˆ γ ) = γ and ℓ G n − 1 ( ˆ γ ) = ℓ G ( γ ). All w e ha ve left is to show tha t if ℓ G ( γ ) ≤ p n − 1 − p n − 2 , then w e ma y insist N n − 1 n − 2 ( ˆ γ ) = 0. F or this, since ℓ G n − 1 ( ˆ γ ) ≤ p n − 1 − p n − 2 w e ha ve ( σ − 1) p n − 1 − p n − 2 ( ˆ γ ) = ι n − 1 n − 2 ( N n − 1 n − 2 ( ˆ γ )) = 0 , GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 17 so N n − 1 n − 2 ( ˆ γ ) = { a n − 2 } · g fo r some g ∈ Γ( m, n − 1) ⊆ k m − 1 E n − 2 b y Prop osition 3.2. W e claim that ˆ γ ′ := ˆ γ − { a n − 1 } · ι n − 1 n − 2 ( g ) has the desired inclusion, norm and length prop erties. T o prov e this claim, notice first that ι n n − 1  { a n − 1 } · ι n − 1 n − 2 ( g )  = 0 b y Exact Seq uence (2.5), and hence ι n n − 1 ( ˆ γ ′ ) = γ . It is also ob vious that N n − 1 n − 2  { a n − 1 } · ι n − 1 n − 2 ( g )  = { a n − 2 } · g by the Pro jection F ormula (2.7), and hence N n − 1 n − 2 ( ˆ γ ′ ) = 0. F or the length condition, notice first that ℓ G n − 1 ( { a n − 1 } · ι n − 1 n − 2 ( g )) = ℓ G n − 2 ( g ) by Prop osition 3.2(4) applied to Γ( m, n − 1 ). In view of the pro p erties of length, together with the fact that a preimage o f γ under ι n n − 1 cannot ha ve G n − 1 -length less than ℓ := ℓ ( γ ) = ℓ G n − 1 ( ˆ γ ), it will b e enough to prov e that ℓ ≥ ℓ G n − 2 ( g ). T o see that this is true, note that we hav e 0 = N n − 1 n − 2  ( σ − 1) ℓ ˆ γ  = ( σ − 1) ℓ ( { a n − 2 } · g ) = { a n − 2 } · ( σ − 1) ℓ g . Applying Prop osition 3.2(4) aga in, we hav e the desired inequalit y .  The previous result g iv es us the fixed subm o dule under one partic- ular subgroup o f G . T o find the fixed submo dule for the r emaining subgroups o f G , we hav e the follo wing Lemma 4.3. If N n n − 1 ( γ ) = 0 and γ ∈ ( k m E n ) H i , then ther e exists ˆ γ ∈ k m E i such that ι n i ( ˆ γ ) = γ and ℓ G i ( ˆ γ ) = ℓ G ( γ ) . A dditional ly, if ℓ G ( γ ) ≤ p i − p i − 1 we may insist N i i − 1 ( ˆ γ ) = 0 . Pr o of. The base case of this result is the previous lemma. F or the inductiv e step, let γ ∈ ( k m E n ) H i with N n n − 1 ( γ ) = 0, and supp ose w e hav e the result for i + 1. Since ( k m E n ) H i ⊆ ( k m E n ) H i +1 , there exists ˜ γ ∈ k m E i +1 suc h that ι n i +1 ( ˜ γ ) = γ and ℓ G i +1 ( ˜ γ ) = ℓ G ( γ ). F urthermore, since ℓ G ( γ ) ≤ p i ≤ p i +1 − p i w e ma y insist N i +1 i ( ˜ γ ) = 0. Applying the previous Lemma t o the extens ion E i +1 /E i , this implies that there exists ˆ γ ∈ k m E i suc h that ℓ G i ( ˆ γ ) = ℓ G i +1 ( ˜ γ ), ι i +1 i ( ˆ γ ) = ˜ γ , and so that if ℓ G i ( ˆ γ ) ≤ p i − p i − 1 then w e may assume N i i − 1 ( ˆ γ ) = 0. But then w e also ha v e ℓ G i ( ˆ γ ) = ℓ G ( γ ) and ι n i ( ˆ γ ) = γ as desired.  W e a re no w ready fo r the main result of the sec tion. W e shall state it in some generality and then restrict ourselv es to a sp ecial case in the subseque n t corollary . 18 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW Prop osition 4.4. F or γ ∈ k m E n , if • ℓ H j ( γ ) > 2 p n − j − 1 ; or if • E n /E j is emb e ddable and ℓ H j ( γ ) > p n − j − 1 ; or if • N n n − 1 ( γ ) = 0 and ℓ H j ( γ ) > p n − j − 1 , then ( σ p j − 1) ℓ H j ( γ ) − 1 γ ∈ ι n j ( N n j ( k m E n )) . Pr o of. T o prov e the claim w e pro ceed by induction on j . The base case is j = n − 1. [LMS2, Lemma 2] v erifies that ℓ H n − 1 ( γ ) > 2 give s the desired conclusion, and additionally sho ws that im( σ p n − 1 − 1) ∩ ( k m E n ) H n − 1 = ι n n − 1 ( { ξ p } · k m − 1 E n − 1 )+ ι n n − 1 ( N n n − 1 ( k m E n )) . So suppose that ℓ H n − 1 ( γ ) = 2. In the case that E n /E n − 1 is em b ed- dable, Alb ert’s Theorem [A] shows that ξ p ∈ N n n − 1 ( E × n ). The Pro j ec- tion F ormula (2.7) then giv es ι n n − 1 ( { ξ p } · k m − 1 E n − 1 ) ⊆ ι n n − 1 ( N n n − 1 ( k m E n )) . Hence if E n /E n − 1 is em b eddable and ℓ H n − 1 ( γ ) = 2, w e are done. W e ha ve left to consider the case where ℓ H n − 1 ( γ ) = 2 and N n n − 1 ( γ ) = 0. Considering this equation in K m E , w e hav e N n n − 1 ( ˜ γ ) = p ˜ f for some ˜ f ∈ K m E n − 1 and preimage ˜ γ ∈ K m E n of γ ∈ k m E n . Hence we ha v e N n n − 1 ( ˜ γ − ˜ f ) = 0 as elemen ts of K m E n − 1 , and so Hilb ert 90 fo r K - theory (2.2) implies that there exists ˜ α ∈ K m E n with (4.5) ˜ γ − ˜ f = ( σ p n − 1 − 1) ˜ α . Considering that ℓ H n − 1 ( γ ) = 2, w e can apply ( σ p n − 1 − 1) to Equation (4.5) to give ( σ p n − 1 − 1) ˜ γ = ( σ p n − 1 − 1)  ( σ p n − 1 − 1) ˜ α + ˜ f  = ( σ p n − 1 − 1) 2 ˜ α The elemen t α ∈ k m E n represen ted b y ˜ α therefore has ℓ H n − 1 ( α ) = 3, and so w e app eal to the initial case to sho w ( σ p n − 1 − 1) 2 α = ( σ p n − 1 − 1) γ ∈ ι n n − 1 ( N n n − 1 ( k m E n )) , as desired. Ha ving settled the base case, w e hav e also completed the case n = 1. No w suppose tha t n ≥ 2 a nd the result holds for j + 1, and we GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 19 sho w it also holds for j . F or simplicit y w e let ε = 1 if either E n /E j is embeddable or N n n − 1 ( γ ) = 0, and let ε = 2 if b oth N n n − 1 ( γ ) 6 = 0 and E n /E j is not em b eddable. Le t γ b e an arbitrary elemen t with ℓ H j ( γ ) > εp n − j − 1 , and consider the elemen t δ := ( σ p j − 1) ℓ H j ( γ ) − εp n − j − 1 − 1 γ . It is easy to see that ℓ H j ( δ ) = εp n − j − 1 + 1 and that ( σ p j − 1) εp n − j − 1 δ = ( σ p j − 1) ℓ H j ( γ ) − 1 γ . Hence if we can sho w ( σ p j − 1) εp n − j − 1 δ ∈ ι n j ( N n j ( k m E n )), then w e will b e done. Since ℓ H j ( δ ) = εp n − j − 1 + 1 w e hav e ( σ p j +1 − 1) εp n − 1 − j − 1 +1 δ = ( σ p j − 1) εp n − j − 1 + p δ = 0 and ( σ p j +1 − 1) εp n − 1 − j − 1 δ = ( σ p j − 1) εp n − j − 1 δ 6 = 0 . Hence we ha ve ℓ H j +1 ( δ ) = εp n − 1 − j − 1 + 1. Note that if N n n − 1 ( γ ) = 0 then N n n − 1 ( δ ) = 0, and that if E /E j is em b eddable t hen so to o is E /E j +1 . Hence b y induction it follo ws that ( σ p j +1 − 1) εp n − 1 − j − 1 δ = ι n j +1 ( N n j +1 ( α )) for some α ∈ k m E n , or equiv alen tly (4.6) ( σ p j − 1) εp n − 1 − j δ = ( σ p j − 1) p n − j − p α. Unfortunately , α do es not g enerate a submo dule long enough to pro- vide our desired equalit y . Instead of b eing length p n − j − 1 we ha ve ℓ H j ( α ) = p n − j − p + 1: ( σ p j − 1) p n − j − p α = ( σ p j − 1) εp n − 1 − j δ 6 = 0 and ( σ p j − 1) p n − j − p +1 α = ( σ p j − 1) εp n − 1 − j +1 δ = 0 . W e use induction to sho w that the H j +1 -fixed part of the F p [ H j +1 ]- submo dule h ( σ p j − 1) α i is generated by some ι n j +1 ( N n j +1 ( β )), whic h will ultimately pro vide the desired r esult. With this goal in mind, w e com- pute ℓ H j +1  ( σ p j − 1) α  . F irst, we hav e ( σ p j +1 − 1) p n − j − 1 − 2 ( σ p j − 1) α = ( σ p j − 1) p n − j − 2 p +1 α 6 = 0 , where t he inequalit y follow s from t he fa ct that ℓ H j ( α ) = p n − j − p + 1 > p n − j − 2 p + 1. W e a lso ha v e ( σ p j +1 − 1) p n − j − 1 − 1 ( σ p j − 1) α = ( σ p j − 1) p n − j − p +1 α = 0 , 20 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW again using ℓ H j ( α ) = p n − j − p + 1 . Hence we ha ve ℓ H j +1  ( σ p j − 1) α  = p n − j − 1 − 1. Pro vided p 6 = 3 or j 6 = n − 2 we ha ve p n − j − 1 − 1 > 2 p n − 1 − j − 1 , so b y induction w e hav e ( σ p j +1 − 1) p n − j − 1 − 2 ( σ p j − 1) α = ι n j +1 ( N n j +1 ( β )) = ( σ p j +1 − 1) p n − j − 1 − 1 β for some β ∈ k m E j +1 . Equiv alently , this means (4.7) ( σ p j − 1) p n − j − 2 p ( σ p j − 1) α = ( σ p j − 1) p n − j − p β . Hence, recalling Equation (4.6) for equality ⋆ below, w e ha v e the desired result: ι n j ( N n j ( β )) = ( σ p j − 1) p n − j − 1 β = ( σ p j − 1) p − 1 ( σ p j − 1) p n − j − p β = ( σ p j − 1) p − 1 ( σ p j − 1) p n − j − 2 p ( σ p j − 1) α = ( σ p j − 1) p n − j − p α ⋆ = ( σ p j − 1) εp n − 1 − j δ . Finally , supp ose t ha t p = 3 and j = n − 2. In this case ℓ H n − 2 ( α ) = 7, so t hat ( σ 3 n − 2 − 1) 6 α ∈ ( k m E n ) H n − 2 . W e also kno w tha t ( σ 3 n − 2 − 1) 6 α = ( σ 3 n − 1 − 1) 2 α = ι n n − 1 ( N n n − 1 ( α )), so that ( σ 3 n − 2 − 1) 6 α ∈ k er ( N n n − 1 ). Hence Lemma 4 .3 giv es ( σ 3 n − 2 − 1) 6 α = ι n n − 1  N n n − 1 ( α )  = ι n n − 2 ( h ) for some h ∈ k m E n − 2 , and so there exists g ∈ Γ( m, n ) so that N n n − 1 ( α ) = { a n − 1 } · g + ι n − 1 n − 2 ( h ) . No w [MSS1 , Prop. 7] pro vides an elemen t χ ∈ k 1 E n with ℓ H n − 1 ( χ ) ≤ 2 and so that N n n − 1 ( χ ) = a n − 1 . No t e that g ∈ k m − 1 E n − 1 giv es ( σ 3 n − 2 − 1) 6  { χ } · ι n n − 1 ( g )  = ( σ 3 n − 1 − 1) 2  { χ } · ι n n − 1 ( g )  =  ( σ 3 n − 1 − 1) 2 { χ }  · ι n n − 1 ( g ) = 0 . Set α ′ = ( σ 3 n − 2 − 1)  α − { χ } · ι n n − 1 ( g )  . Since ℓ H n − 2 ( α ) = 7 and ℓ H n − 2 ( { χ } · ι n n − 1 ( g )) = 6, this leav es ( σ 3 n − 2 − 1) 5 α ′ = ( σ 3 n − 2 − 1) 6 α ; from t his it fo llo ws that ℓ H n − 1 ( α ′ ) = 2. W e a lso ha v e N n n − 1 ( α ′ ) = ( σ 3 n − 2 − 1)  { a n − 1 } · g + ι n − 1 n − 2 ( h ) − { a n − 1 } · g  = 0 . Hence b y induction there exists some elem en t β with ( σ 3 n − 1 − 1) α ′ = ι n n − 1 ◦ N n n − 1 β = ( σ 3 n − 1 − 1) 2 β = ( σ 3 n − 2 − 1) 6 β , GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 21 whic h give s ι n n − 2 ( N n n − 2 ( β )) = ( σ 3 n − 2 − 1) 8 β = ( σ 3 n − 2 − 1) 2 ( σ 3 n − 2 − 1) 6 β = ( σ 3 n − 2 − 1) 2 ( σ 3 n − 1 − 1) α ′ = ( σ 3 n − 2 − 1) 6 α = ( σ 3 n − 2 − 1) 3 ε δ .  Corollary 4.8. F or γ ∈ k m E n , let i b e minimal such that γ ∈ ι n i ( k m E i ) . If N n n − 1 ( γ ) = 0 and ℓ G ( γ ) > p i − 1 , then ( σ − 1) ℓ G ( γ ) − 1 γ ∈ ι n 0 ( N i 0 ( k m E i )) . Note: When i < n , the condition N n n − 1 ( γ ) = 0 is trivial. Pr o of. In the case i = n , the r esult follows by taking j = 0 in the previous prop osition. F or i < n , c ho ose ˆ γ ∈ k m E i with ι n i ( ˆ γ ) = γ ; b y Lemma 4 .3 w e can insist ℓ G i ( ˆ γ ) = ℓ G ( γ ). Then ℓ G i ( ˆ γ ) > p i − 1 , and since E i /E 0 is em b eddable t he previous prop osition applied to the extension E i /E 0 giv es ( σ − 1) ℓ G i (ˆ γ ) − 1 ˆ γ ∈ ι i 0 ( N i 0 ( k m E i )) . Therefore ( σ − 1) ℓ G ( γ ) − 1 γ = ι n i  ( σ − 1) ℓ G i (ˆ γ ) − 1 ˆ γ  ⊆ ι n i  ι i 0  N i 0 ( k m E i )  = ι n 0 ( N i 0 ( k m E i )) as desired.  W e are now ready to giv e the “sparse” F p [ G ]-decomp osition of k m E n pro vided by Theorem 1.1. Pr o of of The or em 1 . 1. Using the no tation and results from the pro of of Corollary 2.12, w e only need to v erify that V i +1 = V p n for ev ery i satisfying 2 p n − 1 + 1 ≤ i ≤ p n − 1. This means that w e must sho w that for an y x ∈ im( σ − 1 ) i − 1 ∩ ( k m E n ) G , w e also ha ve x ∈ im( σ − 1) p n − 1 . Cho ose an α x with ( σ − 1) i − 1 α x = x . Then ℓ G ( α x ) = i , and since i > 2 p n − 1 w e ma y apply Prop osition 4.4 (with j = 0) to conclude that x = ( σ − 1) i − 1 α x = ι n 0 ( N n 0 ( α )) = ( σ − 1) p n − 1 α for some α ∈ k m E . Hence x ∈ V p n as desired.  22 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW 5. Proof of Theorem 1.2 W e are no w prepared to prov e the main result of this pap er. Though the mac hinery dev elop ed thus far a pplies to all extensions E /F with Gal( E /F ) ≃ Z /p n Z and ξ p ∈ E — a ssuming that p > 2 is prime — the main theorem relies critically on t he existenc e o f an exceptional elemen t a n of E /F whic h satisfies a σ − 1 n ∈ E × p . More sp ecifically , we use t his condition to construct mo dules X i whic h app ear in the theorem; this is the only place where the em b eddable condition is used. Let a n b e an a r bit r a ry exceptional elemen t of E / F ; Prop osition 2.10 give s a t so that N n n − 1 ( a t n ) ∈ a n − 1 E × p n − 1 . W e define the mo d- ule X as { a t n } · ι n n − 1 (Γ( m, n )), and claim that our em b eddable con- dition implies X ≃ Γ( m, n ) as F p [ G ]-mo dules (the F p [ G ]-action on Γ( m, n ) is induced from its F p [ G n − 1 ]-action). Since Prop osition 3 .2 sho ws Γ( m, n ) = ⊕ n − 1 i =0 Z i , where Z i ⊆ ι n i ( k m E i ) is a direct sum of cyclic submo dules o f dimension p i , the F p [ G ]-isomorphism X ≃ Γ( m, n ) will b e enough to show that the X i satisfy the necess a ry conditions. T o show X ≃ Γ( m, n ), first no tice that the Pro jection F ormula ( 2.7) sho ws that N n n − 1 ( X ) = { a n − 1 } · Γ( m, n ). T o see t ha t ker( N n n − 1 ) ∩ X = { 0 } , notice that for nonzero g ∈ Γ( m, n ) w e ha v e N n n − 1  { a t n } · ι n n − 1 ( g )  = { a n − 1 } · g 6 = 0 b y Prop osition 3.2(4). Finally , the action of σ comm utes with N n n − 1 and is trivial o n a n − 1 (b y (2.9 ) ) as w ell as a n (b y our em- b eddabilit y condition together with Prop osition 2.10). Hence N n n − 1 giv es an F p [ G ]-isomorphism b etw een X and { a n − 1 } · Γ( m, n ). Prop o si- tion 3.2(4) ha s already established tha t { a n − 1 } · Γ( m, n ) ≃ Γ( m, n ) as F p [ G n − 1 ]-mo dules, completing the pro of of the claim. No w let I n b e an F p -basis for ι n 0 ( N n 0 ( k m E n )), and for eac h 0 ≤ i < n let I i b e an F p -basis for a complemen t of ι n 0 ( N i +1 0 ( k m E i +1 )) within ι n 0 ( N i 0 ( k m E i )). F or eac h x ∈ I i , 1 ≤ i ≤ n , choose an elemen t α x ∈ k m E i so that x = ι n 0 ( N i 0 ( α x )), and define Y i = P x ∈I i h α x i . As in the pro of of Corollary 2.1 2, the generator α x corresp onding to x ∈ I i has h α x i G = h ι n 0 ( N i 0 ( α x )) i = h ( σ − 1) p i − 1 α x i = h x i . By construction, the elemen ts of ∪ i I i are F p -indep enden t, and so the Exclusion Lemma (2.1 1) shows n X i =0 X x ∈I i h α x i = n M i =0 M x ∈I i h α x i . GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 23 Since ι n 0 ◦ N i 0 has the same action on ι n i ( k m E i ) as ( σ − 1) p i − 1 , the mo dules Y i satisfy the appropria te conditions. W e ha v e left to sho w that the X i mo dules ar e indep enden t from the Y i mo dules. The Exclusion Lemma (2.1 1) sa ys w e can che c k in- dep endence b y lo oking at the inte rsection of the corresp onding fixed mo dules. Recall, ho w eve r, that X G ∩ ke r( N n n − 1 ) = { 0 } , whereas Y G i ⊆ ι n 0 ( N i 0 ( k m E i )) ⊆ ker( N n n − 1 ). Hence w e conclude that J = n − 1 M i =0 X i ! + n M i =0 Y i ! = n − 1 M i =0 X i ! ⊕ n M i =0 Y i ! . Our goal is to sho w tha t k m E n = J. T o do t his, recall the notation V ℓ = im  ( σ − 1) ℓ − 1  ∩ ( k m E n ) G . W e shall prov e that f or each 0 ≤ i ≤ n and 1 ≤ j ≤ p i +1 − p i , (5.1) V p i + j ⊆ im  ( σ − 1) p i +1 − 1  ∩ J G . Inasm uc h as the r igh t side of this express ion is visibly in V p i +1 , and since w e hav e V p i +1 ⊆ V p i + j automatically , this condition will ensure that V p i + j = V p i +1 . According to Corollary 2.14, this implies that the mo dule structure of k m E n will contain o nly cyclic summands of dimension p k , 0 ≤ k ≤ n . Condition (5.1) will also sho w that V p i = im  ( σ − 1) p i − 1  ∩ J G = M k ≥ i X G k ⊕ M k ≥ i Y G k , from whic h our construction of the summands X i and Y i , together with Corollary 2 .12, will sho w that k m E n ≃ J . T o verify this condition, supp ose that f = ( σ − 1) p i + j − 1 γ ∈ ( k m E n ) G . No w if p i + j > p n − p n − 1 , then this implies ℓ G ( γ ) > 2 p n − 1 . Hence taking j = 0 in Prop osition 4.4 sho ws that f ∈ ι n 0 ( N n 0 ( k m E n )) = im  ( σ − 1) p n − 1  ∩ ( k m E n ) G . In t his case recall t ha t Y G n = hI n i = ι n 0 ( N n 0 ( k m E n )) b y construction, and so f ∈ Y G n ⊆ im  ( σ − 1) p n − 1  ∩ J G as desired. Otherwise w e hav e p i + j ≤ p n − p n − 1 , meaning that ( σ − 1) p n − p n − 1 γ = ι n n − 1 ( N n n − 1 ( γ )) = 0. Hence from Exact Sequence (2.5) w e m ust b e in t he case that N n n − 1 ( γ ) = { a n − 1 } · ι n n − 1 ( g ), where g ∈ Γ( m, n ). By construction of the mo dule X , there exists a unique x ∈ X — p ossibly zero — so that N n n − 1 ( x ) = N n n − 1 ( γ ). Moreo v er, since X ≃ 24 N. LEMIRE, J. MIN ´ A ˇ C, A. SCHUL TZ, AND J. SW A LLOW Γ( m, n ) we mus t ha v e ℓ G ( x ) = ℓ G n − 1 ( g ). Notice that since 0 = N n n − 1  ( σ − 1) ℓ ( γ ) γ  = ( σ − 1) ℓ ( γ )  { a n − 1 } · ι n n − 1 ( g )  and { a n − 1 } · Γ( m, n ) ≃ Γ( m, n ) b y Prop osition 3.2 , w e m ust then hav e ℓ G ( x ) = ℓ G n − 1 ( g ) ≤ ℓ G ( γ ). Hence the elemen t γ − x has trivial image under the map N n n − 1 , and moreo v er ℓ G ( γ − x ) ≤ max { ℓ G ( γ ) , ℓ G ( x ) } = ℓ G ( γ ). Supp ose first that ℓ G ( γ − x ) < ℓ G ( γ ). In this case it follo ws that ℓ G ( x ) = ℓ G ( γ ), and indeed that f = ( σ − 1) p i + j − 1 x . Hence w e ha ve f ∈ im  ( σ − 1) p i + j − 1  ∩ X G . But notice that since X is a direct sum of cyclic submo dules of dimension p k , where 0 ≤ k ≤ n − 1, this in turn implies that f ∈ im  ( σ − 1) p i +1 − 1  ∩ X G ⊆ im  ( σ − 1) p i +1 − 1  ∩ J G . Finally , w e ar e left with the case t ha t ℓ G ( γ − x ) = ℓ G ( γ ). In this case w e hav e γ − x ∈ ker( N n n − 1 ) ∩  ( k m E n ) H i +1 \ ( k m E n ) H i  . Hence Lemma 4 .3 and the fact t ha t ι n i ( k m E i ) ⊆ ( k m E n ) H i implies that γ − x ∈ im( ι n i +1 ) \ im( ι n i ), and Corolla r y 4.8 sho ws that ( σ − 1) p i + j − 1 ( γ − x ) = ι n 0 ( N i +1 0 ( α )) for some α ∈ k m E i +1 , so that f = ( σ − 1) p i + j − 1  x + ( γ − x )  = ( σ − 1) p i + j − 1 x + ι n 0 ( N i +1 0 ( α )) . Considering that ι n 0 ◦ N i +1 0 is represen ted by ( σ − 1) p i +1 − 1 on ι n i +1 ( k m E i +1 ), it is easy to see that ι n 0 ( N i +1 0 ( α )) ∈ im  ( σ − 1) p i +1 − 1  ∩ J G . On the o t her hand, since ℓ G ( x ) ≤ ℓ G ( γ ) w e see that ( σ − 1) p i + j − 1 x ∈ im  ( σ − 1) p i + j − 1  ∩ X G ; since X is composed of cyclic indecomp osables of prime-p o we r dimension, it therefore follo ws that ( σ − 1) p i + j − 1 x ∈ im  ( σ − 1) p i +1 − 1  J G . Com bining these tw o observ ations, w e hav e f ∈ im  ( σ − 1) p i +1 − 1  ∩ J G as desired. GALOIS COHOMOLOGY FOR EMBEDDABLE CYCLIC EXTENSIONS 25 Reference s [A] A.A. Alb ert. 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Dep ar tment of Ma thema tics, Middlesex College, University o f Western Ont ario, Londo n, Ont ario N6 A 5B7 CAN ADA E-mail addr ess : nlemi re@uwo .ca Dep ar tment of Ma thema tics, Middlesex College, University o f Western Ont ario, Londo n, Ont ario N6 A 5B7 CAN ADA E-mail addr ess : minac @uwo.c a Dep ar tment o f Ma thema tics, University of I llinois a t Urbana-Champ aign, 1409 W. Green Street, Urbana, IL 61801 USA E-mail addr ess : acs@m ath.ui uc.edu Dep ar tment of Ma thema tics, Da vidso n College, Box 7046, Da vidson, Nor th Car o lina 28035-7046 U SA E-mail addr ess : joswa llow@d avidson.edu