Formal groups and ring structures for certain periodic cohomology theories

Pub . Ma . UAB Vol . 26 Nó 3 Des . 1982 FORMAL GROUPS AND RINGSTRUCTURES FOR CERTAIN PERIODIC COHOMOLOGY THEORIES The pu pose o his alk is o epo on some esul s con- ce ning a classi ica ionp oblem o a special kind o 'coho- mology heo ies . Fo he beginning, howe e , I would like o conside a di e en and pe haps mo e conc e e p oblem which may se eas mo i a ion o he es . Le K (-) deno eo dina ycomplex K- heo y . We conside i as a E/2-g aded heo y de ined on he ca ego y CW * o poin- ed spaces o he homo opy ype o a CW-complex . Recall ha K0 (S 0 - Z , K 1 (S0 ) = 0 and ha he e is a na u alequi alen ce ~ : KO (X) - -K O (S ^ X),  he Bo isomo phism . K * (-)  is usually conside ed as a mul iplica i e heo y, he p oduc being in- ducedby he enso p oduc ope a iono complex ec o bund- les . We may ask he ollowing Ques ion 1 : A e he ep oduc s in K (-) di e en om he o dina y one and, i so, can one desc ibe he se P od(K) o all isomo phism classes o such p oduc s in some easo- nable way ? 0 U s Wü gle No e ha all p oduc swe conside he e a e assumed o be wi h uni , associa i e and commu a i e in he g aded sense .Mo eo e wo p oduc s U,U' : K (X)® K (X)~ K (X) a e calledisomo phic i he e is an isomo phism e : K (-)-~ K (-) o cohomology heo ies wi h alues in he ca ego yAb o abelian g oups (an addi i e isomo phism) such ha he ollowing diag am commu es : e®e  e u K (-)® K  K (-) To answe he ques ion abo e one cou`ld ce ainly y o cons uc elemen s U EK 0 (BUABU) wi h app op ia e p ope ies and hen de e mine he se o all such elemen s . He e, howe e , we will adop a di e en poin o iew . Le A be an ung aded commu a i e ing wi h uni . Fo any such ing A we conside he se C(A) o all isomo phism classes [T] o Z/2-g aded mul iplica i e cohomology heo ies T (-) wi h coe icien ing T (S0 ) o he o m TOS O ) = A, T 1 (S 0 ) = 0 (Z/2 -g aded ing heo ies wi h coe icien s A o sho ) . . Clea ly, [K]E= - C(Z) . Using hisno a ion we ask he ollowing- unp ecise- ques ion : 208 Ques ion 2 : Gi en a ing A, can one desc ibe he se C(A) o a leas some in e es ing subse s o C(A) in an explici way? O cou se, his is jus heclassi ica ion p oblem o Z/2 - g aded ing heo ies wi hcoe icien s A . Now we ema k ha he e is a connec ion be ween Ques ion 1 a id Ques ion 2 . Pu A=Z and lé CK (a) deno e he subse o CM) whose elemen s a e all isomo phism classes [T] E C(Z) wi h he p ope y ha T *(-) is addi i elyisomo phic o K*(-), .i .e . *  * T  K (-) as Z/2 -g aded cohomology heo ies wi h alues in he ca ego y Ab .  Suppose[TIC CK (E),  le e : T * (X)>  K *(X)  be an addi i e equi alence and suppose a : T * (X)® T * (X) - ' T * (X)  is a p oduc on T * (-) .  Then eoa o (e -1 ® e -1 ) :  K * (X)® K * (X) -- > - K * (X)  de- * ines a p oduc on K  Mo eo e , di e en equi alences e and isomo phic p oduc s on T * (-) p oduce isomo phicp oduc son K * (-) andone sees easily ha he e is a bijec ion (*)  CK M) -s P od(K) de ined by a -' eoao(e -1 ®e -1 ) . Thisleads us o s udy he p ob- lem aisedby Ques ion 2 in mo e de ail . * Obse e ha any X/2 - g aded ing heo y T (-) wi hcoe i- cien san ung aded ing A is au oma ically a complex-o ien able heo y, i .e . he canonicalcomplex line bundle n m o e CP - is * T (-)-o ien able . This ollows immedia ely om [1], p .399.Le m : Cp w XCP - --' CP .b e he classi ying map o he bundle n .xn . and le x ET 0 (CP .) be an Eule . class o n . (a C- o ien a ion * o T (-)) . Then, as is well known, he o mal powe se ies * F(x 1 ,x 2 ) := m (x)E AQx 1 ,x 2 j is a one-dimensional commu a i e o mal g oup law on A (a o mal g oup on A o sho ), whe e * * x i= p i (x) ET (CP . x CP_) . Now o malg oups co esponding o di e en C - o ien a ions o he same heo y a e isomo phic and isomo phic heo ieswi h he same coe icien ing p o- duceisomo phic o mal g oups, so i we associa e o any 7L/2- * g aded ing heo y T (-) wi h coe icien s A i s o mal g oup we ge a map (D  :  C (A)  -  FG (A) whe e FG(A) deno es hese o (s ic ) isomo phism classes o o mal g oupso e A . We will use his map o ge an answe o Ques ion 2 in some pa icula cases . Suppose i s ha A = k is a ield . I hecha ac e is ic o k is 0, classical esul simply ha C(k) consis s o only one elemen , namely H (- ;k) . Fo ields o posi i e cha ac e is ic, howe e , he si ua ion changes . Fi s we ha e : Theo em 1 : Le k be á ieldo cha ac e is ic p> 2 . Then he más (D : C(k) -->- FG(k)  is á biiec ion . Rema k : Fo p=2 we ha eonlypa ial esul s . In his case, hemap D is su jec i e bu no injec i e . Di icul ies a ise om he ac ha all elemen s n C(k) di e en om H ** (- k) a enon-commu a i e . Fo mal g oupso e ields o posi i e ch ac e is ic a e a he wellunde s ood (see o example he book [3]) . In pa - icula , he e is an impo an isomo phism in a ian o such o malg oups F, hei heigh h F E = U{-} . B ie ly, h F = n n i [p] F (x) = axp + e ms o highe o de , a q¿ 0, and h F= i [p] F (x) = 0 . Le FG(k) n deno e he subse o FG(k) o o - mal g oups o heigh n andpu C(k) n = (D-1(FG(k)n) . Then FG(k)= U n=w FG(k ) n and C(k)  = U n=- C (k ) n . The nex heo em ells us ha n=1  n=1 Z/2 - g aded ing heo ieswi hcoe icien s k and o mal g oups o equalheigh a e e ys ongly ela ed, in ac hey only di e by hei mul iplica i e s uc u e : * Theo em 2 : Le p be any p_ ime and suppose T 1 (-), T 2 (-) a e E/2- g aded in heo ies wi hcoe icien s k, a ield o cha - *  * ac e is ic p . Then T 1 (-) and T 2 (-) a e isomo phic as coho - mology heo ieswi h alues in he ca ego y o k- ec o - spaces i andonl i hei o malg oups a e o he same heigh . Recall om [21,[41 ha o any in ege n he /2-g aded e sion * wi hcoe icien s k, K(n) (- ;k), ep esen s an elemen o C(k) n . Fo n = - we se K(w) * (- ;k) = H ** (- ;k) . No e also ha *  * K(1)  (- ;& p )  = K  (- ;F p ) .  Using he same a gumen which lead o he bijec ion (*) we ge om heo ems 1 and 2 he Co olla y 3 : Le k be á ieldo cha ac e is ic p>2 . Then o all nE N U {-} he e a e biiec ions C(k) n- ~ P od(K(n)*(-,k)) -* in log n (x)  =  EP -1 x p  E w X] i>0 p ime p and any posi i e o he n- h Mo a aK- heo y FG(k) n . I shouldbe no ed ha o FG(k) n , he e a e se e almo e o less explici desc ip ionsa ailable (see e .g . [3]) . Le us ecall e y b ie ly one o hem' Conside he powe se ies andpu Fn (x,y) = logn l (log n (x) + log n(y)) . Fn (x,y) is a o mal g oup o e Z(P) . F n (x,y), i s educ ion mod p, is de inedo e I p and so o e e e y ield o cha ac e is ic p . Le k sep  be a sepa able closu e o k and S n= Au k  (F n ) he au omo phism _  sep g oup o Fn o e k seP . A classical esul o Dieudonné-Lubin ellsus ha S n is isomo phic o he g oup o uni s o he maximal o de in he cen aldi isionalgeb a D n o in a ian 1/n and ank n 2 o e 0p . Le P be he Galois g oup Gal(k sep :k) . Then P ac s on S (by ac ing on he coe icien s o powe n se ies)and he e is a bijec ion FG(k) n -, H 1 (P,S n ) . This bijec ion oge he wi h he ac ha o mal g oups o in ini e heigh o e a ing o p ime cha ac e is ic a e iso- mo phic o he addi i e o mal g oup imply he ollowing Co olla y 4 : I k is á sepa able closed ield o odd cha ac e is ic and n<- o i k is án a bi a e ield ó posi i e cha ac e is ic and n=-, hen , ug o isomo nhism , K(n) (-,k) is he only Z/2 - a aded ing heo y wi h coe icien s k and i o mal Q ouP o heigh n . I n . = 1, S . 1 is isomo phic o _ he g oup 2 p o p-adic uni s .I k=Y is a ini e ield, P is opologically gene a ed by p he F obenius homomo phism and one ob ains H (Pa p )= 7i p . So co olla y 3 implies a bijec ion C(Yp )~ P od(K* (-,F p ) )-, ~P o global e sion o Theo em 2 . Fo mo e gene al ings A we ha e only e y pa ial esul s o o e o he momen and he ques ion seems o be di icul . To end his alk,le me jus desc ibe some esul s o he case A = E . This wiil be enough o answe ou ini ial Ques ion 1 . Le P deno e he se o all p imes and le F(x,y) be a o mal g oup o en 2E . De ine he heigh unc ion o F, h F : P -- ' N U{-}, by se ing h F (p) = heigh o F mod p o e F p . I is an iso- mo phism in a ian o F . Using his no ion we ge some so Theo em 5 : Le T 1 (-) and T 2 (-) be Z/2 - ci aded ing heo ies wi h coe icien s 2Z and o mal g oupsF 1 es p . F 2 . Then T 1 (-) and T 2 (-) a e addi i elyisomo phic i and onlYi h F  (p)  = h F2 (p)  o all P imes p . 1 We do no know i he map (D : C(Z) - FG(Z)  is sú jec i e o injec i e in gene alal hough we ha esomepa ial esul s which we will no desc ibehe e .Howe e , i we es ic ou s a en ion o he subse CK (Z) o C(Z), we can be mo e p ecise . Le FG(Z) 1 be he se o all isomo phism classes o o mal g oups F o e Z o heigh 1 a any p ime, i .e . h F (p) = 1 o all p, andle (D J< deno e he es ic ion o 4) o CK (E) . Theo em 6 : The e a e biiec ions 4)K 1 P od(K)-}  CK (Z)  FG(Z)  --->  II Z * PEP p One mayask wha all he e new p oduc s on K (-) des- c ibed by heo em 6 a e good o . I u ns ou ha he e a e in e es ing connec ionsbe ween hem andcha ac e is ic classes cx E H (BU,Q) associa ed o ce ain in eg al Hi zeb uch gene a (i .e . ing homomo phisms) ' ZC Q which can be desc ibed in e ms o Riemann-Rock ela ions . Also, o any exo ic p oduc on K (-) hei co esponds a se o "exo icAdamsope a ions"wi hin e es ing p ope ies . Re e ences [11  Dold,A . : Che n classes in gene alcohomology . Symposia Ma hema ica ol . V (1970) [21  Johnson,D .C ., Wilson,S . : BP-ope a ions and Mo a a's ex ao dina y K- heo ies . Ma h . Z . 144, 55-75 (1975) [31  Hazewinkel,M . : Fo malg oups and applica ions . Academic P ess, 1978 [41  Wü gle ,U . : On p oduc s in a amily o cohomology heo ies associa ed o he in a ian p ime ideals o u * (BP) . Commen . Ma h . Hel . 52, 457-481 (1977) Ma hema isches Ins i u de Uni e si á Sidle s asse 5 CH-3012 Be n