Formal groups and ring structures for certain periodic cohomology theories

Pub . Mat . UAB Vol . 26 Nó 3 Des . 1982 FORMAL GROUPS AND RINGSTRUCTURES FOR CERTAIN PERIODIC COHOMOLOGY THEORIES The purpose of this talk is to report on some results concerning a classificationproblem for a special kind of'cohomology theories . Forthe beginning, however, I would like to consider a different and perhaps more concrete problem which may serveas motivation for the rest . Let K (-) denoteordinarycomplex K-theory . We consider it as a E/2-graded theory defined on the category CW * of pointed spaces of the homotopy type of a CW-complex . Recallthat K0 (S 0 - Z , K 1 (S0 ) = 0 and that there is a naturalequivalen ce ~ : KO (X) - -K O (S ^ X),  the Bott isomorphism . K * (-)  is usually considered as a multiplicative theory, the productbeing inducedby the tensor product operationof complex vector bundles . We may askthe following Question 1 : Are thereproducts in K (-) differentfrom the ordinary one and, if so, can one describe the set Prod(K) of all isomorphism classes of such products in some reasonable way ? 0 Urs Würgler Note that all productswe considerhere are assumed to be with unit, associative and commutative in the graded sense .Moreover two products U,U' : K (X)® K (X)~ K (X) are calledisomorphic if there is an isomorphism e : K (-)-~ K (-) of cohomology theories with values in the categoryAb of abelian groups (an additive isomorphism) such that the following diagram commutes : e®e  e u K (-)® K  K (-) To answer the question above one cou`ld certainly try to constructelements U EK 0 (BUABU) with appropriate properties and then determine the set of all such elements . Here, however, we will adopt a different point of view . Let A be an ungraded commutative ring with unit . Forany such ring A we consider the set C(A) of all isomorphism classes [T] of Z/2-graded multiplicative cohomology theories T (-) with coefficient ring T (S0 ) of the form TOS O ) = A, T 1 (S 0 ) = 0 (Z/2 -graded ring theories with coefficients A for short) . . Clearly, [K]E= - C(Z) . Usingthisnotation we ask the followingunprecisequestion : 208 Question 2 : Given a ring A, can one describe the set C(A) or at least some interesting subsets of C(A) in an explicit way? Of course, this is just theclassification problem for Z/2 - gradedring theories withcoefficients A . Now we remark that there is a connection between Question 1 arid Question 2 . Put A=Z and lét CK (a) denote the subset of CM) whose elements are all isomorphism classes [T] E C(Z) with the property that T *(-) is additivelyisomorphic to K*(-), .i .e . *  * T  K (-) as Z/2 -graded cohomology theories with values in the category Ab .  Suppose[TIC CK (E),  let e : T * (X)>  K *(X)  be an additive equivalence and suppose a : T * (X)® T * (X) - ' T * (X)  is a product on T * (-) .  Then eoa o (e -1 ® e -1 ) :  K * (X)® K * (X) -- > - K * (X)  de- * fines a product on K  Moreover, different equivalences e and isomorphic products on T * (-) produce isomorphicproductson K * (-) andone sees easilythatthere is a bijection (*)  CK M) -s Prod(K) defined by a -' eoao(e -1 ®e -1 ) . Thisleads us to study the problem raisedby Question 2 in more detail . * Observethat any X/2 - gradedringtheory T (-) withcoefficientsan ungraded ring A is automatically a complex-orientable theory, i .e . the canonicalcomplex line bundle n m over CP - is * T (-)-orientable . Thisfollows immediately from [1], p .399.Let m : Cp w XCP - --' CP .b e the classifying map of the bundle n .xn . and let x ET 0 (CP .) be an Euler . class of n . (a Corientation * of T (-)) . Then, as is well known, the formal power series * F(x 1 ,x 2 ) := m (x)E AQx 1 ,x 2 j is a one-dimensional commutative formal group law on A (a formal group on A for short), where * * x i= pr i (x) ET (CP . x CP_) . Now formalgroups corresponding to different C - orientations of the same theory are isomorphic and isomorphictheorieswith the same coefficient ring produceisomorphic formal groups, so if we associate to any 7L/2- * gradedringtheory T (-) with coefficients A its formal group we get a map (D  :  C (A)  -  FG (A) where FG(A) denotes theset of (strict) isomorphism classes of formal groupsover A . We will use this map to get an answer to Question 2 in some particular cases . Supposefirstthat A = k is a field . If thecharacteristic of k is 0, classicalresultsimply that C(k) consists of only one element, namely H (- ;k) . For fields of positive character istic, however, the situation changes . Firstwe have : Theorem 1 : Let k be á fieldof characteristic p> 2 . Then the más (D : C(k) -->- FG(k)  is á biiection . Remark : For p=2 we haveonlypartialresults . In this case, themap D is surjective butnot injective . Difficulties arise from the factthat all elements nf C(k) different fr om H ** (- k) arenon-commutative . Formal groupsover fields of positive chracteristic are ratherwellunderstood (see for example the book [3]) . In particular, there is an important isomorphism invariant for such formalgroups F, theirheightht F E = U{-} . Briefly, ht F = n n if [p] F (x) = axp + terms of higher order, a q¿ 0, and ht F= if [p] F (x) = 0 . Let FG(k) n denote the subset of FG(k) of formal groups of height n andput 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 next theorem tells us that n=1  n=1 Z/2 - graded ringtheorieswithcoefficients k and formal groups of equalheight are verystrongly related, in fact they only differ by their multiplicative structure : * Theorem 2 : Let p be any p_rime and suppose T 1 (-), T 2 (-) are E/2graded rin theories withcoefficients k, a field of char - *  * acteristic p . Then T 1 (-) and T 2 (-) are isomorphic as coho - mologytheorieswithvalues in the category of kvector - spaces if andonl if their formalgroups are of the same height . Recall from [21,[41 that forany integer n the /2-gradedversion * withcoefficients k, K(n) (- ;k), represents an element of C(k) n . For n = - we set K(w) * (- ;k) = H ** (- ;k) . Note also that *  * K(1)  (- ;&r p )  = K  (- ;F p ) .  Using the same argument which lead to the bijection (*) we get from theorems 1 and 2 the Corollary 3 : Let k be á fieldof characteristic p>2 . Then for all nE N U {-} there are biiections C(k) n- ~ Prod(K(n)*(-,k)) -* in log n (x)  =  EP -1 x p  E w X] i>0 prime p and any positive of the n-th MoravaK-theory FG(k) n . It shouldbe noted that for FG(k) n , there are severalmore or less explicitdescriptionsavailable (see e .g . [3]) . Let us recall very briefly one of them' Consider the power series andput Fn (x,y) = logn l (log n (x) + log n(y)) . Fn (x,y) is a formal group over Z(P) . F n (x,y), its reduction mod p, is definedover Ir p and so over every field of characteristic p . Let k sep  be a separable closure of k and S n= Autk  (F n ) the automorphism _  sep group of Fn over k seP . A classicalresult of Dieudonné-Lubin tellsus that S n is isomorphic to the group of units of the maximal order in the centraldivisionalgebra D n of invariant 1/n and rank n 2 over 0p . Let P be the Galois group Gal(k sep :k) . Then P acts on S (by acting on the coefficients of power n series)and there is a bijection FG(k) n -, H 1 (P,S n ) . This bijection together with the fact that formal groups of infinite height over a ring of prime characteristic are isomorphic to the additive formal group imply the following Corollary 4 : If k is á separable closed field of odd charac teristic and n<- or if k is án arbitrare field óf positive characteristic and n=-, then , ug to isomornhism , K(n) (-,k) is the only Z/2 - araded ring theory with coefficients k and i formal QrouP of height n . If n . = 1, S . 1 is isomorphic to _ the group 2 p of p-adic units .If k=Y r is a finite field, P is topologically generated by p the Frobenius homomorphism and one obtains H (Pa p )= 7i p . So corollary 3 implies a bijection C(Ypr )~ Prod(K* (-,F pr ) )-, ~P of global version of Theorem 2 . For more general rings A we have only very partial results to offer for the moment and the question seems to be difficult . To end this talk,let me just describe some results for the case A = E . This wiil be enough to answer our initial Question 1 . Let P denote the set of all primes and let F(x,y) be a formal group oven 2E . Define the height function of F, htF : P -- ' N U{-}, by setting ht F (p) = height of F mod p over F p . It is an isomorphism invariant of F . Using this notion we get some sort Theorem 5 : Let T 1 (-) and T 2 (-) be Z/2 - ciraded ring theories with coefficients 2Z and formal groupsF 1 res p . F 2 . Then T 1 (-) and T 2 (-) are additivelyisomorphic if and onlYif ht F  (p)  = htF2 (p)  forall Primes p . 1 We do not know if the map (D : C(Z) - FG(Z)  is súrjective or injective in generalalthough we havesomepartial results which we will not describehere .However, if we restrict our s attention to the subset CK (Z) of C(Z), we can be more precise . Let FG(Z) 1 be the set of all isomorphism classes of formal groups F over Z of height 1 at any prime, i .e . ht F (p) = 1 for all p, andlet (D J< denote the restriction of 4) to CK (E) . Theorem 6 : There are biiections 4)K 1 Prod(K)-}  CK (Z)  FG(Z)  --->  II Z * PEP p One mayask what all there new products on K (-) described by theorem 6 are good for . It turns out that there are interesting connectionsbetweenthem andcharacteristic classes cx E H (BU,Q) associated to certain integral Hirzebruch genera (i .e . ring homomorphisms) ' ZC Q which can be described in terms of Riemann-Rock relations . Also, to any exotic product on K (-) theircorresponds a set of "exoticAdamsoperations"withinteresting properties . References [11  Dold,A . : Chern classes in generalcohomology . Symposia Mathematica vol . V (1970) [21  Johnson,D .C ., Wilson,S . : BP-operations and Morava's extraordinary K-theories . Math . Z . 144, 55-75 (1975) [31  Hazewinkel,M . : Formalgroups and applications . Academic Press, 1978 [41  Würgler,U . : On products in a family of cohomology theories associated to the invariant prime ideals of u * (BP) . Comment . Math . Helv . 52, 457-481 (1977) Mathematisches Institut der Universitát Sidlerstrasse 5 CH-3012 Bern