UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS JEAN-YVES BUTOWIEZ AND PAUL TURNER Abstract.We investigate the relationship between multiplicative unstable* * cohomolgy oper- ations G0(-) ! E0(-) and formal group laws for a certain important class* * of theories. As an application we study additive multiplicative idempotents. 1.Introduction The Adams operations in K-theory furnish a splendid example of unstable cohom* *ology operations with the additional pleasing property that they are multiplicative. * *Such operations may be regarded as self ring maps of the homotopy ring space 1 K = BU xZ which * *represents the functor K0(-): Spaces! Rings. More generally given a suitably behaved multi* *plicative cohomology theory G*(-) there is an infinite loop space 1 G which is a ring up * *to homotopy such that for any space X there is an isomorphism of rings G0(X) ~=[X; 1 G]. Gi* *ven another such theory E*(-) the multiplicative natural transformations G0(-) ! E0(-) corr* *espond to maps of homotopy ring spaces 1 G ! 1 E. Recall from [6] the notion of a detecting category. For a homology theory E** *(-), an E- detecting category for a space X is a subcategory C of the homotopy category cl* *osed under finite products such that there is an isomorphism o :colimE*(Z) ! E*(X): CX where CX is the category with objects the homotopy classes of maps Z ! X for Z * *an object of C and whose morphisms are certain homotopy commutative diagrams. See Sectio* *n 2 for details. The aim of this paper is to study HRing (1 G; 1 E) = maps of homotopy ring spaces1 G ! 1 E = unstable multiplicative operationsG0(-) ! E0(-) for the case when the category of finite products of CP 1 is an E-detecting cat* *egory for 1 G. Landweber exact theories such as K-theory and Elliptic cohomology satisfy the r* *elevant criteria. It has been described variously in [8] , [9] and [3] how to construct stable * *cohomology oper- ations from formal group laws and their strict isomorphisms. Stable multiplicat* *ive operations have been extensively studied for Brown-Peterson cohomology in [2] and [10]. Un* *stably one can construct multiplicative families of operations (see [4] and [12]) using formal* * group laws and their homomorphisms. Alternatively one can study unstable multiplicative operat* *ions on the 0th-cohomology where there is a far more intricate connection with formal group* * laws which 1 2 JEAN-YVES BUTOWIEZ AND PAUL TURNER it is the aim of this paper to describe. We separate out the statement of the * *main theorem for the case of two periodic theories both because of the simplified statement * *it allows and the importance of such theories. For background on unstable operations we refer the* * reader to the work of Boardman, Johnson and Wilson [4] and Wilson [13]. We will be working with one dimensional commutativePgraded formal group laws.* * We demand that a formal group law F (x; y) = x +F y = aijxiyj 2 R*[[x; y]] is graded b* *y decreeing |aij| = -2(i + j - 1). Notice that the coefficients of a graded formal group l* *aw lie in R*0 , that is they arePall in non-positive degrees. A homomorphism from +F to +G will* * be a power series OE(x) = OEixi 2 R*[[x]] where |OEi| = -2(i - 1) such that OE(x) +F OE(* *y) = OE(x +G y). The category of such is denoted FGL(R*)op. Given a formal group law +F over R** * and a homomorphism :R* ! S* we write +F for the induced formal group law over S*. I* *t is well known that for a complex oriented cohomology theory E*(-) one obtains a formal * *group law by pulling back an orientation class generating E*(CP 1) ~=E*[[xE ]] using the * *multiplication on CP 1. The following construction is a version of a construction due Thomason [11] e* *xtending the work of Grothendieck. Let F :C ! Cat be a functor from a category C to the cat* *egory of small categories Cat. The Grothendieck Construction on F is the category Gr(F )* *constructed as follows. Objects: Pairs (c; x) where c is an object in C and x is an object in F (c) Morphisms: Pairs (; OE): (c; x) ! (d; y) where :c ! d is a morphism in C an* *d OE: F ()x ! y is a morphism in F (d). The composition (0; OE0) O (; OE) is deemed to be (0O* * ; OE0O F (0)OE). The appropriate category for two periodic theories which we call FG is a quot* *ient of Gr(FGL) where FGL: N PRing ! Cat is the functor taking R* to the category F GL(R*)opand* * N PRing is the category of non-positively graded rings. The non-periodic case is more s* *ubtle and requires a new category VFG more complicated but similar in spirit to FG (see Definition* * 3.8). Of course for two periodic theories VFG = FG. Having defined VFG appropriately the main t* *heorem is as follows. Theorem 3.12 Let E and G be multiplicative complex oriented cohomology theori* *es with formal group laws FE and FG respectively. Suppose that o E* is in even degrees and is a unique factorisation domain. o G* is in even degrees. o is a E-detecting category for 1 G with nice duality Then there is a bijection of sets HRing (1 G; 1 E) $ VFG((G*0 ; FG ); (E*0 ; FE )) For the case G = E this is an equivalence of monoids. 2.Detecting categories and operations Let us recall the notion of a detecting category in the sense of [6] and used* * implicitly in the work of Kashiwabara [7]. Let E*(-) be a homology theory and let C be a full su* *bcategory of the homotopy category which includes the point space pt and which is closed * *under finite products. For a space X let CX be the category with objects the homotopy class* *es of maps UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS * * 3 Z ! X for Z an object of C and whose morphisms are homotopy commutative diagrams f Z ___________//_@@Z0 " @@@ """ ff@@OO@"""ff0""" X where f :Z ! Z0 is a map in C. There is a homomorphism o :colimE*(Z) ! E*(X): CX and C is said to be an E-detecting category for X when o is an isomorphism. Our* * particular interest will be when C is the category of finite products of CP 1 (with zero c* *opies being a point) which we denote by . Examples of such are provided when E is compl* *ex oriented and G is Landweber exact (see [7]). When CX is a directed category a typical element of the colimit can be writt* *en as a pair (ff; x) where ff 2 CX and x 2 E*(Z). The equivalence relation on such pairs is * *generated by: (ff; x) ~ (ff0; x0) if there exists a morphism f in CX such that in E-homology * *f*(x) = x0. We will require our spaces to exhibit a nice duality duality between homology* * and cohomo- logy. The following standard lemma is useful. Lemma 2.1. Suppose that C is a detecting category for X and that E*(X) and E** *(Z) are free over E* for all Z 2 C. Then 1. E*(X) ~=limE*(Z) ~=ModE*(colimE*(Z); E*) CX CX 2. E0(X) ~=limE0(Z) ~=Mod0E(colimE*(Z); E*) CX * CX From now on we will assume that E*(-) and G*(-) are cohomology theories and t* *hat C is an E-detecting category for 1 G and further that E*(1 G) and E*(Z) are free ove* *r E* for all Z 2 C. We will refer to this as nice duality. Note that for 2 E0(1 G) ~=Mod0E*(colimE*(Z); E*) and (ff; x) 2 colimE*(Z) w* *e have CX CX < ; (ff; x)> = < (ff); x> where < ; -> and < (ff); -> are the linear functionals corresponding by duality* * to and (ff) respectively. Note also that since additive operations correspond to primitives* * in E0(1 G) we have that is additive if and only if < ; -> annihilates decomposables in E*(1* * G). The following Proposition justifies the name detecting category. Proposition 2.2. Under the above assumption let and 0be cohomology operation* *s G0(-) ! E0(-). If (ff) = 0(ff) for all ff 2 G0(Z), Z 2 C, then = 0. Proof.For all ff 2 G0(Z), Z 2 C we have (ff) = 0(ff)) < (ff); -> = < 0(ff); -> ) < (ff); x> = < 0(ff); x> 8x 2 E*(Z) ) < ; (ff; x)> = < 0; (ff; x)> * * __ This implies < ; -> = < 0; -> so = 0. * * |__| Corollary 2.3. If (ff) = 0 for all ff 2 G0(Z), Z 2 C, then is trivial. 4 JEAN-YVES BUTOWIEZ AND PAUL TURNER Now we turn to the construction of operations for which we can also use detec* *ting categor- ies. It can be seen that to define :G0(-) ! E0(-) it suffices to define a fa* *mily of maps Z :G0(Z) ! E0(Z) one for each Z 2 C such that for any morphism f :Z ! Z0 the f* *ollowing diagram commutes. Z (1) G0(Z)O_____//E0(Z)OOO f*|| f*|| | | G0(Z0) __0_//_E0(Z0) Z For the case when E and G are complex oriented theories and C = some i* *mportant simplifications can be made. Note that all morphisms in are compositions* * of ones of the form 1 x 1 x . .x.1 x fi x 1 x . .x.1 where fi is one of i: pt ! CP 1; p: CP 1 * *! pt; : CP 1 ! CP 1xCP 1; m: CP 1xCP 1 ! CP 1. When G is multiplicative then the colimit colim* *E*(Z) CX also has the structure of a coalgebraic ring (Hopf ring) given by (ff; x) * (ff0; x0) = (ff x 1 + 1 x ff0; x x0) P P (ff; x) = (ff; x0) (ff; x00) if(x) = x0 x00 (ff; x) O (ff0; x0) = (ff x ff0; x x0) This is of course true whenever E has K"unneth isomorphisms for objects of C.* * These colimit Hopf rings were first considered by Kashiwabara in [7] and the following result* * is due to him. Recall that for a complex oriented cohomology theory G*(-) we have G*((CP * *1)xk) ~= G*[[x1; : :;:xk]]. Proposition 2.4. Let be an E-detecting category for the multiplicative t* *heory G. Let ff 2 G0((CP 1)xk) and x 2 E*((CP 1)xk) for some k. Then there exists a pol* *ynomial q 2 G0((CP 1)xk) such that (q; x) ~ (ff; x) in colimE*(Z). C1 G So in particular when defining operations it is only necessary to check Diagr* *am (1) on polynomials in G0(Z0). 3. The main theorem The Grothendieck construction is explained in the introduction. Consider Gr(F* *GL) where FGL: N PRing ! Cat is the functor taking a non-positively graded ring G* to the* * category F GL(G*)op. Explicity we have Objects: Pairs (G*; +G ) with G* 2 N PRing and +G a formal group law over G* Morphisms: Pairs (; OE): (G*; +G ) ! (E*; +E ) where :G* ! E* is a degree * *zero ring homomorphism and OE(x) 2 E*[[x]] satisfying OE(x +E y) = OE(x) +G OE(y). Unravelling the definition of composition, paying careful attention to compos* *ition in F GL(G*)op we have (0; OE0) O (; OE) = (0O ; (0OE) O OE0) We introduce an equivalence relation on morphisms by (2) (; OE) ~ (0; OE0) iff(u)OE(x) = 0(u)OE0(x) and = 0on G0 UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS * * 5 where u 2 G-2 is the periodicity unit. One can check that ~ respects compositio* *n (see Lemma 3.7 below) allowing us to make the definition we require. Definition 3.1.FG = Gr(FGL)= ~ Here now is the statement of the main theorem for two periodic theories, the * *proof of which is a corollary to the general statement given in Theorem 3.12. Theorem 3.2. Let E and G be multiplicative two periodic complex oriented cohom* *ology theories with formal group laws FE and FG respectively. Suppose that o E* is in even degrees and is a unique factorisation domain o G* is in even degrees o is a E-detecting category for 1 G with nice duality Then there is a bijection of sets HRing (1 G; 1 E) $ FG((G*0 ; FG ); (E*0 ; FE )) For the case G = E this is an equivalence of monoids. Remark 3.3. The map :G*0 ! E*0 coincides with the induced map *: ss*(1 G) ! ss*(1 E) in degree zero, but not in general in other degrees. Example 3.4. Adams operations Stable Adams operations have been investigated a* *t the price of enlarging coefficients, but Wilson [13] realised that these were poten* *tially unstable operations. To start with consider complex K-theory where we know very well wha* *t happens. According to Theorem 3.2 we have HRing (1 K; 1 K) $ FG((K*0 ; FK ); (K*0 ; FK )) $ {(; OE) | :Z[u] ! Z[u]; OE(x +K y) = OE(x) +K OE(* *y)}= ~ where x +K y = x + y + uxy. Now degree zero ring maps Z[u] ! Z[u] are in one-t* *o-one correspondence with Z and it can be shown that there are no non-trivial formal * *group law homomorphisms OE(x +K y) = OE(x) +K OE(y) unless = id. When = id there is a * *formal group law homomorphism for each integer k given by the k-series [k](x) = x+K . * *.+.Kx. When = -id we get homomorphisms -[k](x) for each k. The relation ~ identifies (-id;* * -[k](x)) with (id; [k](x)) so regarding Z as a monoid under multiplication we have an eq* *uivalence of monoids HRing (1 K; 1 K) ~=Z: This is as expected and of course the operation corresponding to k 2 Z is Adams* * k (it is enough to have additivity and k(i) = ik for i a line bundle or equivalently s* *etting y = i - 1 that k(y) = u[k](u-1y)). Now let MP be 2-periodic complex cobordism. We have Z = End(FMP ) FG((MP *0 ; FMP ); (MP *0 ; FMP )) $ HRing (1 MP; 1 MP ) which gives rise to operations which seem deserving of the name Adams operation* *s (once again each operations arises from the k-series [k](x)). Compare this with Wilson [13]* *. Similarly for the two periodic theories En we get Ando's pk (see [1]) arising from [pk](x) 2* * End(FEn). In fact the above description shows En admits all the Adams operations. In the above examples the work of Kashiwabara [7] shows that is an app* *ropriate detecting category. 6 JEAN-YVES BUTOWIEZ AND PAUL TURNER Example 3.5. Unstable genera for families Hopkins defines a genus for families* * to be a multiplicative map of spectra M ! E for M some cobordism theory [5]. Unstably o* *ne could consider multiplicative maps 1 M ! 1 E. For example let MP be 2-periodic compl* *ex cobordism, then the Todd genus may be thought of as a ring homomorphism MP 0(pt* *) ! K0(pt) and we can ask how many multiplicative operations MP 0(-) ! K0(-) are th* *ere `lifting' the Todd genus? Now HRing (1 MP; 1 K) $ FG((MP *0 ; FMP ); (K*0 ; FK )) $ {(; OE) | :MP *0 ! K*0 ; OE(x +K y) = OE(x) +MP OE(y)}= ~ Note that in degree zero : MP 0(pt) ! K0(pt) agrees with the associated operat* *ion. Thus a lift of the Todd genus is defined by (u) and the right hand side above consists* * of formal group law homomorphisms arising in the K-theory case (see Example 3.4). Thus there is* * a lift of the Todd genus for each integer k. In fact such an operation is just the composite * * k O T d where T d: MP 0(-) ! MP 0(-) Td Z ~=K0(-). Example 3.6. Chern character As a final example let HPQ be 2-periodic rational* * homology. According to Theorem 3.2 we have HRing (1 K; 1 HPQ) $ FG((F *0; FK ); (HPQ *0; FHPQ )) $ {(; OE) | :Z[u] ! Q[u]; OE(x + y) = OE(x) +K OE* *(y)}= ~ Analysing the relevant formal group law homomorphisms we see that the right han* *d side is in one-to-one correspondence with Q (here is it necessary to pay careful attent* *ion to ~). The correspondence is given by: ff 7! (i; OE) where i is the inclusion and OE(x) = * *u-1(effux- 1). The Chern Character is the operation corresponding to 1 2 Q. In order to discuss the general case our first task is to introduce a new cat* *egory of formal group laws to replace the Grothendieck construction used in the 2-periodic case* *. Firstly though we need to discuss Verschiebung operators on graded rings. For j 2 N we define* * the Ver- schiebung Vj as a functor from graded rings to graded rings as follows. Let R* * *= kRk be a graded ring and set (VjR*)kj= Rk and (VjR*)i= {0} fori 62 jZ: If j = 0, we set (V0R*)0 = kRk and (V0R*)i= {0} for i 6= 0. Evidently, we have * *V1 = id. For a homomorphism : R* ! S* we write Vj() meaning now considered as a homomorphi* *sm from VjR* to VjS*. For a formal power series F in R*[[x1; : :;:xn]] we write Vj* *F meaning F now considered in VjR*[[x1; : :;:xn]]. Similarly using infix notation for a formal * *power series x +F y in R*[[x; y]] we write x +VjF y meaning +F now considered in VjR*[[x; y]]. Now * *we can define a category V which generalises Gr(FGL). Recall that N PRing is the category of no* *n-positively graded rings. Take: Objects: Pairs (G*; +G ) with G* 2 N PRing and +G a formal group law over G* Morphisms: Triples (j; ; OE): (G*; +G ) ! (E*; +E ) where o j 2 N, o :VjG* ! E* is a degree zero ring homomorphism and UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS * * 7 * * P o OE(x) 2 E*[[x]] satisfying OE(x +E y) = OE(x) +VjG OE(y). The grading on O* *E(x) = i1 OEixi is given by |OEi| = 2j - 2i. Composition of morphism is according to the following rule. (j0; 0; OE0) O (j; ; OE) = (j0j; 0O Vj0; (0Vj0OE) O OE0) Now we wish to introduce an equivalence relation on morphisms. (j; ; OE) ~ (j0; 0; OE0) iff(a)OE(x)k = 0(a)OE0(x)k; 8a 2 G-2k; 8k and * *= 0on G0 Lemma 3.7. ~ respects composition in V. Proof.Suppose we have morphisms = (j; ; OE): +G ! +G0 and i= (ri; i; i): +G0 * *! +G00 for i = 1; 2 such that 1 ~ 2. We claim 1 O ~ 2 O . We need to check that for a* * in G-2k, 1 O (a)[1(OE) O 1]k = 2 O (a)[2(OE) O 2]k assuming that, for every b in G0-2lwe have (3) 1(b) 1(x)l= 2(b) 2(x)l We can assume that 1 6= 0; 2 6= 0, and that there exists b 2 G0-2l(l > 0) suc* *h that 1(b) 6= 0 (or 2(b) 6= 0). Then by (3) the two series 1 and 2 have the same valuation, * *and writing 1 = ff1xi+ O(xi+1) and 2 = ff2xi+ O(xi+1), we get 1(b)ff1l= 2(b)ff2land ff2 1* * = ff1 2 whenever |b| = -2l. From this follows that ff1 j 1(OE) O 1(x) = (___) 2(OE) O 2(x) ff2 * * __ and the desired equality follows. The other cases of the proof are left to the * *reader. |__| The previous Lemma allows us to define a new category VFG as the quotient of * *V by the relation ~. Definition 3.8.VFG = V= ~ So we have the same objects as before but the morphism sets are now VFG((j; ; OE); (j0; 0; OE0)) = V((j; ; OE); (j0; 0; OE0))= ~ : Before stating the main result we will need a few technical results. Let den* *ote the lexico- graphical ordering of k-tupels of integers. For a k-tupel I = (i1; : :i:k) writ* *e x_I= xi11. .x.ikk. Lemma 3.9. Let R* be a graded ring which is a unique factorisation domain. S* *uppose we have fixed a basis of irreducible homogeneous elements. Let X f(x_) = x_I+ Jx_J 6= 0 JI be a formal power series in R*[[x1; : :;:xk]] which is homogeneous and symmetri* *c and such that f(x1; : :;:xk)f(y1; : :;:yk) is invariant by any permutation of the 2k variable* *s. Then there exists a homogeneous power series ffl(x) 2 R*[[x]] such that f(x1; : :;:xk)k-1 = ffl(x1) . .f.fl(xk) 8 JEAN-YVES BUTOWIEZ AND PAUL TURNER Proof.Rewrite the equality f(x1; : :;:xk)f(y1; : :;:yk) = f(y1; x2; : :;:xk)f(x* *1; y2; : :;:yk) as follows X f(x_)(y_I+ Jy_J) = JI X X ((y1; x2; : :;:xk)I + J(y1; x2; : :;:xk)J)((x1; y2; : :;:yk)I + J(x1; * *y2; : :;:yk)J) JI JI Dividing by y1i1and then setting y1 = 0 gives X f(x_)(y2i2. .y.kik+ J(y2; : :;:yk)J) = X J(i2;:::;ik) X (x2i2. .x.kik+ J(x2; : :;:xk)J)((x1; y2; : :;:yk)I + J(x1; y2; : * *:;:yk)J) J(i2;:::;ik) JI Evidently the left hand side is divisible by y2i2. Dividing by y2i2and then s* *etting y2 = 0 we obtain X f(x_)(y3i3. .y.kik+ J(y3; : :;:yk)J) = X J(i3;:::;ik) X (x2i2. .x.kik+ J(x2; : :;:xk)J)(x1i1y3i3. .y.kik+ J(x1; y3; * *: :;:yk)J) J(i2;:::;ik) J(i1;i3;:::;ik) Continue like this with y3; y4; : :;:yk to get f(x_) = g(x2; : :;:xk)ffl1(x1) for some ffl1 2 R*[[x]] with ffl1(x) = xi1+ O(xi1+1) and some formal power seri* *es g satisfying symmetry conditions analogous to the ones stated for f. Now take g and repeat the above. Iterate until finally one has (4) f(x_)k-1 = fflk(xk)fflk-1(xk-1) . .f.fl1(x1) for series fflr(x) = xir+ O(xir+1). By symmetry we have ffl1(x1)fflr(xr) = ffl1(xr)fflr(x1) for 2 r k. Thus ff* *l1(x) = fflr(x) for * * __ 2 r k. Equation (4) implies that ffl1 is homogeneous. * * |__| Corollary 3.10. Under the hypotheses of the previous lemma there exists 2 R* a* *nd '(x) 2 R*[[x]] such that f(x1; : :;:xk) = '(x1) . .'.(xk) Moreover if f is homogeneous of degree 0 then we can assume deg(') 0. Proof.Using Lemma 3.9 there exists ffl(x) such that f(x1: :x:k)k-1 = ffl(x1) . .f.fl(xk): In Frac(R*)[[x]] define __ ffl(x) i X aj j OE(x) = ____= x + __x j>ibj with aj,_bj homogeneous and aj prime_to_bj. Set_L = lcm{bj} 2 R-2l which is wel* *l defined since OE(x) 2 R*[[x]]. By expanding OE(x1) . .O.E(xk) and looking at coefficien* *ts it can be seen that Lk divides . Set * = ___2 R Lk UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS * * 9 and __ * '(x) = LOE(x) 2 R [[x]] so f(x1: :x:k) = '(x1) . .'.(xk): Q When deg(f) = 0 and deg(') < 0 write = u pffp (ffp 2 N; u 2 (R*)x ). Then de* *g(u) > 0 and deg(u') 0. So to obtain the statement about degrees we can replace ' by u'* * and by * * __ u-k. * *|__| Corollary 3.11. In VFG if (j; ; OE) ~ (j0; 0; OE0) then (a)OE(x1) : :O:E(xk) = 0(a)OE0(x1) : :O:E0(xk) 8a 2 G-2k; 8k Proof.Suppose that OE0 6= 0 and set u(x_) = f(x_)=g(x_) where f(x_) = (a)OE(x1)* * : :O:E(xk) and g(x_) = 0(a)OE0(x1) : :O:E0(xk). Note that u(x; x; : :;:x) = 1 since (j; ; OE)* * ~ (j0; 0; OE0). Let N 2 N be large enough so that v(x_) = xN1xN2. .x.Nku(x1; : :;:xk) 2 Frac(E*)[[x* *1; : :;:xk]] and use Corollary 3.10 to write v(x_) = aeffi(x1) . .f.fi(xk). Now (xN )k = v(x; x;* * : :;:x) = aeffi(x)k so * * __ ffi(x) = xN for some and aek = 1. This gives u(x_) = 1 so f(x_) = g(x_). * * |__| We can now state the main theorem in its full generality. Theorem 3.12. Let E and G be multiplicative complex oriented cohomology theor* *ies with formal group laws FE and FG respectively. Suppose that o E* is in even degrees and is a unique factorisation domain. o G* is in even degrees. o is a E-detecting category for 1 G with nice duality. Then there is a bijection of sets HRing (1 G; 1 E) $ VFG((G*0 ; FG ); (E*0 ; FE )) For the case G = E this is an equivalence of monoids. Proof.We will construct a function o :HRing (1 G; 1 E) ! VFG((G*0 ; FG ); (E*0 ; FE )) and show it to be a bijection. Let :1 G ! 1 E be a multiplicative map or equi* *valently a multiplicative operation :G0(-) ! E0(-). Now either there exists a 2 G-2k * *(some k 1) such that (ax1: :x:k) 6= 0 2 E0((CP 1)xk) or no such a exists. For the * *latter case let :G* ! E* be the ring homomorphism determined by on G0 and zero on non-ze* *ro degrees and set o( ) to be the class of (1; ; 0). For the former case let a 2 G* *-2k be such that (ax1: :x:k) 6= 0. Suppose we have fixed a basis of irreducible homogeneous ele* *ments for E*. Write X (ax1: :x:k) = ax_I+ Jx_J 6= 0 JI Using Corollary 3.10 there exists a 2 E* and OEa(x) 2 E*[[x]] such that (ax1: :x:k) = aOEa(x1) . .O.Ea(xk): 10 JEAN-YVES BUTOWIEZ AND PAUL TURNER By computing (abx1. .x.k+l) 2 E0((CP 1)xk+l) for (ax1. .x.k) 6= 0 and (bx1* *. .x.l) 6= 0 we see that the power series OEa above is independant of a and we can drop the * *subscript. The same computation shows that ab = ab a+b = a + b: Since (ax1: :x:k) is homogeneous of degreee zero we can assume that OE is of* * degree 2j 0 and using the assignment a 7! a we can define a degree zero ring homomorphism * *:VjG*0 ! E*0 . Taking a 2 G-2k such that (ax1. .x.k) 6= 0 and using the naturality of O* *E with respect to the multiplication m: CP 1 x CP 1 ! CP 1 one can see that OE satisfies OE(x +E y) = OE(x) +VjG OE(y) and so is a homomorphism of formal group laws of the required type. Define o( )* * to be the class of (j; ; OE) in VFG. The only ambiguity here is if the degree of OE is ne* *gative and we have a choice of invertible elements to force a positive OE. However any two choices* * evidently give rise to the same class in VFG. To show o is injective suppose o( ) = o( 0) so (j; ; OE) ~ (j0; 0; OE0). Coro* *llary 3.11 implies (ax1: :x:k) = 0(ax1: :x:k) for all a 2 G-2k for all k. By Proposition 2.2 and* * Proposition 2.4 we then have = 0. To show o is surjective let (j; ; OE) 2 VFG((G*0 ; FG ); (E*0 ; FE )). We nee* *d to construct a multiplicative such that o( ) = (; OE). For a 2 G-2k set (k)(ax1: :x:k) = (a)OE(x1) . .O.E(xk): Prescribing naturality with respect to projections and diagonals and using G0((* *CP 1)k) = limnG0((CP n)k) we have defined (k):G0((CP 1)k) ! E0((CP 1)k). Following the d* *iscussion in Section 2 in order to define :G0(-) ! E0(-) it remains only to show that D* *iagram (1) commutes on polynomials for f = 1 x 1 x . .x.1 x fi x 1 x . .x.1 where fi is th* *e inclusion i: pt ! CP 1 or the multiplication m: CP 1x CP 1 ! CP 1. For the inclusion this* * is because the induced map is zero on elements of positive valuation. For the multiplicati* *on this follows easily from the fact that OE is a formal group law homomorphism. The operation * *we have thus defined is multiplicative by construction and additive since < ; -> annihilates* * indecomposables: indeed for (ff; x) and (ff0; x0) in the augmentation ideal we have < ; (ff; x) * (ff0;=x0)>< ; (ff x 1 + 1 x ff0; x x0)> = < (ff x 1 + 1 x ff0); x x0> = < (ff) x 1 + 1 x (ff0)); x x0> = 0 Evidently o( ) ~ (j; ; OE) and thus o is a bijection. The statement about mon* *oids is easy to check. * * __ * * |__| Remark 3.13. When G and E are two periodic this result reduces to Theorem 3.2* * since in this case FG = VFG. To show that the inclusion FG ! VFG surjects observePthat (* *j; ; OE) ~ (1; 0; OE0) where for a 2 G-2k we set 0(a) = uk(1-j)(a) and OE0(x) = uj-1OEix* *iwhere u 2 E-2 is the periodicity unit. UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS * *11 Example 3.14. It is easy to see Wilson's unstable Adams operations in MU emerg* *ing from the above theorem. We have HRing (1 MU; 1 MU) $ {(j; ; OE) | :VjMU *0 ! MU *0 ; OE(x +MU y) = OE(x) +MU OE(y)}* *= ~ For k 2 Z the triple (1; Id; [k](x)) defines an operation k. To compare this * *with Wilson's analysis observe that HRing (1 MU; 1 MUQ) $ {(j; ; OE) | :VjMU *0 ! MUQ *0 ; OE(x +MU y) = OE(x) +MU OE(y)}* *= ~ and letting (a) = kna for a 2 MU-2n consider the triple (1; ; [k](x)_k) which d* *efines the opera- tion which Wilson shows is in fact defined integrally. To see this happening he* *re observe that (1; ; [k](x)_k) ~ (1; Id; [k](x)). Example 3.15. Let us consider unstable multiplicative operations Ell0(-) ! HPQ* *0(-). We have HRing (1 Ell; 1 HPQ) $ {(j; ; OE) | :VjEll*0 ! HPQ *0 ; OE(x + y) = OE(x) +Ell OE(y)}=* * ~ When j = 1 associated to each degree zero ring map :Z[1_2][ffi; ffl] ! Q[[u]] * *is an exponential map OE satisfying OE(x + y) = OE(x) +Ell OE(y). Taking (ffi) = u2 and (ffl) = u* *4 we get have that +Ell is the (graded) L-genus and it is well known that OE(x) = u-1 tanh(ux). So* * we have a Chern character-like operation c: Ell0(-) ! HPQ0(-). So for ffix2 2 Ell0(CP 1) * * Ell*[[x]] we have c(ffix2) = tanh2(ux). Similarly for (ffi) = -u2=8 and (ffl) = 0 we get hav* *e that +Ell is the (graded) ^A-genus and OE(x) = u-1(eux=2- e-ux=2). As with the Chern charact* *er these are not in fact genuinely unstable at all. 4. Additive multiplicative idempotents As an application we study additive multiplicative idempotents 1 G ! 1 G unde* *r the conditions of the main theorem. For a discussion of idempotents in the unstabl* *e setting we refer the reader again to the work of Boardman, Johnson and Wilson [4]. We call* * an additive multiplicative idempotent :1 G ! 1 G trivial if either = Id or is of the * *form (5) 1 G ' 10G x G0 ! G0 ! 10G x G0 ' 1 G Proposition 4.1. Suppose the assumptions of Theorem 3.12 hold and further that * *G0 Q and G* is generated as a ring by the coefficients of the formal group law. Then* * there are no non-trivial additive multiplicative idempotents 1 G ! 1 G. Proof.Let :1 G ! 1 G be an additive multiplicative idempotent not of the form* * given by (5). We shall show = Id. Let (j; ; OE) ~ o( ) with OE(x) = Lxi+ O(xi+1). S* *ince 2 = we have (j; ; OE) O (j; ; OE) ~ (j; ; OE) that is (j2; O Vj; VjOE O OE) ~ (j; ; OE) 12 JEAN-YVES BUTOWIEZ AND PAUL TURNER Taking a 2 G-2k such that (a) 6= 0 we use the relation just given to establish * *that i = 1 (so OE(x) = Lx + O(x2)) and furthermore that (6) ( O (a))(L) = (a) and (7) OE O OE(x) = (L)OE(x) (we have omitted the VjPas it corrects degree yet clutters the notation). Writing OE(x) = Lx + i>1OEixi it follows from (7) that for i > 1 we have OE* *i2 ker(). Now define OE(x) * 1 OE0(x) = ____ 2 G [__][[x]] L L 0(a) = (a)Lk fora 2 G-2k Using (6) it can be seen that 0is idempotent. By consideration of degree (recal* *l that deg(OE) = 2j 0 and that G* is in even non-positive degrees) we see that the only two pos* *sibilities for j are 0 and 1. Case 1: j = 1. In this case we must have |L| = 0. Now since G0 Q we have = * *Id: G0 ! G0 so in particular (L) = L and we can extend 0to G*[_1_L]. Working over G*[_1_* *L] we have that OE0is a strict isomorphism from +G to +0G and G*[_1_L] is generated by L; 1_Lan* *d the coefficients of +0G . It follows that 0surjects and hence is the identity. Thus ker(0) = {0}* * and OE0(x) = x. Moreover (j; ; OE) ~ (1; 0; OE0) so = Id. Case 2: j = 0. In this case we must have |L| = -2. Since V0G* is concentrated* * in degree zero we have O (a) = (a) and (L) = 1. The argument is now very similar to the * *above. * * __ * * |__| References [1]Matthew Ando. Isogenies of formal group laws and power operations in the co* *homology theories En. Duke Journal of Mathematics, 79:423-485, 1995. [2]Shoro Araki. Multiplicative operations in BP cohomology. Osaka Journal of M* *athematics, 12:343-356, 1975. [3]J. Michael Boardman. Stable operations in generalized cohomology. In I. Jam* *es, editor, Handbook of algeb- raic topology, pages 585-686. Elsevier Science, North-Holland, 1995. [4]J.Michael Boardman, David C. Johnson, and W.Stephen Wilson. Unstable operat* *ions in generalized co- homology. In I.M James, editor, Handbook of Algebraic topology, pages 687-8* *28. Elsevier Science, North Holland, 1995. [5]Michael Hopkins. Topological modular forms, the witten genus, and the theor* *em of the cube. In Proceedings of the International Congress of Mathematicians (Zurich 1994), pages 553-56* *5. Birkhauser, 1995. [6]John R. Hunton and Paul R. Turner. The homology of spaces representing exac* *t pairs of homotopy functors. To appear in Topology. [7]Takuji Kashiwabara. Hopf rings and unstable operations. Journal of Pure and* * Applied Algebra, 94:183-193, 1994. [8]Peter S. Landweber. BP*BP and typical formal groups. Osaka Journal of Mathe* *matics, 12:357-363, 1975. [9]Haynes Miller. The elliptic character and the Witten genus. In Algebraic to* *pology (Evanston, IL, 1988), volume 96 of Contemporary Mathematics, pages 281-289. American Mathematical* * Society, 1989. [10]Douglas Ravenel. Multiplicative operations in BP*BP. Pacific Journal of Mat* *hematics, 57:539-543, 1975. [11]R. W. Thomason. Homotopy colimits in the category of small categories. Math* *ematical Proceedings of the Cambridge Philosophical Society, 85:91-109, 1979. [12]Paul Turner. Unstable BP-operations and typical formal groups. Journal of P* *ure and Applied Algebra, 110:91-100, 1996. UNSTABLE MULTIPLICATIVE COHOMOLOGY OPERATIONS * *13 [13]W. Stephen Wilson. Brown-Peterson Homology: An Introduction and Sampler, v* *olume 48 of Regional Conference Series in Mathematics. American Mathematical Society, 1982. E-mail address: jbutowiez@lemel.fr E-mail address: pt@maths.abdn.ac.uk