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