Stable Operations in Generalized Cohomology
J. Michael Boardman
January 1995
[To appear in Handbook of Algebraic Topology,
ed. I. M. James, Elsevier (Amsterdam, 1995)]
TABLE OF CONTENTS
1 Introduction 1
2 Notation and five examples 3
3 Generalized cohomology of spaces 6
4 Generalized homology and duality 13
5 Complex orientation 16
6 The categories 21
7 Algebraic objects in categories *
*28
8 What is a module? 35
9 Ecohomology of spectra 43
10 What is a stable module? 50
11 Stable comodules 56
12 What is a stable algebra? 64
13 Operations and complex orientation 71
14 Examples of ring spectra for stable operations *
* 73
15 Stable BP cohomology comodules 83
Index of symbols 87
References 89
1 Introduction
For any space X, the Steenrod algebra A of stable cohomology operations act*
*s on
the ordinary cohomology H*(X; Fp) to make it an Aalgebra. Milnor discovered [2*
*2]
that it is useful to treat H*(X; Fp) as a comodule over the dual of A, which be*
*comes
a Hopf algebra. Adams extended this program in [1, 3] to multiplicative general*
*ized
cohomology theories E*(), under appropriate hypotheses. The coefficient ring *
*E*
is now graded, and E*(X) is an E*algebra.
Our purpose is to describe the structure of the stable operations on E*() *
*in a
manner that will generalize in [9] to unstable operations. Unlike some treatmen*
*ts, we
JMB  1  23 Feb 1995
Stable cohomology operations
impose no finiteness or connectedness conditions whatever on the spaces and spe*
*ctra
involved, only a single freeness condition on E. We emphasize universal proper*
*ties
as the appropriate setting for many results. An early version of some of the id*
*eas is
presented in [8], which is limited to ordinary cohomology, MU, and BP .
For general E, the stable operations form the endomorphism ring A = E*(E; o)
of E (in our notation). For each x 2 E*(X), we have the E*module homomorphism
x*: A ! E*(X) given by x*r = rx. The key idea is (roughly) that given an E*
module M, we define SM as the set of all E*module homomorphisms A ! M; this
is to be thought of as the set of candidates for the values of all operations o*
*n a typical
element of M.
Generally, we encode the action of A on a stable module M as the function
aeM : M ! SM given by (aeM x)r = rx. There is an E*module structure on SM
(different from the obvious one) that makes aeM a homomorphism of E*modules. T*
*his
is not yet enough; composition of operations makes the functor S what is known *
*as
a comonad, and we need (M; aeM ) to be a coalgebra over this comonad. When M is
an E*algebra, so is SM, and we can similarly define stable algebras.
This work serves as more than just a pattern for the promised unstable theo*
*ry of
[9]. To compare unstable structures with the analogous stable structures, we s*
*hall
there construct suitable natural transformations; this is far easier to do when*
* both
theories are developed in the same manner. Much of the basic category theory is*
* the
same for either case; we keep it all here for convenience. Finally, we need sp*
*ecific
stable results for later use.
Outline In x2, we introduce five assorted ring spectra E, which will serve thr*
*ough
out as our examples. We review some elementary category theory and set up notat*
*ion.
In xx3, 4, we study E(co)homology in enough detail to suggest what categor*
*ies to
use. In x9, we consider (co)homology in the stable homotopy category of spectra*
*. It
is essential for us to work in the correct categories, in order to make our cat*
*egorical
machinery run smoothly; otherwise it does not run at all. We therefore take pai*
*ns in
x6 to say precisely what our categories are.
In x7, we discuss the various kinds of algebraic object, such as group, mod*
*ule, and
ring, that we need in general categories. In x8, we rework the definition of a *
*module
over a ring until we find a way that will generalize to the unstable context.
In x10, we discuss stable modules from several points of view. We introduce*
* the
comonad S, and define a stable module as an Scoalgebra. Theorem 10.16 shows th*
*at
E*(X) is (more or less) a stable module.
In x11, we make the homology E*(E; o) a coalgebra (in a sense), provided on*
*ly
that it is a free E*module. A stable module then becomes a comodule over it;
indeed, Thm. 11.13 shows that the theories of stable modules and stable comodul*
*es are
entirely equivalent. Theorem 11.14 provides a useful universal property of E*(E*
*; o).
Theorem 11.35 shows that our structure on E*(E; o) agrees with that introduced *
*by
Adams [1].
Everything mentioned so far works for spectra X, too. In x12, we take accou*
*nt of
the multiplication present on E*(X) when X is a space by making SM an E*algebra
whenever M is. This leads to the definition of a stable algebra. Again, there i*
*s an
JMB  2  23 Feb 1995
x2. Notation and five examples
equivalent comodule version.
All our examples of Ecohomology come with a complex orientation. This has
standard implications for the structure of E*(C P 1) etc., which we review in x*
*5. In
x13, we study the consequences for operations.
In x14, we present the structure on E*(E; o) in detail for each of our five*
* examples
E. We do not actually construct the operations, which are all well known. It is*
* clear
that many other examples are available.
In x15, we study the special case of BP cohomology in greater depth. For a
general introduction, see Wilson [37]. Stable BP operations are well establis*
*hed; a
short early history would include Landweber [17], Novikov [28], Quillen [30], A*
*dams
[3], Zahler [41, 42], MillerRavenelWilson [21], and more recently, Ravenel's *
*book
[31]. We review Landweber's filtration theorem, for imitation in [9].
An index of symbols is included at the end.
Acknowledgements We thank Dave Johnson and Steve Wilson for making this
paper necessary. As noted, it serves chiefly as a platform for [9]. It incorp*
*orates
several suggestions of Steve Wilson, especially the use of corepresented functo*
*rs in
x8. We also thank Nigel Ray for pointing out some useful references.
2 Notation and five examples
Our five examples of commutative ring spectra E are:
H(F p) The EilenbergMacLane spectrum, for a fixed prime p 2, which represen*
*ts
ordinary cohomology H*(; Fp) and is a ring spectrum (see e. g. Switze*
*r [34,
13.88]);
BP The BrownPeterson spectrum, for a fixed prime p 2 (which is suppress*
*ed
from the notation), a ring spectrum by Quillen [29];
MU The unitary (or complex) cobordism Thom spectrum, which is a ring spec
trum (see e. g. Switzer [34, 13.89]);
KU The complex Bott spectrum (often written K), which represents topologi*
*cal
complex Ktheory and is a ring spectrum [ibid., 13.90];
K(n) The Morava Ktheory spectrum, for a fixed prime p > 2 (again suppressed
from the notation), and any n 0. (We take p > 2 in order to ensure th*
*at
the multiplication is commutative as well as associative; see Morava [*
*26],
and especially ShimadaYagita [33, Cor. 6.7] or W"urgler [38, Thm. 2.1*
*4].
See [16] for background information.)
In particular, K(0) = H(Q ) (for any p), and K(1) is a summand of KU
theory mod p.
Indeed, all our ring spectra are understood to be commutative. Each E defines
a multiplicative cohomology theory E*(X) and homology theory E*(X), which we
discuss in xx3, 4. They have the same coefficient ring E*.
Because we deal almost exclusively in cohomology, we assign the degree n to
cohomology classes in En(X) and elements of En; this forces homology classes in
JMB  3  23 Feb 1995
Stable cohomology operations
En(X) to have degree n. Note that under this convention, elements of BP *and
MU* are given negative degrees.
For any space X, E*(X) and E*(X) are E*modules. We therefore adopt E* as o*
*ur
ground ring throughout, and all tensor products and groups Hom (M; N) are taken
over E* unless otherwise specified. Except for (co)homology, we generally follo*
*w the
practice of [25] in writing a graded group with components Mn as M rather than *
*M*.
When we do write M* (e. g. E* as above), we mean the whole graded group, not a
typical component.
All our rings and algebras are associative and are presumed to have a unit *
*element
1, which is to be preserved by homomorphisms. Dually, coalgebras are assumed to
be coassociative.
Summations are often understood as taken over all available values of the i*
*ndex.
We do not attempt to give each construct a unique symbol. For example, all *
*mul
tiplications are named OE, which we decorate as OES etc. only as needed to dist*
*inguish
different multiplications. All actions are named and all coactions are named a*
*e. To
compensate, we generally specify where each equation takes place.
Signs We follow the convention that a minus sign should be introduced whenever
two symbols of odd degree become transposed for any reason. As explained in [7*
*],
this is a purely lexical convention, which depends only on the order of appeara*
*nce
of the various symbols, not on their meanings. The principle is that consistenc*
*y will
be maintained provided one starts from equations that conform and performs only
"reasonable" manipulations on them. The main requirement is that each symbol
having a degree should appear exactly once in every term of an equation.
Category theory Our basic reference is MacLane's book [20], which also provides
most of our notation and terminology.
In any category A, the set of morphisms from X to Y is denoted A(X; Y ), or*
* occa
sionally Mor (X; Y ). If A is a graded category (always assumed additive), An(X*
*; Y )
denotes the abelian group of morphisms from X to Y of degree n. Unmarked ar
rows are intended to be the obvious morphisms. We write p1: X x Y ! X and
p2: X x Y ! Y for the projections from the product X x Y to its factors, and du*
*ally
i1: X ! X q Y and i2: Y ! X q Y for coproducts. We also write q: X ! T for the
unique morphism to a terminal object T .
We denote by I: A ! A the identity functor of A. We sometimes find it usefu*
*l to
write a natural transformation ff between functors F; F 0: A ! B as
ff: F ! F 0: A ! B :
If A and B are graded, we can have deg(ff) = m 6= 0; in this case, we require
ffY OF f = (1)m deg(f)F 0f OffX for each morphism f: X ! Y . In contrast, o*
*ur
graded functors invariably preserve degree.
If ff0: F 0! F 00is another natural transformation, we have the composite n*
*atural
transformation ff0Off: F ! F 00. There is the identity natural transformation 1*
*: F !
F . Given also G: B ! C, we denote the composite functor as GF : A ! C (never
G OF ), and define the natural transformation Gff: GF ! GF 0: A ! C by (Gff)X =
G(ffX). Similarly, given fi: G ! G0, we define fiF : GF ! G0F : A ! C by (fiF )*
*X =
JMB  4  23 Feb 1995
x2. Notation and five examples
fi(F X). We also have fiff = fiF 0OGff = G0ff OfiF : GF ! G0F 0: A ! C (or G0f*
*f OfiF
in the graded case).
We make incessant use of Yoneda's Lemma [20, III.2].
Adjoint functors It should be no surprise that we have numerous pairs of adjoi*
*nt
functors. Suppose given a functor V : B ! A (which is usually, but not necessar*
*ily,
some forgetful functor) and an object A in A.
Definition 2.1 We call an object M in B Vfree on A, with basis i: A ! V M, a
morphism in A, if for each B in B, any morphism f: A ! V B in A "extends" to
a unique morphism g: M ! B in B, called the left adjunct of f, in the sense that
V g Oi = f: A ! V B.
In the language of [20, III.1], i is a universal arrow, which induces the b*
*ijection
B(M; B) ~= A(A; V B). The free object M is unique up to canonical isomorphism,
but there is no guarantee that one exists. In the favorable case when we have a*
* free
object F A for each A in A, with basis jA: A ! V F A, there is a unique way to *
*define
F h for each morphism h in A to make j natural; then F becomes a functor and the
isomorphism
B(F A; B) ~=A(A; V B); (2:2)
is natural in both A and B. Explicitly, we recover f: A ! V B from g: F A ! B as
f = V g OjA: A ! V F A ! V B in A. (2:3)
For any B, we define fflB: F V B ! B in B as extending 1: V B ! V B. Then
ffl: F V ! I is also natural, and we may construct the left adjunct g of f as
g = fflB OF f: F A ! F V B ! B in B. (2:4)
The fact that this is inverse to eq. (2.3) is neatly expressed by the pair of i*
*dentities
(i) V ffl OjV= 1: V ! V : B ! A
(2:5)
(ii)fflF OF j= 1: F ! F : A ! B
We summarize the basic facts about adjoint functors from [20, Thm. IV.1.2].
Theorem 2.6 The following conditions on a functor V : B ! A are equivalent:
(i)V has a left adjoint F : A ! B;
(ii)V is a right adjoint to some functor F : A ! B;
(iii)There is a functor F : A ! B and an isomorphism (2.2), natural in A a*
*nd
B;
(iv)For all A in A, we can choose a Vfree object F A and a basis jA of it;
(v) There is a functor F : A ! B with natural transformations j: I ! V F a*
*nd
ffl: F V ! I that satisfy eqs. (2.5). 
In view of the symmetry in (v), or between (i) and (ii), we have the dual r*
*esult,
which we do not state. Nevertheless, we do give the dual to Defn. 2.1.
Definition 2.7 The object N in B is Vcofree on A, with cobasis p: V N ! A,
a morphism in A, if for each B in B, any f: V B ! A in A "lifts" uniquely to a
morphism g: B ! N in B, called the right adjunct of f, in the sense that p OV g*
* = f.
JMB  5  23 Feb 1995
Stable cohomology operations
3 Generalized cohomology of spaces
In this section and the next, we review multiplicative cohomology theories *
*E*()
and their associated homology theories E*() in sufficient depth to decide what*
* ob
jects our categories should contain. We also establish much of our notation.
Spaces We find we have to work mostly with unbased spaces. The most convenient
spaces are CWcomplexes. We denote by T the onepoint space. It is sometimes us*
*eful
to allow also spaces that are homotopy equivalent to CWcomplexes, so that we c*
*an
form products and loop spaces directly. A pair (X; A) of spaces is assumed to b*
*e a
CWpair (or homotopy equivalent, as a pair, to one).
Ungraded cohomology For our purposes, an ungraded cohomology theory is a
homotopyinvariant contravariant functor h() that assigns to each space X an
abelian group h(X), and satisfies the usual two axioms:
(i)h() is halfexact: If X = A [ B, where A and B are wellbehaved
subspaces (e.g. subcomplexes of a CWcomplex X), and y 2 h(A)
and z 2 h(B) agree in h(A \ B), there exists x 2 h(X) (not in
general unique) that lifts both y and z; (3.1)
(ii)h()`is strongly additive: For any topologicalQdisjoint union X =
ffXff, the inclusions Xff X induce h(X) ~= ffh(Xff).
For a space X with basepoint o 2 X, we may define the reduced cohomology
h(X; o) by the split short exact sequence
0 ! h(X; o) ! h(X) ! h(o) ! 0: (3:2)
We recover the absolute cohomology h(X) by constructing the disjoint union X+ of
X with a (new) basepoint; by (ii), h(X+ ) ~=h(X) h(o) and the inclusion X X+
induces an isomorphism
h(X+ ; o) ~=h(X) : (3:3)
For a good pair (X; A) of spaces, we may define the relative cohomology as
h(X; A) = h(X=A; o); (3:4)
and these groups behave as expected. We generalize eq. (3.2).
Lemma 3.5 If A is a retract of X, we have the split short exact sequence
0 ! h(X; A) ! h(X) ! h(A) ! 0:
If A has a basepoint o, we have also the split short exact sequence
0 ! h(X; A) ! h(X; o) ! h(A; o) ! 0: 
With no basepoints, we have to be a little careful in representing h(). Le*
*t Ho be
the homotopy category of spaces that are (homotopy equivalent to) CWcomplexes.
Theorem 3.6 Let h() be an ungraded cohomology theory as above. Then:
(a) h() is represented in Ho by an Hspace H, with a universal class 2 h(H*
*; o)
h(H) that induces an isomorphism Ho(X; H) ~=h(X) of abelian groups by f 7! h(f)
for all X;
JMB  6  23 Feb 1995
x3. Generalized cohomology of spaces
(b) For any cohomology theory k(), operations : h() ! k() correspond to
elements 2 k(H).
Proof What Brown's representation theorem [10, Thm. 2.8, Ex. 3.1] actually pro
vides is a based connected space (H0; o), which represents h(; o) on based con*
*nected
spaces (X; o) only. Then West [35] shows that h(; o) is represented on all ba*
*sed
spaces by the product space
H = h(T ) x H0; (3:7)
where we treat the group h(T ) as a discrete space. By eq. (3.3), H also repre*
*sents
h() in the unbased category Ho .
The map !: T ! H that corresponds to 0 2 h(T ) furnishes H with a (homotopi
cally welldefined) basepoint, and the exact sequence (3.2) shows that 2 h(H; *
*o).
Yoneda's Lemma represents the addition
+
Ho (X; H xH) ~=h(X) x h(X) ! h(X) ~=Ho (X; H)
by an addition map : H x H ! H which makes H an Hspace, and also gives (b).
(Lemma 7.7(a) will tell us much more about H.) 
Example: KU For finitedimensional spaces X, the ungraded cohomology theory
KU(X) is defined (e. g. Husemoller [15]) as the Grothendieck group of complex v*
*ector
bundles over X. The class of the vector bundle is denoted [], and every element
of KU(X) has the form []  [j]. The trivial nplane bundle is denoted simply n.
Addition is defined from the Whitney sum of vector bundles, [] + [j] = [ j], a*
*nd
multiplication from the tensor product, [][j] = [ j]. In particular, KU(T ) ~=*
*Z ,
as a ring.
Let (X; o) be a based connected space, still finitedimensional. Because an*
*y vector
bundle over X has a stable inverse j such that j is trivial, every element of
KU(X; o) can be written in the form []n for some nplane vector bundle , provi*
*ded
n is large enough. The bundle has a classifying map X ! BU(n) BU, which
leads to the representation Ho(X; BU) ~=KU(X; o). As in the proof of Thm. 3.6, *
*this
extends to an isomorphism Ho(X; ZxBU) ~=KU(X), valid for all finitedimensional
spaces X.
To extend KU() to all spaces as an ungraded cohomology theory, we must def*
*ine
KU(X) = Ho (X; ZxBU). It remains true that any vector bundle over X defines
an element [] 2 KU(X), but in general, not all elements of KU(X) have the form
[]  [j].
Splittings All our splittings depend on the following elementary result.
Lemma 3.8 Assume that : h() ! h() is an idempotent cohomology operation,
O = . Then the image h() also satisfies the axioms (3.1).
Proof For (i), given y 2 h(A) and z 2 h(B) that agree in h(A \ B), the half
exactness of h yields an element x 2 h(X) that lifts y and z. Because is idemp*
*otent,
x 2 h(X) also lifts y and z, to show that (i) holds.
For (ii), we need only the naturality of . Given elements xff= x0ff2 h(Xff*
*),
axiom (ii) for h provides x0 2 h(X) that lifts each x0ff. Then x = x0 2 h(X) li*
*fts
each xff, and is unique because h satisfies (ii). 
JMB  7  23 Feb 1995
Stable cohomology operations
We immediately deduce the standard tool for constructing splittings. Theor*
*em
3.6(b) allows us to write the identity operation as .
Lemma 3.9 Let be an additive idempotent operation on the ungraded cohomology
theory h(). Then:
(a)  is also idempotent;
(b) We can define ungraded cohomology theories
h0(X) = Ker[: h(X) ! h(X)] = Im[  : h(X) ! h(X)]
and
h00(X) = Ker[  : h(X) ! h(X)] = Im[: h(X) ! h(X)];
(c) We have the natural direct sum decompositon h(X) = h0(X) h00(X);
(d) If the Hspaces H0 and H00represent h0() and h00() as in Thm. 3.6(a), *
*then
H0x H00represents h(). 
For future use in [9], we extend this result to certain nonadditive idempot*
*ent
operations. To emphasize the nonadditivity, we retain the parentheses in ().
Lemma 3.10 Assume the nonadditive operation on the ungraded cohomology the
ory h() satisfies the axioms:
(i)(0) = 0;
(3.11)
(ii)(x + y  (y)) = (x) for any x; y 2 h(X).
Then:
(a) and  are idempotent;
(b) We can define the kernel cohomology theory h0() = Ker = Im () as a
subgroup of h();
(c) We can define the coimage cohomology theory h00(X) = Coim = h(X)=h0(X)
as a quotient of h(X), with projection ss: h(X) ! h00(X);
(d) We have the natural short exact sequence of ungraded cohomology theories
ss 00
0 ! h0(X) ! h(X) ! h (X) ! 0; (3:12)
_
(e) induces a nonadditive operation : h00(X) ! h(X) which splits (3.12) and
induces the bijection of sets h00(X) = Coim ~=Im [: h(X) ! h(X)];
(f) The short exact sequence (3.12) is represented by a fibration of Hspace*
*s and
Hmaps
H0 ! H ! H00
in which H ! H00admits a section (not an Hmap) and H ' H0x H00as spaces.
Remark Note the asymmetry of the situation. It is necessary to distinguish (cf*
*. [20,
VIII.3]) between the coimage of , which is a quotient group of h(X), and the im*
*age
of , which in interesting cases is only a subset of h(X), not a subgroup (other*
*wise
Lemma 3.9 would be available).
Proof For (a), we put x = (y) in (ii) to see that is idempotent. If we put x *
*= 0
instead, we see that (y  (y)) = 0, which implies that  is idempotent.
JMB  8  23 Feb 1995
x3. Generalized cohomology of spaces
For (b), we have just seen that Im() Ker . The opposite inclusion is triv*
*ial,
because if (x) = 0, we can write x = ()(x).
To see that h0(X) is a subgroup, we first note that 0 2 h0(X) by (i). Take*
* any
z 2 h0(X), which we may write as z = y  (y). Then by (ii), x + z 2 h0(X) if and
only if x 2 h0(X). Therefore by Lemma 3.8 (which did not require to be additiv*
*e),
h0() is a cohomology theory.
This_allows us to define the coimage h00(X) in (c) as an abelian group. By *
*(ii) and
(b), in (e) is well defined and provides the inverse bijection to Im h(X) ! *
*h00(X).
By Lemma 3.8, Im and hence h00() satisfy the axioms (3.1), and h00is a cohomo*
*logy
theory. Then (d) combines (b) and (c).
For_(f), we represent ss by a fibration H ! H00, which is an Hmap of Hspa*
*ces.
Then is represented by a section. It follows from the short exact sequence (*
*3.12)
that the fibre of ss represents h0. 
Graded cohomology A graded cohomology theory E*() consists of an ungraded
cohomology theory En() for each integer n, connected by natural suspension iso
morphisms
: En(X) ~=En+1(S1 x X; o x X) (3:13)
of abelian groups, much as in ConnerFloyd [12, x4]. By Lemma 3.5, there is a s*
*plit
short exact sequence
0 ! En+1(S1xX; oxX) ! En+1(S1xX) ! En+1(oxX) ! 0: (3:14)
For a based space (X; o), induces, with the help of eq. (3.4), the commutative
diagram of split exact sequences
En(X; o) _____________En(X) _______________En(o)
pp
pp  
~ pp ~= ~ 
= pp ? =  (3:15)
p? ! ?
S1xX n+1 1
En+1(X; o) _______En+1 _______; o ______E (S xo; o)
oxX
whose bottom row comes from Lemma 3.5, where the suspension of X is
S1xX S1xX OE 1
X = S1 ^ X = _______~= _______ S xo :
S1 _ X oxX
We deduce the more commonly used reduced suspension isomorphism : En(X; o) ~=
En+1(X; o). In view of eq. (3.3), we recover eq. (3.13) as a special case.
By iteration of eq. (3.13), or analogy, there are kfold suspension isomorp*
*hisms for
all k > 0
k: En(X) ~=En+k(Sk x X; o x X) : (3:16)
Theorem 3.17 Any graded cohomology theory E*() is represented in Ho by an
spectrum n 7! E_n, consisting of Hspaces E_n equipped with universal elements
n 2 En(E_n; o) En(E_n) and isomorphisms (in Ho ) of Hspaces E_n' E_n+1.
JMB  9  23 Feb 1995
Stable cohomology operations
Proof Theorem 3.6 provides the Hspaces E_nand elements n. Then as a functor of
X, the sequence (3.14) is represented by the fibration of Hspaces
1 {o}
E_n+1 ! E_Sn+1! E_n+1
(which is not to be confused with the path space fibration). In particular,
En+1(S1xX; oxX) ~=Ho (X; E_n+1); (3:18)
and eq. (3.13) is represented by the desired isomorphism E_n' E_n+1. 
Similarly, k in eq. (3.16) is represented by the iterated homotopy equivale*
*nce
E_n ' kE_n+k.
We find it more convenient to work with the left adjunct E_n ! E_n+1 of the
isomorphism. We introduce a sign, which is suggested by x9.
Definition 3.19 For each n, we define the based structure map fn: E_n ! E_n+1
by
f*nn+1 = (1)nn in En+1(E_n; o). (3:20)
Theorem 3.17 gives a 11 correspondence between cohomology classes and maps.
We suspend in both senses and compare.
Lemma 3.21 Given a based space X, suppose that the class x 2 En(X; o) corre
sponds to the based map xU : X ! E_n. Then the map fn OxU : X ! E_n ! E_n+1
corresponds to the class (1)nx 2 En+1(X; o) (see diag. (3.15)).
Proof In E*(X; o), we have (xU )*f*nn+1 = (1)n(xU )*n = (1)nx. 
Multiplicative graded cohomology The cohomology theory E*() is multiplica
tive if E*(X) is naturally a commutative graded ring (with unit element 1X and *
*the
customary signs) and eq. (3.13) is an isomorphism of E*(X)modules of degree 1,
where we use the projection p2: S1 x X ! X to make (3.14) a short exact sequence
of E*(X)modules. Explicitly, (xy) = (1)i(p*2x)y for x 2 Ei(X) and y 2 E*(X).
The coefficient ring is defined as E* = E*(T ).
The natural ring structure on E*(X) is equivalent to having natural cross p*
*roduct
pairings
x: Ek(X) x Em (Y ) ! Ek+m (X x Y )
that are biadditive, commutative, associative, and have 1T 2 E*(T ) as the unit*
*. They
may be defined in terms of the ring structure as x x y = (p*1x)(p*2y); converse*
*ly, given
x; y 2 E*(X), we recover xy = *(x x y), using the diagonal map : X ! X x X.
By means of X ~= T x X, E*(X) becomes a module over E* = E*(T ), and we
may rewrite the xproduct more usefully as
x: E*(X) E*(Y ) ! E*(X x Y ); (3:22)
where the tensor product is taken over E*. On the rare occasion that this is an
isomorphism, it is called the cohomology K"unneth isomorphism.
Definition 3.23 We define the canonical generator u1 2 E1(S1; o) E1(S1) as
corresponding to 1T 2 E1(S1xT; oxT ) ~=E1(S1; o), by taking X = T in eq. (3.13).
JMB  10  23 Feb 1995
x3. Generalized cohomology of spaces
Then by naturality, for any x 2 En(X) we have
x = u1 x x in En+1(S1xX; oxX). (3:24)
Similarly, kx = uk x x in eq. (3.16), where the canonical generator uk 2 Ek(Sk;*
* o)
corresponds to k1T.
Theorem 3.25 A multiplicative structure on the graded cohomology theory E*()
is represented by multiplication maps OE: E_kx E_m ! E_k^ E_m ! E_k+m and a unit
map j: T ! E_0, such that:
(a) The cross product of x 2 Ek(X) and y 2 Em (Y ) is
xxy OE
x x y: X x Y ! E_kx E_m ! E_k+m ; (3:26)
(b) The unit element of E*(X) is 1X = j Oq: X ! T ! E_0;
(c) Given v 2 Eh, the module action v: Ek() ! Ek+h() is represented by the
map
vx1 OE
v: E_k~=T x E_k! E_h x E_k! E_k+h; (3:27)
(d) The structure map E_n ! E_n+1of Defn. 3.19 is
(1)nu1^1 OE
fn: E_n = S1 ^ E_n! E_1^ E_n! E_n+1: (3:28)
Proof We take k x m 2 Ek+m (E_kxE_m ) as OE and 1T 2 E0(T ) as j; then (a) and
(b) follow by naturality. By definition, vx corresponds to v x x 2 Ek+h(T xX).
Thus by eq. (3.26), scalar multiplication by v in E*(X) is represented by eq. (*
*3.27);
equivalently, we use the identity vx = (v1)x in E*(X). By eq. (3.24), the map (*
*3.28)
takes n+1 to (1)nn and is therefore fn. 
From now on, we shall assume that E*() is multiplicative. We shall have mu*
*ch
more to say (in Cor. 7.8) about the spaces E_n, once we have the language.
Example: KU The key to making a graded cohomology theory out of KU()
is Bott periodicity, in the following form. (See AtiyahBott [6] or Husemoller*
* [15,
Ch. 10] for an elegant proof that is close to our point of view.) It gives us e*
*verything
we need to build a periodic graded cohomology theory.
Theorem 3.29 (Bott) The Hopf line bundle over C P 1~=S2 induces an isomor
phism
([]  1) x : KU(X) ~=KU(S2xX; oxX)
for any space X.
Definition 3.30 We define the graded cohomology theory KU*() as having the
representing spaces KU__2n = Z x BU and KU__2n+1= U for all integers n, so that
KU2n(X) = Ho(X; ZxBU) = KU(X) and KU2n+1(X) = Ho(X; U).
In odd degrees, we use the suspension isomorphism
KU2n+1(X) ~=KU2n+2(S1xX; oxX) ~=Ho (X; (Z xBU)) (3:31)
represented by U ' BU = (Z xBU). In even degrees, rather than specify
: KU2n(X) ~= KU2n+1(S1xX; oxX) directly, we use the double suspension iso
morphism 2: KU2n(X) ~=KU2n+2(S2xX; oxX) provided by Thm. 3.29.
JMB  11  23 Feb 1995
Stable cohomology operations
The ring structure on KU(X) makes KU*(X) multiplicative, with the help of
eq. (3.31). (The only case that presents any difficulty is
KU2m+1 (X) x KU2n+1(X) ! KU2(m+n+1)(X);
which requires another appeal to Thm. 3.29.)
The coefficient ring is clearly Z[u; u1], where we define u 2 KU2 = KU(T *
*) = Z
as the copy of 1. To keep the degrees straight, all we have to do is insert app*
*ropri
ate powers un everywhere. (It is traditional to simplify matters by setting u *
*= 1,
thus making KU*() a Z=2graded cohomology theory; however, this strategy is not
available to us, as it would allow only operations that preserve this identific*
*ation.)
For example, Thm. 3.29 provides the canonical element
u2 = u1([]  1) in KU2(S2; o) KU2(S2). (3:32)
The skeleton filtration The cohomology E*(X) is usually uncountable for infi
nite X, which makes K"unneth isomorphisms (3.22) unlikely without some kind of
completion. This suggests that it ought to be given a topology.
Given any space X (which we take as a CWcomplex), the skeleton filtration *
*of
E*(X) is defined by
F sE*(X) = Ker[E*(X) ! E*(Xs1)] = Im[E*(X; Xs1) ! E*(X)] (3:33)
for s 0, where Xn denotes the nskeleton of X, and this filtration is natural.*
* It is
a decreasing filtration by ideals,
E*(X) = F 0E*(X) F 1E*(X) F 2E*(X) : : :
Moreover, it is multiplicative,
(F sE*(X))(F tE*(X)) F s+tE*(X) (for all s,t), (3:34)
because Xs1xX [ X xXt1 contains the (s+t1)skeleton of X x X, as in [34,
Prop. 13.67].
When X is connected, with basepoint o, we recognize F 1E*(X) from the exact
sequence (3.2) as the augmentation ideal
F 1E*(X) = E*(X; o) = Ker[E*(X) ! E*(o) = E*]: (3:35)
Filtered modules We need to be somewhat more general.
Definition 3.36 Given any E*module M filtered by submodules F aM, the asso
ciated filtration topology on M has a basis consisting of the cosets x + F aM, *
*for all
x 2 M and all indices a.
For this to be a topology, we need the directedness condition, that given F*
* aM
and F bM, there exists c such that F cM F aM \ F bM.
We consider the projections M ! M=F aM. We observe that M is Hausdorff if
and only if the induced homomorphism M ! limaM=F aM is monic, and that M is
complete (in the sense that all Cauchy sequences n 7! xn 2 M converge) if and o*
*nly
if it is epic. (A Cauchy sequence is one that satisfies xm xn ! 0. However, it*
*s limit
is unique only if M is Hausdorff.)
JMB  12  23 Feb 1995
x4. Generalized homology and duality
Definition 3.37 We define the completion of the filtered module M as Mc =
lima M=F aM. The projections M ! M=F aM lift to define the completion map
M ! cM.
We shall observe in x6 that cM has a canonical filtration that makes it com*
*plete
Hausdorff.
In particular, we have the skeleton topology on E*(X). It is of course dis*
*crete
when X is finitedimensional. Since E*(X)=F sE*(X) E*(Xs1), Milnor's short
exact sequence [24, Lemma 2]
0 ! lims1Ek1(Xs) ! Ek(X) ! limsEk(Xs) ! 0 (3:38)
may be written in the form
0 ! F 1Ek(X) ! Ek(X) ! limsEk(X)=F sEk(X) ! 0; (3:39)
T sk
where F 1Ek(X) = s F E (X) and we recognize the limit term as the completion
of Ek(X). Thus the skeleton filtration is always complete, but examples show th*
*at
it need not be Hausdorff. The elements of F 1Ek(X) are called phantom classes. *
*In
this case, the completion is simply the quotient of Ek(X) by the phantom classe*
*s.
Remark The terminology is unfortunate, but standard. The word "complete" is
sometimes understood to include "Hausdorff", which would leave us with no word *
*to
describe our situation. Here, completion is really Hausdorffification.
4 Generalized homology and duality
Associated to each of our multiplicative cohomology theories E*() is a mul*
*ti
plicative homology theory E*(), whose coefficient ring E*(T ) we can identify *
*with
E*(T ) = E*. In this section, we study the relationship between them. We shall *
*see
in x9 that the situation is quite general. In line with a suggestion of Adams [*
*1], we
have two main tools: a K"unneth isomorphism, Thm. 4.2, and a universal coeffici*
*ent
isomorphism, Thm. 4.14. (With our emphasis on cohomology, we never write E* for
E* or En for En, as is often done.)
Homology too has external cross products
x: E*(X) E*(Y ) ! E*(X x Y ); (4:1)
that make E*(X) an E*module. This is more often than (3.22) an isomorphism.
Theorem 4.2 Assume that E*(X) or E*(Y ) is a free or flat E*module. Then
the pairing (4.1) induces the K"unneth isomorphism E*(X xY ) ~=E*(X) E*(Y ) in
homology.
Proof See Switzer [34, Thm. 13.75]. Assume that E*(Y ) is flat. The idea is *
*that
as X varies, (4.1) is then a natural transformation of homology theories, which*
* is an
isomorphism for X = T and therefore generally. 
This is particularly useful for E = K(n) or H(F p), for then all E*modules*
* are
free. When E*(X) is free (or flat), we can define the comultiplication
* ~=
: E*(X) ! E*(X x X)  E*(X) E*(X); (4:3)
JMB  13  23 Feb 1995
Stable cohomology operations
which, together with the counit ffl = q*: E*(X) ! E*(T ) = E* induced by q: X !*
* T ,
makes E*(X) an E*coalgebra.
The homology analogue of Milnor's exact sequence (3.38) is simply [24, Lemm*
*a 1]
En(X) = colimsEn(Xs) : (4:4)
Duality Our only real use of homology is the Kronecker pairing
<; >: E*(X) E*(X) ! E*;
which is E*bilinear in the sense that = v = (1)hi for x *
*2 Ei(X),
z 2 E*(X), and v 2 Eh. We convert it to the right adjunct form
d: E*(X) ! DE*(X) (4:5)
by defining (dx)z = . Here, DM denotes the dual module Hom *(M; E*) of any
E*module M, defined by (DM)n = Hom n(M; E*). (But we still like to write the
evaluation as <; >: DM M ! E*.) This is the correct indexing to make DM an
E*module and d a homomorphism of E*modules. It is reasonable to ask whether d
is an isomorphism. We shall give a useful answer in Thm. 4.14.
There is an obvious natural pairing iD : DM DN ! D(M N), defined by
= (1)deg(x) deg(g) in E*. (4:6)
All these structure maps fit together in the commutative diagram
dd iD
E*(X) E*(Y ) _____DE*(X) DE*(Y ) ___D(E*(X) E*(Y ))
 6
 
x Dx (4:7)
 
? 
E*(X x Y ) ________________________________DE*(XxdY )
which, algebraically, states that = : Its significance i*
*s that if
any four of the maps are isomorphisms, so is the fifth.
We need more. We need a topology on DE*(X) to match the topology on E*(X).
There is an obvious candidate. (We stress that the homology E*(X) invariably has
the discrete topology.)
Definition 4.8 Given any E*module M, we define the dualfinite filtration on
DM = Hom *(M; E*) as consisting of the submodules F LDM = Ker[DM ! DL],
where L runs through all finitely generated submodules of M. It gives rise by
Defn. 3.36 to the dualfinite topology on DM.
This filtration is obviously Hausdorff, and we see it is complete by writin*
*g DM =
limL DL, the inverse limit of discrete E*modules. It certainly makes d contin*
*uous,
because any finitely generated L E*(X) lifts to E*(Xs) for some s, by eq. (4.4*
*).
The profinite filtration The skeleton filtration is adequate for discussing sp*
*aces
of finite type (those having finite skeletons), but not all our spaces have fin*
*ite type.
We need a somewhat coarser topology that has better properties and a better cha*
*nce
of making d in (4.5) a homeomorphism.
JMB  14  23 Feb 1995
x4. Generalized homology and duality
Definition 4.9 Given a CWcomplex X, we define the profinite filtration of E**
*(X)
as consisting of all the ideals
F aE*(X) = Ker[E*(X) ! E*(Xa)] = Im[E*(X; Xa) ! E*(X)];
where Xa runs through all finite subcomplexes of X. We call the resulting filtr*
*ation
topology (see Defn. 3.36) the profinite topology.
The particular indexing set is not important and we rarely specify it. The *
*ideals
F aE*(X) do form a directed system: given F aand F b, there exists Xc such that
F c F a\ F b, namely Xc = Xa [ Xb.
This is our preferred topology on E*(X), for all spaces X. It is natural i*
*n X:
given a map f: X ! Y , f*: E*(Y ) ! E*(X) is continuous, because for each finite
Xa X, there is a finite Yb Y for which fXa Yb, so that f*(F b) F a. Indeed,
it is the coarsest natural topology that makes E*(X) discrete for all finite X.
Of course, it coincides with the skeleton topology when X has finite type. *
*However,
it has one elementary property that the skeleton topology lacks.
`
Lemma 4.10Q For any disjoint union X = ffXff, the profinite topology makes
Ek(X) ~= ffEk(Xff) a homeomorphism. 
Definition 4.11 For any space X, we define its completed Ecohomology E*(X)^
as the completion of E*(X) with respect to the profinite filtration.
A result of Adams [2, Thm. 1.8] shows that the profinite topology is always*
* com
plete, that
E*(X) ! limaE*(X)=F aE*(X) limaE*(Xa)
is surjective, which allows us to identify canonically
"
E*(X)^ = E*(X)= F aE*(X) ~=limE*(Xa) (4:12)
a a
for all spaces X. This completed cohomology is not at all new; it was discussed*
* at
some length by Adams [ibid.].
T As before, the topology on E*(X) need not be Hausdorff. The intersection
aF aE*(X) (which contains F 1E*(X)) need not vanish, and its elements are
called weakly phantom classes. In practice, one hopes there are none, so t*
*hat
E*(X)^ = E*(X).
Strong duality We note that the morphism d in eq. (4.5) remains continuous with
the profinite topology on E*(X).
Definition 4.13 We say the space X has strong duality if d: E*(X) ! DE*(X) in
(4.5) is a homeomorphism between the profinite topology on E*(X) and the dual
finite topology on DE*(X) (see Defn. 4.8).
Theorem 4.14 Assume that E*(X) is a free E*module. Then X has strong dualit*
*y,
i. e. d: E*(X) ~= DE*(X) is a homeomorphism between the profinite topology on
E*(X) and the dualfinite topology on DE*(X). In particular, E*(X) is complete
Hausdorff.
JMB  15  23 Feb 1995
Stable cohomology operations
This is best viewed as a stable result, and will be included in Thm. 9.25.
K"unneth homeomorphisms The cohomology K"unneth homomorphism (3.22) is
rarely an isomorphism, but our chances improve if we complete it. Generally, gi*
*ven
E*modules M and N filtered by submodules F aM and F bN, we filter M N by the
submodules
F a;b(M N) = Im[(F aM N) (M F bN) ! M N]
(4:15)
= Ker[M N ! (M=F aM) (N=F bN)]
where the second form follows from the right exactness of . (Often, but not alw*
*ays,
F aM N and M F bN are submodules of M N.) We construct the completed
tensor product M b N as the completion of M N with respect to this filtration.
The filtration makes xmultiplication (3.22) continuous, because given Zc *
*Z =
X x Y , the inverse image of F cE*(Z) contains F a;b(E*(X)E*(Y )), provided Zc
Xa x Yb. We may therefore complete it to
x: E*(X) b E*(Y ) ! E*(X x Y )^ (4:16)
and ask whether this is an isomorphism. Again, we need more than a bijection.
Definition 4.17 If the pairing (4.16) is a homeomorphism and E*(X xY )^ =
E*(X xY ), we call the resulting homeomorphism E*(X xY ) ~= E*(X) b E*(Y ) a
K"unneth homeomorphism. (Note that we require E*(XxY ) to be already Hausdorff.)
Similarly, iD : DM DN ! D(M N) is continuous. We therefore complete
diag. (4.7) to
dd iD
E*(X) b E*(Y ) _____DE*(X) b DE*(Y ) ___D(E*(X) E*(Y ))
 6
 
x Dx (4:18)
 
? 
E*(X xY ) _________________________________DE*(XxYd)
Theorem 4.19 Assume that E*(X) and E*(Y ) are free E*modules. Then we have
the K"unneth homeomorphism E*(X xY ) ~=E*(X) b E*(Y ) in cohomology.
Proof The hypotheses, with the help of Thms. 4.2 and 4.14, make (4.18) a diagr*
*am
of homeomorphisms. (For iD , we may appeal to Lemma 6.15(e).) 
5 Complex orientation
All five of our examples of cohomology theories E*() are equipped with a c*
*omplex
orientation. This will provide Chern classes and a good supply of spaces with *
*free
Ehomology.
The Chern class of a line bundle Denote by M() the Thom space of a vector
bundle . A complex orientation (for line bundles) assigns to each complex line *
*bundle
JMB  16  23 Feb 1995
x5. Complex orientation
over any space X a natural Thom class t() 2 E2(M()), such that for the line
bundle 1 over a point, t(1) = u2 2 E2(S2).
Remark We assume here a specific homeomorphism between S2 and the onepoint
compactification of C , as determined by some orientation convention. In some c*
*on
texts, it is useful to allow the slightly more general normalization t(1) = u2,*
* where
2 E* may be any invertible element; but then 1t() is a Thom class in the stri*
*cter
sense. We have no need here of this extra flexibility.
For our purposes, a closely related concept is more useful.
Definition 5.1 Given E, a line bundle Chern class assigns to each complex line
bundle over any space X a class x() 2 E2(X), called the (first) EChern class *
*of
, that satisfies the axioms:
(i)It is natural: Given a map f: X0 ! X and a line bundle over X, for the
induced line bundle f* over X0 we have x(f*) = f*x() in E2(X0);
(ii)It is normalized: For the Hopf line bundle over C P 1 ~=S2, we have
x() = u2 2 E2(S2), the canonical generator of E*(S2).
It is easy to see that x() = i*t() satisfies the axioms, where i: X M() de*
*notes
the inclusion of the zero section. (Conversely, Connell [11, Thms. 4.1, 4.5] sh*
*ows that
every line bundle Chern class arises in this way, from a unique complex orienta*
*tion.)
For E = KU, it is obvious from eq. (3.32) that
x() = u1([]  1) 2 KU2(X) (5:2)
is a line bundle Chern class.
Complex projective spaces Of course, Chern classes need not exist for general
E. As the Hopf line bundle over C P 1 = BU(1) is universal, it is enough to ha*
*ve
x = x() 2 E2(C P 1). We start with CP n.
Lemma 5.3 (Dold) Assume that the Hopf line bundle over C P nhas the Chern
class x = x() 2 E2(C P n), where n 0. Then:
(a) E*(C P n) = E*[x : xn+1 = 0], a truncated polynomial algebra over E*;
(b) We have the duality isomorphism d: E*(C P n) ~=DE*(C P n);
(c) E*(C P n) is the free E*module with basis {fi0; fi1; fi2; : :;:fin}, wh*
*ere fii 2
E2i(C P n) is defined as dual to xi.
Proof See Adams [3, Lemmas II.2.5, II.2.14] or Switzer [34, Props. 16.29, 16.3*
*0].
The idea is that the presence of x forces the AtiyahHirzebruch spectral sequen*
*ces for
both E*(C P n) and E*(C P n) to collapse. (There is of course no topology on E**
*(C P n)
to check.) One has to verify that xn+1 = 0 exactly. In terms of the skeleton fi*
*ltration,
x 2 F 2E*(C P n). Then by eq. (3.34), xn+1 2 F 2n+2E*(C P n) = 0. 
The result for C P 1 follows immediately, by eq. (4.4) and Thm. 4.14, and a*
*lso
clarifies exactly how nonunique a complex orientation is. Similarly named elem*
*ents
correspond under inclusion.
JMB  17  23 Feb 1995
Stable cohomology operations
Lemma 5.4 (Dold) Assume that we have the Chern class x = x() 2 E2(C P 1).
Then:
(a) E*(C P 1) = E*[[x]], the algebra of formal power series in x over E*, fi*
*ltered
by powers of the ideal (x);
(b) We have strong duality d: E*(C P 1) ~=DE*(C P 1);
(c) E*(C P 1) is the free E*module with basis {fi0; fi1; fi2; fi3; : :}:, w*
*here fii 2
E2i(C P 1) is dual to xi for i 0. 
Chern classes of a vector bundle We proceed to BU by way of CP 1 = BU(1)
BU. A useful intermediate step is the torus group T (n) = U(1)x: :x:U(1), for w*
*hich
BT (n) = BU(1) x : :x:BU(1). We have K"unneth isomorphisms
E*(BT (n)) ~=E*(C P 1) E*(C P 1) : : :E*(C P 1)
in homology by Thm. 4.2, and
E*(BT (n)) = E*[[x1; x2; : :;:xn]] ~=E*(C P 1) b : :b:E*(C P 1)(5:5)
in cohomology by Thm. 4.19, where xi= p*ix() = x(p*i).
Lemma 5.6 Assume E has a line bundle Chern class. Then:
(a) E*(BU) = E*[[c1; c2; c3; : :]:], where ci 2 E2i(BU) restricts to the ith*
* elemen
tary symmetric function of the xj 2 E*(BT (n)) for any n i, and E*(BU(n)) =
E*[[c1; c2; : :;:cn]] is the quotient of this with ci= 0 for all i > n;
(b) We have strong duality d: E*(BU) ~= DE*(BU) and d: E*(BU(n)) ~=
DE*(BU(n)), and in particular, E*(BU) and E*(BU(n)) are Hausdorff;
(c) E*(BU) = E*[fi1; fi2; fi3; : :]:, where fii is inherited from fii 2 E2i(*
*C P 1) by
C P 1 = BU(1) BU and fi0 7! 1, and E*(BU(n)) E*(BU) is the E*free submod
ule spanned by all monomials of polynomial degree n in the fii.
Proof See Adams [3, Lemma II.4.1] or Switzer [34, Thms. 16.31, 16.32]. 
From this it is immediate, as in ConnerFloyd [12, Thm. 7.6], Adams [3, Lem*
*ma
II.4.3], or Switzer [34, Thm. 16.2], that general Chern classes exist. The axi*
*oms
determine them uniquely on BT (n), and this is enough.
Theorem 5.7 Assume E has a complex orientation. Then there exist uniquely E
Chern classes ci() 2 E2i(X), for i > 0 and any complex vector bundle over any
space X, that satisfy the axioms:
(i)Naturality: ci(f*) = f*ci() 2 E2i(X0) for any vector bundle over X and
any map f: X0 ! X;
(ii)For any nplane bundle , ci() = 0 for all i > n;
(iii)For any line bundle , c1() = x();
(iv)For any vector bundles and j over X, we have the Cartan formula
k1X
ck( j) = ck() + cki()ci(j) + ck(j) in E*(X). 
i=1
JMB  18  23 Feb 1995
x5. Complex orientation
The unitary groups We study the unitary group U by means of the Bott map
b: (Z x BU) ! U, one of the structure maps of the spectrum KU. The Hopf line
bundle over CP n1defines the unbased inclusion
C P n1 CP 1 = BU(1) BU ~=1 x BU Z x BU: (5:8)
Its fibre over the point A 2 CP n1is Hom C(A; C), where we also regard A as a *
*line
in Cn.
When we apply Bott periodicity as in Thm. 3.29, we obtain the element
([]1) x [] = [() ? ]  n in KU(S2xC P n1),
where ? denotes the orthogonal complement bundle having the fibre Hom C(A? ; C)
over A 2 CP n1. The nplane bundle ()? is, by design, trivial over D2xC P n1
for any 2disk D2 S2, and its clutching function
h: S1 x CP n1! U(n) (5:9)
induces the Bott map b, restricted as in (5.8). Here, S1 C is to be regarded a*
*s the
circle group. We read off that (for suitable choices of orientation) h(z; A): C*
*n ! Cn
is the wellknown map that preserves A? and on A is multiplication by z; explic*
*itly,
on any vector Y 2 Cn, it is
h(z; A)Y = Y + (z1) X in Cn, (5:10)
where X is any unit vector in A. (From the grouptheoretic point of view, the i*
*mage
of h is the union of all the conjugates of U(1) U(n).)
In [40], Yokota used (essentially) this map h and the multiplication in U(n*
*) to
construct explicit cell decompositions of SU(n) and hence U(n), and deduce their
ordinary (co)homology. The method works equally well for E(co)homology.
Lemma 5.11 Assume that E has a line bundle Chern class. Then E*(U(n)) is a f*
*ree
E*module with a basis consisting of all the Pontryagin products fli1fli2: :f:l*
*ik, where
n > i1 > i2 > : :i:k 0, k 0 (we allow the empty product 1), fli = h*(zxfii) 2
E2i+1(U(n)) with h as in eq. (5.9), and z 2 E1(S1) is dual to u1.
Proof Because we are decomposing U(n) rather than SU(n), we use a slightly dif
ferent (and simpler) decomposition. We regard U(n) as a principal right U(n1)
bundle over S2n1, with projection map ss: U(n) ! S2n1 given by ssg = gen, whe*
*re
en = (0; 0; : :;:0; 1) 2 C n and we recognize U(n1) as the subgroup of U(n) th*
*at
fixes en. Given g 2 U(n)  U(n1), so that ssg 6= en, it is easy to solve eq. *
*(5.10),
as in [40], for a unique pair (z; A) such that h(z; A)en = ssg, which allows us*
* to write
g = h(z; A)g0 for some g0 2 U(n1). Moreover, z 6= 1 and A =2C P n2; in other
words, ss Oh identifies the top cell of S1 x CP n1with S2n1  en.
It follows that the map
hx1
S1 x CP n1x U(n1) ! U(n) x U(n1) ! U(n)
induces the isomorphism in the commutative square
E*(S1xC P n1) E*(U(n1)) ___________E*(U(n))
 
 
? ~ ?
=
E*(S1xC P n1; K) E*(U(n1)) ______E*(U(n); U(n1))
JMB  19  23 Feb 1995
Stable cohomology operations
where K = S1 x CP n2[ 1 x CP n1. From Lemma 5.3, we deduce that both vertical
arrows are split epic and obtain the decomposition
E*(U(n)) ~=E*(U(n1)) fln1E*(U(n1))
of E*(U(n)) as the direct sum (with a shift) of two copies of E*(U(n1)), as the
multiplication by fln1 is an embedding. The result now follows by induction on*
* n,
starting from U(1) = S1.
Alternatively, we apply the AtiyahHirzebruch homology spectral sequence to*
* the
map h, to deduce that the spectral sequence for E*(U(n)) collapses whenever that
for E*(C P n1) does. 
Corollary 5.12 Assume that E has a line bundle Chern class, and that E* has
no 2torsion. Then E*(U) = (fl0; fl1; fl2; : :):, an exterior algebra on the ge*
*nerators
fli= b*(zxfii), where b: (Z xBU) ! U denotes the Bott map and fii2 E2i(Z xBU)
is inherited from CP 1 by the inclusion (5.8).
Proof We let n ! 1 in the Lemma and use eq. (4.4). The homotopy commutativity
of U gives fljfli= fliflj and hence fl2i= 0. 
The formal group law Conspicuous by its absence is any formula for ci( j).
For line bundles, the universal example is p*1p*2 over CP 1xC P 1, where denot*
*es
the Hopf line bundle. In view of eq. (5.5), there must be some formula
X
x( j) = x() + x(j) + ai;jx()ix(j)j = F (x(); x(j)) (5:13)
i;j
that is valid in the universal case, and therefore generally, where
X
F (x; y) = x + y + ai;jxiyj in E*[[x; y]] (5:14)
i;j
is a welldefined formal power series with coefficients ai;j2 E2i2j+2for i > *
*0 and
j > 0. (In the common case that the series is infinite, it may be necessary to *
*interpret
eq. (5.13) in the completion E*(X)^ of E*(X).) By use of the splitting princip*
*le
(working in BT (n)) and heavy algebra, one can in principle determine formulae *
*for
ci( j) for general complex vector bundles.
The series F (x; y) is known as the formal group law of E (or more accurate*
*ly, of
its Chern class x()). It satisfies the three identities:
(i)F (x; y) = F (y; x);
(ii)F (F (x; y); z) = F (x; F (y; z)); (5.15)
(iii)F (x; 0) = x .
The first two reflect the commutativity and associativity of . The last comes f*
*rom
ffl ~= for a trivial line bundle ffl, and shows that F (x; y) has no terms of*
* the form
ai;0xi other than x.
In the case E = KU, we can write down
x( j) = x() + x(j) + ux()x(j) in KU*(X) (5:16)
directly from eq. (5.2), since x( j) = u1([][j]  1); in other words, the for*
*mal
group law for KU is F (x; y) = x + y + uxy.
JMB  20  23 Feb 1995
x6. The categories
6 The categories
In this section we introduce the major categories we need, based on the dis*
*cussion
in x3. We also fix some terminology and notation. Our basic reference for categ*
*ory
theory is MacLane [20]. The ground ring throughout is our coefficient ring E*,*
* a
commutative graded ring.
Aop denotes the dual category of any category A. It has a morphism fop: Y !*
* X
for each morphism f: X ! Y in A. If A is graded (and therefore additive), deg(f*
*op) =
deg (f) and composition in Aop is given by fop Ogop = (1)deg(f) deg(g)(g Of)op.
Setdenotes the category of sets. Cartesian products serve as products and d*
*isjoint
unions as coproducts. The onepoint set T is a terminal object, and the empty s*
*et is
an initial object.
Ho denotes the homotopy category of unbased spaces that are homotopy equiva*
*lent
to a CWcomplex. This will be our main category of spaces. Milnor proved [23,
Prop. 3] that it admits products X xY , with never any need to retopologize. Th*
*e one
point space T is a terminal object. Arbitrary disjoint unions serve as coproduc*
*ts; in
particular, any space is the disjoint union of connected spaces. We identify Ek*
*(X) =
Ho (X; E_k) according to Thm. 3.17.
Of course, any equivalent category will serve as well. We reserve the opti*
*on of
taking any specific space to be a CWcomplex, extending constructions to the re*
*st of
Ho by naturality.
Ho0denotes the homotopy category of based spaces as in Ho, where the basepo*
*int
o is assumed to be nondegenerate; all maps and homotopies are to preserve the
basepoint. Although this category is more common, we use it only rarely. Miln*
*or
proved [23, Cor. 3] that the loop space X of such a space X again lies in the c*
*ategory.
Finite cartesian products remain products, but the onepoint space T becomes a
zero object and arbitrary wedges (onepoint unions) serve as coproducts. The ex*
*act
sequence (3.2) identifies Ek(X; o) = Ho0(X; E_k).
Stabdenotes the stable homotopy category (in any of various equivalent vers*
*ions,
e. g. [3]). It is an additive category, and has the point spectrum as a zero o*
*bject.
Arbitrary wedges of spectra serve as coproducts. It is equipped with a stabiliz*
*ation
functor Ho 0! Stab, which we suppress from our notation. There is a biadditive
smash product functor ^: StabxStab ! Stab, which (up to coherent isomorphisms) *
*is
commutative and associative, has the sphere spectrum T +as a unit, and is compa*
*tible
with the smash product in Ho 0. We define the suspension X = S1 ^ X, which is
therefore compatible with : Ho0! Ho0.
Stab*denotes the graded stable homotopy category; it has the same objects as
Stab , with maps of any degree as morphisms. It is a graded additive category. *
*We
write Stabn(X; Y ) = {X; Y }n for the group of maps of degree n (in the convent*
*ions of
x2). Given a fixed choice of one of the two isomorphisms S1 ' T +in Stab* of de*
*gree
1, we define the canonical natural desuspension isomorphism
X = S1 ^ X ' T +^ X ' X (6:1)
JMB  21  23 Feb 1995
Stable cohomology operations
of degree 1 for any spectrum X. (We do not give it a symbol.) Composition with *
*it
yields isomorphisms, for any n 0:
{X; nY } ~={X; Y }n; {X; Y }n ~={nX; Y };
which express Stab* in terms of Stab and .
However, there is a difficulty with smash products. Given maps f: X ! X0 and
g: Y ! Y 0of degrees m and n, the diagram
f^Y
X ^ Y ______X0^Y
 
X^g  (1)mn X0^g
? f^Y 0 ?
X ^ Y 0______X0^Y 0
commutes only up to the indicated sign (1)mn , owing to the necessity of shuff*
*ling
suspension factors. Consequently, the graded smash product is a functor defined*
* not
on Stab*xStab *, but on a new graded category (which might be called Stab*Stab *
**)
with the biadditivity and signs built in. All we need to know is how to compose*
*: given
also f0: X0 ! X00of degree m0and g0: Y 0! Y 00, we have
0n 0 0 00 00
(g0^ f0) O(g ^ f) = (1)m (g Og) ^ (f Of): X ^ Y ! X ^ Y :(6:2)
From the topological point of view, this is the source of the principle of *
*signs
(see x2). For example, a map f: X ! Y of degree n induces, for any W and Z, the
homomorphisms of graded groups of degree n:
f*: Stab*(W; X) ! Stab*(W; Y ) given by f*g = f Og;
(6:3)
f*: Stab*(Y; Z) ! Stab*(X; Z) given by f*g = (1)n deg(g)g Of.
Ab denotes the category of abelian groups. It is the prototypical abelian c*
*ategory
and needs no review here.
Ab* denotes the graded category of graded abelian groups, graded by all int*
*egers
(positive and negative).
Mod denotes the additive category of (necessarily graded) E*modules, in w*
*hich
the morphismsQare E*moduleLhomomorphisms of degree 0. Degreewise direct prod
ucts ffMffand sums ffMffserve as products and coproducts. It is equipped wi*
*th
the biadditive functor : Mod x Mod ! Mod (taken over E*), which is associativ*
*e,
commutative, and has E* as unit (up to coherent isomorphisms).
We note that the homology functor E*(): Ho ! Mod preserves arbitrary co
products, i. e. is strongly additive.
Mod * denotes the graded category of E*modules, in which homomorphisms of
any degree are allowed. That is, Mod *(M; N) is the graded group whose component
Mod n(M; N) in degree n is the group of E*module homomorphisms f: M ! N of
degree n, with components fi: Mi ! Ni+n that satisfy fi+h(vx) = (1)nhv(fix) for
x 2 Mi and v 2 Eh. The sign must be present if the algebra is to imitate the to*
*pology.
JMB  22  23 Feb 1995
x6. The categories
Moreover, Mod *(M; N) is an E*module in the obvious way, with vf defined by
(vf)x = v(fx) = f(vx) for v 2 E*. Given E*module homomorphisms g: L0 ! L
and h: M ! M0, we define Hom (g; h): Mod *(L; M) ! Mod *(L0; M0) by
Hom (g; h)f = Mod *(g; h)f = (1)deg(g)(deg(f)+deg(h))h Of Og: L0!(6M0;:*
*4)
to make it a homomorphism of E*modules. Similarly for tensor products: given
morphisms f: L ! L0and g: M ! M0, we define the morphism f g: LM ! L0M0
in Mod * by
(f g)(x y) = (1)deg(g) deg(x)fx gy :
If also f0: L0! L00and g0: M0 ! M00, composition is given, like (6.2), by
0) deg(g)0 0 00 00
(g0f0) O(gf) = (1)deg(f (g Og) (f Of): L M ! L M : (6:5)
We imitate the suspension isomorphisms (3.13) and (3.16) algebraically by i*
*ntro
ducing suspension functors into Mod and Mod *.
Definition 6.6 Given an E*module M and any integer k, we define the kfold
suspension kM of M by shifting everything up in degree by k: (kM)i is a formal
copy of Mik, consisting of the elements kx for x 2 Mik.
To make the function k: M ! kM an isomorphism of E*modules of degree k,
we must define the action of v 2 Eh on kM by
v(kx) = (1)hkk(vx) in kM. (6:7)
Further, k: M ~=kM becomes a natural isomorphism I ~=k of degree k of func
tors on Mod * if we define kf: kM ! kN by (kf)(kx) = (1)knk(fx) on a
morphism f: M ! N of any degree n. (Here, denotes both a natural isomorphism
and a functor.)
Algdenotes the categoryQof commutative E*algebras. It admits arbitrary deg*
*ree
wise cartesian products ffAffas products. The tensor product A B of algebras
serves as the coproduct of A and B, and E* is the initial object.
Categories of filtered objects The discussion in xx3, 4 strongly suggests that*
* for
cohomology, we need filtered versions of Mod , Mod *, and Alg.
FMod denotes the category of complete Hausdorff filtered E*modules and co*
*n
tinuous E*module homomorphisms of degree 0. An object M is an E*module M,
equipped with a directed system of E*submodules F aM, and hence a topology as *
*in
Defn. 3.36. (We do not require the indexing set to be the integers, or even cou*
*ntable.)
These are required to satisfy M = limaM=F aM, to make the topology complete
Hausdorff. The category remains an additive category.
The forgetful functor V : FMod ! Mod simply discards the filtration. Conv*
*ersely,
any E*module M may be treated as a discrete filtered module by taking 0 as the
only submodule F aM; this defines an inclusion Mod FMod . Generally, a filter*
*ed
module M is discrete if and only if some F aM is zero.
JMB  23  23 Feb 1995
Stable cohomology operations
We frequently encounter filtered E*modules M that are not complete Hausdor*
*ff.
We defined the completion cM = limaM=F aM of M in Defn. 3.37. The completion
map M ! cM is monic if and only if M is Hausdorff, and epic if and only if M is
complete. Each cM ! M=F aM is epic, because M ! M=F aM is.
We filter cM in the obvious way, by F acM= Ker[Mc ! M=F aM]. This filters t*
*he
completion map and induces isomorphisms M=F aM ~= cM=F acM; it is now obvious
that cM is indeed complete Hausdorff (as the terminology demands) and so an obj*
*ect
of FMod . If M happens to be already complete Hausdorff, M ! cM is an isomorphi*
*sm
in FMod . We make frequent use of the expected universal property: given an obj*
*ect
N of FMod , any continuous E*module homomorphism M ! N factors uniquely
through a morphism cM ! N in FMod . In the language of Defn. 2.1, cM is V free*
* on
M, with the completion map M ! cM as a basis.
If F aM F bM, we can write F bM=F aM = Ker[M=F aM ! M=F bM]. If we
now fix F bM and apply the left exact functor lima, we see that the completion *
*of
F bM, filtered by those F aM contained in it, is just Ker[Mc ! M=F bM] = F bcM,*
* as
expected.
None of the above facts requires the filtration to be countable.
The obvious filtration (4.15) on the tensor product M N is rarely complete,*
* even
when M and N are. We therefore complete it to define the completed tensor produ*
*ct
M b N in FMod . In view of the second form of (4.15), it may usefully be written
M b N = lim[(M=F aM) (N=F bN)]: (6:8)
a;b
This makes it clear that Mc b cN = M b N, that it does not matter whether we
complete M and N first or not. (We continue to write f g rather than f b g for
the completed morphisms, leaving it to the context to indicate that completion *
*is
assumed.)
FMod * denotes the graded category of complete Hausdorff filtered E*module*
*s, in
which continuous E*module homomorphisms of any degree are allowed.
We give the Ecohomology E*(X) of a space X the profinite topology from
Defn. 4.9, and complete it to E*(X)^ as in Defn. 4.11 if necessary; by Lemma 4.*
*10,
the functor E*()^: Hoop ! FMod takes arbitrary coproducts in Ho to products in
FMod . Thus cohomology remains strongly additive in this enriched sense.
As noted in x4, the profinite topology on Ecohomology makes cup and cross
products continuous, which suggests our other main category.
FAlgdenotes the category of complete Hausdorff commutative filtered E*alge*
*bras
A, with multiplication OE: A A ! A and unit j: E* ! A. We filter objects as in
FMod , except that the filtration is now by ideals F aA. Then OE is automatic*
*ally
continuous, and it is sometimes useful to complete it to A b A ! A. We have the
forgetful functor FAlg ! FMod .
Degreewise cartesian products serve as products, and we note that the cohom*
*ology
functor E*()^: Hoop! FAlg takes coproducts in Ho to products in FAlg. The init*
*ial
object is just E* itself. Coproducts in FAlg are less obvious.
JMB  24  23 Feb 1995
x6. The categories
Lemma 6.9 The completed tensor product A b B of algebras serves as the copro*
*duct
in the category FAlg.
Proof We first consider the uncompleted tensor product A B, made into an E*
algebra in the standard way, filtered as in (4.15) by the ideals
F a;b(A B) = Im[(F aAB) (AF bB) ! AB]:
We define continuous injections i: A ! A B and j: B ! A B by ix = x 1 and
jy = 1 y. Given continuous homomorphisms f: A ! C and g: B ! C, where C
is any object in FAlg, there is a unique homomorphism of algebras h: A B ! C
satisfying h Oi = f and h Oj = g, defined by h(x y) = (fx)(gy), thanks to the
commutativity of C. It is also continuous: given F cC C, choose F aA and F bB
such that f(F aA) F cC and g(F bB) F cC; then h(F a;b(A B)) F cC. Because
A B is rarely complete, we complete it, and the homomorphism h, to obtain the
desired unique algebra homomorphism bh: A b B ! C in FAlg. 
Although E*()^ does not in general take products in Ho to coproducts in FA*
*lg, it
does in the favorable cases when we have the K"unneth homeomorphism E*(XxY ) ~=
E*(X) b E*(Y ) as in Defn. 4.17.
The module of indecomposables If (A; OE; j; ffl) is a (completed) algebra with
counit (or augmentation) ffl: A ! E* (which is required_to be a morphism of alg*
*ebras
as in e. g. a Hopf_algebra), the augmentation ideal A = Ker ffl splits_off_as_*
*an E*
module, A ~=E* A_._One_can_define the module_of_indecomposables_QA =A =AA ,
i. e. Coker[OE:A A !A ] (or Coker[OE:A bA !A ] in the completed case). A c*
*leaner
way to write this categorically is
QA = Coker[OE  A ffl  ffl A: A A ! A] in Mod , (6:10)
__
as we_see by using the splitting of A; the homomorphism here is zero onA 1 and
1 A and 1 on E* = E* E*.
Lemma 6.11 The functor Q, defined on (completed) E*algebras with counit, pr*
*e
serves finite coproducts: Q(A B) ~= QA QB (or Q(A b B) ~= QA QB) and
QE* = 0.
Proof For C = A B (and similarly A b B) we have the direct sum decomposition
__ __ __ ____
C = (A 1) (1B ) (A B ):
____ ____ __ __ ____ __ __
Then OE(C_C ) containsA_A 1 from (A 1) (A 1), 1 B B similarly, andA B
from (A 1)__(1B_). The image is the direct sum of these, because the other six
pieces ofC C give nothing new. This allows us to read off the cokernel. 
Coalgdenotes the category of cocommutative E*coalgebras, with comultiplica*
*tion
: A ! A A and counit ffl: A ! E*.
When E*(X) is a free E*module, (4.3) and q*: E*(X) ! E* make it an object *
*in
Coalg.
Lemma 6.12 In the category Coalg:
JMB  25  23 Feb 1995
Stable cohomology operations
(a) The tensor product A B of two coalgebras is again a coalgebra (see e. g*
*. [25,
x2]), and serves as the product;
(b) E* is the terminal object;
L
(c) Arbitrary direct sums ffAffof coalgebras serve as coproducts. 
There are also the slightly more general completed coalgebras A, where A is*
* filtered
as above and we have instead : A ! A b A. If A and B are completed coalgebras,
so is A b B.
The module of primitives If (A; ; ffl; j) is a (completed) coalgebra with uni*
*t (e. g.
a Hopf algebra), where j: E* ! A is required to be a morphism of coalgebras, we*
* can
define, dually to (6.10), the module of coalgebra primitives
P A = Ker[  A j  j A: A ! A A] A (6:13)
in Mod (or FMod , with A b A in place of A A), a submodule of A. The dual of
Lemma 6.11 holds.
Lemma 6.14 The functor P , defined on (completed) coalgebras with unit, pres*
*erves
finite products: P (A B) ~=P A P B (or P (A b B) ~=P A P B) and P E* = 0. 
Dual modules We warn that the completed tensor product b does not make FMod
a closed category (as  b M admits no right adjoint). Nor do we attempt to topo*
*logize
FMod (M; N) in general.
Nevertheless, we found it useful in Defn. 4.8 to filter the dual DM = Mod **
*(M; E*)
of a discrete E*module M by the submodules F LDM = Ker[DM ! DL], where L
runs through all finitely generated submodules of M. Then DM = limLDL in FMod ,
where each DL is discrete; in particular, DM is automatically complete Hausdorf*
*f.
The dual Df: DN ! DM of any homomorphism f: M ! N is continuous, be
cause (Df)1(F LDM) = F fLDN. In the important case when M is free, we obtain
topologically equivalent filtrations by taking only those L that are (i) free o*
*f finite
rank, or (ii) free of finite rank, and a summand of M, or (iii) generated by fi*
*nite
subsets of a given basis of M.
Lemma 6.15 Let M, Mff, and N be discrete E*modules. Then:
(a) The canonical isomorphism D(M N) ~= DM DN = DM x DN is a
homeomorphism;
L Q
(b) The canonical isomorphism D( ffMff) ~= ffDMffis a homeomorphism;
(c) If f: M ! N is epic, then the dual Df: DN ! DM is a topological embeddin*
*g;
(d) The functor D takes colimits in Mod to limits in FMod ;
(e) iD : DM b DN ~=D(M N) in FMod , if M or N is a free E*module.
Proof In (a), D(M N) ! DM x DN is continuous because D is a functor. Given
a basic open set F LD(M N) D(M N), where L M N is finitely generated,
there are finitely generated submodules P M and Q N such that L P Q;
then F PDM F QDN F LD(M N) shows that we have a homeomorphism. More
generally, we get (b).
JMB  26  23 Feb 1995
x6. The categories
In (c), we can lift any finitely generated submodule L N to a finitely gen*
*erated
submodule K M such that fK = L. Then F LDN = DN \ F KDM in DM.
If C = Coker[f: M ! N], we have DC = Ker[Df: DN ! DM] as an E*module.
By (c), the topology on DC is correct, so that D sends cokernels to kernels. T*
*his,
with (b), gives (d).
In (e), we may assume M is free. Equality is obvious for M = E* and therefo*
*re,
by additivity, for M free of finite rank. By (d) and (6.8), the general case is*
* the limit
in FMod of the isomorphisms DL DN ~= D(L N) as L runs through the free
submodules of M of finite rank that are summands of M. 
The evaluation e: DL L ! E*, which we write as e(r c) = , is standa*
*rd.
The dual concept, of a homomorphism E* ! DLL for suitable L, is far less known,
even for finitedimensional vector spaces.
Lemma 6.16 Let L be a discrete free E*module. We can define the universal e*
*le
ment u = uL 2 DL b L by the property that for any r 2 DL = Mod *(L; E*), the
homomorphism
DL b: DL b L ! DL E* ~=DL
takes u to r. It induces the following isomorphisms of E*modules:
(a) Mod *(L; M) ~=DL b M for any discrete E*module M, by f 7! (DL f)u,
with inverse r x 7! [c 7! (1)deg(c) deg(x) x];
(b) FMod *(DL; N) ~=N b L for any object N of FMod , by g 7! (g L)u, with
inverse y c 7! [r 7! (1)e y], where e = deg (r) deg(c) + deg(r) deg(y)*
* +
deg (c) deg(y);
(c) FMod *(DL; E*) ~= E* L ~= L, by g $ c, where c = (g L)u and gr =
(1)deg(r) deg(c).
Remark We are not claiming to have isomorphisms in FMod . Indeed, for reasons
already mentioned, we do not even topologize FMod *(DL; N) etc. In any case, t*
*he
obvious E*module structures are the wrong ones for our applications.
Proof In terms of an E*basis {cff: ff 2 } of L, u is given by
X
u = uL = (1)deg(cff)c*ff cff2 DL b L;
ff
where c*ffdenotes the linear functional dual to cff, given by = 1 a*
*nd = 0
for fi 6= ff. In effect, (a) generalizes the definition of u, and is clearly an*
* isomorphism
when L has finite rank, with inverse as stated.
For general L, we let K run through all the free submodules of L of finite *
*rank.
The functor Mod *(; M) automatically takes the colimit L = colimK K to a limit.
On the right, the functor  b M preserves the limit DL = limK DK by (6.8).
Similarly, (b) is obvious when L has finite rank and N is discrete. For gen*
*eral L
and discrete N, any continuous homomorphism DL ! N must factor through some
DK, so that on the left, we have the colimit colimK Mod *(DK; N). On the right,*
* we
also have a colimit, N L = colimK N K (as no completion is needed). This gives
(b) for discrete N and general L. For general N, we observe that both sides pre*
*serve
the limit N = limbN=F bN, with the help of (6.8).
JMB  27  23 Feb 1995
Stable cohomology operations
In the special case (c) of (b), the defining property of u implies by natur*
*ality that
gr = for any r 2 DL and any g: DL ! E*. 
It will be convenient to rearrange the signs in (b).
P deg(y ) deg(c )
Corollary 6.17 The general element ff(1) ff yffffcff2 N b L of deg*
*ree
k corresponds to the general morphism DL ! N of degree k given by
X
r 7! (1)k deg(r) yff: 
ff
7 Algebraic objects in categories
It has been known for a long time (e. g. Lawvere [19]) how to define algebr*
*aic ob
jects in general categories. We are primarily interested in abelian group objec*
*ts and
generalizations, especially E*module and E*algebra objects, where E* is a fix*
*ed com
mutative graded ring. We review the material on categories we need from MacLane*
*'s
book [20, Chs. VI, VII].
Group objects Let C be any category having a terminal object T and (enough)
finite products. (Recall that T is the empty product.)
A group object in C is an object G equipped with a multiplication morphism
: G x G ! G, a unit morphism !: T ! G, and an inversion morphism : G ! G,
that satisfy the usual axioms, expressed as wellknown commutative diagrams (wh*
*ich
may be viewed in [32, x1]). Then for any object X, C(X; G) becomes a group (as *
*we
see generally in Lemma 7.7), whose unit element is ! Oq: X ! T ! G. In the group
C(G; G), is the inverse of 1G .
An abelian group object G has commutative (another diagram); in this case,
we call the addition and ! the zero morphism. Then the group C(X; G) is abelia*
*n.
If H is another group object in C, a morphism f: G ! H is a morphism of gro*
*up
objects if it commutes with the three structure morphisms; as is standard for s*
*ets
and true generally (again by Lemma 7.7), it is enough to check . Thus we form t*
*he
category Gp(C) of all group objects in C; one important example is Gp(Ho ).
Example In the category Set, one writes the structure maps of an abelian group
object as (x; y) = x + y, !(a) = 0, and (x) = x, where T = {a}. Then the
axioms take the form (x + y) + z = x + (y + z), x + 0 = x, x + (x) = 0, and
x + y = y + x, the usual axioms for an abelian group.
Example An (abelian) group object A in Coalg is a cocommutative Hopf algebra
over E*, with (commutative) multiplication OE: A A ! A and unit j: E* ! A; the
canonical antiautomorphism O: A ! A is by [25, Defn. 8.4] the inversion . (Reca*
*ll
from Lemma 6.12(a) that A A is the product in Coalg.)
Dually, a cogroup object in C is simply a group object G in the dual catego*
*ry Cop.
That is, we use coproducts instead of products, an initial object I instead of *
*T , and
reverse all the arrows; so that G is equipped with a comultiplication G ! G q G,
counit G ! I, and inversion G ! G, satisfying the evident rules.
JMB  28  23 Feb 1995
x7. Algebraic objects in categories
Example A commutative Hopf algebra A over E* may be regarded as a cogroup
object in Alg with comultiplication : A ! A A, counit ffl: A ! E*, and invers*
*ion
O: A ! A. (As in Lemma 6.9, A A is the coproduct.)
Example In the based homotopy category Ho 0, the circle S1, and hence the susp*
*en
sion X, are wellknown cogroup objects.
In any additive category, we have abelian group objects for free.
Lemma 7.1 In a (graded) additive category C:
(a) Every object admits a unique structure as abelian group object and as ab*
*elian
cogroup object;
(b) Every morphism is a morphism of abelian (co)group objects;
(c) The (graded) abelian group structure on C(X; Y ) resulting from the group
object Y or the cogroup object X is the given one.
Proof The zero object is terminal, which forces ! = 0. The sum G G serves as
both product and coproduct. The axioms force = p1 + p2 and = 1G : G ! G,
and these choices work. The dual of an additive category is again additive. 
The product GxH of two group objects is another group object, with the obvi*
*ous
multiplication
x
: G x H x G x H ~=G x G x H x H ! G x H; (7:2)
unit ! x !: T ~=T x T ! G x H, and inversion x : G x H ! G x H. This serves
as the product in the category Gp (C). The trivial group object T , with the un*
*ique
structure morphisms, serves as the terminal object.
This allows one to define group objects in Gp (C), as follows. To say that*
* G is
an object of Gp(C) means that it is equipped with a multiplication G , unit !G *
*, and
inversion G that make it a group object in C. In diag. (7.2) we made G x G an o*
*bject
of Gp (C). Then G is a group object in Gp (C) if it is equipped also with morp*
*hisms
: G x G ! G, !: T ! G, and : G ! G in Gp (C) that satisfy the axioms. The
following useful result is well known.
Proposition 7.3 Let G be a group object in the category Gp (C). Then the two
group structures on G coincide and are abelian.
Proof Lemma 7.7 will show that it is sufficient to consider the case C = Set, *
*where
the result is a standard exercise (e. g. [20, Ex. III.6.4]). 
Module objects A graded group object M in C is a function n 7! Mn that assigns
to each integer n (positiveQor negative) an abelian group object Mn in C. (Note*
* that
the infinite product nMn and coproduct are irrelevant.)
An E*module object in a (graded) category C is a graded group object n 7! *
*Mn
that is equipped with morphisms v: Mn ! Mn+h of abelian group objects (of degree
h) for all v 2 E* and all n, where h = deg(v), subject to the axioms:
(i)(v+v0) = v + v0 in the group C(Mn; Mn+h ), for v; v02 Eh;
(ii)(vv0) = v Ov0 for all v; v02 E*; (7.4)
(iii)1 = 1: Mn ! Mn.
JMB  29  23 Feb 1995
Stable cohomology operations
It follows that the inversion = (1) = 1 in C(Mn; Mn).
In an additive category, Lemma 7.1 shows that all we need is a graded object
n 7! Mn equipped with morphisms v: Mn ! Mn+h that satisfy the axioms (7.4). If
C is graded, we often (but not always) have only a single object M, with Mn = M*
* for
all n; then the definition reduces to a graded ring homomorphism : E* ! End *C(*
*M).
In a graded category, the concept of E*module object is selfdual, thanks *
*to the
commutativity of E* (provided we watch the signs and indexing): n 7! Mn is an
E*module object in C, with v acting by v: Mn ! Mn+h , if and only if n 7! Mn
is an E*module object in Cop, with v acting by (v)op: Mn+h ! Mn in Cop. (But
we note that this observation fails in general in ungraded additive categories,*
* because
the required signs are absent.)
Algebra objects A (commutative) monoid object in C is an object G equipped
with a multiplication morphism OE: G x G ! G and a unit morphism j: T ! G that
satisfy the axioms for associativity, (commutativity,) and unit. Apart from the*
* lack
of inverses and a change in notation, this is like a group object.
A graded monoid object is a graded object n 7! Mn, equipped with multiplica*
*tions
OE: Mk xMm ! Mk+m and a unit j: T ! M0, that satisfy the axioms for associativi*
*ty
and unit. (There is a problem in defining commutativity for graded monoid objec*
*ts,
because extra structure is needed to handle the signs.)
An E*algebra object in C is an E*module object that is also a graded mono*
*id
object, with the two structures related by three commutative diagrams that inte*
*rpret
the two distributive laws and (vx)y = v(xy) = x(vy). It is commutative if yx =
xy, interpreted as another diagram. Here, the sign (1)n becomes ((1)n).
It is often useful to replace the vaction v: Mn ! Mn+h in an E*algebra ob*
*ject
by the simpler morphism jv = v Oj: T ! Mh, so that j1 = j; the diagram
jxMn vxMn
T x Mn _______M0x Mn ______Mhx Mn
Q Q  
Q Q~= OE OE
Q Q  
Qs ? v ?
Mn __________Mn+h
shows that we can recover v from jv as the composite
jvxMn h n OE n+h
v: Mn ~=T x Mn ! M x M ! M : (7:5)
Equivalently, we have interpreted the identity vx = (v1)x.
General algebraic objects Other kinds of algebraic object can be defined simi
larly, provided they are (or can be) described in terms of operations ff: Gxn(f*
*f)! G
subject to universal laws, where Gxn = G x G x : :x:G, with n factors. Frequent*
*ly,
our algebraic object lies in the dual category Cop and is the corresponding coa*
*lgebraic
object in C. Our general results extend without difficulty (except notationally*
*) to the
dual and graded variants, and we omit details.
The following observation is quite elementary but extremely useful.
JMB  30  23 Feb 1995
x7. Algebraic objects in categories
Lemma 7.6 Let G be an algebraic object in C that is equipped with operations
ff: Gxn(ff)! G, and V : C ! D be a functor.
(a) If V preserves (enough) finite powers of G, then V G is an algebraic obj*
*ect in
D of the same kind, equipped with the operations
V ff
ff: (V G)xn(ff)~=V (Gxn(ff)) ! V G;
(b) If f: G ! H is a morphism of algebraic objects in C, where H is another
algebraic object of the same kind, and V preserves (enough) powers of G and H,
then V f: V G ! V H is a morphism of algebraic objects in D;
(c) If : V ! W is a natural transformation, where W : C ! D is another funct*
*or
that preserves (enough) powers of G, then G: V G ! W G is a morphism of algebra*
*ic
objects in D. 
More precisely, V and W do not need to preserve all finite powers, only the*
* powers
of G and H that actually appear in the operations and laws (including the termi*
*nal
object T , if used).
Example As S1 is a cogroup object in Ho 0, (a) shows that the loop space X on
any based space X becomes a group object in Ho 0, and hence in Ho . If X is alr*
*eady
a group object in Ho 0, (a) provides a second group object structure on X; but *
*by
Prop. 7.3, these two group structures coincide and are abelian.
One common case where this lemma applies trivially is when V is an additive*
* func
tor between additive categories. There are other functors of interest that auto*
*mati
cally preserve products: for any object X in C, the corepresented functor C(X; *
*): C !
Set preserves products by definition, and dually, C(; X) = Cop(X; ): Cop ! Set
takes coproducts in C to products in Set. Then Lemma 7.6 gives parts (a), (b), *
*and
(c) of the following.
Lemma 7.7 Let G and H be fixed objects in the category C, and V and W be the
contravariant represented functors C(; G); C(; H): Cop ! Set (or dually, cova*
*riant
corepresented functors C(G; ); C(H; ): C ! Set).
(a) If G is a (co)algebraic object in C, then for any object X in C, V X is *
*naturally
an algebraic object in Set of the same kind;
(b) With G as in (a), then for any morphism f: X ! Y in C, V f: V Y ! V X (or
V f: V X ! V Y ) is a morphism of algebraic objects in Set;
(c) Any morphism f: G ! H of (co)algebraic objects in C induces a natural mo*
*r
phism C(X; f): V X ! W X (or C(f; X): W X ! V X) of algebraic objects in Set;
(d) Conversely, if V X has a natural algebraic structure, it is induced as i*
*n (a)
by a unique (co)algebraic structure on G of the same kind, provided the necessa*
*ry
(co)powers of G exist in C;
(e) Any natural transformation of algebraic objects V X ! W X (or W X ! V X)
in Set is induced as in (c) by a unique morphism f: G ! H of (co)algebraic obje*
*cts
in C.
JMB  31  23 Feb 1995
Stable cohomology operations
Proof In (d), we may identify C(X; G)xn with C(X; Gxn). Then by Yoneda's Lemma,
each natural transformation ff: C(; G)xn ! C(; G) is induced by a unique mor
phism, which we also call ff: Gxn ! G; the uniqueness shows that the same laws
apply, thus making G an algebraic object. Part (e) is similar. 
This allows us to clarify Thm. 3.17.
Corollary 7.8 We have the E*algebra object n 7! E_n in the category Ho ; in
particular, each E_n is an abelian group object in Ho . Moreover, each equival*
*ence
E_n ' E_n+1 is an isomorphism of group objects.
Proof We apply (d) and (e) to the cohomology functors En(): Hoop! Set, repre
sented according to Thm. 3.17 by the spaces E_n. Part (e) also gives the last a*
*ssertion;
by Prop. 7.3, the group structure on E_n+1 is well defined. 
Symmetric monoidal categories The theory presented so far is not general
enough. In order to express the multiplicative structures, we need symmetric
monoidal categories. We review the few basic facts we need from MacLane [20,
Ch. VII].
A (symmetric) monoidal category (C; ; K) is a category C equipped with a bi
functor : C x C ! C and unit object K = KC. (But if C is graded, we need a more
general kind of bifunctor that is biadditive and includes signs, with composit*
*ion as
in eq. (6.5).) It is understood (but suppressed from our notation) that the spe*
*cifica
tion includes [ibid.] coherent natural isomorphisms for associativity, (commuta*
*tivity,
with signs if C is graded) and K X ~=X ~=X K.
As examples, we have (Ab ; Z; Z), (Mod ; ; E*), (FMod ; b; E*), (Stab; ^;*
* T +),
the graded versions of all these, and the dual (Cop; ; K) of any symmetric mono*
*idal
category. The original example was (C; x; T ), for any category C that admits f*
*inite
products (including the empty product T ).
Example We define the symmetric monoidal category (SetZ; x; T ) of graded sets*
*.`For
this purpose, the graded set n 7! An is best treated as the disjoint union A = *
* nAn,
equipped with the degree function A ! Z given by deg(An) = n. The product A x B
is given the degree function deg((x; y)) = deg(x) + deg(y). The unit object is *
*the set
T consisting of one point in degree zero.
The purpose (for us) of a (symmetric) monoidal category is to extend the de*
*finition
of monoid object. A (commutative) monoid object in (C; ; K) is an object M of
C that is equipped with a multiplication morphism OE: M M ! M and a unit
morphism j: K ! M (both of degree 0 if C is graded) that satisfy the usual axio*
*ms
for associativity, (commutativity,) and left and right unit. In (Set; x; T ), t*
*his reduces
to the usual concept of (commutative) monoid; more generally, in (C; x; T ), it*
* reduces
to the concept of (commutative) monoid object as before.
A graded monoid object in (C; ; K) is a graded object n 7! Mn in C equipped
with multiplications OE: Mk Mm ! Mk+m and unit j: K ! M0 (with degree 0)
that satisfy the axioms for associativity and twosided unit. (Again, we defer*
* the
discussion of commutativity.) Morphisms of monoids are defined in the obvious w*
*ay.
JMB  32  23 Feb 1995
x7. Algebraic objects in categories
A (symmetric) monoidal functor (F; iF ; zF ): (C; ; KC) ! (D; ; KD ) between
(symmetric) monoidal categories consists of a functor F : C ! D, together with a
natural transformation iF : F X F Y ! F (X Y ) and a morphism zF : KD ! F KC
in D. Of course, iF and zF are required to respect the isomorphisms for associa*
*tivity,
(commutativity,) and unit. If M is a (commutative) monoid object in C, F M will*
* be
one in D, equipped with the obvious multiplication
iF(M;M) FOE
OE: F M F M ! F (M M) ! F M
and unit F j OzF : KD ! F KC ! F M.
We do not require iF and zF to be isomorphisms (but if they are, so much the
better). One example is the duality functor
(D; iD ; zD ): (Mod op; ; E*) ! (FMod ; b; E*)
defined by DM = Mod *(M; E*) and filtered in Defn. 4.8, where zD : E* ~=DE* is *
*ob
vious and iD was originally defined in eq. (4.6) and completed later for diag. *
*(4.18). By
Lemma 6.15(e), iD is sometimes an isomorphism. Another example is the symmetric
monoidal functor
(C(X; ); i; z): (C; x; T ) ! (Set; x; T )
used in Lemma 7.7 to map an algebraic object in C to the corresponding algebraic
object in Set; in this case, i and z are automatically isomorphisms.
Monoidal functors compose in the obvious way. Given another (symmetric)
monoidal functor (G; iG ; zG ): (D; ; KD ) ! (E; ; KE), the composite (symmetri*
*c)
monoidal functor (GF; iGF ; zGF ): (C; ; KC) ! (E; ; KE) uses the natural trans*
*for
mation
iG GiF
iGF : GF X GF Y ! G(F X F Y ) ! GF (X Y )
and morphism
zG GzF
zGF : KE ! GKD ! GF KC :
Given two (symmetric) monoidal functors
(F; iF ; zF ); (G; iG ; zG ): (C; ; KC) ! (D; ; KD );
a natural transformation : F ! G is called monoidal if there are commutative di*
*a
grams
XY
F X F Y ______GX GY KD
  
   @
i (X;Y ) i (X;Y )  @ zG
 F  G zF @
  
? (XY ) ? ? @@R
F (X Y ) ______G(X Y ) F KC ______GKCKC
Thus if X is a monoid object in C, X: F X ! GX will be a morphism of monoid
objects in D.
We adapt Lemma 7.7 to monoidal functors.
Lemma 7.9 Given a graded monoid object n 7! Cn in the (graded) monoidal cate
gory (Cop; ; K), write (F M)n = C(Cn; M) for any object M in C. Then:
JMB  33  23 Feb 1995
Stable cohomology operations
(a) We can make F a monoidal functor
(F; iF ; zF ): (C; ; K) ! (SetZ; x; T ); (7:10)
(b) If the graded monoid object n 7! Dn defines similarly the monoidal funct*
*or
G, then a morphism h: C ! D in Cop of graded monoid objects induces a monoidal
natural transformation : F ! G.
Proof Let the multiplications and unit of C be OE: Ck Cm ! Ck+m and j: K ! C0
(in Cop). We defined F M as a graded set. Given f 2 (F M)k and g 2 (F N)m , we
define iF (f; g) 2 F (M N)k+m = C(Ck+m ; M N) as the composite
OEop k m fg
Ck+m ! C C ! M N in C. (7:11)
The morphism zF : T ! (F K)0 = C(C0; K) has jop: C0 ! K as its image. In (b), we
define (M)n: (F M)n = C(Cn; M) ! C(Dn; M) = (GM)n as composition in C with
hop: Dn ! Cn. The necessary verification is routine. 
Additive symmetric monoidal categories We need a slightly more general cat
egorical structure, arranged in two layers. If the category C is both monoidal*
* and
additive, it will be appropriate to use the monoidal structure (C; ; K) to defi*
*ne mul
tiplication, but to return to the additive structure of C to define addition. *
*In this
situation, we require the bifunctor to be biadditive. Rather than strive for g*
*reat
generality, we limit attention to the cases we actually need. (We do not attemp*
*t to
define the tensor product of E*module objects.)
Because C is additive, an E*module object reduces simply to a graded object
n 7! Mn equipped with morphisms v: Mn ! Mn+h for all v 2 E* and all n (where
h = deg(v)) that satisfy the axioms (7.4). Further, we can now define commutati*
*ve
graded monoid objects n 7! Mn, including the expected sign.
Definition 7.12 A (commutative) E*algebra object in the (possibly graded) ad*
*di
tive (symmetric) monoidal category (C; ; K) is a graded object n 7! Mn equipped
with:
(i)morphisms v: Mn ! Mn+h , for all n, h, and v 2 Eh, that make it an
E*module object in C;
(ii)morphisms (OE; j) that make it a graded (commutative) monoid object;
in such a way that the diagrams commute up to the indicated sign:
OE OE
Mk Mm ________Mk+m Mk Mm ________Mk+m
 
v1 v 1v (1)kh v (7:13)
? OE ? ? OE ?
Mk+h Mm ______Mk+m+h Mk Mm+h ______Mk+m+h
In the commutative case, the two diagrams are equivalent.
Example An E*algebra object in (Ab ; Z; Z) is just an E*algebra.
JMB  34  23 Feb 1995
x8. What is a module?
We can again simplify the structure by replacing the vactions v by the sin*
*gle
morphism jv = v Oj: K ! Mh for each v 2 Eh; as in eq. (7.5), we recover v from
jv as the composite
jvMn h n OE n+h
v: Mn ~=K Mn ! M M ! M :
Lemma 7.14 Let n 7! Cn be a (commutative) E*algebra object in the (graded)
additive (symmetric) monoidal category (Cop; ; K). Then the functor (7.10) beco*
*mes
a (symmetric) monoidal functor
(F; iF ; zF ): (C; ; K) ! (Mod ; ; E*) (or (Mod *; ; E*)).
Proof For fixed L, the functor C(; L): Cop ! Ab (or Ab *) takes the E*module
object C in Cop to the E*module F L, by Lemma 7.7(a). The action of v 2 Eh on *
*F L
is the composition Mor ((v)op; L): F L ! F L with (v)op: Cn+h ! Cn (including s*
*igns
as in eq. (6.4) if C is graded). As L varies, F takes values in Mod by Lemma 7*
*.7(b);
diags. (7.13) show that iF : F L x F N ! F (LN) is E*bilinear, allowing us to *
*write
iF : F L F N ! F (LN). We define zF : E* ! F K on v 2 Eh as
(v)op jop
zF v: Ch ! C0 ! K in C, (7:15)
to make it an E*module homomorphism. 
8 What is a module?
In this section, we study the relationship between the category RMod of l*
*eft R
modules and the category Ab of abelian groups from several points of view, in o*
*rder
to abstract and generalize it to cover all our main objects of interest in a un*
*iform
manner. The central theme is the classical construction by Eilenberg and Moore *
*[13]
(or see MacLane [20, Ch. VI]) of a pair of adjoint functors by means of algebra*
*s in
categories, except that the less familiar (but equivalent) dual formulation, in*
* terms
of comonads, turns out to be appropriate.
This will serve as a pattern for our definitions. There are of course vari*
*ants for
graded categories and graded objects. GradedLcategories can be handled by repla*
*cing
the graded group A*(X; Y ) by the group nAn(X; Y ), or sometimes even the dis*
*joint
union of the sets An(X; Y ). Graded objects can be handled by working in the ca*
*tegory
AZ of graded objects n 7! Xn in A. We omit details.
The ring R is usually not commutative. Like all our rings, it is understoo*
*d to
have a multiplication OE and a unit element 1R; we define the unit homomorphism
j: Z ! R by j1 = 1R. The associativity and unit axioms on R take the form of th*
*ree
commutative diagrams in Ab:
OER
R R R ______R R Z R ______RjRR R Z ______RRRj
 
  Q ~  Q ~ 
ROE OE Q = OE Q = OE
  QQ  Q Q 
  Qs ? Qs ? (8:1)
? OE ? R R
R R __________R
(i) (ii) (iii)
JMB  35  23 Feb 1995
Stable cohomology operations
In this section (only), all tensor products and Hom groups are taken over the
integers Z.
First Answer The standard definition of a left Rmodule (e. g. [25, Defn. 1.2])
equips an abelian group M with a left action M : R M ! M in Ab. It is required
to satisfy the usual two axioms, which we express as commutative diagrams:
OEM Z M ______RjMM
R R M ______R M
 Q Q ~= 
(i) RM M (ii) Q M (8:2)
? M ? Q ?
R M __________M QQsM
Second Answer We make our First Answer more functorial by introducing the
functor T = R : Ab ! Ab . We define natural transformations OE: T T ! T and
j: I ! T on A by OEA = OER A: R R A ! R A and (jA)x = 1 x 2 R A.
The action on M is now a morphism M : T M ! M, and the axioms (8.2) take the
cleaner form jM
OEM M ______TM
T T M ______TM
@ = 
(i) TM M (ii) @ M (8:3)
? M ? @ 
T M _______M @RM?
Third Answer We have so far attempted to describe a module structure over a
ring without first properly defining a ring structure. In particular, we have n*
*ot yet
mentioned the fact that R is itself an Rmodule, as is evident by comparing axi*
*oms
(8.2) with two axioms of (8.1). The function of the other axiom (8.1)(iii) is t*
*o ensure
that R is a free module on one generator 1R: given x 2 M, there is a unique mod*
*ule
homomorphism f: R ! M that satisfies f1R = x, since fr = f(r1R) = rf1R = rx.
The three axioms on R translate into commutative diagrams of natural transf*
*or
mations in Ab:
OET T ______TTjT T ______TTTj
T T T______TT  
@ =  @ = 
(i) TOE OE(ii) @ OE (iii) @ OE (8:4)
? OE ? @@R ? @@R ?
T T _______T T T
Thus a ring structure on R is equivalent to what is known as a monad (or triple)
structure (OE; j) on the functor T . By analogy, we call OE the multiplication *
*and j the
unit of the monad T . We recognize an Rmodule as being precisely what is known*
* as
a Talgebra, namely, an object M equipped with an action morphism M : T M ! M
that satisfies the axioms (8.3).
Fourth Answer More generally, the first two axioms of (8.4) show that for any
abelian group A, the action OEA: T T A ! T A makes T A an Rmodule, which we
call F A; this defines a functor F : Ab ! RMod . We thus have the factorizat*
*ion
JMB  36  23 Feb 1995
x8. What is a module?
T = V F , where V : RMod ! Ab denotes the forgetful functor. We similarly fac*
*tor
OE = V fflF : T T = V (F V )F ! V F = T , where ffl: F V ! I is defined on *
*the R
module M as fflM = M : R M ! M; by axiom (8.3)(i), fflM lies in RMod . In th*
*is
formulation, axiom (8.4)(i) simply defines the natural transformation V fflfflF*
* : T T T =
V (F V F V )F ! V F = T , while the other two reduce to the identities (2.5) re*
*lating
j and ffl.
All this works in any category A, as an application of Thm. 2.6(v).
Theorem 8.5 (EilenbergMoore) Given a monad (T; OE; j) in A, let B be the ca*
*t
egory of Talgebras, V : B ! A the forgetful functor, and F : A ! B the functor
that assigns to each A in A the Talgebra F A = (T A; OEA). Then F is left adj*
*oint
to V , B(F A; M) ~= A(A; V M) for any M in B, and F A is Vfree on A with basis
jA: A ! T A = V F A (in the language of Defn. 2.1).
Proof We have already outlined most of the proof in the special case when A = *
*Ab
and T = R , and can apply Thm. 2.6. For further details, see EilenbergMoore
[13, Thm. 2.2] or MacLane [20, Thm. VI.2.1]. 
The image of F is known as the Kleisli category of all Vfree objects.
Fifth Answer The problem with our answers so far is that they rely heavily on
the tensor product, which really has little to do with modules. While tensor pr*
*oducts
are (as we shall see) convenient for computation, they are simply not available*
* in the
nonadditive context of [9].
We therefore replace the functor T = R  by its equivalent right adjoint H*
* =
Hom (R; ): Ab ! Ab. The right adjoint of OE: T T ! T is the comultiplication *
* : H !
HH, which is given on A as the homomorphism
A: Hom (R; A) ! Hom (R; Hom (R; A))
that sends f: R ! A to s 7! [r 7! f(rs)]. The right adjoint of j: I ! T is the *
*counit
ffl: H ! I, where fflA: Hom (R; A) ! A is simply evaluation on 1R. The axioms (*
*8.4)
dualize to
H ______HH H ______HH
H ________HH
@ =  @ = 
(i)  H (ii) @ Hffl(iii) @ fflH (8:6)
? H ? @ @R ? @@R ?
HH ______HHH H H
which state that (H; ; ffl) is what is known as a comonad in Ab.
Similarly, we replace the action M on a module M by the right adjunct coac*
*tion
aeM : M ! HM = Hom (R; M). This is given explicitly by (aeM x)r = rx, which also
shows us how to recover the action from aeM . The way to think of Hom (R; M) is*
* as
the set of all possible candidates for the Raction on a typical element of M; *
*then aeM
JMB  37  23 Feb 1995
Stable cohomology operations
selects for each x 2 M the action r 7! rx. The action axioms (8.3) become
aeM M ______HMaeM
M ________HM
  @ = 
(i) aeM  M (ii) @ fflM (8:7)
? HaeM ? @@@R ?
HM ______HHM M
which state that M is what is called a coalgebra over the comonad H. Occasional*
*ly, it
is useful to evaluate the right side of (i) on a typical r 2 R, to yield the co*
*mmutative
square
rM
M _____________________M
ae ae
 M  M (8:8)
? Hom(r*;M) ?
Hom (R; M) _____________Hom(R; M)
where rM : M ! M denotes the action of r on M and r*: R ! R denotes right
multiplication by r.
A homomorphism f: M ! N of Rmodules is now a morphism in Ab for which
we have the commutative square
aeM
M ________HM
 
f Hf (8:9)
 
? aeN ?
N ________HN
This description successfully avoids all tensor products. It too works quit*
*e gener
ally.
Theorem 8.10 Given a comonad H in A, let C be the category of Hcoalgebras,
V : C ! A the forgetful functor, and C: A ! C the functor that assigns to each *
*A in
A the Hcoalgebra HA with the coaction A: HA ! HHA. Then C is right adjoint
to V , A(V M; A) ~=C(M; CA) for all M in C, and CA = (HA; A) is Vcofree on A
with cobasis fflA: HA = V CA ! A (in the language of Defn. 2.7).
Proof This is just Thm. 8.5 in the dual category Aop. 
Sixth Answer The previous answer is certainly elegant, but we shall need an
alternate description of Rmodules that does not use and ffl. The key to achi*
*eving
this is not to take adjuncts of everything.
Given an element x 2 M, we put f = aeM x: R ! M (given by fr = rx). Then
commutativity of the square
aeR
R ________HR
 
f Hf (8:11)
 
? aeM ?
M ________HM
JMB  38  23 Feb 1995
x8. What is a module?
expresses the law (sr)x = s(rx). In other words, f: R ! M is a homomorphism of
Rmodules. The law 1Rx = x is expressed as f1R = x.
Seventh Answer The first level of abstraction in category theory is to avoid d*
*ealing
with the elements of a set. The next level is to avoid dealing with the objects*
* in a
category. We have not yet used the fact that H is a corepresented functor. Gi*
*ven
any functor F : Ab ! Ab , Yoneda's Lemma (dualized) yields a 11 correspondence
between natural transformations : H ! F and elements (R)idR 2 F R, where
R: Hom (R; R) = HR ! F R and idR 2 HR denotes the identity morphism of R.
For example, : H ! HH corresponds to aeR 2 HHR, the coaction on the Rmodule
R, and ffl: H ! I corresponds to 1R2 R = IR. We note that aeR1R = idR.
To this end, we replace the object M by the corepresented functor FM =
Hom (M; ): Ab ! Ab . (We already did this for M = R, to get FR = H.) We
replace the coaction morphism aeM : M ! HM by the equivalent natural transforma
tion aeM : FM ! FM H: Ab ! Ab; explicitly, aeM N: FM N ! FM HN is
H Hom(aeM ;1)
aeM N: Hom (M; N) ! Hom (HM; HN) ! Hom (M; HN) : (8:12)
The axioms (8.7) translate into equivalent commutative diagrams of natural tran*
*s
formations
aeM
FM ________FMH F ______aeM
M FM H
  
  @ 
(i) aeM FM (ii) @ = FM ffl (8:13)
  @ 
  @@R ?
? aeM H ?
FM H ______FM HH FM
We observe that if we take M = R, these reduce to axioms (8.6)(i) and (ii).
Eighth Answer In our applications, we do not have the luxury of starting out
with a comonad; we have to construct it. Consequently, we are not able to invo*
*ke
Thm. 8.10 directly. Instead, we generalize our Sixth Answer. We have to treat
modules and rings together.
We assume that A is a category of sets with structure in the sense that we *
*are
given a faithful forgetful functor W : A ! Set. We assume given:
(i)A functor H: A ! A;
(ii)An object R in A that corepresents H in the sense that W HM =
A(R; M), naturally in M;
(iii)An element 1R of the set W R; (8.14)
(iv)A morphism aeR: R ! HR in A, which we call the precoaction on R,
such that W aeR: W R ! W HR = A(R; R) in Set carries 1R 2 W R
to the identity morphism idR: R ! R of R.
We impose no further axioms at this point. In fact, we call any morphism aeM : *
*M !
HM a precoaction on M, and a morphism f: M ! N a morphism of precoactions
JMB  39  23 Feb 1995
Stable cohomology operations
if it makes diag. (8.9) commute. To see what it takes to make aeM a coaction,*
* we
consider the function
W aeM : W M ! W HM = A(R; M) in Set .
Definition 8.15 Given an object M of A, a coaction on M is a precoaction
aeM : M ! HM such that for any element x 2 W M, the morphism f = (W aeM )x: R !
M in A satisfies:
(i)f makes diag. (8.11) commute, i. e. is a morphism of precoactions;
(ii)W f: W R ! W M sends 1R2 W R to x 2 W M.
We do not assume yet that aeR is itself a coaction. Lemma 8.20 will show th*
*at in
the presence of suitable additional structure, this definition does agree with *
*previous
notions of what a coaction should be.
Ninth Answer We generalize our Seventh Answer to the category A as above. We
convert everything to corepresented functors. We make no claims to elegance, on*
*ly
that the machinery does what we need.
We replace an object M by the corepresented functor FM = A(M; ): A ! Set,
and a precoaction aeM : M ! HM by the equivalent natural transformation aeM : *
*FM !
FM H: A ! Set. Explicitly, aeM N: FM N ! FM HN is (cf. (8.12))
H A(aeM ;HN)
aeM N: A(M; N) ! A(HM; HN) ! A(M; HN) : (8:16)
In particular, we convert the precoaction aeR to the natural transformation ae*
*R: W H !
W HH, where aeRN: W HN ! W HHN is
H A(aeR;HN)
aeRN: A(R; N) ! A(HR; HN) ! A(R; HN) : (8:17)
Similarly, if g: M ! N is a morphism of precoactions, we obtain the natural
transformation Fg: FN ! FM and from diag. (8.9) the commutative square
aeN
FN ______FN H
 
Fg FgH (8:18)
? aeM ?
FM ______FM H
We now assume that H is equipped with natural transformations:
(i) : H ! HH such that W : W H ! W HH is the natural transfor
mation aeR of (8.17);
(8.19)
(ii)ffl: H ! I such that W fflR: W HR = A(R; R) ! W R sends idR to
1R.
We assume no further properties of and ffl. In particular, (i) implies (and *
*by
naturality is equivalent to) the statement that
W R
A(R; R) = W HR ! W HHR = A(R; HR)
takes idR to the morphism aeR.
JMB  40  23 Feb 1995
x8. What is a module?
Lemma 8.20 Assume we have a category A equipped with W , H, R, , and ffl,
satisfying the axioms (8.14) and (8.19). Then given an object M of A, a precoa*
*ction
aeM : M ! HM is a coaction in the sense of Defn. 8.15 if and only if it makes d*
*iags. (8.7)
commute.
Proof Since W is faithful, we may apply W to diags. (8.7) and work with diagra*
*ms
of sets. Thus (i) becomes
WaeM =
W M _________W HM ________A(R;oM)e
 
WaeM WHaeM A(R;aeM )
= ? W M ? = ?
A(R; M) ________WHM _______W HHM ______A(R;oHM)e
We evaluate on any x 2 W M and put f = (W aeM )x: R ! M. The upper route gives
aeM Of: R ! HM, while the lower route gives Hf OaeR: R ! HM by axiom (8.19)(i).
These agree if and only if f is a morphism of precoactions as in diag. (8.11).
For diag. (8.7)(ii) we consider
WaeM WHf
W M ______W HM ______WoHRe
QQ =  
Q Q WfflM WfflR
Qs ? Wf ?
W M ________WoRe
The element f 2 W HM = A(R; M) lifts to idR2 W HR = A(R; R), which by axiom
(8.19)(ii) maps to 1R2 W R. Thus (W f)1R = x is exactly what we need. 
As in our Seventh Answer, we convert the objects in diags. (8.7) to corepre*
*sented
functors.
Corollary 8.21 The precoaction aeM : M ! HM is a coaction (in the sense of
Defn. 8.15) if and only if the associated natural transformation aeM : FM ! FM*
* H: A !
Set makes diags. (8.13) commute. 
Now we can recover the full strength of Thm. 8.10.
Lemma 8.22 Assume that aeR: R ! HR is a coaction in the sense of Defn. 8.15,*
* and
that and ffl satisfy axioms (8.19). Then:
(a) and ffl make H a comonad in A;
(b) A precoaction aeM : M ! HM makes M an Hcoalgebra if and only if it is a
coaction in the sense of Defn. 8.15.
Proof The first two axioms of (8.6) are just axioms (8.13) for M = R, which we*
* have
by Cor. 8.21. For the third, we have to show that W fflHN OW N: W HN ! W HN
is the identity. We evaluate on g 2 W HN = A(R; N). From (8.17), (W N)g =
Hg OaeR. We consider the diagram in fig. 1, which commutes merely because ffl: *
*H ! I
is natural. We start from idR 2 A(R; R), which maps to Hg OaeR 2 A(R; HN),
1R 2 W R, idR2 A(R; R) (by axiom (8.14)(iv)), and hence to g 2 A(R; N).
JMB  41  23 Feb 1995
Stable cohomology operations
Figure 1: Diagram for the comonad H
A(R; R) ________WHR= __________WRWfflR
A(R;aeR) WHaeR WaeR
? ? ?
A(R; HR) ______WHHR= ________WHRWfflHR_____oA(R;eR)=
A(R;Hg) WHHg WHg A(R;g)
? ? ? ?
A(R; HN) ______WHHN= _______WWHNfflHN______oA(R;eN)=
Part (b) is then a restatement of Lemma 8.20. 
Change of categories Now assume A0 is a second category, equipped similarly
with W 0, H0, 0 etc. satisfying axioms (8.14) and (8.19). We assume that A and
A0 are connected by a somewhat forgetful functor V : A ! A0 such that W 0V = W .
Then given an object M of A, there is an obvious natural transformation !V : FM*
* !
FV MV : A ! Set, defined on N in A as V : A(M; N) ! A0(V M; V N).
We assume that H and H0 are related by a natural transformation : V H !
H0V : A ! A0. If aeM : M ! HM is a precoaction on M in A, we give V M the
precoaction
V aeM M 0
aeV M: V M ! V HM ! H V M in A0. (8:23)
This we convert to the commutative diagram of natural transformations
aeM
FM _________FMH
 ! H
 ?V

!V FV MV H

 FV M
? aeV MV ?
FV MV ______FVMH0V
Because W H is corepresented by R, the natural transformation W 0: W H =
W 0V H ! W 0H0V is determined by a certain morphism u: R0! V R in A0(which will
be obvious in applications); explicitly, given M in A, W 0M: W HM = W 0V HM !
W 0H0V M is
V 0 A0(u;V M) 0 0
W 0M: A(R; M) ! A (V R; V M) ! A (R ; V M) :
Lemma 8.24 Assume that u satisfies:
(i)u: R0! V R is a morphism of precoactions (this uses (8.23));
(ii)W 0u: W 0R0! W 0V R = W R sends 1R0to 1R.
Then : V H ! H0V is a natural transformation of comonads, in the sense that we
JMB  42  23 Feb 1995
x9. Ecohomology of spectra
have commutative diagrams
V
V H ______V HH
  V H
 ?H  @
  V ffl
(i)  H0V H (ii)  @
  @
 H0 ? @R
? 0V ? 0 ______ffl0V
V
H0V ______H0H0V H V
Proof We apply W 0and expand all the definitions. 
9 Ecohomology of spectra
In this section, we adapt the results and techniques of xx3, 4 to the grade*
*d stable
homotopy category Stab*of spectra. Our general reference is Adams [3]. Many res*
*ults
become simpler and most are well known, apart from the topological embellishmen*
*ts.
Cohomology Any based space (X; o) may be regarded as a spectrum, via the sta
bilization functor Ho 0! Stab. Given a spectrum E, whether X is a based space
or a spectrum, we define the reduced Ecohomology of X as E*(X; o) = {X; E}* =
Stab *(X; E), the graded group of morphisms in Stab*from X to E that has the co*
*m
ponent Ek(X; o) = {X; E}k in degree k. The universal class 2 E0(E; o) is thus *
*the
identity map of E.
The suspension isomorphism E*(X; o) ~=E*(X; o) is that induced by the canon
ical desuspension map X ' X of degree 1 in Stab* given by (6.1) (with signs as
in eq. (6.3)). Equivalently, given x 2 Ek(X; o), the class x 2 Ek+1(X; o) is t*
*he
composite of the maps X ! E and E ' E (with no sign).
This cohomology is the only kind available in the stable context. For compa*
*tibility
with the unstable notation of x3, we always write the cohomology of a spectrum *
*X,
redundantly but unambiguously, as E*(X; o).
The skeleton filtration of E*(X; o) can be defined exactly as unstably, in *
*eq. (3.33).
It is quite satisfactory for spectra of finite type (those with each skeleton f*
*inite),
which include many of our examples, but is wildly inappropriate for nonconnect*
*ive
spectra such as KU. We therefore give E*(X; o) the profinite filtration and top*
*ology,
exactly as in Defn. 4.9. If necessary, we complete it as in Defn. 4.11 to the c*
*ompleted
cohomology E*(X; o)^.
A map r: E ! E in Stab* of degree h induces the stable cohomology operation
r*: Ek(X; o) ! Ek+h(X; o). It commutes with suspension up to the sign (1)h as *
*in
fig. 2.
Spaces For a space X, it is more useful, whether or not X is based, to work wi*
*th
the absolute Ecohomology of X defined by E*(X) = E*(X+ ; o), as suggested by
eq. (3.3). The absolute theory is thereby included in the reduced theory. In pa*
*rtic
ular, the coefficient group of Ecohomology is E* = E*(T ) = E*(T +; o) = ssS*(*
*E; o).
Conversely, every graded cohomology theory on spaces has this form.
JMB  43  23 Feb 1995
Stable cohomology operations
Figure 2: Operations and suspension
r*
Ek(X; o) _________Ek+h(X;o)
 
~= (1)h ~=
? ?
r*
Ek+1(X; o) ______Ek+h+1(X;o)
Theorem 9.1 Let E*() be a graded cohomology theory on Ho in the sense of x3.
Then:
(a) There is a spectrum E, unique up to equivalence, that represents E*() as
above;
(b) Any sequence of cohomology operations rk: Ek(X) ! Ek+h(X), that are de
fined and natural for spaces X and commute with suspension up to the sign (1)h
as in fig. 2, is induced by a map of spectra r: E ! E of degree h.
Sketch proof The representing spaces E_n provided by Thm. 3.17 and the structu*
*re
maps fn: E_n ! E_n+1from Defn. 3.19 are used to construct the spectrum E for (a*
*).
In (b), Thm. 3.6(b) provides a representing map rk: E_k! E_k+h for each operati*
*on
rk. We take X = E_kin fig. 2 and evaluate on the universal class k. By Lemma 3.*
*21,
the class (1)k+h rkk corresponds to the upper route fk+h Ork in the square
rk
E_k ______E_k+h

f f 0
 k  k+h in Ho (9:2)
? rk+1 ?
E_k+1______E_k+h+1
Meanwhile, by Defn. 3.19, rk+1O fk corresponds to the class (1)krk+1k. Thus the
square commutes, and we may take the maps rk as the raw material for constructi*
*ng
the desired map of spectra r: E ! E. (However, r need not be unique.) A similar
construction gives the uniqueness in (a). Further details depend on the choice*
* of
implementation of Stab*. 
Stabilization In Thms. 3.17 and 9.1 we have two ways to represent Ecohomology,
in the categories Ho and Stab*. Thus for any space X, we may identify:
(i)The cohomology class x 2 Ek(X);
(ii)The map of spectra xS: X+ ! E, of degree k, defined by x = x*S;
(iii)The map of spaces xU : X ! E_k, defined by x = x*Uk.
We compare the two maps by taking x = k in (ii).
Definition 9.3 For each integer k, we define the stabilization map of spectra
oek: E_k! E by oe*k = k 2 Ek(E_k; o) Ek(E_k). It has degree k.
JMB  44  23 Feb 1995
x9. Ecohomology of spectra
It follows immediately that for any x 2 Ek(X), xS is the composite
x+U oek
xS: X+ ! E_+k! E_k! E in Stab*. (9:4)
If x is based, i. e. x 2 Ek(X; o), we can simplify this to
xU oek *
xS: X ! E_k ! E in Stab . (9:5)
In practice, we normally omit the suffixes S and U and write x for all three. *
*(On
occasion, this can cause some difficulty with signs, as x and xS have degree k,*
* while
xU is a map of spaces and has no degree.)
Lemma 9.6 The structure maps fk: E_k ! E_k+1and the stabilization maps oek a*
*re
related by the commutative square
fk
E_k ______E_k+1
  *
' oek+1 in Stab
? oe ?
E_k ________Ek
in which we use the canonical desuspension map (6.1).
Proof The upper route in the square corresponds to the class f*kk+1 = (1)k k 2
E*(E_k; o). If we write g: E_k ' E_k for the desuspension, the lower route cor*
*re
sponds to (oek Og)* = (1)kg*oe*k = (1)kg*k = (1)k k. 
These maps display E as the homotopy colimit in Stab* of the based spaces E*
*_n.
The relevant Milnor short exact sequence (cf. diag. (3.38)) is
0 ! limn1Ek1(E_n; o) ! Ek(E; o) ! limnEk(E_n; o) ! 0 :(9:7)
Moreover, the profinite topology makes the map from Ek(E; o) an open map and
therefore a homeomorphism whenever it is a bijection. (Take the basic open set
F aE*(E; o) defined by some finite subspectrum Ea E. This inclusion lifts (up *
*to
homotopy) to a map of spectra (of degree n) Ea ! E_n;b E_nfor some n and some
finite subcomplex E_n;bof E_n. Then the image of F aE*(E; o) contains F bE*(E_n*
*).)
The maps oen also relate the stable and unstable operations in Thm. 9.1(b).*
* Sup
pose the stable operation r of degree h is represented stably in Stab* by a map*
* of
spectra rS: E ! E of degree h, and unstably in Ho by the maps rk: E_k! E_k+h.
These maps are related by the commutative square
rU
E_k______E_k+h
  *
oek oek+h in Stab (9:8)
? rS ?
E ________E
because by the definition of oen, both routes represent the class rk 2 E*(E_k; *
*o).
Cohomologically,
oe*kr = (1)khr Ooek = (1)khrk in Ek+h(E_k; o). (9:9)
JMB  45  23 Feb 1995
Stable cohomology operations
(Without the sign, r 7! rk is not in general an E*module homomorphism.)
Ring spectra Now let E be a ring spectrum, i. e. a commutative monoid object in
the symmetric monoidal category (Stab; ^; T +), with multiplication OE: E ^ E !*
* E
and unit j: T +! E. (All our ring spectra are assumed commutative.)
Given x 2 E*(X; o) and y 2 E*(Y; o), we define their cross product x x y 2
E*(X ^ Y; o) as
x^y OE
x x y: X ^ Y ! E ^ E ! E :
These products are biadditive, commutative, associative, and have j 2 E*(T +; o*
*) as
the unit in the sense that under the isomorphism
E*(T +^ X; o) ~=E*(X; o) (9:10)
induced by X ' T +^ X, j x x corresponds to x.
The coefficient group E* = E*(T +; o) = ssS*(E; o) becomes a commutative ri*
*ng,
using xproducts and T + ' T +^ T + for multiplication; its unit element is 1T =
j 2 E0(T +; o). Then E*(X; o) becomes a left E*module if we define vx 2 E*(X; *
*o)
for v 2 E* and x 2 E*(X; o) as corresponding to v x x 2 E*(T +^ X; o) under the
isomorphism (9.10); expanded, this is
v^x OE
vx: X ' T +^ X ! E ^ E ! E :
Rearranging slightly, we see that scalar multiplication by v on E*(; o) is rep*
*resented
by the map
v^E OE *
v: E ' T +^ E ! E ^ E ! E in Stab , (9:11)
as in eq. (3.27). The map v corresponds to the class v. We apply Lemma 7.7(d).
Lemma 9.12 The actions (9.11) make the ring spectrum E an E*module object in
the graded category Stab*, which represents the E*module structure on cohomolo*
*gy
E*(; o). 
Now that xproducts are known to be E*bilinear, we can write them in the m*
*ore
familiar and useful form
x: E*(X; o) E*(Y; o) ! E*(X ^ Y; o) : (9:13)
Together with the definition z: E* = E*(T +; o), they make Ecohomology a symme*
*tric
monoidal functor
(E*(; o); x; z): (Stab*op; ^; T +) ! (Mod *; ; E*) :(9:14)
For spaces X and Y , we have X+ ^ Y + ~=(X x Y )+ , and we recover the unst*
*able
xpairing (3.22) as a special case of (9.13). The reduced diagonal map + : X+ !
(X xX)+ ~=X+ ^X+ and projection q+ : X+ ! T +make X+ a commutative monoid
object in Stabop, so that E*(X) = E*(X+ ; o) becomes a commutative monoid object
in Mod , i. e. a commutative E*algebra. We have a multiplicative graded cohomo*
*logy
theory in the sense of x3.
JMB  46  23 Feb 1995
x9. Ecohomology of spectra
The stable and unstable multiplication maps are related by the commutative *
*di
agram, similar to eq. (9.5),
OEU
E_k x E_m ______E_k+m
 =
? ?
E_k^ E_m ______E_k+m in Stab* (9:15)
oek^oem oek+m
? OES ?
E ^ E _________E
However, there is a technical difficulty in extending Thm. 9.1 to make E a ring
spectrum.
Theorem 9.16 Assume there are no weakly phantom classes in the groups E0(E; *
*o),
E0(E^E; o) and E0(E^E^E; o). Then any natural multiplicative structure that is
defined on E*(X) for all spaces X (as in x3) is induced by a unique ring spectr*
*um
structure on E.
Proof Theorem 3.25 provides a compatible family of unstable multiplica*
*tions
OEU : E_kx E_m ! E_k+m and the unit jU : T ! E_0. We immediately recover jS from
jU by taking x = 1 2 E*(T ) in eq. (9.4), but there is a problem with OES. We *
*may
regard E ^ E as the homotopy colimit in Stab*of the spaces E_n^ E_nand obtain t*
*he
Milnor short exact sequence
0 ! limn1E1(E_n^E_n; o) ! E0(E^E; o) ! limnE0(E_n^E_n; o) ! 0
analogous to (9.7). It shows that there exists a lifting OES that makes diag. *
*(9.15)
commute for all k and m, but it is not unique in general. Our hypotheses simpli*
*fy
the diagrams for E0(E; o), E0(E^E; o), and the analogue for E0(E^E^E; o) to the
limit term only, to ensure respectively that OES: (i) has jS as a unit; (ii) is*
* unique and
commutative; and (iii) is associative. 
Homology The companion homology theory to E*() is easily defined (see G. W.
Whitehead [36] or Adams [3]) in the stable context. The reduced Ehomology of a
spectrum or based space X is simply
E*(X; o) = {T +; E ^ X}* = ssS*(E ^ X; o); (9:17)
the stable homotopy of E ^ X. (We observe that ssS*(; o) is itself the homolo*
*gy
theory given by taking E = T +, but we do not wish to write it T*+(; o).) It h*
*as the
component Ek(X; o) = {T +; E ^ X}k = ssSk(E ^ X; o) in degree k. Again, we ha*
*ve
the suspension isomorphism Ek(X; o) ~=Ek+1(X; o), induced by (the inverse of) t*
*he
canonical desuspension (6.1).
For a space X, we have the absolute Ehomology
E*(X) = E*(X+ ; o) = {T +; E ^ X+ }*;
as suggested by eq. (3.3) for cohomology, and it satisfies axioms dual to (3.1)*
*. The
coefficient group is
E*(T ) = E*(T +; o) = {T +; E ^ T +}* ~={T +; E}* = E* = ssS*(E; o);
JMB  47  23 Feb 1995
Stable cohomology operations
the same as Ecohomology. (But we note that Ek(T ) ~=Ek.)
When E is a ring spectrum, it too is a symmetric monoidal functor
(E*(; o); x; z): (Stab*; ^; T +) ! (Mod *; ; E*); (9:18)
with an obvious xproduct pairing
x: E*(X; o) E*(Y; o) ! E*(X ^ Y; o); (9:19)
if we use the above identification z: E* ~=E*(T +; o). We can ask whether eq. (*
*9.19)
is an isomorphism. The following two results provide all the homology isomorphi*
*sms
we need.
Theorem 9.20 Assume that E*(X; o) or E*(Y; o) is a free or flat E*module. T*
*hen
the pairing (9.19) induces the K"unneth isomorphism E*(X ^ Y; o) ~= E*(X; o)
E*(Y; o) in homology.
Proof The proof of Thm. 4.2 works just as well for spectra. 
Lemma 9.21 For E = H(F p), K(n), MU, BP , or KU, E*(E; o) is a free E*modul*
*e.
Remark For E = KU, this is a substantial result of AdamsClarke [4, Thm. 2.1].
Proof For E = H(F p) or K(n), all E*modules are free. For E = MU or BP , the
result is well known [3]. For KU, we defer the proof until we have a good descr*
*iption
of KU*(KU; o), in x14. 
The homology version of the Milnor short exact sequence (9.7) is simply
E*(E; o) = colimnE*(E_n; o); (9:22)
analogous to eq. (4.4). More generally, from the definition (9.17),
E*(X; o) = colimaE*(Xa; o) (9:23)
for any X, where Xa runs over all finite subspectra of X.
Strong duality The Kronecker pairing <; >: E*(X; o) E*(X; o) ! E* is easily
constructed for spectra E and X, directly from the definitions. As in x4, it m*
*akes
sense to ask whether the right adjunct form
d: E*(X; o) ! DE*(X; o) (9:24)
is an isomorphism, or better, a homeomorphism. Again, one theorem is all we nee*
*d.
It includes the unstable result Thm. 4.14.
Theorem 9.25 Assume that E*(X; o) is a free E*module. Then X has strong du
ality, i. e. d in (9.24) is a homeomorphism between the profinite topology on E*
**(X; o)
and the dualfinite topology on DE*(X; o). In particular, E*(X; o) is complete *
*Haus
dorff.
Emodules To establish Thm. 9.25, we must take Emodules seriously. An E
module is a spectrum G equipped with an action map G : E ^ G ! G in Stab that
satisfies the usual two axioms (8.3), using the functor T = E ^ . Everything *
*is
formally identical to the Rmodule case, with the monoid object R in the symmet*
*ric
monoidal category (Ab ; ; Z) replaced by E in (Stab*; ^; T +). We form the cate*
*gory
EMod of Emodules, and the graded version EMod *.
JMB  48  23 Feb 1995
x9. Ecohomology of spectra
Theorem 9.26 The forgetful functor V : EMod * ! Stab*has the free functor E*
* ^
: Stab*! EMod * as a left adjoint, and for any spectrum X and Emodule G, we
have a natural homeomorphism
G*(X) = Stab*(X; V G) ~=EMod * (E ^ X; G) : (9:27)
Proof Theorem 8.5 provides the isomorphism. We make it trivially a homeomor
phism by topologizing EMod * (E ^ X; G), not as a subspace of G*(E ^ X), but by
filtering it by the submodules
F aEMod *(E^X; G) = Ker[EMod * (E^X; G) ! EMod *(E^Xa; G)];
where Xa runs through the finite subspectra of X. 
Corollary 9.28 Let g: E ^ X ! E ^ Y be an Emodule morphism (not nec
essarily of the form E ^ f). Then for any Emodule G, g*: EMod *(E^Y; G) !
EMod * (E^X; G) is continuous.
Proof The right adjunct of g is a map f: X ! E ^ Y of spectra. Given a finite
Xa X, we choose a finite Yb Y such that fXa factors through E ^ Yb; then by
taking left adjuncts, g restricts to a morphism of Emodules E ^ Xa ! E ^ Yb. *
*It
follows that g*(F b) F a, in the notation of the Theorem. 
The desired theorem follows directly, as in Adams [3, Lemma II.11.1].
Proof of Thm. 9.25 We choose a basis of E*(X; o) consisting ofWmaps Snff! E ^ X
of degree zero, and use them as the components of a map f: W = ffSnff! E ^ X.
By Thm. 9.26, the left adjunct of f is a morphism of Emodules g: E ^ W ! E ^ X.
By construction, g induces an isomorphism g*: E*(W; o) ~= E*(X; o) on homotopy
groups, and is therefore an isomorphism in Stab. It follows formally that g is *
*also an
isomorphism in EMod . We factor d to obtain the commutative diagram
~= ssS*(;o)
d: E*(X; o)______EMod *(E ^ X; E) ______DE*(X;o)
Mor(g;E) Dg*
~ ? ssS(;o) ?
d: E*(W; o)______EMod=*(E ^ W; E) ______DE*(W;o)*
Theorem 9.26 provides the two marked homeomorphisms. By Cor. 9.28, Mor (g; E) is
a homeomorphism. It is clear from Lemma 4.10 that W has strong duality. We have
a diagram of homeomorphisms. 
K"unneth homeomorphisms As the K"unneth pairing (9.13) is continuous, we can
complete it to
x: E*(X; o) b E*(Y; o) ! E*(X ^ Y; o)^; (9:29)
and the symmetric monoidal functor (9.14) to another one,
(E*(; o)^; x; z): (Stab*op; ^; T +) ! (FMod *; b; E*);(9:30)
for completed cohomology. As in Thm. 4.19, we combine Thm. 9.25 with Thm. 9.20
to deduce K"unneth homeomorphisms.
Theorem 9.31 Assume that E*(X; o) and E*(Y; o) are free E*modules. Then the
pairing (9.29) induces the cohomology K"unneth homeomorphism
E*(X ^ Y; o) ~=E*(X; o) b E*(Y; o) : 
JMB  49  23 Feb 1995
Stable cohomology operations
10 What is a stable module?
In this section, we give various interpretations of what it means to have a*
* module
over the stable operations on Ecohomology, with a view to future generalizatio*
*n in
[9] to unstable operations. We are primarily interested in the absolute cohomo*
*logy
E*(X) = E*(X+ ; o) of a space X, and state most results for this case only. Nev*
*erthe
less, we sometimes need the more general reduced cohomology E*(X; o) of a spect*
*rum
X.
An operation r: E*(; o) ! E*(; o) is stable if it is natural on Stab*. I*
*t is
automatically additive, Stab* being an additive category, but need not be an E*
module homomorphism.
Recall from x3 (or x9) that the profinite filtration makes E*(X) (or E*(X; *
*o))
a filtered E*module. When Hausdorff, it is an object of FMod *. We remind t*
*hat
all tensor products are taken over the coefficient ring E* = E*(T ) = E*(T +; o)
unless otherwise indicated, where T denotes the onepoint space and T + the sph*
*ere
spectrum.
First Answer Since Ecohomology E*(; o) is represented in Stab* by the spec
trum E, Yoneda's Lemma identifies the ring A of all stable operations with the
endomorphism ring End (E) = {E; E}* = E*(E; o) of E. Its unit element is , the
universal class of E. It acts on E*(X) = E*(X+ ; o) by composition,
X : A E*(X) = E*(E; o) E*(X) ! E*(X) : (10:1)
In particular, for each v 2 Eh we have the scalar multiplication operation x 7!*
* vx
on E*(X), which by Lemma 9.12 is represented by the map of spectra v: E ! E
of degree h in eq. (9.11) or the element v 2 Eh(E; o). This defines an embeddin*
*g of
rings (usually not central)
: E* ! E*(E; o) = A; (10:2)
which we used already in eq. (10.1) to make A an E*bimodule under composition
and X a homomorphism of E*modules.
Notation Standard notation for tensor products is ambiguous here, and will soon
become hopelessly inadequate for coping with the future plethora of bimodules a*
*nd
multimodules. When it is necessary to convey detailed information about the many
E*actions involved, we rewrite X as
X : E1*(E2; o) 2 E2*(X) ! E1*(X); (10:3)
which we call the E*action scheme of X . Here, Ei denotes a copy of E tagged
for identification, and i indicates a tensor product that is to be formed using*
* the
two E*actions labeled by i. If desired, we can add information about the degre*
*es by
writing
X : E1i(E2; o) 2 E2j(X) ! E1i+j(X):
For example, the composition
O *
A A = E*(E; o) E*(E; o) ! E (E; o) = A (10:4)
has action scheme E1*(E2; o) 2 E2*(E3; o) ! E1*(E3; o). We promise to use this
overelaborate notation sparingly.
JMB  50  23 Feb 1995
x10. What is a stable module?
The important special case X = T of the action (10.1) gives
T: A ~=A E* ! E*; (10:5)
which encodes the action of A on the coefficient ring E* = E*(T ).
The action (10.1) satisfies the usual two laws:
(sr)x = s(rx); x = x; (10:6)
for any operations s and r and any x 2 E*(X). This suggests that a stable module
structure on a given E*module M should consist of an action M : AM ! M that
satisfies these laws and is a homomorphism of left E*modules. Because the ten*
*sor
product is taken over E*, this implies that M extends the given module action *
*of E*
on M.
Unfortunately, this description is inadequate even for finite X. In the cla*
*ssical case
E = H(F p), A is the Steenrod algebra over Fp, which is generated by the Steenr*
*od
operations subject to explicitly given Adem relations. In general, A is uncount*
*able,
which suggests that we should make use of the profinite topology on it. We desc*
*ribed
a filtration for tensor products in eq. (4.15). However, the tensor product in*
* the
action (10.1) is formed using the right E*action on A, for which we have not d*
*efined
a filtration; worse, the usual E*module structure on the tensor product is not*
* the
one that makes X an E*module homomorphism. We have to find something else.
Second Answer In [1, 3], Adams suggested that for suitable ring spectra E, one
could avoid the various limit problems and infinite products that are inherent *
*in
cohomology by replacing the action (10.1) by the dual coaction on homology. Sta*
*bly,
the only difference between homology operations and cohomology operations is the
possibility of weakly phantom cohomology operations; in practice, these usually*
* do not
exist. Unstably, however, the difference is vast. Our ignorance of unstable hom*
*ology
operations in general forces us to learn to live with cohomology. We therefore *
*dualize
only partially. We defer the details until x11.
If E*(E; o) is a free E*module, we can convert the action X in (10.1) int*
*o a
coaction (after completion)
aeX : E*(X) ! E*(X) b E*(E; o) (10:7)
(whose action scheme is E2*(X) ! E1*(X) b1E1*(E2; o)). There is much structure
on E*(E; o), as explicated in [1, 3]. Dual to the composition (10.4) with unit *
*(10.2)
in E*(E; o), there is a coassociative comultiplication with counit
= S: E*(E; o) ! E*(E; o) E*(E; o); ffl = fflS: E*(E; o) ! E*
**;
JMB  51  23 Feb 1995
Stable cohomology operations
on E*(E; o). The action axioms (10.6) on X translate into the diagrams
aeX
E*(X) ________________E*(X) b E*(E; o)
 
(i) aeX 1 S
? ae ?
X 1
E*(X) b E*(E; o) ______E*(X) b E*(E; o) b E*(E; o)
(10:8)
aeX
E*(X) ______E*(X)b E*(E; o)
(ii)  1fflS
? ?
E*(X)^ ________E*(X)=b E*
These are in effect the usual axioms for a comodule coaction over E*(E; o) on E*
**(X)^,
the only novelty being the two distinct E*actions on E*(E; o).
Historically, the original example was developed by Milnor [22] in the case*
* E =
H(F p), to give a description of the Steenrod operations that is both elegant a*
*nd more
informative; we summarize it in x14. Even in this case, the completed tensor pr*
*oduct
is needed in the coaction (10.7) when X is infinitedimensional. For finite spa*
*ces or
spectra X, one can use SpanierWhitehead duality to switch between homology and
cohomology. This leads to Adams's coaction on homology [1, Lecture 3], except t*
*hat
he used a left coaction in an attempt to make the E*actions easier to track. I*
*t turns
out that in cohomology, the right coaction, even with its notational difficulti*
*es, is
both more customary and more convenient.
Third Answer We rewrite our Second Answer in a more categorical form in order
to allow generalization. We still leave the details to x11.
As the target of aeX is complete, we lose nothing if we complete the cohomo*
*logy
E*() to E*()^ everywhere. We define the functor S0: FMod * ! FMod * by S0M =
M b E*(E; o). Then we can use S and fflS to define natural transformations
0SM = M S: S0M ! S0S0M; ffl0SM = M fflS: S0M ! M:
The coalgebra properties of S and fflS will supply the necessary axioms (8.6) *
*to make
S0 a comonad in FMod *.
We rewrite the coaction (10.7) as a morphism ae0X: E*(X)^ ! S0(E*(X)^) in t*
*he
category FMod *. This converts the axioms (10.8) into diags. (8.7), which then*
* state
that E*(X)^ is precisely an S0coalgebra in FMod *.
We have condensed our answer down to the single word S0coalgebra.
Fourth Answer We are not done rewriting yet. The problem with our Third
Answer is that it still depends heavily on the tensor product, an essentially b*
*ilinear
construction that is simply unavailable for operations that are not additive (n*
*ot that
this has stopped us from trying).
We therefore go back to our First Answer and convert X to adjoint form, as
suggested by x8. We treat x 2 E*(X) as a map of spectra x: X+ ! E, and note that
the E*module homomorphism x*: A = E*(E; o) ! E*(X) is continuous. (There is
the usual sign, x*r = (1)deg(x) deg(r)r Ox = (1)deg(x) deg(r)rx, from eq. (6.*
*3).)
JMB  52  23 Feb 1995
x10. What is a stable module?
For convenience, we assume that A is Hausdorff and work in FMod *. Given a*
*ny
complete Hausdorff filtered E*module M (i. e. object of FMod ), we define
SM = FMod *(A; M) = FMod *(E*(E; o); M) : (10:9)
Then for any space X, we define the coaction
aeX : E*(X) ! S(E*(X)^) = FMod *(A; E*(X)^) (10:10)
on x 2 E*(X) by aeX x = x*: A = E*(E; o) ! E*(X)^, completing as necessary. In
the important special case X = T , we find
aeT: E* = E*(T ) ! FMod *(A; E*(T )) = SE*(T ) = SE*: (10:11)
Similarly, we have aeX : E*(X; o) ! S(E*(X; o)^) for spectra and based spaces X.
Theorem 10.12 Assume that the E*module A = E*(E; o) is Hausdorff (as is tr*
*ue
for E = H(F p), MU, BP , KU, or K(n) by Lemma 9.21 and Thm. 9.25). Then we
can make the functor S defined in eq. (10.9) a comonad in the category FMod of
complete Hausdorff filtered E*modules.
Now that we have a suitable comonad, the definition of stable module is cle*
*ar.
This is the answer that will generalize satisfactorily.
Definition 10.13 A stable (Ecohomology) module is an Scoalgebra in FMod *,
i. e. a complete Hausdorff filtered E*module M that is equipped with a morphism
aeM : M ! SM in FMod * (10:14)
that is E*linear and continuous and satisfies the coaction axioms (8.7). We t*
*hen
define the action of r 2 Ah = Eh(E; o) on x 2 Mk by rx = (1)kh(aeM x)r 2 M.
A closed submodule L M is called (stably) invariant if aeM restricts to a*
*eL: L !
SL. Then the quotient M=L also inherits a stable module structure.
The group SM may be thought of as the set of all candidates for the action *
*of A
on a typical element of M. Then aeM selects for each x 2 M an appropriate acti*
*on on
x. The axioms (8.7) translate into the usual action axioms (10.6). If we evalua*
*te the
first only partially, we obtain the commutative square
r
M _______M
 
aeM aeM (10:15)
? !rM ?
SM ______SM
where the natural transformation !r is defined on f 2 SM as
(!rM)f = (1)deg(r) deg(f)f Or*: A ! M;
using r*: A = E*(E; o) ! E*(E; o) = A. It may be viewed as the analogue of
diag. (8.8).
Theorem 10.16 Assume that the E*module A = E*(E; o) is Hausdorff (as is tr*
*ue
for E = H(F p), MU, BP , KU, or K(n) by Lemma 9.21 and Thm. 9.25). Then:
JMB  53  23 Feb 1995
Stable cohomology operations
(a) We can factor aeX (defined in eq. (10.10)) through E*(X)^ as aeX : E*(X*
*)^ !
S(E*(X)^), to make E*(X)^a stable module for any space X (and similarly E*(X; o*
*)^
for spectra);
(b) ae is universal: given an object N of FMod *, any transformation
X: E*(X; o) ! FMod *(N; E*(X; o)^)
(or bX: E*(X; o)^ ! FMod *(N; E*(X; o)^)) of any degree, that is defined for al*
*l spec
tra X and natural on Stab*, is induced from aeX by a unique morphism f: N ! A in
FMod * as the composite
aeX * * *
X: E*(X; o) ! SE (X; o)^= FMod (A; E (X; o)^)
Hom(f;1) * (10:17)
! FMod (N; E*(X; o)^) :
Proofs of Thms. 10.12 and 10.16 The discussion in x8 is intended to suggest th*
*at
these two proofs are interlaced. The main proof is in seven steps. Lemma 9.12
provides the E*module object E in Stab*. We find it useful to write idA for t*
*he
identity map A ! A, considered as an element of SA.
Step 1. We introduce an E*module structure (different from the obvious one*
*) on
the graded group SM defined by eq. (10.9); by hypothesis, A is an object of FMo*
*d *
and S is defined. By Lemma 7.6(a), the additive functor
E*(;o)^ *op Mor(;M) *
FMod *(E*(; o)^; M): Stab*! FMod ! Ab
takes the E*module object E to an E*module object in Ab *, i. e. makes SM an
E*module. (By Lemma 7.1(a), the additive structure on SM must be the obvious
one.) As M varies, Lemma 7.7(b) shows that SM is functorial, and we have a func*
*tor
S: FMod * ! Mod *. We enrich it later, in Step 3, to take values in FMod *.
Step 2. We show that aeX is an E*module homomorphism. Given a spectrum (or
space) X, the cohomology functor E*(; o)^: Stab*op! FMod *induces the natural
transformation of additive functors
Stab*(X; ) ! FMod *(E*(; o)^; E*(X; o)^): Stab*! Ab* :
We apply this to the E*module object E in Stab*; then Lemma 7.6(c) shows that *
*aeX
is a homomorphism of E*modules.
Step 3. In order to make S and aeX take values in FMod *, we must filter *
*SM.
If M is filtered by the submodules F aM, we filter SM in the obvious way by the
F a(SM) = S(F aM), which are E*submodules because S is a functor. We trivially
have the exact sequence
0 ! SF aM ! SM ! S(M=F aM);
which we use to rewrite the filtration in the more useful form
F aSM = Ker[SM ! S(M=F aM)]: (10:18)
(In fact, there is a short exact sequence in all our examples. However, we do *
*not
exploit this fact because (a) it requires a stronger hypothesis on E, but more *
*impor
tantly, (b) it does not generalize correctly.)
JMB  54  23 Feb 1995
x10. What is a stable module?
It is not difficult to see directly that SM is complete Hausdorff. Because*
* M is
complete Hausdorff, we have the limit M = limaM=F aM, which is automatically
preserved by S. This yields by eq. (10.18) the inclusion
" #
SM M M
lima_______~=limIm SM ! S _____ limS _____= SM
F aSM a F aM a F aM
in Ab*. But this inclusion is visibly epic and therefore an isomorphism, which *
*makes
SM complete Hausdorff.
We have now defined S as a functor taking values in FMod *as required. Our
choice of the profinite topology on E*(X; o) and the naturality of ae make it c*
*lear that
aeX is continuous and factors as asserted in Thm. 10.16(a).
Step 4. We convert the object E*(X)^ of FMod * to the corepresented functor
FX = FMod *(E*(X)^; ): FMod * ! Ab* (and similarly E*(X; o)^ for spectra X). As
suggested by eq. (8.16), we also convert the coaction aeX to the natural transf*
*ormation
aeX : FX ! FX S: FMod * ! Ab*. Given M, the homomorphism
aeX M: FX M = FMod *(E*(X)^; M) ! FMod *(E*(X)^; SM) = FX SM (10:19)
is defined by (aeX M)f = Sf OaeX : E*(X)^ ! S(E*(X)^) ! SM.
Step 5. We define the natural transformation : S ! SS by taking X = E in
eq. (10.19), so that
M: SM = FMod *(A; M) ! FMod *(A; SM) = SSM (10:20)
is given on the element f: A ! M of SM as the composite
aeE * Sf
( M)f: A = E*(E; o) ! SE (E; o) = SA ! SM :
(In terms of elements, this is r 7! [s 7! f(r*s) = (1)deg(r) deg(s)f(sr)].) W*
*hen we
substitute the E*module object E for X in diag. (10.19), Lemma 7.6(c) shows th*
*at
M takes values in Mod *. Naturality in M shows that is filtered and takes va*
*lues
in FMod *, as required.
Step 6. The other required natural transformation,
fflM: SM = FMod *(A; M) ! M; (10:21)
is defined simply as evaluation on the universal class 2 A, i. e. (fflM)f = f.*
* Once
again, naturality in M shows that fflM is filtered, but we have to verify that *
*fflM
is an E*module homomorphism. (All proofs involving ffl are necessarily somewh*
*at
computational, because the definition is.) Additivity is clear. Take any v 2 Eh*
*. By
Lemma 9.12, the structure map v: E ! E induces (v)* = v in E*(E; o). Given an
element f: A = E*(E; o) ! M of SM, we defined vf = f O(v)* in Step 1; then
ffl(vf) = ffl(f O(v)*) = f(v)* = f(v) = vf = vfflf;
using the given E*linearity of f.
Step 7. We show that S is a comonad and that E*(X)^ is an Scoalgebra. Natu
rality of ae with respect to the map of spectra x: X+ ! E for any x 2 E*(X) sho*
*ws
that aeX is a coaction on E*(X)^ in the sense of Defn. 8.15, using R = A = E*(E*
*; o),
JMB  55  23 Feb 1995
Stable cohomology operations
aeR = aeE , and 1R = . By Lemma 8.20, aeX makes E*(X)^ (or E*(X; o)^) an S
coalgebra; we constructed and ffl to satisfy the conditions (8.19). Finally,*
* S is a
comonad by Lemma 8.22(a).
Yoneda's Lemma gives Thm. 10.16(b) for . Because E*(; o) is represented
by E, is classified by the element f = (E) 2 FMod *(N; A) and so given by
eq. (10.17). If we are given b instead, we compose with E*(X; o) ! E*(X; o)^ to
obtain . Conversely, any factors through b by naturality. 
11 Stable comodules
Although the Fourth Answer of x10, in terms of stable modules, is the clean*
*est and
most general, the Second Answer, in terms of stable comodules, is usually avail*
*able
and more practical in the cases of interest. (One could argue that this feature*
* is what
makes these cases interesting.) At least for E = MU or BP , such comodules are
called cobordism comodules. This is the context for Landweber theory, as develo*
*ped
in [17, 18] and discussed in x15 for BP .
Rather than develop the Second and Third Answers from scratch, we deduce
them from the Fourth Answer by comparing the algebraic structures on E*(E; o) a*
*nd
E*(E; o). This section is entirely algebraic in the sense that the only spectr*
*um we
study in any depth is E. In Thm. 11.35 we show that the structure maps jR, S, *
*and
fflS on E*(E; o) agree with those of Adams.
We assume later in this section that E*(E; o) is a free E*module, which is*
* true for
our five examples by Lemma 9.21. The duality d: E*(E; o) ~=DE*(E; o) in Thm. 9.*
*25
allows us to identify the following, with only slight abuse of notation:
(i)The cohomology operation r on E*() (or E*(; o));
(ii)The class r 2 E*(E; o), which we also write simply as r;
(11.1)
(iii)The map of spectra r: E ! E, a morphism in Stab*;
(iv)The E*linear functional : E*(E; o) ! E*.
The degree of r is the same in any of these contexts (once we remember that Ei(*
*E; o)
has degree i).
The bimodule algebra E*(E; o) As E*(E; o) is better understood and smaller
than E*(E; o), (iv) is the preferred choice in (11.1). There is much structure*
* on
E*(E; o). First, like all Ehomology, it is a left E*module.
When we apply the additive functor E*(; o) to the E*module object E in
Lemma 9.12, we obtain by Lemma 7.6(a) the E*module object E*(E; o) in Mod ,
equipped with the E*module homomorphism (v)* of degree h for each v 2 Eh. To
extract a bimodule as commonly understood, we define the right action by
c . v = (1)hm (v)*c for v 2 Eh, c 2 Em (E; o),
to ensure that v0(c . v) = (v0c) . v, with no signs. Nevertheless, we find it *
*technically
convenient to keep all functions and operations on the left and work with (v)*.
The ring spectrum structure (OE; j) on E induces the multiplication
x OE*
OE = OES: E*(E; o) E*(E; o) ! E*(E ^ E; o) ! E*(E; o)
JMB  56  23 Feb 1995
x11. Stable comodules
and left unit
j*
j = jS: E* ~=E*(T +; o) ! E*(E; o)
for E*(E; o). In particular, we have the unit element 1 = j1 2 E0(E; o).
The equation vc = v(1c) = (v1)c = (jv)c describes the left E*action in ter*
*ms of
OE and j, and implies that j is a ring homomorphism. We shall see presently tha*
*t the
right action is similarly determined by its effect on 1.
Definition 11.2 We define the right unit function jR: E* ! E*(E; o) on v 2 E**
* =
E*(T +; o) by jRv = v*1, using the homology homomorphism v*: E* ~=E*(T +; o) !
E*(E; o) induced by the map v: T +! E in Stab*.
We summarize all this structure. We recall that in general, the left and ri*
*ght units
and E*actions on E*(E; o) are quite different.
Proposition 11.3 In E*(E; o), for any ring spectrum E:
(a) E*(E; o) is an E*bimodule;
(b) The unit element 1 = j1 = jR1 is well defined;
(c) The multiplication OE makes E*(E; o) a commutative E*algebra with respe*
*ct
to the left or right E*action;
(d) j: E* ! E*(E; o) and jR: E* ! E*(E; o) are ring homomorphisms;
(e) The left action of v 2 E* is left multiplication by v1;
(f) The right action of v 2 E* is right multiplication by jRv.
Proof For (c), we apply the Ehomology symmetric monoidal functor (9.18) to the
commutative monoid object E in Stab, to obtain the commutative monoid object
E*(E; o) in Mod , i. e. commutative E*algebra, with respect to the left E*act*
*ion.
We trivially have (b), because the map j: T + ! E is 1T 2 E0(T ). For (f),*
* we
apply Ehomology to (9.11), which expresses v in terms of the multiplication. T*
*his
implies that jR is a ring homomorphism. 
Remark There is a wellknown conjugation O: E*(E; o) ! E*(E; o) which inter
changes the left and right E*actions. We avoid it because it does not generali*
*ze to
the unstable situation.
The functor S0 Duality and Lemma 6.16(b) provide the natural isomorphism
S0M = M b E*(E; o) ~=FMod *(E*(E; o); M) = SM (11:4)
for any complete Hausdorff filtered E*module M, with action scheme
(S0M)2 = M1 b1E1*(E2; o) ~=FMod 1*(E1*(E2; o); M1) = (SM)2 :
The functors S and S0 are those of x10. Moreover, this is an isomorphism of fil*
*tered
E*modules in FMod if we filter S0M as in (4.15), which is the same as filteri*
*ng it
by the submodules S0F aM. (We remind that E*(E; o), like all homology, invariab*
*ly
carries the discrete topology.) Explicitly, with the help of Prop. 11.3, the is*
*omorphism
of E*actions is expressed by
= for r 2 E*(E; o), v 2 E*, c 2 E*(E; o).(11:5)
JMB  57  23 Feb 1995
Stable cohomology operations
In view of the proliferation of E*actions, one must be careful in applying*
* du
ality; the correct way to establish all properties of S0 is to deduce them from*
* the
corresponding properties of S in x10 by applying the isomorphism (11.4). (Once *
*our
equivalences are well established, we shall normally omit the 0everywhere.)
The coalgebra structure on E*(E; o) The comonad structure ( S; fflS) on S in
Thm. 10.12 corresponds under (11.4) to a comonad structure on S0 consisting of
natural transformations 0M: S0M ! S0S0M and ffl0M: S0M ! M. By naturality
and the case M = E*, 0M must take the form M for a certain welldefined
comultiplication
= S: E*(E; o) ! E*(E; o) E*(E; o) (11:6)
(with action scheme E1*(E3; o) ! E1*(E2; o)2E2*(E3; o)). It is not cocommutative
(in any ordinary sense). Similarly, ffl0M must have the form
M fflS: S0M = M b E*(E; o) ! M E* ~=M
for some welldefined counit
ffl = fflS: E*(E; o) ! E*: (11:7)
(Here and elsewhere, the isomorphism M E* ~=M always involves the usual sign,
x v 7! (1)deg(x) deg(v)vx.) Both S and fflS are morphisms of E*bimodules.
Lemma 11.8 Assume that E*(E; o) is a free E*module. Then the homomorphisms
S and fflS in diags. (11.6) and (11.7) make E*(E; o) a coalgebra over E*.
Proof By taking M = E*, the comonad axioms (8.6) for S0 translate into the coa*
*s
sociativity of S,
S
E*(E; o) ________________E*(E;o) E*(E; o)
 
 
 S 1 S (11:9)
 
? 1 ?
E*(E; o) E*(E; o) ______E*(E;o)S E*(E; o) E*(E; o)
and the two counit axioms on fflS:
S S
E*(E; o) ______E*(E;o) E*(E; o) E*(E; o) ______E*(E;o) E*(E; o)
H H ~ H ~
H H= ffl 1 H H = 1ffl
H H  S H H  S
Hj ? HHj ?
E* E*(E; o) E*(E; o) E*
(i) (ii)
(11:10)
These commutative diagrams express precisely what we mean by saying that E*(E; *
*o)
is a coalgebra. 
Comodules We are now ready to convert Defn. 10.13 of a stable module and
Thm. 10.16, by means of the isomorphism (11.4). The coaction aeM : M ! SM
JMB  58  23 Feb 1995
x11. Stable comodules
in (10.14) on a stable module M corresponds to a coaction aeM : M ! S0M =
M b E*(E; o).
Definition 11.11 A stable (Ecohomology) comodule structure on a complete
Hausdorff filtered E*module M (i. e. object of FMod ) consists of a coaction a*
*eM : M !
M b E*(E; o) that is a continuous morphism of filtered E*modules (i. e. morphi*
*sm in
FMod , with action scheme M2 ! M1 b1E1*(E2; o)) and satisfies the axioms
aeM M ______MabE*(E;eo)M
M _______________M bE*(E; o) Q
  Q Q ~= 
aeM M S Q MfflS
? ae ? QQs ? (11:12)
M 1
M bE*(E; o) ______MbE*(E; o) bE*(E; o) M E*
(i) (ii)
Theorem 11.13 Assume that E*(E; o) is a free E*module (which is true for E*
* =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Then given a complete Hausdorff
filtered E*module M (i. e. object of FMod ), a stable module structure on M i*
*n the
sense of Defn. 10.13 is precisely equivalent under (11.4) to a stable comodule *
*structure
on M in the sense of Defn. 11.11.
Proof The axioms (11.12) are just the axioms (8.7) interpreted for S0. 
Theorem 11.14 Assume that E*(E; o) is a free E*module (which is true for E*
* =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Then:
(a) For any space (or spectrum) X, there is a natural coaction
aeX : E*(X) ! E*(X) b E*(E; o) (11:15)
(or aeX : E*(X; o) ! E*(X; o) b E*(E; o)) in FMod that makes E*(X)^(or E*(X; o*
*)^) a
stable comodule, which corresponds by Thm. 11.13 to the coaction aeX of Thm. 10*
*.16;
(b) ae is universal: given a discrete E*module N, any transformation
X: E*(X; o) ! E*(X; o) b N (or bX: E*(X; o)^ ! E*(X; o)^ bN),
that is defined and natural for all spectra X, is induced from aeX by a unique *
*morphism
f: E*(E; o) ! N of E*modules, as
aeX * 1f *
X: E*(X; o) ! E (X; o) b E*(E; o) ! E (X; o) b N : (11:16)
Proof For (a), we combine Thm. 11.13 with Thm. 10.16(a).
However, (b) is not a translation of Thm. 10.16(b), although the proof is s*
*imilar.
Because E*(; o) is represented in Stab* by E, is determined by the value (E) 2
E*(E; o) b N, which corresponds to the desired homomorphism f: E*(E; o) ! N
under the isomorphism E*(E; o) b N ~=Mod *(E*(E; o); N) of Lemma 6.16(a). 
Remark The universal property (b) shows that diags. (11.12), with M = E*(X; o),
may be viewed as defining S and fflS in terms of ae. Three applications of the*
* unique
ness in (b) then show that S is coassociative and has fflS as a counit.
JMB  59  23 Feb 1995
Stable cohomology operations
Remark From a purely theoretical point of view, one should write the coaction
(11.15) as aeX : E*(X)^ ! E*(X)^ bE*(E; o), using three completions, in order t*
*o stay
inside the category FMod of filtered modules at all times. This seems excessiv*
*e. The
way we are writing aeX , using just the b (and that only when necessary) and le*
*aving
the other completions implicit, conveys exactly the same algebraic and topologi*
*cal
information after completion. But we warn that in using diag. (11.12)(ii), M b *
*E* ~=
M is valid if and only if M is complete Hausdorff. In particular, E*(X) can onl*
*y be
a stable comodule if it is already Hausdorff.
Linear functionals Theorem 11.13 establishes the equivalence between stable
modules and stable comodules. For applications, we need to make this correspon
dence explicit. Given a stable comodule M, we recover the action of r 2 E*(E; o*
*) on
the stable module M from aeM as
aeM M *
r: M ! M b E*(E; o) ! M E ~=M; (11:17)
by means of the isomorphism (11.4), whose details are supplied by Lemma 6.16(b).
To make everything explicit, we choose x 2 Mk and write
X
aeM x = (1)deg(xff) deg(cff)xff cff in M b E*(E; o),(11:18)
ff
where the sum may be infinite. (We introduce signs here to keep other formulae
cleaner. It is noteworthy that in the explicit formulae of x14, these signs are*
* invariably
+1.) Then from Cor. 6.17, the corresponding element x* 2 FMod k(E*(E; o); M) *
*is
given by X
x*r = (1)k deg(r) xff;
ff
and conversely. We rewrite this more conveniently as
X
rx = xff in M, for all r (11:19)
ff
(with no signs at all!), where we emphasize that the cffand xffdepend only on x*
*, not
on r. The sums here may be infinite, but will converge because (11.18) does.
The statement that aeM is an E*homomorphism may be expressed as
X
r(vx) = xff in M, for all r, (11:20)
ff
for any v 2 E*, with the help of (11.5).
It is important for our purposes not to require the cffto form a basis of E*
**(E; o),
or even be linearly independent; but if they do form a basis, the xffare unique*
*ly
determined by (11.19) as xff= c*ffx, where c*ffdenotes the operation dual to cf*
*f.
As fflS is dual to : E* ! E*(E; o), we see that
<; > = fflS: E*(E; o) ! E*; (11:21)
which is obvious by comparing diag. (11.12)(ii) with (11.17). In other words, *
*the
functional fflS corresponds to the identity operation in the list (11.1). In p*
*ractice, fflS
is always easy to write down; it is S that causes difficulties. Of course, S *
*is dual
to composition (10.4), as we make explicit later in eq. (11.34).
JMB  60  23 Feb 1995
x11. Stable comodules
The cohomology of a point Our first test space is the onepoint space T . We
have enough to determine the stable structure of E*(T ) completely.
Proposition 11.22 Let r be a stable operation on Ecohomology and v 2 E*.
Assume that E*(E; o) is a free E*module. Then in the stable comodule E*(T ) = *
*E*:
(a) The action of the operation r is given by
rv = in E*(T ) = E*; (11:23)
(b) The coaction aeT: E* ! E* E*(E; o) ~=E*(E; o) coincides with the right *
*unit
jR: E* ! E*(E; o);
(c) If we write E* = ssS*(E; o) and regard r: E ! E as a map of spectra, the
induced homomorphism r*: E* ! E* on stable homotopy groups is given by r*v =
.
Proof If we regard v as a map v: T +! E and use Defn. 11.2, we find
rv = v*r = = = ;
which is (a). We compare eq. (11.19) with eq. (11.18) to rewrite this as aeTv =*
* 1jRv,
which gives (b). Parts (a) and (c) are equivalent, because both rv and r*v are*
* the
same morphism r Ov: T +! E ! E in Stab*. 
The cohomology of spheres Our second test space is the sphere Sk. By definitio*
*n,
stable operations commute up to sign with the suspension isomorphism, as in fig*
*. 2
in x9. In view of the multiplicativity of E and eq. (3.24), this reduces to
aeSuk = uk 1 in E*(Sk; o) E*(E; o) (11:24)
for all integers k (positive or negative), where Sk denotes the ksphere spectr*
*um
and uk 2 Ek(Sk; o) ~= E0 the standard generator. Equivalently, from eqs. (11.1*
*8)
and (11.19), the action of any operation r is given by
ruk = uk in E*(Sk; o). (11:25)
Both formulae then hold in E*(Sk) for the space Sk, which exists for k 0. Als*
*o,
eq. (11.20) gives r(vuk).
Homology homomorphisms In some applications, it is useful to regard the ele
ment x 2 Ek(X; o) as a map of spectra x: X ! E and compute the homomorphism
induced on Ehomology.
Proposition 11.26 Assume that E*(E; o) is a free E*module. Given x 2 E*(X; *
*o),
suppose that rx is given by eq. (11.19). Then the Ehomology homomorphism
x*: E*(X; o) ! E*(E; o) induced by the map of spectra x: X ! E is given on
z 2 Em (X; o) by X
x*z = (1)deg(cff)(deg(xff)+m) cff: (11:27)
ff
Proof For a general operation r, we have
= =
X o X AE
= vff= r; (1)deg(cff) deg(vff)vffcff ;
ff ff
JMB  61  23 Feb 1995
Stable cohomology operations
where vff= . Since this holds for all r, eq. (11.27) follows by duality*
*. 
Conversely, we can recover aeX x from x* when X is well behaved.
Proposition 11.28 Assume that E*(X) is a free E*module. Take x 2
E*(X). Then under the isomorphism E*(X) b E*(E; o) ~= Mod *(E*(X); E*(E; o))
of Lemma 6.16(a), the element aeX x corresponds to the homomorphism x*: E*(X) !
E*(E; o) of E*modules.
Proof We apply Lemma 6.16(a) to eq. (11.18), using the strong duality E*(X) ~=
DE*(X) of Thm. 9.25, and compare with eq. (11.27). 
Similarly, it is important to know the Ehomology homomorphism r*: E*(E; o)*
* !
E*(E; o) induced by an operation r, regarded as a map r: E ! E of spectra. This
provides a convenient faithful representation of the operations on E*(E; o), as*
* it is
clear that * = idand (sr)* = s*O r*. From diag. (10.15) and the isomorphism (11*
*.4)
we deduce the commutative square
r
M ________________M
 
aeM aeM (11:29)
 
? ?
Mr*
M b E*(E; o) ______Mb E*(E; o)
We need to know how to pass between and r*. From the identity *
* =
= <; r*c> and eq. (11.21), we easily recover the functional from*
* r* as
r* fflS *
: E*(E; o) ! E*(E; o) ! E : (11:30)
The following result gives the reverse direction.
Lemma 11.31 Let r 2 E*(E; o) be an operation and assume that E*(E; o) is a f*
*ree
E*module. Then:
(a) The diagram
r*
E*(E; o)________________E*(E; o)
 
 S  S (11:32)
? 1r ?
E*(E; o) E*(E; o) ______E*(E;o)* E*(E; o)
commutes; in other words, r* is a morphism of left E*(E; o)comodules;
(b) r*: E*(E; o) ! E*(E; o) is the unique homomorphism of left E*modules th*
*at
satisfies eq. (11.30) and is a morphism of left E*(E; o)comodules as in (a);
(c) The homomorphism r* is given in terms of the functional as
S
r*: E*(E; o)!E*(E; o) E*(E; o)
1 (11:33)
! E*(E; o) E* ~=E*(E; o) :
JMB  62  23 Feb 1995
x11. Stable comodules
Proof After applying M b , diag. (11.32) corresponds under eq. (11.4) to the *
*square
rM
SM _______SM
 
 M  M
 S  S
? rSM ?
SSM ______SSM
where r: S ! S is defined on f 2 SM = FMod *(A; M) by (rM)f =
(1)deg(r) deg(f)f Or*. (Note that r only takes values in Ab *, because it fail*
*s to pre
serve the preferred E*module structure on SM.) Since S is corepresented by A,
commutativity of this diagram reduces to the equality
aeE Or* = Sr* OaeE : A = E*(E; o) ! SE*(E; o) = SA
in SSA, which expresses the naturality of ae. (Explicitly, (sr)* = r* Os* for*
* all
s 2 A, which is the associativity t(sr) = (ts)r for fixed r.)
If we compose diag. (11.32) with
1 fflS: E*(E; o) E*(E; o) ! E*(E; o) E* ~=E*(E; o)
and use eq. (11.30) and diag. (11.10)(ii), we obtain (c). This also establishe*
*s the
uniqueness of r* in (b). 
To summarize, eqs. (11.30) and (11.33) express r* and in terms of ea*
*ch
other, with the help of S and fflS. Conversely, these equations may be viewed*
* as
characterizing S and fflS in terms of the r* and for all r.
We can at last make explicit how S is dual to the composition (10.4). It *
*is
immediate from eq. (11.30) that = Or*. We substitute eq. (11.33*
*) to
obtain
S
: E*(E; o) ! E*(E; o) E*(E; o)
1 (11:34)
! E*(E; o) E* ~=E*(E; o) ! E*
(Note that we cannot simply write here, which is undefined unless
happens to be right E*linear.)
Remark From a more sophisticated point of view, several of our formulae may be
explained by noting that S makes E*(E; o) a stable comodule, provided we use t*
*he
right E*module action. The comodule axioms are (11.9) and (11.10)(ii). Then *
*by
comparing eq. (11.33) with eq. (11.17), we see that the action of r on E*(E; o)*
* is just
r*, and diag. (11.32) becomes a special case of diag. (11.29), which in turn co*
*mes from
diag. (8.8).
Compatibility It is clear that eqs. (11.30) and (11.33) determine fflS and S
uniquely, and that eq. (11.23) determines jR. We now show that they agree with
the homomorphisms introduced by Adams [3, III.12].
Theorem 11.35 Assume that E*(E; o) is a free E*module (which is true for E*
* =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Suppose that E*(E; o) is equipped
with S and fflS that satisfy eqs. (11.30) and (11.33), and jR as in Defn. 11.2*
*. Then:
JMB  63  23 Feb 1995
Stable cohomology operations
(a) jR must be
ssS*(j^E;o)
E* = ssS*(E; o) ~=ssS*(T +^ E; o) ! ssS*(E ^ E; o) = E*(E; o);
(b) fflS must be
ssS*(OE;o)
fflS: E*(E; o) = ssS*(E ^ E; o) ! ssS*(E; o) = E*;
(c) S must be
E*(j^E)
S: E*(E; o) ~=E*(T +^E; o) ! E*(E^E; o) ~=E*(E; o) E*(E; o);
where we use the twisted K"unneth isomorphism with action scheme
E1*(E2 ^ E3; o) ~=E1*(E2; o) 2 E2*(E3; o):
Proof The definition jRv = v*1 expands to
j + E^v
T +! E ' E ^ T ! E ^ E :
We prove (a) by rearranging this as
v + j^E
T +! E ' T ^ E ! E ^ E :
Given r 2 E*(E; o) and c 2 E*(E; o), we may construct as the composi*
*te
c r^E OE
: T +! E ^ E ! E ^ E ! E :
If we take r = and compare with (11.30), we obtain (b).
The commutative diagram
(j^E)* ~=
E*(E; o)___________E*(E^ E; o) ______E*(E;oo)e E*(E; o)
  
  
r* (E^r)* 1r*
  
? (j^E)* ? ~ ?
E*(E; o)___________E*(E^ E; o) ______E*(E;oo)e=E*(E; o)
H H 
H H = 
H H OE* 1ffl
H Hj ? ss S
E*(E; o)
shows, with the help of eq. (11.30), that r* is the composite of 1 with A*
*dams's
, which appears as the top row. (Here, both K"unneth isomorphisms are twisted.)
Since this holds for all r, comparison with eq. (11.33) gives (c). 
12 What is a stable algebra?
In x10, we gave four answers for the structure of a module over the ring A =
E*(E; o) of stable operations in Ecohomology, to encode the algebraic structure
JMB  64  23 Feb 1995
x12. What is a stable algebra?
present on the E*module E*(X) or E*(X; o) for a space or spectrum X. When X is
a space, E*(X) is an E*algebra. In this section, we enrich each answer and the*
*orem
to include this multiplicative structure.
The organizing principle of this section is to make everything symmetric mo*
*noidal.
We have three symmetric monoidal categories in view: (Stab*; ^; T +), (Mod *; *
*; E*),
and the filtered version (FMod *; b; E*). We also have three symmetric monoid*
*al
functors: Ecohomology (9.14), completed Ecohomology (9.30), and Ehomology
(9.18).
In this section, we generally assume that E*(E; o) is a free E**
*module;
then Thm. 9.31 provides K"unneth homeomorphisms E*(E^E; o) ~= A b A and
E*(E^E^E; o) ~=A b A b A.
First Answer For a spectrum X, we have the action (10.1)
X : A E*(X; o) ! E*(X; o):
Given an operation r, we would like to have an external Cartan formula
X
r(xxy) = r0ffx x r00ffy in E*(X ^ Y; o) (12:1)
ff
for suitable choices of operations r0ffand r00ff(and signs). For a space X, thi*
*s leads to
the corresponding internal Cartan formula,
X
r(xy) = (r0ffx)(r00ffy) in E*(X). (12:2)
ff
For the universal example X = Y = E, with x = y = , eq. (12.1) reduces to
X
OE*r = r0ffx r00ff in E*(E ^ E; o).
ff
This requires OE*r to lie in the image of the cross product (9.13)
x: E*(E; o) E*(E; o) ! E*(E ^ E; o);
which rarely happens. However, the pairing becomes an isomorphism if we use the
completed tensor product and so allow infinite sums. This is another reason to *
*topol
ogize E*(X).
From this point of view, a stable algebra should consist of a filtered E*a*
*lgebra M
equipped with a continuous E*linear action M : A M ! M that satisfies eq. (12*
*.2)
for all r. We must not forget the unit 1M of the algebra M, for which eq. (11*
*.23)
requires r1M = 1M .
In the classical case E = H(F p), there is a good finite Cartan formula, an*
*d this de
scription is adequate for many applications. For MU and BP , however, this appr*
*oach
is not very practical and must be reworked.
Second Answer We have the coaction aeX : E*(X) ! E*(X) b E*(E; o) from (10.7).
We shall find that the rather opaque Cartan formula (12.1) translates (for spac*
*es)
JMB  65  23 Feb 1995
Stable cohomology operations
into the commutative diagram
aeXaeY
E*(X) E*(Y ) _________(E*(X)bE*(E; o)) (E*(Y ) bE*(E; o))
 
 
 
 ?

x E*(X) b E*(Y ) b (E*(E; o)E*(E; o)) (12:3)
 
 xOE
 
? aeXxY ?
E*(X x Y ) ____________________E*(XxY ) b E*(E; o)
By taking Y = X, we deduce that aeX is a homomorphism of E*algebras. This
includes the units, which come from 1 2 E0(T ), since aeT is given by Prop. 11.*
*22(b).
Explicitly, given x 2 E*(X) and y 2 E*(Y ), assume that rx and ry are given*
* as
in eq. (11.19) by
X X
rx = xff; ry = yfi; (for all r)
ff fi
for suitable elements xff2 E*(X), yfi2 E*(Y ), and cff; dfi2 E*(E; o). Evaluati*
*on of
diag. (12.3) on x y using eq. (11.18) yields the external Cartan formula
X X
r(xxy) = (1)deg(dfi) deg(xff) xffxyfi in E*(X x Y()^12:4)
ff fi
for all r. This works too for x 2 E*(X; o), y 2 E*(Y; o), and x x y 2 E*(X ^ Y;*
* o),
where X and Y are based spaces or spectra. For a space X, we can take Y = X and
deduce the internal Cartan formula
X X
r(xy) = (1)deg(dfi) deg(xff) xffyfi in E*(X)^, for(all1r*
*.2:5)
ff fi
All this makes it clear what the definition of a stable comodule algebra sh*
*ould be.
The following lemma makes it reasonable. As in x10, we defer most proofs until *
*we
have our preferred definitions, at the end of the section.
Lemma 12.6 Assume that E*(E; o) is a free E*module. Then the comultiplicati*
*on
= S: E*(E; o) ! E*(E; o) E*(E; o) and counit ffl = fflS: E*(E; o) ! E* are
homomorphisms of E*algebras.
As an immediate corollary of 1 = 1 1, we have
S(vw) = v w in E*(E; o) E*(E; o)
for any v 2 E* and w 2 jRE*. If we combine this with eq. (11.33), we obtain
r*(vw) = v jR in E*(E; o)
for any stable operation r. What makes these formulae useful is that the eleme*
*nts
vw always span E*(E; o) Q as a Q module. Thus in the important case when E*
has no torsion, these innocuous equations are powerful enough to determine S a*
*nd
r* completely.
JMB  66  23 Feb 1995
x12. What is a stable algebra?
Definition 12.7 We call a stable comodule M in the sense of Defn. 11.11 a sta*
*ble
(Ecohomology) comodule algebra if M is an object of FAlg and its coaction aeM *
* is a
morphism in FAlg.
In detail, M is a complete Hausdorff filtered E*algebra equipped with a co*
*ac
tion aeM : M ! M b E*(E; o) that is a continuous homomorphism of E*algebras and
satisfies the coaction axioms (11.12), which are now diagrams in FAlg.
Theorem 12.8 Assume that E*(E; o) is a free E*module (which is true for E =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Then:
(a) For any space X, the coaction aeX in (11.15) makes E*(X)^ a stable comod*
*ule
algebra;
(b) ae is universal: given a discrete commutative E*algebra B, any multipli*
*cative
transformation X: E*(X; o) ! E*(X; o) b B (or bX: E*(X; o)^ ! E*(X; o) b B)
that is defined for all spectra X and natural on Stab*is induced from aeX by a *
*unique
E*algebra homomorphism f: E*(E; o) ! B as
aeX * 1f *
X: E*(X; o) ! E (X; o) b E*(E; o) ! E (X; o) b B :
Third Answer We restate our Second Answer in terms of the functor S0M =
M b E*(E; o) introduced in x11. What we have really done is construct the symme*
*tric
monoidal functor
(S0; iS0; zS0): (FMod ; b; E*) ! (FMod ; b; E*) (12:9)
where iS0: S0M b S0N ! S0(M b N) is given by
M b E*(E; o) b N b E*(E; o)~= M b N b(E*(E; o) E*(E; o))
MNOE
! M b N b E*(E; o)
and zS0: E* ! S0E* is just jR: E* ! E*(E; o). We saw iS0 in diag. (12.3).
We can now reinterpret Lemma 12.6 as saying that the natural transformations
0: S0! S0S0 and ffl0: S0! I are monoidal, thus making S0 a comonad in FAlg. Th*
*en
diag. (12.3) simply states that ae is monoidal. Since E* = E*(T ) by definitio*
*n, the
other needed axiom reduces to aeT = zS0, which we have by Prop. 11.22(b).
Fourth Answer We enrich the object SM = FMod *(A; M) in x10 to include the
multiplicative structure.
Theorem 12.10 Assume that E*(E; o) is a free E*module (which is true for E*
* =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Then we can make S a symmetric
monoidal comonad in FMod and hence a comonad in FAlg.
The definition of stable algebra is now clear.
Definition 12.11 A stable (Ecohomology) algebra is an Scoalgebra in FAlg, i*
*. e. a
complete Hausdorff filtered E*algebra M equipped with a continuous homomorphism
aeM : M ! SM of E*algebras that satisfies the coaction axioms (8.7).
JMB  67  23 Feb 1995
Stable cohomology operations
If a closed ideal L M is invariant (see Defn. 10.13), then M=L inherits a *
*stable
algebra structure.
Theorem 12.12 Assume that E*(E; o) is a free E*module (which is true for E*
* =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Then given a complete Hausdorff
filtered E*algebra M (i. e. object of FAlg), a stable comodule algebra structu*
*re on
M in the sense of Defn. 12.7 is equivalent to a stable algebra structure on M i*
*n the
sense of Defn. 12.11.
Theorem 12.13 Assume that E*(E; o) is a free E*module (which is true for E*
* =
H(F p), BP , MU, KU, or K(n) by Lemma 9.21). Then:
(a) The natural transformation ae: E*()^ ! S(E*()^) defined on spaces by
diag. (10.10) (or ae: E*(; o)^ ! S(E*(; o)^) for spectra) is monoidal and mak*
*es
E*(X)^ a stable algebra for any space X;
(b) ae is universal: given a cocommutative comonoid object C in FMod , any m*
*ul
tiplicative transformation
X: E*(X; o) ! FMod *(C; E*(X; o))
(or bX: E*(X; o)^ ! FMod *(C; E*(X; o)^)) that is defined for all spectra X and
natural on Stab is induced from aeX by a unique morphism f: C ! A of comonoids
in FMod as
aeX * * *
X: E*(X; o) ! S(E (X)^) = FMod (A; E (X)^)
Hom(f;1) *
! FMod (C; E*(X)^) :
Proof of Thms. 12.10 and 12.13 In proving Thm. 10.12, we made A = E*(E; o) an
E*module object. We add the necessary monoidal structure to S = FMod *(A; ) in
five steps.
Step 1. We construct the symmetric monoidal functor
(S; iS; zS): (FMod *; b; E*) ! (Mod *; ; E*) : (12:14)
We start from the ring spectrum E, with multiplication OE: E ^ E ! E, unit j: T*
* +!
E, and vaction v: E ! E, and note that it is automatically an E*algebra objec*
*t in
the symmetric monoidal category (Stab*; ^; T +) in the sense of Defn. 7.12. We *
*apply
the Ecohomology functor (9.14) to make A an E*algebra object in FMod *op, wi*
*th
the comultiplication
OE* *
A : A = E*(E; o) ! E (E ^ E; o) ~=A b A
and counit fflA = j*: A = E*(E; o) ! E*(T +; o) = E*. Then Lemma 7.14 produces
the desired functor (12.14), with zS: E* ! SE* given on v 2 E* by eq. (7.15) as
zSv = j* O(v)* = v*: A = E*(E; o) ! E*(T +; o) = E*: (12:15)
This identifies zS with jR. Then S takes monoid objects in FMod *(i. e. objec*
*ts of
FAlg) to monoid objects in Mod * (i. e. E*algebras).
JMB  68  23 Feb 1995
x12. What is a stable algebra?
Step 2. To prove that ae: E*(; o) ! S(E*(; o)^) is monoidal, we need to c*
*heck
commutativity of the diagram in Mod
aeXaeY
E*(X; o) E*(Y; o) _________S(E*(X;o)^) S(E*(Y; o)^)
 
 iS
 ?

x S(E*(X; o)^ bE*(Y; o)^) (12:16)

 
 Sx
? ae ?
E*(X ^ Y; o) _________________S(E*(X^XY;^o)^)Y
By naturality, it is enough to take X = Y = E and evaluate on the universal ele*
*ment
. By construction, aeE = idA2 SA. By the definition (7.11) of iS, the upper ro*
*ute
gives A 2 S(AA), which by definition corresponds under Sx to OE* 2 SE*(E^E; o)
as required.
Because E* = E*(T +; o), the other needed diagram reduces to zS = aeT, whic*
*h we
have by eq. (12.15).
Step 3. For later use, we combine diag. (12.16) (still in the case X = Y = *
*E) with
the commutative square
aeE
E*(E; o) __________SE*(E;o)
 
OE* SOE*
 
? ae ?
E*(E ^ E; o) ______SE*(E^EE;^o)E
and the definition of A to obtain the following commutative diagram, which inv*
*olves
only A,
aeE
A ____________________________SA
 
 
 A S A (12:17)
 
? aeEaeE iS ?
A b A _______SA b SA ______S(A b A)
Step 4. The monoidality of is a formal consequence of that of ae. The two
commutative diagrams to check are
M N
SM SN _________SSM SSN
 zS
  * _______ *
 iS(SM;SN) E SE
   
 ?  
iS(M;N) S(SM b SN) zS SzS
   (12:18)
   
 SiS(M;N) ? E* ?
  SE* ______SSE*
? (M bN) ?
S(M b N) __________SS(Mb N) (ii)
(i)
JMB  69  23 Feb 1995
Stable cohomology operations
where we again leave some tensor products uncompleted.
As (i) is natural in M and N, we may work with the universal example M = N =
A and evaluate on idA idA. The upper route gives the element
A aeEaeE iS
A ! A b A ! SA b SA ! S(A b A)
of SS(A b A). The lower route gives
aeE S A
A ! SA ! S(A b A);
which we just saw in diag. (12.17) is the same.
Since zS = aeT, (ii) reduces to axiom (8.7)(i) for the Scoalgebra E* = E*(*
*T ).
Step 5. We next check that ffl is monoidal; this too is formal. As ever, th*
*ere are
two diagrams:
SM SN E*
 Q
 Q fflffl @ =
(i) iS(M;N) Q Q (ii) zS @ (12:19)
? QQs ? @ @@R
ffl ffl
S(M b N) ______M b N SE* ______E*
Again we take M = N = A in (i) and evaluate on idA idA. The lower route gives
A = , by the definition of A . This agrees with ffl ffl, since ffl idA= .*
* For (ii),
it is clear from eq. (12.15) that fflzSv = v.
In Thm. 12.13(b), we are given a comonoid object C, equipped with morphisms
C : C ! C b C and fflC : C ! E* in FMod *. Let us write (V; iV ; zV ) for the *
*symmet
ric monoidal functor with V = FMod *(C; ) that results from Lemma 7.9. Theo
rem 10.16(b) provides the unique morphism f: C ! A in FMod * that induces V from
S as in eq. (10.17).
We compare diag. (12.16) and a similar diagram with V in place of S. Evalua*
*tion
of the universal case X = Y = E on shows that (f f) O C = A Of: C ! A b A.
Since T + takes 1 2 E* = E*(T ) to the unit element zV 2 V E*, eq. (10.17) shows
that fflC = fflA Of. 
Comodule algebras We can now fill in the missing proofs on comodule algebras.
By construction, the isomorphism S0M ~= SM in (11.4) transforms the symmetric
monoidal structure (12.9) on S0 into the symmetric monoidal structure (12.14) o*
*n S.
Also, ae0is monoidal and we have diag. (12.3).
Proof of Lemma 12.6 If we replace S by S0in the four diagrams (12.18) and (12.*
*19)
for M = N = E*, we obtain exactly the diagrams we need. 
Proof of Thm. 12.12 The isomorphism S0M ~=SM is now an isomorphism of alge
bras, and the two definitions agree. 
Proof of Thm. 12.8 For (a), we combine Thm. 12.13(a) with Thm. 12.12.
In (b), Thm. 11.14(b) provides the unique homomorphism f: E*(E; o) ! B of
E*modules that induces from ae as in eq. (11.16); it corresponds to the eleme*
*nt
(E) under the isomorphism E*(E; o) b B ~=Mod *(E*(E; o); B) of Lemma 6.16(a).
JMB  70  23 Feb 1995
x13. Operations and complex orientation
If we evaluate (T +) on 1, we see that f1 = 1. The multiplicativity of is expr*
*essed
as a diagram resembling (12.3) with B in place of E*(E; o). We evaluate it in *
*the
universal case X = Y = E on and again use Lemma 6.16(a) to convert elements
of E*(E ^ E; o) b B to module homomorphisms E*(E; o) E*(E; o) ! B, with the
help of E*(E ^ E; o) ~=D(E*(E; o) E*(E; o)) from Thms. 9.20 and 9.25. The upper
route yields OEB O(f f): E*(E; o) E*(E; o) ! B. Since x = OE 2 E*(E ^ E; o),
the lower route yields
x OE* f
E*(E; o) E*(E; o) ! E*(E ^ E; o) ! E*(E; o) ! B :
Thus f is multiplicative and so is an E*algebra homomorphism. 
13 Operations and complex orientation
In this section, we show how a complex orientation on E determines the elem*
*ents
bi2 E*(E; o) from our point of view. We assume that E*(E; o) is free, so that x*
*x11, 12
apply. We pay particular attention to the plocal case, and the main relations *
*that
apply there.
Complex projective space We recall from Defn. 5.1 that a complex orientation
for E yields a first Chern class x() 2 E2(X) for each complex line bundle over
any space X. As the Hopf line bundle over C P 1 is universal, we need only stu*
*dy
x = x() 2 E2(C P 1). Thus C P 1 is our third test space. Since E*(C P 1) =
E*[[x]] by Lemma 5.4, the coaction ae on E*(C P 1) is completely determined by *
*aex,
multiplicativity, E*linearity, and continuity.
Definition 13.1 Given a complex orientation for E, we define the elements bi 2
E2(i1)(E; o) for all i 0 by the identity
1X
aex = b(x) = xi bi in E*(C P 1) b E*(E; o) ~=E*(E; o)[[x]], (13:2)
i=0
where b(x) is a convenient formal abbreviation that will rapidly become essenti*
*al.
Equivalently, according to eq. (11.19), the action of any operation r 2 A*
* =
E*(E; o) on x is given as
1X
rx = xi in E*(C P 1) = E*[[x]] . (13:3)
i=0
Remark Our indexing convention is taken from [32]. We warn that biis often wri*
*tten
bi1 (e. g. in [3]), as its degree suggests; the latter convention is appropria*
*te in the
current stable context, where b0 = 0 (see below), but less so in the unstable c*
*ontext
of [9], where (our) b0 does become nonzero.
Since the Hopf bundle is universal, eqs. (13.2) and (13.3) carry over by na*
*turality
to the Chern class x() of any complex line bundle over any space X (except that
when X is infinitedimensional and E*(X) is not Hausdorff, the infinite series *
*force
us to work in the completion E*(X)^).
JMB  71  23 Feb 1995
Stable cohomology operations
Proposition 13.4 The elements bi2 E2(i1)(E; o) have the following propertie*
*s:
(a) b0 = 0 and b1 = 1, so that b(x) = x1 + x2b2 + x3b3 + : :;:
(b) The Chern class x 2 E2(C P 1; o), regarded as a map of spectra x: CP 1 !*
* E,
induces x*fii= bi2 E*(E; o), where fii2 E2i(C P 1) is dual to xi(as in Lemma 5.*
*4(c));
(c) Sbk is given by
Xk
Sbk = B(i; k) bi in E*(E; o) E*(E; o),
i=1
wherePB(i; k) denotes the coefficient of xk in b(x)i, or, in condensed notation*
*, Sb(x) =
ib(x)i bi;
(d) fflSbi= 0 for all i > 1, so that fflSb(x) = x.
Proof We prove (a) by restricting to C P 1~= S2 and comparing with eq. (11.24).
Part (b) is an application of Prop. 11.26, using eq. (13.3). For (c) and (d), w*
*e take
M = E*(C P 1) in diags. (11.12) and evaluate on x. 
The formal group law Now CP 1 = K(Z ; 2) is an Hspace, whose multiplication
map : CP 1x CP 1 ! CP 1 may be defined by * = p*1 p*2 for the Hopf bundle
. We therefore have from eq. (5.13)
X
*x = F (xx1; 1xx) = xx1 + 1xx + ai;jxixxj; (13:5)
i;j
where F (x; y) denotes the formal group law (5.14). When we apply ae and write *
*x for
x x 1 and y for 1 x x, we obtain from eq. (13.2) and naturality
X
b(F (x; y)) = FR(b(x); b(y)) = b(x) + b(y) + b(x)ib(y)jjRai;j(13:6)
i;j
in E*(E; o)[[x; y]], which is difficult to express without using the formal not*
*ations
b(x) and F (x; y). On the right, FR(X; Y ) is another convenient abbreviation.*
* (In
the language of formal groups, the series b(x) is an isomorphism between the fo*
*rmal
group laws F and FR.)
The plocal case The above rather formidable machinery does simplify in common
situations. When the ring E* is plocal, most of the bi are redundant.
Lemma 13.7 Assume that E* is plocal. Then if k is not a power of p, the ele*
*ment
bk 2 E*(E; o) can be expressed in terms of E*, jRE*, and elements of the form b*
*pi.
Proof Considerithejcoefficient of xiyj in eq. (13.6), where i+j = k. On the le*
*ft, there
is a term kibk from bk(x+y)k, and all other terms involve only the lower b's. *
*On the
i j
right, no b beyond bi or bj occurs. If ki is not divisible by p and so is a un*
*it in E*,
we deduce an inductive reduction formula for bk. This can be done whenever k is*
* not
a power of p, by choosing i = pm and j = k  pm , where m satisfies pm < k < pm*
*+1 .

We therefore reindex the b's.
JMB  72  23 Feb 1995
x14. Examples of ring spectra for stable operations
Definition 13.8 When E* is plocal, we define b(i)= bpifor each i 0.
We still need to use the internal details of Lemma 13.7 to express each b(*
*k)
inductively in terms of the b(i), ai;j, and jRai;j.
The main relations In the plocal case, it is appropriate to study instead of *
* the
much simpler pth power map i: CP 1 ! CP 1 constructed from . In cohomology,
it must induce
X
i*x = [p](x) = px + gixi+1 in E*(C P 1) ~=E*[[x]] (13:9)
i>0
for suitable coefficients gi2 E2i (which are usually written ai; but we need t*
*o avoid
conflict with certain other elements also known as aithat appear in x14). The f*
*ormal
power series [p](x) is known as the pseries of the formal group law. The bund*
*le
interpretation is i* = p , so that
X
x(p ) = px() + gix()i+1 in E*(Z)^ (13:10)
i>0
for any line bundle over any space Z. (Again, completion is not necessary for
finitedimensional Z, or if the series [p](x) happens to be finite.)
When we apply ae, we obtain
X
b([p](x)) = [p]R(b(x)) = pb(x) + b(x)i+1jRgi in E*(E; o)[[x]],(13:11)
i>0
P i+1
where [p]R(X) denotes the formal power series pX + i(jRgi)X . We extract the
relations we need.
Definition 13.12 For each k > 0, we define the k th main stable relation in E*
**(E; o)
as
(Rk) : L(k) = R(k) in E*(E; o), (13:13)
k
where L(k) and R(k) denote the coefficient of xp in b([p](x)) and [p]R(b(x)) r*
*espec
tively.
The results of x14 will show that, despite appearances, the relations (Rk) *
*contain
all the information of eq. (13.6), with the understanding that the latter is us*
*ed only
to express (inductively) each redundant bj in terms of the b(i), E*, and jRE*, *
*in
accordance with Lemma 13.7.
14 Examples of ring spectra for stable operations
In x10, we developed a comonad S that, for favorable E, expresses all the s*
*tructure
of stable Ecohomology operations. In x11, we described an equivalent comonad S0
in terms of structure on the algebra E*(E; o). In this section, we give the com*
*plete
description of E*(E; o) for each of our five examples, namely E = H(F p), MU, B*
*P ,
KU, and K(n). (The first splits into two, and we break out the degenerate speci*
*al
case H(Q ) = K(0) merely for purposes of illustration.)
JMB  73  23 Feb 1995
Stable cohomology operations
All the results here are well known, but serve as a guide for [9]. Our purp*
*ose is to
exhibit the structure of the results, not to derive them. As Milnor discovered *
*[22] in the
case E = H(F p), the most elegant and convenient formulation of stable cohomolo*
*gy
operations is the Second Answer of xx10, 12, consisting of the multiplicative (*
*i. e.
monoidal) coaction (10.7)
aeX : E*(X) ! E*(X) b E*(E; o)
for each space X (or on E*(X; o), for a spectrum X).
The point is that the knowledge of aeX on a few simple test spaces and test*
* maps
is sufficient to suggest the complete structure of E*(E; o). The test spaces s*
*tudied
so far include the point T in Prop. 11.22, the sphere Sk in eq. (11.24), and co*
*mplex
projective space CP 1 in eq. (13.2).
In each case, we specify (when not obvious):
(i)The coefficient ring E*;
(ii)The E*algebra E*(E; o);
(iii)jR: E* ! E*(E; o), the right unit ring homomorphism;
(iv) : E*(E; o) ! E*(E; o) E*(E; o), the comultiplication;
(v) ffl: E*(E; o) ! E*, the counit.
(See Prop. 11.3 for E*(E; o) and jR. By construction and Lemma 12.6, and ffl *
*are
homomorphisms of E*algebras and of E*bimodules.) In most cases, the results a*
*llow
us to express the universal property of E*(E; o) very simply.
Example: H(F 2) We take E = H = H(F 2), the EilenbergMacLane spectrum
representing ordinary cohomology with coefficients F2. The main reference is M*
*il
nor [22], and many of our formulae, diagrams and results can be found there.
The appropriate test space is R P 1 = K(F 2; 1), an Hspace with multiplication
: RP 1 x RP 1 ! R P 1, and we use a mod 2 analogue of complex orientation.
We have H*(R P 1) = F2[t], with a polynomial generator t 2 H1(R P 1), and
*t = tx1 + 1xt is forced. By analogy with Prop. 13.4(a), we must have
X
aet = t1 + tici in H*(R P 1) b H*(H; o) ~=H*(H; o)[[t]]
i>1
for certain coefficients ci2 H*(H; o). The analogue of eq. (13.6) is simply
X X X
(t+u)1 + (t+u)ici= t1 + tici+ u1 + uici
i>1 i>1 i>1
i j
in H*(H; o)[[t; u]]. Because the left side contains the terms ijtijujci, we m*
*ust have
ci= 0 unless i is a power of 2. Imitating Defn. 13.8, we write i= c2i2 H2i1(H;*
* o)
for i > 0, so that now
1X i
aet = t1 + t2 i in H*(R P 1) b H*(H; o) ~=H*(H; o)[[t]] . (14:1)
i=1
Because H_1 = RP 1, this formula is valid for every t 2 H1(X), for all spaces X*
*. It
is reasonable to define also 0 = c1 = 1. Milnor proved that this is all there i*
*s.
JMB  74  23 Feb 1995
x14. Examples of ring spectra for stable operations
Theorem 14.2 (Milnor) For the EilenbergMacLane ring spectrum H = H(F 2):
(a) H*(H; o) = F2[1; 2; 3; : :]:, a polynomial algebra over F2 on the genera*
*tors
i2 H2i1(H; o) for i > 0;
(b) In the complex orientation for H(F 2), b(i)= 2ifor all i > 0, b(0)= 1, a*
*nd
bj = 0 if j is not a power of 2;
(c) is given by
k1Xi
k = k 1 + 2ki i+ 1 k in H*(H; o) H*(H; o);
i=1
(d) fflk = 0 for all k > 0.
Proof Milnor proved (a) in [22, App. 1]. The complexified Hopf line bundle ov*
*er
R P 1 has Chern class t2. We compare aet2 with eq. (13.2) and read off (b). (Fo*
*r i not
a power of 2, this is a stronger statement than Lemma 13.7 provides.) For (c) a*
*nd
(d), we substitute M = H*(R P 1) into diags. (11.12) and evaluate on t. 
Corollary 14.3 Let B be a discrete commutative graded F 2algebra. Assume
that the operation : H*(X; o) ! H*(X; o) b B is multiplicative (i. e. monoidal)*
* and
natural on Stab*. Then on t 2 H1(R P 1; o) = H1(R P 1), has the form
1X i
t = t1 + t2 0i in H*(R P 1) b B ~=B[[t]],
i=1
i1)
where the elements 0i2 B(2 determine uniquely for all X and may be chosen
arbitrarily.
Proof We combine the universal property Thm. 12.8(b) of H*(H; o) with the uni
versal property of the polynomial algebra F2[1; 2; : :]:. 
Example: H(F p) (for p odd) We take E = H = H(F p), the EilenbergMacLane
spectrum that represents ordinary cohomology with coefficients Fp. The main ref*
*er
ence is still Milnor [22].
We have a complex orientation, therefore by Defn. 13.8 the elements b(i)2
H2(pi1)(H; o); b(i)is normally written i for i 0, where 0 = b(0)= 1. As in
the previous example, eq. (13.6) simplifies to show that bj = 0 whenever j is n*
*ot a
power of p, so that for the Chern class x = x() 2 H2(X) of any complex line bun*
*dle
over X, eq. (13.2) reduces to
1X i
aeX x = x1 + xp i in H*(X) b H*(H; o) . (14:4)
i=1
We need one more test space, the infinitedimensional lens space L = K(F p;*
* 1),
which contains S1 and is another Hspace. The cohomology H*(L) = Fp[x](u) has
an exterior generator u 2 H1(L) which restricts to u1 2 H1(S1). As the polynomi*
*al
generator x 2 H2(L) is the Chern class of a certain complex line bundle, aeLx i*
*s given
by eq. (14.4). This leaves only aeLu, which must take the form
X X
aeLu = xiai+ uxici
i i
JMB  75  23 Feb 1995
Stable cohomology operations
for certain welldefined coefficients ai; ci 2 H*(H; o). By restricting to S1 *
* L and
comparing with eq. (11.24), we see that c0 = 1 and a0 = 0.
The multiplication on L induces *u = ux1 + 1xu and *x = xx1 + 1xx.
Expansion of *aeLu = aeLxL *u = (aeLu)x1 + 1x(aeLu) yields
X X
(xx1 + 1xx)i ai+ (ux1 + 1xu)(xx1 + 1xx)i ci
i X iX X X
= (xix1)ai+ (uxix1)ci+ (1xxi)ai+ (1xuxi)ci :
i i i i
For i > 0, there is no term with u x xi on the right, but there is on the left,*
* which
forces ci= 0 for i > 0. When we take coefficients of xix xj, we find as in Lemm*
*a 13.7
that ai= 0 unless i is a power of p. Again we reindex, defining oi= api2 H2pi1*
*(H; o)
for all i 0, so that now
1X i
aeLu = u1 + xp oi in H*(L) b H*(H; o) . (14:5)
i=0
Again, the elements n and on give everything.
Theorem 14.6 (Milnor) For the EilenbergMacLane ring spectrum H = H(F p)
with p odd:
(a) As a commutative algebra over Fp,
H*(H; o) = Fp[1; 2; 3; : :]: (o0; o1; o2; : :):;
with polynomial generators i= b(i)2 H2(pi1)(H; o) for i 1 and exterior genera*
*tors
oi2 H2pi1(H; o) for i 0;
(b) : H*(H; o) ! H*(H; o) H*(H; o) is given by
k1Xi
k = k1 + pkii+ 1k
i=1
k1Xi
ok = ok1 + pkioi+ 1ok;
i=0
(c) fflk = 0 for all k > 0 and fflok = 0 for all k 0.
Proof Part (a) is Thm. 2 of Milnor [22]. Parts (b) and (c) comprise Thm. 3 [ib*
*id.],
but also follow by substituting aeL into diags. (11.12) and evaluating on x and*
* u.
(Proposition 13.4 also gives k and fflk.) 
We have the analogue of Cor. 14.3.
Corollary 14.7 Let B be a discrete commutative graded Fpalgebra. Assume that
the operation : H*(X; o) ! H*(X; o) b B is multiplicative and natural on Stab*.
Then on H*(L) = Fp[x] (u), has the form
1X i 1X i
x = x1 + xp 0i; u = u1 + xp o0i;
i=1 i=0
i1) 0 (2pi1)
where the elements 0i2 B2(p and oi 2 B determine uniquely for all X
and may be chosen arbitrarily. 
JMB  76  23 Feb 1995
x14. Examples of ring spectra for stable operations
Example: H(Q ) We take E = H = H(Q ), the EilenbergMacLane spectrum that
represents ordinary cohomology with rational coefficients Q . There are no inte*
*resting
stable operations.
Theorem 14.8 For the EilenbergMacLane ring spectrum H = H(Q ), we have
H*(H; o) = H*(H(Q ); o) = Q. 
Example: MU Our main reference is Adams [3, II.x11]. The coefficient ring is
MU* = Z[x1; x2; x3; : :]:, with polynomial generators xn in degree 2n that are*
* not
canonical. We have complex orientation, almost by definition, and therefore the
elements bn 2 MU2n2(MU; o).
The good description of MU* was given by Quillen [30, Thm. 6.5], as the uni*
*versal
formal group: it is generated as a ring by the coefficients ai;j2 MU* that app*
*ear
in the formal group law (5.14), subject to the relations (5.15). Hence the ele*
*ments
jRai;jdetermine jR.
Theorem 14.9 For the unitary cobordism ring spectrum MU:
(a) As a commutative MU*algebra, MU*(MU; o) = MU*[b2; b3; b4; : :]:, with p*
*oly
nomial generators bi2 MU2(i1)(MU; o) for i > 1;
(b) jRai;j2 MU*(MU; o) is determined by eq. (13.6);
(c) is given by
Xk
bk = bk1 + B(i; k)bi in MU*(MU; o) MU*(MU; o),
i=2
where B(i; k) denotes the coefficient of xk in b(x)i;
(d) fflbk = 0 for all k 2.
Proof Part (a) is standard. In (b), the coefficient of xiyj in eq. (13.6) pro*
*vides an
inductive formula for jRai;j. Proposition 13.4 provides (c) and (d). 
As MU*(MU; o) is a polynomial algebra, Cor. 14.3 carries over to this case.
Corollary 14.10 Let B be a discrete commutative MU*algebra. Assume that
the operation : MU*(X; o) ! MU*(X; o) b B is multiplicative and natural on Stab*
**.
Then on x 2 MU2(C P 1), has the form
1X
x = x1 + xib0i in MU*(C P 1) b B ~=B[[x]],
i=2
where the elements b0i2 B2(i1)determine uniquely for all X and may be chosen
arbitrarily. 
In other words, there are no relations over MU* between the bi. The dual
MU*(MU; o) is known as the LandweberNovikov algebra. The results for are
no longer amenable to explicit expression as in Thms. 14.2 and 14.6.
Example: BP The main reference is still Adams [3, II.x16]. The coefficient *
*ring
is now BP *= Z(p)[v1; v2; v3; : :]:, a polynomial algebra on Hazewinkel's gener*
*ators vi
JMB  77  23 Feb 1995
Stable cohomology operations
of degree 2(pi1) for i > 0. (One could instead use Araki's generators [5] or *
*any
other system of polynomial generators, with only slight modifications.)
We still have complex orientation, but because BP *is plocal, we need only*
* the
generators b(i)from Defn. 13.8, where b(0)= 1. Moreover, it is sufficient to wo*
*rk with
the pseries (13.9), because its coefficients gi generate BP *as a Z(p)algebra*
* (as we
shall see in more detail in x15). We write wi= jRvi2 BP*(BP; o).
Theorem 14.11 For the BrownPeterson ring spectrum BP :
(a) As a commutative BP *algebra, BP*(BP; o) = BP *[b(1); b(2); b(3); : :]:*
*, with poly
nomial generators b(i)= bpi2 BP2(pi1)(BP; o) for each i > 0;
(b) The nth main relation (Rn) in eq. (13.13) provides an inductive formula *
*for
wn = jRvn 2 BP*(BP; o);
(c) is given by
pkX
b(k)= b(k)1 + B(i; pk)bi in BP*(BP; o) BP*(BP; o),
i=2
k i
where B(i; pk) denotes the coefficient of xp in b(x) (and Lemma 13.7 is used *
*to
express b(x) and bi in terms of the b(j)and BP *);
(d) fflb(k)= 0 for all k > 0. 
We shall find that the generators b(i)are better suited to [9] than Quillen*
*'s original
generators ti, or their conjugates hi, which were used in [8]. We have the anal*
*ogue of
Cor. 14.10.
Corollary 14.12 Let B be a discrete commutative BP *algebra. Assume the op
eration : BP *(X; o) ! BP *(X; o) b B is multiplicative and natural on Stab*. T*
*hen
on x 2 BP 2(C P 1), has the form
1X
x = x1 + xib0i in BP *(C P 1) b B ~=B[[x]]
i=2
for certain elements b0i2 B2(i1). The elements b0(i)= b0pifor i 1 determine
uniquely for all X and may be chosen arbitrarily. 
Example: KU We take E = KU = K, the complex Bott spectrum, which we
constructed in xx3, 9. Its coefficient ring is the ring Z[u; u1] of Laurent po*
*lynomials
in u 2 KU2, and one writes v = jRu. The complex orientation (5.2) furnishes
elements bi 2 KU*(KU; o), of which b1 = 1. We computed its formal group law
F (x; y) = x + y + uxy in eq. (5.16); thus eq. (13.6) reduces to
b(x + y + uxy) = b(x) + b(y) + b(x)b(y)v : (14:13)
This is small enough for explicit calculation. The coefficient of xyiyields the*
* relation
(i+1)bi+1+ iubi= biv (14:14)
since on the left,
bj(x + y + uxy)j bjyj + jbjyj1x(1 + uy) mod x2 .
JMB  78  23 Feb 1995
x14. Examples of ring spectra for stable operations
(Compare [3, Lemma II.13.5].) This includes the special case 2b2 + u = v for i *
*= 1.
Generally, for i > 1 and j > 1, the coefficient of xiyj yields the relation
min(i;j)Xi+jk i
bibj = ukbi+jkv1; (14:15)
k=0 i k
which serves to reduce any product of b's to a linear expression. Thus the gen*
*eral
expression c in our generators may be assumed linear in the b's. Further, for *
*large
enough m, cvm will have no negative powers of v; if we use eq. (14.14) to remov*
*e all
the positive powers of v, c takes the form
c = uq(1u1 + 2u2b2 + 3u3b3 + : :+:nunbn)vm (14:16)
for some integers i, n, and q. This suggests part (a) of the following.
Lemma 14.17 In KU*(KU; o):
(a) Every element can be written in the form (14.16);
(b) The element c in eq. (14.16) is zero if and only if i= 0 for all i.
This, with eq. (14.14), is a complete description of KU*(KU; o). We shall g*
*ive a
proof in [9].
Theorem 14.18 For the complex Bott spectrum KU:
(a) As a commutative algebra over KU* = Z[u; u1], KU*(KU; o) has the gener
ators:
v = jRu 2 KU2(KU; o);
v1 = jRu1 2 KU2(KU; o);
bi2 KU2i2(KU; o) for i > 1;
subject to the relations (14.14) and (14.15);
(b) As a KU*module, KU*(KU; o) is spanned by the monomials vn and bivn, for
all i > 1 and n 2 Z, subject to the relations (14.14) (multiplied by any vn);
(c) is given by
Xk
bk = bk1 + B(i; k)bi in KU*(KU; o) KU*(KU; o),
i=2
where B(i; k) denotes the coefficient of xk in b(x)i;
(d) ffl is given by fflbi= 0 for all i > 1.
Proof Parts (a) and (b) follow from Lemma 14.17. Parts (c) and (d) are included
in Prop. 13.4. 
Although we no longer have a polynomial algebra, we still have part of Cor.*
* 14.10.
Corollary 14.19 Let B be a discrete commutative KU*algebra. Then any oper
ation : KU*(X; o) ! KU*(X; o) b B that is multiplicative and natural on Stab* is
uniquely determined by its values on KU*(C P 1). 
JMB  79  23 Feb 1995
Stable cohomology operations
The module KU*(KU; o) What makes the description (14.16) unsatisfactory is
that m is not unique; we can always increase m and use eq. (14.14) to remove th*
*e extra
v's to obtain another expression of the same form that looks quite different. F*
*or exam
ple, (b3+ ub2)=2 = (2b4+ 3ub3+ u2b2)v1 2 KU*(KU; o), in spite of the denominat*
*or
2. It is notoriously difficult to write down stable operations in KU*()h(equiv*
*alently,_i
linear functionals KU*(KU; o) ! KU*) other than 1 = id and 1[] = (the
complex conjugate bundle). Following Adams [3], we develop an alternate descrip*
*tion
from which the freeness of KU*(KU; o) will follow easily.
First, we note that Lemma 14.17 implies that KU*(KU; o) has no torsion, whi*
*ch
allows us to work rationally and consider
KU*[v; v1] KU*(KU; o) KU*[v; v1] Q :
The key idea is that if we localize at a prime p, we have available (algebraica*
*lly)
the Adams operation k for any invertible k 2 Z(p). Rationally, we have k for all
nonzero k 2 Q. It is characterized by the properties that it is additive, multi*
*plicative,
and satisfies k[] = [k ] = []k for any line bundle .
To compute ku, we rewrite eq. (3.32) as uu2 = []  1 and apply k. As stabil*
*ity
requires ku2 = u2, and u22= 0, we find
(ku)u2 = []k  1 = (1 + uu2)k  1 = kuu2;
Hence ku = ku. Then eq. (11.23) becomes
= ku = ku : (14:20)
The linear functional : KU*(KU; o) ! KU* Q is multiplicative because
k is, as can be seen by expanding k(x) by eq. (12.4). (These are precisely the
multiplicative linear functionals.)
We apply to eq. (14.14) to obtain, by induction starting from b1 = 1,
k n1
= k1 u : (14:21)
n
Alternatively, for any n > 1 we can write formally
(v  u)(v  2u) : :(:v  (n1)u) * 1
bn = ______________________________2 KU [v; v ] Q (14:22)
n!
and replace v by ku everywhere.
Lemma 14.23 An element c 2 KU*[v; v1] Q lies in KU*(KU; o) if and only if
2 KU* Z(p)for all primes p and integers k > 0 such that p does not divi*
*de
k.
From this we deduce the freeness of KU*(KU; o).
Proof of Lemma 9.21 for E = KU Denote by Fm;n the free KU*module with basis
{vm ; vm+1 ; : :;:vn}. It is enough to show that for any m, KU*(KU; o)\(Fm;m Q*
* ) is a
free KU*module; then any basis extends to a basis of KU*(KU; o)\(Fm1;m+1 Q ),
and thence by induction to a basis of KU*(KU; o). We may multiply by vm and work
with F0;2minstead.
JMB  80  23 Feb 1995
x14. Examples of ring spectra for stable operations
We therefore work in degree zero and take any element
c = 0 + 1w + 2w2 + : :+:n1wn1 (14:24)
in KU0(KU; o) \ (F0;n1 Q), where each i2 Q and we write w = u1v. We have
only to find a common denominator that guarantees i2 Z for all i.
Given any prime p, we choose n distinct positive integers k1, k2, . . . kn,*
* not divisible
by p; then by eq. (14.20),
n1X
= ikij2 Z(p):
i=0
We solve these n linear equations for the i in terms of the , which requ*
*ires
division by the Vandermonde determinant
Y
(p) = det(ki1j) = (ki kj) :
i;j 1j n, the obviousQchoices kj = j yield i 2 Z(p),
because then p does not divide (p). We take = pn (p). 
Before we establish Lemma 14.23, we need a result [3, Lemma II.13.8] which
explains the role of the b's.
Lemma 14.25 Let c be an element of KU*[v; v1] Q . Then c is a KU*linear
combination of the elements 1, v = b1v, b2v, b3v; : :i:f and only if 2 K*
*U* for
all integers k > 0 .
Proof Necessity is clear from eq. (14.21). We may reduce sufficiency to the ca*
*se when
c has degree 0 and write c as a Laurent series in w = u1v. By taking k very la*
*rge, it
is clear that c has no negative powers of w; this allows us to write (see eq. (*
*14.22))
Xn w Xn
c = i = 0 + ibiv
i=0 i i=1
P n ikj
for some n and suitable coefficients i2 Q . By eq. (14.21), = i=0i i .*
* By
induction on k from 1 to n+1, 2 Z yields k (1)k0 mod Z . But n+1 = 0.
Therefore 0 2 Z, and i2 Z for all i. 
Proof of Lemma 14.23 Again, necessity is clear. For sufficiency, we assume gi*
*ven
c in the form eq. (14.24). Let m be the maximum exponent of any prime in the
denominators of the i, so that pm i2 Z(p)for all i and all primes p. Then pm 2
Z (p)for all integers k > 0 and all primes p.
If p does not divide k, we have = km 2 Z (p)by hypothesis.
If k = pq, we have instead km = qm pm 2 Z (p), by our choice of m.
Thus for each k > 0, 2 pZ(p)= Z . Then Lemma 14.25 shows that
cwm 2 KU*(KU; o). 
Example K(n) The coefficient ring is now the plocal ring K(n)* = Fp[vn; v1n],
still with deg(vn) = 2(pn1), where p is odd. We write wn = jRvn, as we did for
BP . We have a complex orientation, and therefore elements b(i)for i 0, where
JMB  81  23 Feb 1995
Stable cohomology operations
b(0)= 1. Although the formal group law remains complicated, it is well known [3*
*2,
Thm. 3.11(b)] that over Fp, the pseries (13.9) reduces to exactly
n *
i*x = vnxp in K(n) [[x]], (14:26)
n pn *
* pn
so that eq. (13.11) simplifies drastically to b(vnxp ) = b(x) wn. The coeffici*
*ent of x
n+i
yields wn = vn, and the coefficient of xp then yields
n pi1
bp(i)= vn b(i) in K(n)*(K(n); o). (14:27)
Lemma 14.28 Assume that k is not a power of p. Then:
(a) bk 2 K(n)2k2(K(n); o) can be expressed in terms of vn and the b(i);
(b) bk = 0 if k < pn.
Proof Part (a) comes from Lemma 13.7. For (b), we trivially have ai;j= 0 whene*
*ver
i + j < pn; in this range, eq. (13.6) behaves exactly as in Thm. 14.6 for H(F p*
*). 
We need one more test space. The infinite lens space L is not appropriate,*
* as
n 1
K(n)*(L) = K(n)*[x : xp = 0], where x is inherited from CP . (Because i is tr*
*ivial
n
on L, we must have xp = 0, which makes the structure of the AtiyahHirzebruch
n1
spectral sequence clear.) Instead,nwe use the finite skeleton Y = L2p , the*
* orbit
space of the unit sphere in C p under the action of the group Z=p S1 C . The
spectral sequence for K(n)*(Y; o) collapses because it can support no different*
*ial,
n 1
to give K(n)*(Y ) = (u) K(n)*[x : xp = 0], where u 2 K(n) (Y ) restricts
to u1 2 K(n)1(S1). (This fails to define u uniquely, because we can replace u *
*by
n1
u0= u + hvnuxp for any h 2 Fp.)
We know aeY x is given by eq. (13.2). We write
pn1X pn1X
aeY u = xiai+ uxici; (14:29)
i=0 i=0
which defines elements ai; ci2 K(n)*(K(n); o). (They are independent of the cho*
*ice
of u.) By restriction to S1 Y , we see that a0 = 0 and c0 = 1.
Unfortunately, Y is no longer an Hspace. The multiplication on L restricts*
* (after
a noncanonical deformation) to a partial multiplication on skeletons : L2k+1xL*
*2m !
L2(k+m)+1= Y , whenever k + m = pn  1. Clearly, K(n)*(L2k+1) = (u) K(n)*[x :
xk+1 = 0], with the coaction ae obtained from aeY by truncation; and similarly *
*for
K(n)*(L2m), except that uxm = 0 also.
As x is inherited from L, we have *x = xx1 + 1xx, for lack of any other pos*
*sible
terms in degree 2. For u, we must have
*u = ux1 + 1xu + vnuxkxxm
n1
for some 2 Fp. (The third term disappears if we replace u by u + (1)kvnuxp ,
but in any case is harmless.) We apply ae to , bearing in mind that wn = vn, and
carry out exactly the same algebra as for E = H(F p); the coefficients of u x x*
*j and
xix xj show that cj = 0 for all j > 0 and that ah = 0 for h not a power of p. We
therefore reindex, as usual.
Definition 14.30 We define a(i)= api2 K(n)2pi1(K(n); o), for 0 i < n.
JMB  82  23 Feb 1995
x15. Stable BP cohomology comodules
n
There is no a(n)because u does not lift to the next skeleton L2p +1. In the*
* new
notation, eq. (14.29) becomes
n1X i
aeY u = u1 + xp a(i) in K(n)*(Y ) K(n)*(K(n); o). (14:31)
i=0
Having odd degree, the a(i)satisfy a2(i)= 0.
Theorem 14.32 (Yagita) For the Morava Ktheory ring spectrum K(n):
(a) The commutative K(n)*algebra K(n)*(K(n); o) has the generators:
a(i)2 K(n)2pi1(K(n); o), for 0 i < n;
b(i)2 K(n)2(pi1)(K(n); o), for i > 0;
subject to the relations (14.27);
(b) jR is given by jRvn = wn = vn 2 K(n)*(K(n); o);
(c) is given by:
k1X i
a(k)= a(k)1 + bp(ki)a(i)+ 1a(k) for 0 k < n;
i=0
pk1X
b(k)= b(k)1 + B(i; pk)bi+ 1b(k) for k > 0;
i=2
k i
where B(i; pk) denotes the coefficient of xp in b(x) (and we use Lemma 14.28 *
*to
express b(x) and bi in terms of the b(i)and vn);
(d) ffla(k)= 0 for 0 k < n and fflb(k)= 0 for k > 0.
Proof The whole theorem is essentially due to Yagita [39], who used different *
*gen
erators. We proved (b) above. For (c) and (d), we substitute aeY in diags. (11.*
*12) as
usual and evaluate on u and x. 
Corollary 14.33 Let B be a discrete commutative K(n)*algebra. Then any op
eration : K(n)*(X; o) ! K(n)*(X; o) b B that is multiplicative and natural on S*
*tab*
is uniquely determined by its values on K(n)*(C P 1) and K(n)*(Y ). 
Remark For low k, the formula for b(k)simplifies by Lemma 14.28(b) to
k1X i
b(k)= b(k) 1 + bp(ki) b(i)+ 1 b(k) for 0 < k n .
i=1
15 Stable BP cohomology comodules
In this section we study stable modules in the case E = BP in more detail. *
*We find
it more practical to work with stable comodules, which by Thm. 11.13 are equiva*
*lent.
This is the context in which Landweber showed [17, 18] that the presence of a s*
*table
comodule structure on M imposes severe constraints on its BP *module structure.
JMB  83  23 Feb 1995
Stable cohomology operations
We recall that BP *= Z (p)[v1; v2; v3; : :]:, a polynomial ring on the Haze*
*winkel
generators vn of degree 2(pn1) (see [14]). It contains the wellknown ideals
In = (p; v1; v2; : :;:vn1) BP * (15:1)
for 0 n 1 (with the convention that I1 = (p; v1; v2; : :):, I1 = (p), and I0*
* =
0.). We show in Lemma 15.8 that they are invariant under the action of the stab*
*le
operations on BP *(T ) = BP *. Indeed, Landweber [17] and Morava [27] showed th*
*at
the In for 0 n < 1 are the only finitely generated invariant prime ideals in B*
*P *.
Nakayama's Lemma The fact that BP *is a local ring with maximal ideal I1
is extremely useful. The advantage is that once we know certain modules are fr*
*ee,
many questions can be answered by working over the more convenient quotient fie*
*ld
BP *=I1 ~=Fp. We say a BP *module M is of finite type if it is bounded above *
*and
each Mk is a finitely generated Z(p)module. (Remember that deg(vi) is negative*
*.)
Lemma 15.2 Assume that f: M ! N is a homomorphism of BP *modules of finite
type, with N free. Then:
(a) f is an isomorphism if and only if f F p: M F p! N F pis an isomorphism;
(b) f is a split monomorphism of BP *modules if and only if f Fp is monic;
(c) If the conditions in (b) hold, both M and Coker f are BP *free;
(d) f is epic if and only if f Fp is epic (even if N is not free).
Proof The "only if" statements are obvious. For the "if" statements, we first *
*con
sider f=p: M=pM ! N=pN. We filter M=pM and N=pN by powers of the ideal
(v1; v2; v3; : :):, so that for the associated graded groups, Gr (f=p): Gr(M=pM*
*) !
Gr (N=pN) is a module homomorphism over the bigraded ring Gr (BP *=(p)) =
F p[v1; v2; v3; : :]:, with Gr (N=pN) free. As M and N are bounded above, thes*
*e fil
trations are finite in each degree. It follows that if f Fp is epic (or monic*
*), so is
f=p.
Then the standard Nakayama's Lemma, applied to Z(p)modules in each degree,
gives (d). If f=p is monic and N is free, we must have Ker f pnM for all n; as*
* M
is of finite type, f must be monic, which gives (a) and some of (b). To see tha*
*t in
(c), M must be free, we lift a basis of M Fp to M and use the liftings to defi*
*ne
a homomorphism of BP *modules g: L ! M, with L free, that makes g Fp an
isomorphism. Then f Og is monic by what we have proved so far, and g is epic by
(d); therefore g must be an isomorphism.
To finish (b) and (c), we choose an Fpbasis of Coker(f Fp), lift it to N,*
* and use
it to define a homomorphism h: K ! N of BP *modules with K free. We use f and
h to define M K ! N, which by (a) is an isomorphism and identifies Coker f with
K. 
The main relations We need to make the structure of BP*(BP; o) more explicit
than in Thm. 14.11. The first few terms of the formal group law for BP in terms*
* of
the Hazewinkel generators are easily found:
F (x; y) x + y  v1xp1y mod (xp; y2). (15:3)
JMB  84  23 Feb 1995
x15. Stable BP cohomology comodules
Also, the pseries for BP begins with
[p](x) = px + (1pp1)v1xp + : ::: (15:4)
All we need to know about [p](x) beyond this is the standard fact *
*(e. g. [32,
Thm. 3.11(b)]) that X
i 2
[p](x) px + vixp mod I1 . (15:5)
i>0
For lack of alternative, bi = 0 whenever i1 is not a multiple of p1, so t*
*hat
b(x) = x + b(1)xp + : :.:The first main relation is well known and readily comp*
*uted
from Defn. 13.12, with the help of eq. (15.4), as
(R1) : v1 = pb(1)+ w1 ; (15:6)
or more easily, as the coefficient of xp1y in eq. (13.6), expanded using eq. (*
*15.3).
Subsequent relations (Rk) are far more complicated and answers in closed form a*
*re
not to be expected. To handle the right side R(k), we introduce the ideal W =
(p; w1; w2; : :): BP*(BP; o), the analogue of I1 for the right BP *action. Th*
*e right
side of eq. (13.11) simplifies by eq. (15.5) to
X i 2
pb(x) + b(x)p wi mod W .
i
i
When we expand b(x)p , all cross terms may be ignored, because they contain a f*
*actor
p 2 W , and we find
k1X i
R(k) pb(k)+ bp(ki)wi+ wk mod W 2. (15:7)
i=1
With slightly more attention to detail, we obtain a sharper, more useful re*
*sult. It
also implies that W = I1 BP*(BP; o), so that W is redundant.
Lemma 15.8 For any n > 0, we have wn vn mod InBP*(BP; o).
Proof We show by induction on n that the relation (Rn) simplifies as stated, s*
*tarting
from eq. (15.6) for n = 1. If the result holds for all i < n, we have wi vi 0 m*
*od In
for i < n. Then R(n) wn from eq. (15.7), as W 2 contains nothing of interest i*
*n this
degree. Meanwhile, the left side L(n) vn by eq. (15.5). 
Recall from Defn. 10.13 and Thm. 11.13 that an ideal J BP *is invariant if*
* it
is a stable subcomodule of BP *= BP *(T ); in view of Prop. 11.22(b), the neces*
*sary
and sufficient condition for this is jRJ JBP*(BP; o). In this case, we have t*
*he
quotient stable comodule BP *=J. For example, Lemma 15.8 shows that the ideals *
*In
are invariant, and we have the stable comodules BP *=In ~=F p[vn; vn+1; vn+2; :*
* :]:(for
n > 0) and BP *=I0 ~=BP *.
Primitive elements The key idea is to explore a general stable comodule M by
looking for comodule morphisms BP *! M from the (relatively) well understood
stable comodule BP *(T ) = BP *. A BP *module homomorphism f: BP *! M is
obviously uniquely determined by the element x = f1 2 M, since fv = f(v1) =
vf1 = vx, and we can choose x arbitrarily. In BP *, we clearly have ae1 = 1 1,*
* which
suggests the following definition.
JMB  85  23 Feb 1995
Stable cohomology operations
Definition 15.9 Given a stable comodule M, we call an element x 2 M stably
primitive if aeM x = x 1.
This is the necessary and sufficient condition for the above homomorphism
f: BP *! M to be a stable morphism. It then induces an isomorphism of stable
comodules BP *= Kerf ~= (BP *)x. In particular, Ker f = Ann (x), the annihilat*
*or
ideal of x, must be an invariant ideal. We are therefore interested in finding *
*primi
tives.
The primitive elements of Mk clearly form a subgroup. Moreover, there is a *
*good
supply of primitives; if M is bounded above, axiom (11.12)(ii) forces every ele*
*ment
x 2 M of top degree to be primitive. (This may be viewed as an algebraic analog*
*ue
of Hopf's theorem, that for a finitedimensional space X, ssk(X) ~=Hk(X; Z) in *
*the
top degree.) If x is primitive, the BP *linearity of aeM gives aeM (vx) = x *
* jRv for
any v 2 BP *. It follows that the comodule structure on BP *=J is unique if the*
* ideal
J is invariant (and none exists otherwise). Landweber [17] located all the prim*
*itive
elements in the stable comodule BP *=In.
Theorem 15.10 (Landweber) For 0 n < 1, the only nonzero primitive elements
in the stable comodule BP *=In are those of the form:
(i)vin, where i 0 and 2 Fp (if n > 0); or
(ii), where 2 Z(p)(if n = 0).
It follows easily as in [17, Thm. 2.7] that the In are the only finitely ge*
*nerated
invariant prime ideals in BP *. This suggests that the BP *=In should be the b*
*asic
building blocks for a general stable comodule. This is the content of Landwebe*
*r's
filtration theorem (cf. [17, Lemma 3.3] and [18, Thm. 3:30]).
Theorem 15.11 (Landweber) Let M be a stable BP cohomology (co)module that
is finitely presented as a BP *module (e. g. BP *(X) for any finite complex X)*
*. Then
M admits a finite filtration by invariant submodules
0 = M0 M1 M2: : :Mm = M;
in which each quotient Mi=Mi1 is generated, as a BP *module, by a single elem*
*ent
xi such that Ann (xi) = Ini for some ni 0.
We outline Landweber's proof [18] for reference. For nonzero M, Ass(M), whi*
*ch
here may be taken as the set of all prime annihilator ideals of elements of M, *
*is a
finite nonempty set of invariant finitely generated prime ideals of BP *. The *
*recipe
for constructing a filtration of M is:
(a) Let In be the maximal element of Ass(M);
(b) Construct the BP *submodule N = 0:In of M, which is defined as {y 2 M*
* :
Iny = 0}, and prove it invariant;
(c) Take a nonzero primitive x1 2 N (e. g. any element of top degree), so *
*that
the maximality of In forces Ann (x1) = In;
(d) Put M1 = (BP *)x1, so that M1 is invariant and isomorphic to BP *=In;
JMB  86  23 Feb 1995
Index of symbols
(e) Replace M by M=M1 and repeat, as long as M is nonzero, making sure that
the process terminates (which requires some care).
Remarks 1. The filtration of M is never a composition series. The module BP *=*
*In
is not irreducible, because we have the short exact sequence
vn * *
0 ! BP *=In ! BP =In ! BP =In+1 ! 0
of stable comodules. Thus we have no uniqueness statement.
2. We cannot expect to arrange n1 n2 : :,:since in (e), Ass(M=M1) need not
be contained in Ass(M).
Index of symbols
This index lists most symbols in roughly alphabetical order (English, then *
*Greek),
with brief descriptions and references. Several symbols have multiple roles.
__
A augmentation ideal in algebra A. E*() Ecohomology, Thm. 3.17.
A etc. generic category. E*()^ completed Ecohomology,
Aop dual category of A, x6. Defn. 4.11.
A = E*(E; o), Steenrod algebra for E,E*() Ehomology, (9.17).
x10. E_n nth space of spectrum E,
Ab , Ab* category of (graded) abelian Thm. 3.17.
groups, x6. e evaluation on DL L, x6.
Alg category of E*algebras, x6. ei basis element of Cn.
a(i) stable element for K(n), (14.31). F free functor, Thms. 2.6, 8.5.
ai;j coefficient in formal group law, F (x; y)formal group law, (5.14).
(5.14). F aM generic filtration submodule,
BG classifying space of group G. Defn. 3.36.
B(i; k) coefficient in b(x)i, Prop. 13.4.FAlgcategory of filtered E*algebras,
BP BrownPeterson spectrum, x2. x6.
b Bott map, Cor. 5.12. F LDM filtration submodule of DM,
bi stable element, Prop. 13.4. Defn. 4.8.
b(i) accelerated bi, Defn. 13.8. FM etc. corepresented functor, x8.
b(x) formal power series, (13.2). FMod , FMod * (graded) category of
C cofree functor, Thm. 8.10. filtered E*modules, x6.
C the field of complex numbers. Fp field with p elements.
C P n, CP 1 complex projective space. FR(X; Y ) right formal group law,
Coalg category of E*coalgebras, x6. (13.6).
ci() Chern class of vector bundle , F sE*(X) skeleton filtration, (3.33).
Thm. 5.7. f etc. generic map or homomorphism.
DM dual of E*module M, Defn. 4.8. f*, f* homomorphism induced by map
d duality homomorphism, (4.5), f, (6.3).
(9.24). fn structure map of spectrum E,
E generic ring spectrum. Defn. 3.19.
E* coefficient ring of E(co)homology,G etc. generic group (object), x7.
xx3, 4. G Emodule spectrum, Thm. 9.26.
JMB  87  23 Feb 1995
Stable cohomology operations
Gp (C) category of group objects in C, E*linear functional defined by
x7. operation r, (11.1).
gi coefficient in pseries, (13.9). S stable comonad, Thm. 10.12.
H generic comonad, (8.6). S0 stable comonad, (11.4).
H, H(R) EilenbergMacLane S (subscript) stable context.
spectrum, xx2, 14. S1 unit circle, as space or group.
Ho , Ho 0homotopy category of (based) Sn unit nsphere.
*
spaces, x6. Stab, Stab (graded) stable homotopy
h() generic ungraded cohomology category, x6.
theory, x3. Set category of sets, x6.
Z
h Yokota clutching function, (5.9). Set category of graded sets, x7.
I identity functor. T monad, (8.4).
In, I1 ideal in BP *, (15.1). T + the onepoint space.
i1, i2 injection in coproduct, x2. T 0sphere, T with basepoint added.
id identity morphism. T (n) torus1group.1
KC unit object in (symmetric) t 2*H (R P1 ), generator of
monoidal category C, x7. H (R P ), (14.1).
K(n) Morava Ktheory, x2. U, U(n) unitary group.
KU complex Ktheory Bott spectrum, U (subscript)unstable2context.
x2, Defn. 3.30. u 2 KU1 , after Defn. 3.30.
L infinite lens space, x14. u 2*E (L), exterior generator of
M etc. generic (filtered) module or E (L),1x14.
algebra. u 2*E (Y ), exterior generator of
E (Y ), x14.
M^, cM completion of filtered M,
u universal element of DL b L,
Defn. 3.37.
Lemma 6.16.
Mod , Mod * (graded) category of u canonical generator of E*(S1),
E*modules, x6. 1 Defn. 3.23.
MU unitary Thom spectrum, x2. u canonical generator of E*(Sn), x3.
o generic basepoint, point spectrum. Vn generic (often forgetful) functor.
op categorical dual, x6. v generic element of E*.
P A the primitives in coalgebra A, v = jRu 2 KU2(KU; o), Thm. 14.18.
(6.13). vn Hazewinkel generator of BP *,
p fixed prime number. K(n)*, x14.
p1, p2 projection from product, x2. W forgetful functor, x8.
[p](x) pseries, (13.9). W ideal in BP*(BP; o), x15.
[p]R(x) right pseries, (13.11). w = u1v 2 KU0(KU; o),
QA the indecomposables of algebra A, Lemma 14.23.
(6.10). wn = jRvn, x14.
Q the field of rational numbers. X etc. generic space or spectrum.
q map to onepoint space T , x2. X+ space X with basepoint adjoined.
R generic ring. x generic cohomology class or module
RMod category of Rmodules, x8. element.
R P 1 real projective space. x 2 E*(C P 1), Chern class of Hopf
r etc. generic cohomology operation. line bundle, Lemma 5.4.
JMB  88  23 Feb 1995
References
x() Chern class of line bundle , numerical coefficient.
Defn. 5.1. addition or multiplication in gener*
*ic
Y skeleton of lens space L, x14. group object, x7.
Z the ring of integers. inversion morphism in generic group
Z =p the group of integers mod p. object, x7.
Z (p) Z localized at p. Hopf line bundle over CP n.
zF morphism for a (symmetric) generic line or vector bundle.
monoidal functor F , x7. i stable element for H(F 2), (14.1).
ff etc. generic index. i stable element for H(F p), (14.4).
ff generic algebraic operation, x7. v action of v on E*module, (7.4).
fii 2 E2i(C P n), Lemma 5.3. ss*(X) homotopy groups of space X.
fli 2 E2i+1(U(n)), Lemma 5.11. ssS
*(X; o)stable homotopy groups of X.
: X ! X x X diagonal map. ae generic coaction.
ffl generic counit morphism. ae
M coaction on module M.
ffl: F V ! Inatural transformation, x2. ae * *
X coaction on E (X) or E (X)^.
i pth power map on CP 1, (13.9). , k suspension isomorphism, (3.13),
iF pairing for (symmetric) monoidal Defn. 6.6.
functor F , x7. X, kX suspension of space X.
j generic monoid unit morphism. k
M, M suspension of module M,
j: I ! V F natural transformation, x2.
Defn. 6.6.
j generic vector bundle.
oek: E_k! E stabilization, Defn. 9.3.
jR right unit, Defn. 11.2.
generic anything. oi stable element for H(F p), (14.5).
complex line bundle, x5. OE generic monoid multiplication.
cohomology operation (usually O canonical antiautomorphism of Hopf
idempotent), x3. k algebra.
2 h(H), universal class, Thm. 3.6. Adams operation, (14.20).
2 E0(E; o), universal class, x9. generic comultiplication.
n 2 En(E_n), universal class, X loop space on based space X.
Thm. 3.17. ! zero morphism of generic group
() exterior algebra. object, x7.
generic action.
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Department of Mathematics, Johns Hopkins University
3400 N. Charles St., Baltimore, MD 21218, U.S.A.
Email: boardman@math.jhu.edu
JMB  91  23 Feb 1995