Unstable Operations in Generalized Cohomology
J. Michael Boardman David Copeland Johnson
W. Stephen Wilson
January 1995
[To appear in Handbook of Algebraic Topology,
ed. I. M. James, Elsevier (Amsterdam, 1995)]
TABLE OF CONTENTS
1 Introduction 2
2 Cohomology operations 9
3 Group objects and Ecohomology 12
4 Group objects and Ehomology 14
5 What is an additively unstable module? 17
6 Unstable comodules 22
7 What is an additively unstable algebra? 32
8 What is an unstable object? 38
9 Unstable, additive, and stable objects *
*43
10 Enriched Hopf rings 47
11 The Ecohomology of a point 59
12 Spheres, suspensions, and additive operations *
*61
13 Spheres, suspensions, and unstable operations *
*63
14 Complex orientation and additive operations 66
15 Complex orientation and unstable operations 68
16 Examples for additive operations 70
17 Examples for unstable operations 78
18 Relations for additive BP operations *
*86
19 Relations in the Hopf ring for BP 92
20 Additively unstable BP objects 102
21 Unstable BP algebras 107
22 Additive splittings of BP cohomology 111
23 Unstable splittings of BP cohomology 115
Index of symbols 121
References 125
JMB, DCJ, WSW  1  23 Feb 1995
Unstable cohomology operations
1 Introduction
A multiplicative generalized cohomology theory E*() on spaces is represent*
*ed by
the spaces E_n of its spectrum, as described in detail in [8, Thm. 3.17]. We d*
*enote
its coefficient ring by E*. Our five examples are ordinary cohomology H*(; Fp*
*),
unitary cobordism MU*(), BrownPeterson cohomology BP *(), complex Ktheory
KU*(), and Morava Ktheory K(n)*(). (They were properly introduced in [8, x2]*
*.)
Recent work [25] shows that a sixth example, the cohomology theory P (n)*(), a*
*lso
satisfies our hypotheses.
We are interested in three kinds of cohomology operation: stable operations*
*, which
form the endomorphism ring E*(E; o) of E (in our notation) and were studied in *
*[8];
unstable operations, defined on En(X) for spaces X and fixed n, which form E*(E*
*_n);
and additive unstable operations r on En() (that satisfy r(x+y) = r(x) + r(y)),
which form the subset P E*(E_n). Since a stable operation restricts to an addi*
*tive
unstable operation on any degree, these are related by
E*(E; o) ! P E*(E_n) E*(E_n) :
Each of these is an E*module in the usual way, by (r+s)(x) = r(x) + s(x) and
(vr)(x) = v r(x) (for any v 2 E*). We can compose, (sr)(x) = (s Or)(x) = s(r(x)*
*),
whenever the sources and targets match. We can also multiply unstable operations
together by (r ^ s)(x) = r(x)s(x).
In the classical case E = H(F p), for which E*(E; o) is the Steenrod algebr*
*a, it is
true that: (a) every additive operation comes from a stable operation; (b) the *
*addi
tive operations generate multiplicatively all the unstable operations. Our diff*
*iculties
stem from the fact that for MU and BP , both (a) and (b) are false. (See [27] *
*for
more discussion of the differences.) We propose to describe completely the alge*
*braic
structure that has to be present on an E*module or E*algebra to make it an un
stable object, with particular attention to the case E = BP . Our definitions l*
*ead to
structure theorems.
Stable BP operations have been available for quite some time and are well *
*estab
lished. Less has been done with unstable BP operations, owing to their complex*
*ity,
but we do have the work [4, 5] of Bendersky, Curtis, Davis, and Miller. The alg*
*ebraic
structure on an additively unstable module is described in [27] and (without pr*
*oofs)
in [6].
Our major task, therefore, is to set up precise algebraic descriptions of t*
*he unstable
structures we need on modules and algebras, along the lines of the stable struc*
*tures
in [8]. Part of the difficulty is that one is forced to work in the unfamiliar *
*context of
nonadditive operations; but the real problem turns out to be Thm. 9.4, that uns*
*table
modules (as distinct from unstable algebras) simply do not exist compatibly wit*
*h our
other objects! When we limit attention to the less exotic additive operations,*
* this
difficulty does not arise and we have both modules and algebras.
In fact, there is a huge amount of data to be codified in an unstable algeb*
*ra.
The key idea is that given an E*algebra M, we define (UM)k for each k as the
set of all algebra homomorphisms E*(E_k) ! M; each such homomorphism may
be thought of as a candidate for the values of all operations on a typical elem*
*ent
of Mk. Apparently merely a graded set, UM becomes an E*algebra for suitable
JMB, DCJ, WSW  2  23 Feb 1995
x1. Introduction
E, thanks to extra structure on the spaces E_n. Then an unstable structure on
M is a homomorphism aeM : M ! UM of E*algebras, which selects for each x 2
Mk the function aeM (x): E*(E_k) ! M; then we define r(x) = aeM (x)r. This is
not enough; in order to compose operations correctly, it is necessary to know t*
*he
Ecohomology homomorphism r*: E*(E_m ) ! E*(E_k) induced by each operation
r: Ek() ! Em (). This extra structure makes the functor U a comonad, and
(M; aeM ) a coalgebra over this comonad. We have a similar construction for add*
*itive
operations, and can compare with the stable constructions of [8].
This is our elegant but extremely terse answer, and we do not believe that *
*it
can be efficiently expressed without using comonads. But it does have the effe*
*ct
that the work consists largely of definitions. In x10, we translate this answe*
*r into
practical language, in the context of Hopf rings, that we can use for computati*
*on.
This includes Cartan formulae for r(x+y) as well as r(xy), and related formulae*
* for
r*(b*c) and r*(bOc) that we use to compute the induced Ehomology homomorphism
r*: E*(E_k) ! E*(E_m ) dual to r*.
Landweber filtrations We recall that BP *= BP *(T ), the BP cohomology of the
onepoint space T , is the polynomial ring Z(p)[v1; v2; v3: :]:, with deg(vn) =*
* 2(pn1)
(under our degree conventions). It contains the wellknown ideals
In = (p; v1; v2; : :;:vn1) BP * (1:1)
for 0 n 1 (with the convention that I1 = (p; v1; v2; : :):, I1 = (p), and I0 *
*= 0).
The significance [8, Lemma 15.8] of In is that it is invariant under the ac*
*tion of
the stable operations on BP *(T ). Indeed, Landweber [15] and Morava [20] show*
*ed
that the In for 0 n < 1 are the only finitely generated invariant prime ideals*
* in
BP *. Landweber used this fact to show (see [16, Thm. 3:30] or [8, Thm. 15.11])*
* that
a stable (co)module M that is finitely presented as a BP *module, including BP*
* *(X)
for any finite complex X, admits a finite filtration by invariant submodules
0 = M0 M1 M2 : : :Mm = M (1:2)
in which each quotient Mi=Mi1is generated (as a BP *module) by a single eleme*
*nt
xi whose annihilator ideal Ann (xi) = Ini for some ni. Thus Mi=Mi1~= BP *=Ini.
The first unstable result on BP cohomology, due to Quillen [22] (see Thm. *
*20.2),
was that for a finite complex X, BP *(X) is generated, as a BP *module, by ele*
*ments
of nonnegative degree. What started this project was the observation that if *
*an
unstable object M is generated by a single element x, there is an unstable oper*
*ation
(see Prop. 1.14 or the Remark following Cor. 20.9) that takes vnx to x, provided
deg (x) is small enough; it follows that vnx 6= 0 and that M cannot be isomorph*
*ic to
BP *=In+1.
The proof of Landweber's theorem depends on the concept of primitive elemen*
*t in
a comodule M. Given any x 2 M, there is the obvious homomorphism of BP *modules
f: BP *! M, defined by fv = vx. It is a morphism of stable modules if and only
if x is primitive, and if so, we have the isomorphism BP *=Ann (x) ~=(BP *)x M*
* of
stable modules. An important example (see [8, Thm. 15.10]) is that the only non*
*zero
primitives in BP *=In, for n > 0, are the (images of the) elements vin, where *
*2 Fp,
JMB, DCJ, WSW  3  23 Feb 1995
Unstable cohomology operations
6= 0, and i 0. For additive unstable operations, the appropriate definition *
*of
primitive becomes more restrictive.
Theorem 1.3 (included in Thm. 20.10) Let M be the BP *module generated by a
single element x with Ann (x) = In, where n > 0. Then M admits an additively
unstable module structure (as defined in x5) if and only if deg(x) f(n)  2, a*
*nd it
is unique.
The only nonzero primitive elements in M are those of the form vinx, where
2 Fp, and deg(vinx) f(n) if i > 0.
Here, and everywhere, we need the numerical function
deg(vn) 2(pn  1) n1 n2
f(n) = ________ = _________= 2(p + p + : :+:p + 1) (1:4)
p  1 p  1
for n > 0; it is reasonable to define also f(0) = 0.
We use this result in Thm. 20.11 to construct a Landweber filtration (1.2) *
*of an
appropriate module M, including BP *(X) for any finite complex X, in which each
quotient Mi=Mi1has the form in Thm. 1.3 (or is BP *free). Once our machinery *
*is
in working order, we are able to give a oneline proof of Thm. 20.3, the weak f*
*orm of
Quillen's theorem.
In our main structure theorem, we do one better by allowing all unstable op
erations instead of only the additive ones. One complication is that the unsta*
*ble
analogue of Thm. 1.3 has to be stated for algebras only, owing to the nonexiste*
*nce of
unstable modules.
Theorem 1.5 (stated precisely as Thm. 21.12) Let M be an unstable BP *algebra
such as BP *(X) for a finite complex X. Then M admits a filtration (1.2) by in
variant ideals Mi, in which each quotient Mi=Mi1 is generated, as a BP *modul*
*e,
by a single element xi such that Ann (xi) = Ini for some ni 0, where deg(xi)
max (f(ni)1; 0).
Splittings of BP cohomology Another application of our machinery yields idem
potent operations that split unstable BP cohomology into indecomposable pieces.
Such splittings were constructed in [26] by means of Postnikov systems. What is*
* new
is that explicit definitions of everything allow us to carry out computations. *
*Our re
sults are logically independent of [26] and rely on it only to recognize the su*
*mmands
as known objects; nevertheless, it is a valuable guide as to what the summands *
*look
like and where to find them. In a sequel [9], two of the authors go on to apply
the structure theorems of [25] to establish analogous (but slightly different) *
*splitting
theorems for the cohomology theory P (n)*(), whose coefficient ring is BP *=In.
For each n 0, we define the ideal
Jn = (vn+1; vn+2; vn+3; : :): BP *: (1:6)
In [26], BaasSullivan theory [2] was used to construct a cohomology *
* the
ory BP *() having coefficients BP *=Jn ~= Z(p)[v1; v2; : :;:vn]. In part*
*icular,
BP <0>*() = H*(; Z(p)). The desired splitting is
Y j
BP k(X) ~=BP k(X) BP k+2(p 1)(X) : (1:7)
j>n
JMB, DCJ, WSW  4  23 Feb 1995
x1. Introduction
The representing spectrum BP is (at least) a BP module spectrum, and
comes equipped with a canonical map of BP module spectra that we shall call
ss: BP ! BP . There is also a canonical map ss: BP ! BP whenever
j > n. (Geometrically, BP allows more singularities than BP .) Everythin*
*g we
need to know about BP is contained in the commutative diagram
vj ss
BP__k+2(pj1)______BP__k __________BP__k
  j3
  j
ss ss jj
  j ss (1:8)
  j
? v ? j
j
BP__k+2(pj1)_____BP__k
of Hspaces and Hmaps, where j > n.
Although the cohomology theory BP *() may be unfamiliar, in the range of
degrees of interest it is easily described in terms of BP cohomology. It is c*
*lear by
construction that ss*: BP *(X) ! BP *(X) kills JnBP *(X).
Theorem 1.9 Assume that k f(n+1), where n 0, and that X is finite
dimensional. Then ss induces a natural isomorphism of BP *modules
OE X j
BP k(X) vjBP k+2(p 1)(X) ~=BP k(X) : (1:10)
j>n
We derive this below as an immediate consequence of Thm. 1.12. It is best p*
*os
sible, as [26] shows that ss* is not surjective in general for k > f(n+1).
Lemma 1.11 (included in_Lemma 22.1) Given k < f(n+1), where n 0, there is
an Hspace splitting n: BP__k! BP__kof ss: BP__k! BP__kwhich naturally
embeds BP k(X) BP k(X) as a summand (as abelian groups).
If also k f(n), the Hspace BP__kdoes not decompose further.
_
Remark The splittings n are not canonical or unique. The ideal Jn, unlike In*
*, is
in no way canonical, but depends on the choice of the polynomial generators of *
*BP *.
Although the BP module structure of BP obviously depends on Jn, it follows*
* from
the Lemma that the resulting Hspace structure_on BP__kis well defined. Even*
* for
fixed Jn, we find there are many choices for n, and no preferred choice is app*
*arent.
We establish Lemma 1.11 in x22 by constructing a suitable idempotent operat*
*ion
n on BP *(). The second assertion implies that the first is best possible. We *
*insert
these splittings into diag. (1.8) to decompose BP cohomology.
Theorem 1.12 Assume n 0. Then:
_ _
(a) For k < f(n+1), the injections n and vjO jfrom Lemma 1.11 induce the
natural abelian group decomposition (1.7), which is maximal if k f(n);
(b) For k = f(n+1), we have instead the natural short exact sequence of abel*
*ian
groups
Y j ss*
0 ! BP k+2(p 1)(X) ! BP k(X) ! BP k(X) ! 0; (1:13)
j>n
JMB, DCJ, WSW  5  23 Feb 1995
Unstable cohomology operations
where none of the groups decomposes_further naturally, and ss* admits a non
additive natural splitting n: BP k(X) ! BP k(X), so that we have eq. (1.7) *
*as a
bijection of sets.
Remark The simplified description of BP <>cohomology in Thm. 1.9 applies ev
erywhere (when X is finitedimensional). These splittings definitely do not pre*
*serve
the BP *module structure. We plan to return to this point in future work.
Proof of Thm. 1.9 For finitedimensional X, the sum in eq. (1.10) is in fact f*
*inite. It
is clear from eq. (1.7) or (1.13) that the sum contains Ker ss*. On the othe*
*r hand,
ss* is a homomorphism of BP *modules which kills Jn. 
Projection to the first factor of the product in eq. (1.7) yields an intere*
*sting oper
ation
0 k0
r: BP k(X) ! BP k (X) BP (X);
where k0= k+2(pn+11) = k+(p1)f(n+1), which roughly has the effect of dividing
0 k0
by vn+1. Precisely, r(vn+1y) = y whenever y 2 BP k (X) BP (X). Given
0 k0
any x 2 BP k(X), we can put y = n+1x; then by Thm. 1.12(a), applied to BP (X),
we have y x mod Jn+1. For convenience, we reindex.
n1)
Proposition 1.14 If k pf(n), there is an operation r: BP k2(p (X) !
BP k(X), which is additive if k < pf(n), with the property that given any eleme*
*nt x 2
BP k(X), where X is finitedimensional, there exists y such that y x mod JnBP *
**(X)
and r(vny) = y. 
Equivalently, we can represent eq. (1.7) by the decomposition of spaces
Y
BP__k' BP__kx BP__k+2(pj1): (1:15)
j>n
Theorem 1.16 Assume n 0. Then:
(a) For k < f(n+1), we have the Hspace decomposition (1.15), which is maxim*
*al
if k f(n);
(b) For k = f(n+1), we have the fibration
Y ss
BP__k+2(pj1)!BP__k! BP__k (1:17)
j>n
of Hspaces and Hmaps, which admits a section (not an Hmap), so that eq. (1.1*
*5)
holds as an equivalence of spaces (but not as Hspaces), and none of the spaces
decomposes further as a product of spaces. (In other words, BP k() is represen*
*ted
by the right side of eq. (1.15), equipped with a different Hspace multiplicati*
*on.)
We use Lemma 1.11 to prove parts (a) of Thms. 1.12 and 1.16 in x22. For par*
*ts
(b), the necessary idempotent n has to be nonadditive, and we construct it in x*
*23.
We need the full strength of our machinery just to prove that n is idempotent.
History Our real motivation for this study is what is called the Johnson Quest*
*ion,
which is stated in [24, p. 745]. Rephrased as a conjecture, it is:
JMB, DCJ, WSW  6  23 Feb 1995
x1. Introduction
Conjecture If x 6= 0 in BPn(X), where X is a space, then vinx 6= 0 for all *
*i > 0.
No counterexamples are known, although examples exist [13, 14, 24] where
vjx = 0 for all j < n. It holds if x reduces nontrivially to homology, therefo*
*re for
n < 2p. We hoped to circumvent our lack of knowledge of unstable homology opera
tions by working instead with the rather better understood unstable BP cohomol*
*ogy
operations and using the (not at all unstable) duality spectral sequence
Ext**BP*(BP *(X); BP *) =) BP*(X)
of Adams [1] (see also [12]). The reason for optimism is that if we substitute
k(BP *=In) for BP *(X), a standard calculation shows that the only surviving Ext
group is Extn = m (BP *=In), with m = f(n)  k  n; so that k f(n)  1 implies
m n  1, almost what we want. If we confine ourselves to additive operations,
we obtain m n  2, off by one more. We can hope to work our way up from
k(BP *=In) to a general BP *(X) by extension and the filtration (1.2).
This is all grounds for our suspicion that for a geometric unstable algebra*
*, i. e.
M = BP *(X) for some space X, the bounds in Thm. 1.5 should be one better (thus
giving us m n in the above discussion). Again, there are no known counter
examples, although spaces are known which have deg(xi) = f(ni), thus showing th*
*at
the bounds cannot be improved by more than one.
Recently, with the help of Mike Hopkins, a new approach to the Johnson Ques*
*tion
has been developed. It requires a much better understanding of the unstable spl*
*ittings
of BP . Now that we have so much explicit information on these splittings, this*
* method
of attack seems promising.
Outline There are two main threads running through this work: the theory of
additive unstable operations, which closely resembles the stable theory of [8],*
* and the
theory of all unstable operations, which is radically different. The comonad t*
*ent is
big enough to accommodate both, as well as the stable theory. We have kept the
additive material in separate sections so that it can be read independently.
In x2, we discuss several classes of cohomology operation. In xx3, 4, we st*
*udy the
E(co)homology of group objects, in preparation for xx5, 7, where we study modu*
*les
and algebras from the additive point of view. In x6, we consider additive opera*
*tions
as linear functionals. In xx12, 14, we study suspensions and complex orientatio*
*n. In
x16, we present the additive structure for each of our five examples E.
It turns out that much of the stable machinery does not extend to all unsta*
*ble
operations, because it relies too heavily on the bilinearity of tensor products*
*. However,
the approach in terms of comonads does work, and in x8 we develop the requisite
comonad U. We also show in x9 that the corresponding comonad for unstable modul*
*es
does not exist and compare the various stable and unstable structures. In x10,*
* we
convert the categorical elegance into machinery we can use; specifically, cohom*
*ology
operations become linear functionals on Hopf rings. In Thm. 10.47, we display i*
*n full
detail the definition of an unstable algebra from this point of view.
In xx11, 13, 15, we revisit the cohomology of a point, sphere, and complex *
*pro
jective space C P 1 from this new Hopf ring point of view. These spaces alone y*
*ield
almost enough generators and relations to specify the Hopf rings for our five e*
*xamples
JMB, DCJ, WSW  7  23 Feb 1995
Unstable cohomology operations
E, as we discuss in detail in x17. The case E = KU is used to determine the str*
*ucture
of KU*(KU; o), as quoted in [8, x14]. From a sufficiently elevated perspective*
*, the
results of x17, the additive results of x16, and the stable results of [8] all *
*fit into a
grand master plan.
In x20, we restrict attention to the case E = BP and use the additive opera*
*tions
to recover Quillen's theorem and prove Thm. 20.11. This relies on the relations
developed in x18. In x21, we use nonadditive operations to improve Thm. 20.11 by
one dimension to Thm. 21.12, which is Thm. 1.5.
In x22, we construct additive idempotent operations n which yield the desir*
*ed fac
torizations (1.7) in all except the top degree. In x23, we finish off Thms. 1.1*
*2 and 1.16
by constructing nonadditive idempotent operations. To do this, it is necessary *
*in x19
to develop the notion of a Hopf ring ideal.
An index of symbols is included at the end.
This work is also notable for what it does not contain. There are no spectr*
*al se
quences, except implicitly in the references. There are no explicit Steenrod op*
*erations,
except in a few examples; in our wholesale approach, most individual operations*
* never
even acquire names. There are no formal indeterminates anywhere; the elements t*
*hat
are sometimes treated as such are really Chern classes x; but when xi= 0, we ca*
*n no
longer take the coefficients of xi.
Notation We make heavy use of the notation and machinery developed in [8].
Topologically, we generally work in the homotopy category Ho of unbased spaces.*
* For
compatibility with the unstable notation, the Ecohomology and Ehomology of a
spectrum X are written E*(X; o) and E*(X; o). Algebraically, our most important
categories are the categories FMod and FAlg of filtered E*modules and algebr*
*as.
These and the other categories we need were introduced in [8, x6]. We make freq*
*uent
use of Yoneda's Lemma. All tensor products are taken over E* unless otherwise
stated.
For reasons discussed in [8], we always give cohomology E*(X) the profinite*
* topol
ogy [8, Defn. 4.9], and complete it as in [8, Defn. 4.11] to E*(X)^ as necessar*
*y. In
contrast, the homology E*(X) is always discrete. Because we emphasize cohomolog*
*y,
we invariably assign the degree i to elements of Ei(X); this forces elements of*
* Ei(X)
to have degree i.
One theorem provides all the duality and K"unneth isomorphisms we need.
Theorem 1.18 Assume that E*(X) is a free E*module. Then we have:
(a) d: E*(X) ~=DE*(X) in FMod , the strong duality homeomorphism;
(b) E*(X xY ) ~=E*(X) E*(Y ), the K"unneth isomorphism in homology;
(c) E*(X xY ) ~=E*(X) b E*(Y ) in FMod , the K"unneth homeomorphism in co
homology, provided E*(Y ) is also a free E*module.
Proof We collect Thms. 4.2, 4.14, and 4.19 from [8]. Indeed, (c) follows from*
* (a)
and (b). 
Acknowledgements The genesis of this paper is that the last two authors had
worked out much of the unstable BP structure theorems, without having a precise
JMB, DCJ, WSW  8  23 Feb 1995
x2. Cohomology operations
definition of unstable algebra, when the first author supplied a suitable frame*
*work,
of which [7] is an early version. In fact, this is an oversimplification: the*
* various
contributions are more intermingled than this might suggest. In the proper cont*
*ext,
several of the proofs simplify significantly. We thank Martin Bendersky for poi*
*nting
out Lemma 19.32, which is vastly simpler than our previous treatment.
The last two authors wish to thank the topologists at the University of Man*
*chester
and the Science and Engineering Research Council (SERC) for support during the
summer of 1985 when this project got its start. The last author wishes to thank
Miriam and Harold Levy for their hospitality and the peace they provided for the
writing of the first draft.
2 Cohomology operations
In this section, we consider several kinds of unstable cohomology operatio*
*n.
Yoneda's Lemma allows us to identify the following:
(i)The cohomology operation r: Ek() ! Em ();
(ii)The cohomology class r = r(k) 2 Em (E_k); (2.1)
(iii)The representing map r: E_k! E_m in Ho .
We write any of these more succinctly as r: k ! m. We use all three interpretat*
*ions.
Some care is needed with degrees and signs, as (i) has degree mk and (ii) has *
*degree
m, while (iii) has no degree at all.
Based operations The following mild but useful condition can be interpreted ma*
*ny
ways. The space T is the onepoint space.
Definition 2.2 We call the operation r based if r(0) = 0 in E*(T ) = E*.
Lemma 2.3 The following conditions on an operation r: k ! m are equivalent:
(a) r(0) = 0 in E*(T ), i. e. r is a based operation;
(b) For any based space (X; o), r restricts to the reduced operation
r: Ek(X; o) ! Em (X; o); (2:4)
(c) As a cohomology class, r 2 Em (E_k; o) Em (E_k);
(d) The map r: E_k! E_m is (homotopically) based.
Proof The short exact sequence [8, eq. (3.2)] shows that (a) and (c) are equiv*
*alent,
also that (a) implies (b); but (c) is the special case of (b) for k 2 Ek(E_k; o*
*). Part
(d) is just a restatement of (a). 
Given any (good) pair of spaces (X; A), we can use (b) to make based operat*
*ions
r: k ! m act on relative cohomology as in [8, eq. (3.4)] by
r m m
Ek(X; A) = Ek(X=A; o) ! E (X=A; o) = E (X; A) : (2:5)
Additive operations An additive operation r: k ! m is one that satis*
*fies
r(x+y) = r(x) + r(y) for any x; y 2 Ek(X). The universal example is
X = E_kx E_k; with x = k x 1, y = 1 x k, x + y = k, (2:6)
JMB, DCJ, WSW  9  23 Feb 1995
Unstable cohomology operations
which gives r(k) = rx1+1xr in E*(E_kxE_k). (The addition map k: E_kxE_k ! E_k
was defined in [8, Thm. 3.6].) This allows us to recognize additive operations *
*three
ways.
Proposition 2.7 The following conditions on an operation r: k ! m are equiva*
*lent,
and define the E*submodule P E*(E_k) E*(E_k; o) E*(E_k):
(a) The operation r: Ek() ! Em () is additive;
(b) The class r 2 Em (E_k) satisfies *kr = p*1r + p*2r in Em (E_kxE_k), i. e.
P E*(E_k) = Ker[*k p*1 p*2: E*(E_k) ! E*(E_k x E_k)]; (2:8)
(c) The map r: E_k! E_m is a morphism of group objects in Ho . 
Corollary 2.9 Assume that E*(E_k) is a free E*module. Then P E*(E_k) is com
plete Hausdorff and so an object of FMod .
Proof In eq. (2.8), E*(E_k) and E*(E_kxE_k) are complete Hausdorff by Thm. 1.1*
*8.

When E*(E_k) is free, the K"unneth homeomorphism for E*(E_kxE_k) makes
E*(E_k) a completed Hopf algebra; then (b) agrees with the primitives in the se*
*nse of
[8, eq. (6.13)], completed. However, we need no hypotheses on E to define P E*(*
*E_k).
On some spaces, all operations are additive.
Lemma 2.10 On the suspension X of any based space (X; o), we have r(x+y) =
r(x) + r(y) in Em (X; o) for any based operation r: k ! m and any elements x; y*
* 2
Ek(X; o).
Proof By [8, Lemma 7.6(c)], r: Ek(X; o) ! Em (X; o) preserves the group struc
ture defined from the cogroup object X in Ho 0. By [8, Prop. 7.3], this struct*
*ure
coincides with the given Ecohomology addition. 
Products of operations Given operations r: k ! m and s: k ! n, the product
operation r ^ s: k ! m + n, defined by (r ^ s)x = (rx)(sx), corresponds to the
cup product in E*(E_k), which may be constructed using the diagonal map : E_k!
E_k x E_k. We often wish to neglect such operations; if r and s are additive, r*
* ^ s is
clearly not additive, but conveys no new information.
The map , together with q: E_k! T , makes E_ka monoid object in the symmetr*
*ic
monoidal category (Ho op; x; T ). We therefore dualize eq. (2.8) and introduce*
* the
quotient E*module
QE*(E_k) = Coker[*  i*1 i*2: E*(E_k x E_k) ! E*(E_k)] (2:11)
of "indecomposables" of E*(E_k), where i1 and i2 are the inclusions (using the *
*base
point). (We shall not need a topology on this module.) When E*(E_k) is a free
E*module, we have by Thm. 1.18(c) a K"unneth homeomorphism for E*(E_kxE_k),
and QE*(E_k) is the quotient of E*(E_k; o) by all finite (or infinite) sums of *
*products
of two based operations.
Looping of operations On restriction to spaces, a stable operation r on E*(; *
*o)
of degree h induces a sequence of additive operations rk: k ! k+h. It is clear *
*from
[8, fig. 2 in x9] that rk+1 determines rk. We generalize this construction to u*
*nstable
operations (but omit the sign, in order to make it a homomorphism of E*modules*
*).
JMB, DCJ, WSW  10  23 Feb 1995
x2. Cohomology operations
Proposition 2.12 Given a based unstable operation r: k ! m, we can define the
looped operation r: k1 ! m1 in any of three equivalent ways:
(a) The operation that makes the diagram commute (with no sign),
~= ~=
Ek1(X) ________Ek(S1xX; oxX) ______oEk((X+e); o)
pp
pp  
pp r r
ppr   (2:13)
p? ? ?
~= ~=
Em1 (X) ________Em (S1xX; oxX) ______oEme((X+ ); o)
which we can express algebraically as
(r)x = rx; (2:14)
(b) The image of r under the E*module homomorphism
(1)k1f*k1
: Em (E_k; o) ! Em (E_k1; o) ~=Em1 (E_k1; o)
induced by the structure map fk1: E_k1 ! E_kof [8, Defn. 3.19];
(c) The map
(1)mkr
r: E_k1' E_k ! E_m ' E_m1;
where we use the right adjunct equivalences to fk1 and fm1 .
Proof For a based space X, diag. (2.13) simplifies by naturality to
~=
Ek1(X; o) ______Ek(X; o)
pp
ppr r (2:15)
? ~ ?
=
Em1 (X; o)______Em (X; o)
If we evaluate on the universal case k1 2 Ek1(E_k1; o) by eq. (2.14), we find
(r)k1 = rk1 = (1)k1rf*k1k = (1)k1f*k1r;
which gives (b). Further, by [8, Lemma 3.21], the class (r)k1 2 E*(E_k1; o)
corresponds, up to the sign (1)m1 , to the lower route in the square
fk1
E_k1 _________E_k

 
(r)  (1)mk r in Ho (2:16)

? fm1 ?
E_m1 ________E_m
which therefore commutes up to sign. We take adjuncts of this to get (c). 
We recall from [8, Defn. 9.3] the stabilization map oek: E_k! E of spectra.
JMB, DCJ, WSW  11  23 Feb 1995
Unstable cohomology operations
Corollary 2.17 Ooe*k= oe*k1: E*(E; o) ! E*(E_k1; o):
Proof Suppose the stable operation r 2 Eh(E; o) restricts to give the additive*
* op
erations rk: k ! k + h and rk1: k1 ! k+h1. By [8, eq. (9.8)], oe*kr = (1)kh*
*rk
and oe*k1r = (1)(k1)hrk1. We compare diag. (2.16) with [8, eq. (9.2)] to s*
*ee that
rk = (1)hrk1. 
Corollary 2.18 The loop construction in Prop. 2.12(b) factors as
: E*(E_k; o) ! QE*(E_k) ! P E*(E_k1) E*(E_k1; o) : (2:19)
Proof It is clear from Prop. 2.12(c), or from eq. (2.14) and Lemma 2.10, that *
*r is
always additive. The construction factors through QE*(E_k) by Prop. 2.12(b) and
naturality of Q, since QE*(E_k1) ~=E*(E_k1; o). (Loosely, there are no produc*
*ts
in E*(X; o).) 
These results allow us to rewrite the Milnor short exact sequence [8, eq. (*
*9.7)] in
the more useful form (which does not change any terms)
0 ! lim1P E*(E_k) ! E*(E; o) ! limP E*(E_k) ! 0 : (2:20)
k k
It remains true that the projection from E*(E; o) is an open map, and therefore*
* a
homeomorphism whenever it is a bijection. The kth component is the E*module
homomorphism
oe*k: E*(E; o) ! P E*(E_k) E*(E_k) (2:21)
induced by the stabilization map oek. It sends a stable operation r to the ind*
*uced
additive operation rk on Ek() (but with a sign; see [8, eq. (9.9)]).
The factorization (2.19) raises two obvious questions:
(a) Can every additive operation be delooped?
(2.22)
(b) Does r = 0 imply that r decomposes?
Both hold precisely when we have an isomorphism : QE*(E_k) ~=P E*(E_k1). We
discuss this further in x4.
3 Group objects and Ecohomology
Before we can discuss additive Ecohomology operations adequately, it is ne*
*cessary
to generalize x2. We extend Prop. 2.7 by defining the primitives P E*(X) for a*
*ny
group object X in the homotopy category Ho. Dually, we extend the definition of*
* the
indecomposables QE*(X) to any based space X.
Coalgebra primitives We start from the definition (2.8) of P E*(E_k).
Definition 3.1 Given any group object (or Hspace) X in Ho , with multiplicat*
*ion
: X xX ! X, we define the E*submodule P E*(X) of coalgebra primitives in E*(X)
as
P E*(X) = {x 2 E*(X) : *x = p*1x + p*2x in E*(X xX) }:
JMB, DCJ, WSW  12  23 Feb 1995
x3. Group objects and Ecohomology
Remark As in Prop. 2.7(c), the class x 2 Ek(X) is primitive if and only if the
associated map x: X ! E_kis a morphism of group objects in Ho .
We note that P E*(X) is defined even if E*(X) is not a (completed) coalgebr*
*a.
Thus P E*(): Gp(Ho )op ! Mod is a functor defined on the dual of the category*
* of
group objects in Ho . We topologize P E*(X) as a subspace of E*(X).
If Y is another group object in Ho, we construct the product group object X*
* x Y
in the obvious way. The onepoint space T is trivially a group object, and is t*
*erminal
in Gp(Ho ). Lemma 6.14 of [8] carries over to this situation.
Lemma 3.2 For the product X x Y of two group objects X and Y in Ho , we have
P E*(X xY ) ~=P E*(X) P E*(Y ) in FMod . Also, P E*(T ) = 0.
In other words, the functor P E*() takes finite products in Gp(Ho ) to cop*
*roducts
(direct sums) in FMod .
Remark No K"unneth formula is needed for this result.
Proof We dualize the proof of [8, Lemma 6.11]. Let us write Z = X x Y for the
product group object and !Y : T ! Y for the unit (or zero) map of Y . We note f*
*irst
that the maps j1 = 1X x!Y : X ~=X xT ! X xY = Z, j2: Y ! Z (defined similarly),
p1: Z = X xY ! X, and p2: Z ! Y are all morphisms of group objects and therefore
send primitives to primitives. Define the map
f: Z = X x Y ~= (X xT ) x (T xY ) ! (X xY ) x (X xY ) = Z x Z
using (1X x!Y ) x (!X x1Y ). Then Z Of = 1Z and PsOf = jsOps (for s = 1; 2),
where Ps: Z xZ ! Z denotes the projection for Z. Any element z 2 P E*(Z) satisf*
*ies
*Zz = P1*z + P2*z, by definition. When we apply f*, we obtain z = p*1x + p*2y, *
*where
x = j*1z 2 E*(X) and y = j*2z 2 E*(Y ) must be primitive. Conversely, any primi*
*tives
x and y determine a primitive z by this formula. We have a homeomorphism because
j*sand p*sare continuous.
We compute P E*(T ) = {v 2 E* : v = v + v} = 0. 
Since the unit map !: T ! X of X is a morphism of group objects, P E*(T ) =*
* 0
implies that P E*(X) E*(X; o).
The space E_k is more than just a group object. By [8, Cor. 7.8], we have *
*the
E*module object n 7! E_nin Ho, on which v 2 Eh acts by the maps v: E_k! E_k+h
that represent scalar multiplication by v. Clearly, v is additive.
Lemma 3.3 Assume that E*(E_k) is Hausdorff for all k. Then:
(a) We have the E*module object n 7! P E*(E_n) in the ungraded category
FMod op, with the action of v 2 Eh given by P (v)*: P E*(E_k+h) ! P E*(E_k);
(b) The object in (a) is related to the stable E*module object E*(E; o) of *
*[8,
Prop. 11.3] by the following diagram, which commutes up to sign for any v 2 Eh,
(v)*
E*(E; o) ________E*(E;o)

 
oe*k(1)hk+h oe*k (3:4)
 
? P(v)* ?
P E*(E_k+h)______P E*(E_k)
JMB, DCJ, WSW  13  23 Feb 1995
Unstable cohomology operations
Proof In (a), the object n 7! E_nis in fact an E*module object in Gp(Ho ). We*
* apply
[8, Lemma 7.6(a)] to the functor P E*(); it preserves finite products by Lemma*
* 3.2.
For (b), we apply Ecohomology to diag. [8, eq. (9.8)], taking r = v. 
Indecomposables Dually, we extend eq. (2.11) to any based space X by defining
the quotient E*module
QE*(X) = Coker[*  i*1 i*2: E*(X xX) ! E*(X)] (3:5)
of "indecomposables" of E*(X). (We shall not need a topology on it.)
4 Group objects and Ehomology
We dualize x2 by defining the indecomposables QE*(E_k) and primitives P E*(*
*E_k)
in Ehomology. This will prove useful because E*(E_k) is usually smaller and m*
*ore
manageable than E*(E_k). As in x3, we need to handle more general X. However,
some properties that were immediate in x2 become less intuitive and have to be
proved.
The structure map fk: E_k ! E_k+1 (see [8, Defn. 3.19]) of the spectrum E
induces the important suspension homomorphism
fk*
E*(E_k) ! E*(E_k; o) ~=E*(E_k; o) ! E*(E_k+1; o); (4:1)
dual (apart from sign) to the looping in Prop. 2.12(b). Again, suspended eleme*
*nts
behave better. We dualize Lemma 2.10.
Lemma 4.2 For any elements x; y 2 Ek(X; o), the induced Ehomology homomor
phisms satisfy
(x + y)* = x* + y*: E*(X; o) ! E*(E_k; o) :
Proof By [8, Lemma 7.6(c)], Ehomology induces a homomorphism
Ho0(X; E_k) ! Mod (E*(X; o); E*(E_k; o))
of groups, where both group structures are induced by the cogroup structure on X
in Ho 0. By [8, Prop. 7.3], they agree with the obvious group structures. 
Indecomposables We dualize Defn. 3.1.
Definition 4.3 Given any group object (or Hspace) X in Ho , we define the E*
module QE*(X) of "indecomposables" of E*(X) as
QE*(X) = Coker[*  p1* p2*: E*(X xX) ! E*(X)]:
It comes equipped with a canonical projection E*(X) ! QE*(X).
When E*(X) is free, we have the K"unneth isomorphism Thm. 1.18(b)*
* for
E*(X xX) and this agrees with the usual definition for the algebra E*(X). We ne*
*ed
one easy example.
Lemma 4.4 Let G be a discrete abelian group. Then QE*(G) ~= E* Z G as an
E*module.
JMB, DCJ, WSW  14  23 Feb 1995
x4. Group objects and Ehomology
Proof We recognize E*(G) as the group algebra of G over E*, with an E*basis
element [g] for each g 2 G. The correspondence we seek is induced by v[g] $ v *
*g,
and is well defined in both directions. 
Lemma 3.2 dualizes without difficulty; again, no K"unneth formula is needed*
*. Then
we will be able to dualize Lemma 3.3.
Lemma 4.5 For the product X x Y of two group objects X and Y in Ho , we have
QE*(X xY ) ~=QE*(X) QE*(Y ). Also, QE*(T ) = 0. In other words, the functor
QE*(): Gp(Ho ) ! Mod preserves finite products. 
We have an immediate application to the Hopf bundle.
Lemma 4.6 Assume E has a complex orientation. Then the inclusion C P 1 !
Z x BU (see [8, eq. (5.8)]) defined by the Hopf line bundle over C P 1 induces*
* an
isomorphism of E*modules
~=
E*(C P 1) ! QE*(Z x BU) ~=E* QE*(BU) :
Proof The second isomorphism comes from Lemmas 4.5 and 4.4. We compare Lem
mas 5.4 and 5.6 of [8]; the generators fii corrrespond, except that fi0 7! (1; *
*0). 
Lemma 4.7 For any ring spectrum E:
(a) n 7! QE*(E_n) is an E*module object in the ungraded category Mod of E*
modules;
(b) The suspension (4.1) factors through QE*(E_k);
(c) The stabilization oek*: E*(E_k; o) ! E*(E; o) factors through QE*(E_k).
Proof The proof of (a) is like Lemma 3.3(a), except that we use the functor QE*
**()
and Lemma 4.5.
For (c), we use oe*k = k to restate the universal example (2.6) as
oekOk = oekOp1 + oekOp2: E_kx E_k! E in Stab*.
We apply Ehomology to see that oek* factors as desired. Similarly for (b), ex*
*cept
that we use Lemma 4.2 with X = (E_kxE_k), x = p1, and y = p2. 
Dually to the short exact sequence (2.20), we may use (b) and (c) to rewrit*
*e [8,
eq. (9.22)] in the more convenient form
E*(E; o) = colimE*(E_k; o) = colimQE*(E_k) : (4:8)
k k
There is a multiplication, analogous to the stable multiplication on E*(E; *
*o).
Lemma 4.9 There is a bilinear multiplication
QOE: QE*(E_k) QE*(E_m ) ! QE*(E_k+m );
which may be defined as a quotient of
x OE*
E*(E_k) E*(E_m ) ! E*(E_kxE_m ) ! E*(E_k+m ) :
JMB, DCJ, WSW  15  23 Feb 1995
Unstable cohomology operations
Proof The only difficulty is to prove that QOE is well defined. We express th*
*e dis
tributive law for the E*algebra object n 7! E_nas the commutative square
OEL
E_k x E_kx E_m ______E_k+mx E_k+m
fx1 g (4:10)
? OE ?
E_k x E_m ____________E_k+m
in which f = k, g = k+m , and OEL has the components OE O(p1 x 1) and OE O(p2 x*
* 1).
(Cohomologically, OEL represents the operation (x; y; z) 7! (xz; yz).) We deduc*
*e the
commutative diagram in homology
x OEL*
E*(E_kxE_k)E*(E_m ) ______E*(E_kxE_kxE_m ) ________E*(E_k+m xE_k+m )
f 1 (fx1) g*
 *  * 
? x ? OE ?
*
E*(E_k) E*(E_m ) __________E*(E_k x E_m)_____________E*(E_k+m )
(4:11)
By Defn. 4.3, we have the exact sequence
k*p1*p2*
E*(E_k x E_k) ! E*(E_k) ! QE*(E_k) ! 0 :
After tensoring with E*(E_m ), this remains exact. We note that diag. (4.10) an*
*d hence
diag. (4.11) also commute if we take f = p1 and g = p1, or f = p2 and g = p2. T*
*hen
diag. (4.11), with these three choices for f and g, shows that its bottom row i*
*nduces
a quotient pairing QE*(E_k) E*(E_m ) ! QE*(E_k+m ).
A second similar step, on the right, uses this pairing to produce QOE. 
Coalgebra primitives We also dualize eq. (3.5) in the obvious way. If X is a b*
*ased
space, we construct the E*module homomorphism
*  i1* i2*: E*(X) ! E*(X x X) : (4:12)
Definition 4.13 Given any based space X, we define the E*submodule of coalge*
*bra
primitives P E*(X) = Ker[*  i1* i2*] E*(X).
Again, the definition is meaningful even without a K"unneth formula for E*(*
*XxX).
The companion result to Lemma 4.4 is elementary.
Proposition 4.14 For any discrete based space X, we have P E*(X) = 0. 
The suspension (4.1) factors, with the help of Lemma 4.7(b), as
E*(E_k; o) ! QE*(E_k) ! P E*(E_k+1) E*(E_k+1; o) : (4:15)
Again we ask whether QE*(E_k) ! P E*(E_k+1) is an isomorphism.
Duality Under reasonable assumptions, the sequence (2.19) is dual to (4.15). O*
*ne
can see from Lemma 4.17 and x17 that this holds for each of our five examples E.
Moreover, in each case there are isomorphisms QE*(E_k) ~=P E*(E_k+1) in (4.15),*
* thus
answering the questions (2.22) affirmatively.
JMB, DCJ, WSW  16  23 Feb 1995
x5. What is an additively unstable module?
Lemma 4.16 Assume that E*(X) is a free E*module.
(a) If X is a group object in Ho (or an Hspace), then d induces a homeomorp*
*hism
d: P E*(X) ~=DQE*(X) in FMod ;
(b) If X is a based space and the image of the homomorphism (4.12) splits off
both E*(X) and E*(X xX), then d induces a bijection d: QE*(X) ~=DP E*(X).
Proof In (a), d induces the commutative diagram
*p*1p*2
0 __________P E*(X) _______E*(X) ______________________E*(X xX)
d d d
? ? D(*p1*p2*) ?
0 _________DQE*(X) ______DE*(X) ____________________DE*(X xX)
whose rows are exact by Defns. 3.1 and 4.3, because D automatically takes coker*
*nels
to kernels. Strong duality for X and X x X from Thm. 1.18 provides two homeomor
phisms d. The third d is therefore also a homeomorphism, because DQE*(X) has the
subspace topology from DE*(X) by [8, Lemma 6.15(c)].
The proof of (b) is analogous, except that we assume the splittings to ensu*
*re
that the bottom row of the relevant diagram is (split) exact, use [8, Lemma 6.1*
*5(a)]
instead, and have no topology to check. 
We clearly need information on when E*(E_k) is free.
Lemma 4.17 For E = H(F p), BP , MU, K(n), or KU:
(a) E*(E_k) and QE*(E_k) are free E*modules for all k;
(b) E*(E_k) and P E*(E_k) are complete Hausdorff for all k.
Proof For E = H(F p) or K(n), all E*modules are free and (a) is trivial.
We consider the remaining three cases together. For odd k, E*(E_k) is an ex*
*terior
algebra over E* by [23] (for BP or MU) or [8, Cor. 5.12] (for KU, when E_k = U),
and (a) is clear.
For even k, we write E_k = Ek x E_0kas in [8, eq. (3.7)], where E_0kdenotes*
* the
zero component and Ek is treated as a discrete group. Then E*(E_0k) is a polyno*
*mial
algebra over E*, by [23] (for BP or MU) or [8, Lemma 5.6(c)] (for KU, when E_0k=
BU), so that E*(E_0k) (and hence E*(E_k)) and QE*(E_0k) are free modules.
To finish (a), we note that by Lemmas 4.5 and 4.4,
QE*(E_k) = (E* Z Ek) QE*(E_0k) :
The first summand is free, because Ek = Z (for KU), or is Zfree (for MU), or is
Z (p)free (for BP ).
Part (b) is immediate from (a) by Thm. 1.18(a) and Cor. 2.9. 
5 What is an additively unstable module?
In this section, we give various interpretations of what it means to have a*
* module
over the additive unstable operations on Ecohomology. All four stable answers *
*in [8]
generalize.
JMB, DCJ, WSW  17  23 Feb 1995
Unstable cohomology operations
We recall from [8, Cor. 7.8] that each E_k is an abelian group object in Ho*
* and
therefore also in Gp(Ho ), and that n 7! E_nis an E*module object in Ho, with *
*v 2 Eh
acting by the map v: E_k! E_k+h. From Prop. 2.7 we have the submodule P E*(E_k)
of additive operations defined on Ek().
We assume throughout that E*(E_k) is a free E*module. Then by Cor. 2.9,
P E*(E_k) is complete Hausdorff and an object of FMod .
First Answer The additive operations r: k ! m act on E*(X) by composition
O: P Em (E_k) x Ek(X) ! Em (X) (5:1)
in Ho. We recover the stable action [8, eq. (10.1)] by using oe*k: E*(E; o) ! P*
* E*(E_k).
This composition is already biadditive. Given x 2 Ek(X) and v 2 Eh, the
commutative square
1xv
P Em (E_k+h) x Ek(X) ______PEm (E_k+h) x Ek+h(X)
 
P(v)*x1 O (5:2)
 
? ?
O
P Em (E_k) x Ek(X) ______________Em (X)
expresses the identity (r . v)x = rvx = r(vx) for operations r: k + h ! m. It s*
*uggests
that we should make the action (5.1) more closely resemble the stable action by
introducing a formal shift and rewriting it with a tensor product as
X : kP Em (E_k) k Ek(X) ! Em (X) : (5:3)
(Here, unlike [8], the action scheme is clearly visible: the notation k indicat*
*es that
the tensor product is to be formed using the two E*actions indexed by k.)
This approach was initiated in [27, x11]. However, it presents even more pr*
*oblems
than in the stable case, and we do not pursue it further here.
Second Answer Our hypotheses ensure that P E*(E_k) is dual to QE*(E_k). We
can convert the action of P E*(E_k) into a coaction
Ek(X) ! E*(X) b QE*(E_k) :
These are clearly not the components of an E*module homomorphism, because the
degree varies.
In x6, as suggested by (5.3), we shall shift degrees by introducing Q(E)k*=
kQE*(E_k), which will allow us to write the coaction as an E*module homomor
phism with components
aeX : Ek(X) ! E*(X) b Q(E)k* (5:4)
and the same action scheme as stably. We shall construct a comultiplication Q(*
* )
and counit Q(ffl) that make Q(E)**a coalgebra and allow us to interpret E*(X) a*
*s a
Q(E)**comodule.
Third Answer We write our Second Answer more functorially. Given any E*
module M, we construct the graded group A0M having the component
(A0M)k = Mi biQ(E)ki= (M b Q(E)k*)k
JMB, DCJ, WSW  18  23 Feb 1995
x5. What is an additively unstable module?
in degree k. In x6 we shall make A0M an E*module. Then M Q( ) and M Q(ffl)
define natural transformations 0: A0 ! A0A0 and ffl0: A0 ! I, which will make *
*A0 a
comonad in FMod and E*(X)^ an A0coalgebra.
Fourth Answer Still imitating the stable case, we eliminate all tensor product*
*s by
converting the First Answer to adjoint form. This will make everything very mu*
*ch
cleaner, evidence that this is the natural answer (although the Second Answer is
undeniably convenient for computation).
Any element x 2 Ek(X) may be regarded as a map x: X ! E_k, which induces the
morphism x*: E*(E_k) ! E*(X)^ in FMod . Generally, given any object M in FMod ,
we define for each integer k the abelian group
AkM = FMod (P E*(E_k); M) (5:5)
of all continuous E*module homomorphisms P E*(E_k) ! M. (There is no need to
shift degrees.) Then we convert the action (5.1) to the coaction
aeX : Ek(X) ! Ak(E*(X)^) = FMod (P E*(E_k); E*(X)^) (5:6)
by defining aeX x = x*P E*(E_k).
We assemble the AkM, as k varies, to form the graded group AM with componen*
*ts
(AM)k = AkM, and the coactions aeX into the single homomorphism aeX : E*(X) !
A(E*(X)^) of graded groups of degree zero.
The destabilization oe*k: E*(E; o) ! P E*(E_k) (see [8, Defn. 9.3]) induces
AkM = FMod (P E*(E_k); M) ! FMod k(E*(E; o); M) = (SM)k; (5:7)
if we also assume that E*(E; o) is Hausdorff. As k varies, we take these as the
components of the stabilization natural transformation oeM: AM ! SM, of degree
zero. It allows us to compare with the stable case.
Theorem 5.8 Assume that E*(E_k) is a free E*module for all k (as is true f*
*or
E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) We can make the functor A, defined in eq. (5.5), a comonad in the catego*
*ry
FMod of complete Hausdorff filtered E*modules;
(b) If E*(E; o) is also Hausdorff, the stabilization oe: A ! S (defined in e*
*q. (5.7))
is a morphism of comonads in FMod .
The relevant definitions are now clear.
Definition 5.9 An additively unstable (Ecohomology) module is an Acoalgebra
in FMod , i. e. a complete Hausdorff filtered E*module M equipped with a morph*
*ism
aeM : M ! AM in FMod that satisfies the coaction axioms [8, eq. (8.7)]. We t*
*hen
define the action of r 2 P Em (E_k) on x 2 Mk by rx = aeM (x)r 2 M (with no sig*
*n).
A closed submodule L M is called (additively unstably) invariant if aeM r*
*estricts
to give aeL: L ! AL. Then the quotient M=L inherits an additively unstable modu*
*le
structure.
JMB, DCJ, WSW  19  23 Feb 1995
Unstable cohomology operations
This is a stronger structure than a stable module (when E*(E; o) is Hausdor*
*ff, so
that stable modules exist). Given a coaction aeM as above, Thm. 5.8(b) shows t*
*hat
the coaction
aeM oeM
M ! AM ! SM (5:10)
makes M a stable module.
One may think of AkM as the set of all candidates for the action of P E*(E_*
*k) on
a typical element of Mk, and aeM as the selection of a candidate for each x 2 *
*Mk. The
coaction axioms translate into the usual action axioms (sr)x = s(rx) and kx = x.
As stably, it is sometimes useful to fix r: k ! m and express the first axiom a*
*s the
commutative square
r
Mk _______Mm
ae ae
 M  M (5:11)
? !rM ?
AkM ______AmM
where !rM denotes composition with P r*: P E*(E_m ) ! P E*(E_k).
Theorem 5.12 Assume that E*(E_k) is a free E*module for all k (as is true *
*for
E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) aeX (defined in eq. (5.6)) factors through E*(X)^ as aeX : E*(X)^ ! A(E**
*(X)^)
to make E*(X)^ an additively unstable module for any space X;
(b) If E*(E; o) is Hausdorff, we recover the stable coaction in [8, Thm. 10.*
*16(a)]
from aeX by diag. (5.10);
(c) ae is universal: given an object N of FMod and an integer k, any add*
*i
tive natural transformation of abelian groups X: Ek(X) ! FMod (N; E*(X)^) (or
bX: Ek(X)^ ! FMod (N; E*(X)^)) that is defined on all spaces X is induced from*
* aeX
by a unique morphism f: N ! P E*(E_k) in FMod , as
aeX k * * *
X: Ek(X) ! A (E (X)^) = FMod (P E (E_k); E (X)^)
Hom(f;1)
! FMod (N; E*(X)^) :
Proof of Thms. 5.8 and 5.12 We prove parts (a) and (b) of both Theorems togeth*
*er,
in the same seven steps as the stable proof of Thms. 10.12 and 10.16 of [8]. As*
* most
steps are more or less repetitions of that proof, except for the insertion of i*
*ndices
everywhere, we indicate only the substantive changes for (a) and the additions *
*needed
to handle oe for the (b) parts. Instead of 2 E*(E; o), we have k 2 P E*(E_k). *
*Instead
of idA, we have the identity map idk: P E*(E_k) = P E*(E_k), considered as an e*
*lement
of AkP E*(E_k). We write aek for aeX when X = E_k.
Step 1. We construct an E*module structure on the graded group AM we de
fined in eq. (5.5). We start with the E*module object n 7! P E*(E_n) in FMod *
* op
from Lemma 3.3(a), with v 2 E* acting by P (v)*. We apply the additive func
tor Mor (; M): FMod op! Ab to obtain by [8, Lemma 7.6(a)] the E*module object
n 7! AnM in Ab, i. e. make AM an E*module.
JMB, DCJ, WSW  20  23 Feb 1995
x5. What is an additively unstable module?
Despite appearances, the square (3.4) does commute in the dual ca*
*tegory
FMod *op, to show that oeM: AM ! SM is an E*module homomorphism.
Step 2. We have defined aeX as a natural transformation of sets. For fixed *
*X, the
cohomology functor E*()^: Ho ! FMod op induces the natural transformation
Ho (X; ) ! FMod (P E*()^; E*(X)^): Gp(Ho ) ! Set:
We apply [8, Lemma 7.6(c)] to the E*module object n 7! E_n to see that aeX is*
* a
morphism of E*module objects, i. e. takes values in Mod .
For Thm. 5.12(b), we note that given x 2 Ek(X), we have (oe(E*(X)^))x*U =
x*UOoe*k= x*S, by [8, eq. (9.4)].
If X is a group object in Ho and x 2 P Ek(X), the associated map x: X !
E_k is a morphism of group objects (as remarked after Defn. 3.1) and so induces
x*: P E*(E_k) ! P E*(X). If E*(X) (and hence P E*(X)) is Hausdorff, aeX restric*
*ts
to define
P aeX : P E*(X) ! AP E*(X) : (5:13)
Step 3. We filter AM exactly as we did SM in [8, x10], by the submodules
F a(AM) = A(F aM), using naturality. The proof that AM is complete Hausdorff
is formally the same as for SM. Our choice of filtrations and the naturality o*
*f ae
clearly make aeX and oeM continuous, so that aeX factors through E*(X)^ and oe *
*takes
values in FMod .
Step 4. Whenever X is a group object in Ho and E*(X) is Hausdorff, we conve*
*rt
the object P E*(X) of FMod to the corepresented functor
FPX = FMod (P E*(X); ): FMod ! Ab
and the coaction P aeX in (5.13) to a natural transformation aePX : F*
*PX !
FPX A: FMod ! Ab. Given M, aePX M: FPX M ! FPX AM is the homomorphism
aePX M: FMod (P E*(X); M) ! FMod (P E*(X); AM) (5:14)
that is defined on f: P E*(X) ! M as the composite
PaeX * Af
(aePX M)f: P E*(X) ! AP E (X) ! AM :
Step 5. To construct = A: A ! AA, we take X = E_kin (5.14) and define
( M)k: FMod (P E*(E_k); M) ! FMod (P E*(E_k); AM)
on the element f: P E*(E_k) ! M of AkM as the composite
Paek * Af
( M)kf: P E*(E_k) ! AP E (E_k) ! AM :
When we substitute the E*module object n 7! E_nfor X in (5.14), [8, Lemma 7.6(*
*c)]
shows that ( M)k: AkM ! AkAM lies in Mod . As k varies, we obtain the natural
transformation : A ! AA. Naturality in M also shows that M is filtered and so
lies in FMod .
Step 6. The other required natural transformation, ffl: A ! I, is defined o*
*n M
simply as the evaluation
(fflM)k = (fflAM)k: AkM = FMod (P E*(E_k); M) ! M (5:15)
JMB, DCJ, WSW  21  23 Feb 1995
Unstable cohomology operations
on k 2 P Ek(E_k). It is continuous by naturality. It is compatible with the s*
*table
version, fflA = fflS Ooe: A ! I, since given f 2 AkM, we have
(fflSM)(oeM)f = ((oeM)f) = foe*k = fk = (fflAM)f :
Step 7. To see that aeX is a coaction on E*(X), we use [8, Lemma 8.20] (ada*
*pted
to graded objects). We use R = P E*(E_n) (really, the graded object n 7! P E*(E*
*_n)),
1R = n, and aeR = P aen. By [8, Lemma 8.22], A is a comonad in FMod .
To see that oe: A ! S is a morphism of comonads, we apply [8, Lemma 8.24]. *
*The
first condition on u = oe*k: E*(E; o) ! P E*(E_k) is the commutative diagram
oe*k
Eh(E; o) _____________________________P Ek+h(E_k)

 
 Paek
 
 ?
aeE FMod (P E*(E_k+h); P E*(E_k))

 
 Hom(oe*k+h;1)
 
? Hom(1;oe* ?
k)
FMod h(E*(E; o); E*(E; o))__________FMod k+h(E*(E; o); P E*(E_k))
A stable operation rS 2 Eh(E; o) restricts to an additive operation rU : k ! k *
*+ h.
On rS, the lower route gives by diag. [8, eq. (9.8)]
oe*kOr*S= (1)hk(rS Ooek)* = (1)hk(oek+h OrU )* = (1)hkr*UOoe*k+h:
This agrees with the upper route, because oe*krS = (1)hkrU by [8, eq. (9.9)]. *
* The
second condition needed is oe*k = k, which holds by the definition of oek.
For Thm. 5.12(c), as in [8, Thm. 10.16(b)], it is enough to consider X. Bec*
*ause
E_k represents Ek(), natural transformations are classified by the elements f*
* =
k: N ! E*(E_k), i. e. morphisms in FMod . The additivity (X)(x+y) = (X)(x) +
(X)(y) of X on the universal example (2.6) yields
*kOf = p*1Of + p*2Of: N ! E*(E_k x E_k):
By Prop. 2.7(b), f factors through P E*(E_k). 
6 Unstable comodules
Although the Fourth Answer of x5 is the cleanest and most general, the Seco*
*nd
Answer, in terms of unstable comodules, is usually the most practical and is av*
*ailable
in the cases of interest. The parallel with the stable theory of [8] is extreme*
*ly close,
in spite of the very different provenance of the two theories. Some of the mach*
*inery
was used in [6]; here we supply the missing definitions.
We assume throughout this section that E*(E_k) and QE*(E_k) are free E*mod*
*ules
for all k, so that we have available all the results of x5.
The bigraded group Q(E)** As noted in x5, tensor products do not work correctly
because the groups QE*(E_k) have the wrong degree; we therefore shift degrees. *
*We
JMB, DCJ, WSW  22  23 Feb 1995
x6. Unstable comodules
also adopt more efficient notation, that hides the details of construction and *
*empha
sizes the algebraic aspects and the formal similarity to stable comodules. (We *
*remind
that homology Ei(X) has degree i under our conventions.)
Definition 6.1 We define the bigraded group Q(E)**as having the components
Q(E)ki= QEi(E_k) (the component of QE*(E_k) in degree i), except that we assign
the degree ki (instead of i) to elements of Q(E)ki. (This is the degree that *
*governs
signs in formulae. We thus have the formal isomorphism k: QE*(E_k) ~=Q(E)k*of
degree k.)
We define the left action of v 2 Eh on kc 2 Q(E)k*, for c 2 QE*(E_k), by v(*
*kc) =
(1)hkkvc, as in [8, eq. (6.7)], to make k: QE*(E_k) ~= Q(E)k*an isomorphism of
E*modules of degree k.
We equip Q(E)k*with the projection
k k
qk: E*(E_k) ! QE*(E_k) ! Q(E)* : (6:2)
We define the stabilization
k Qoek*
Q(oe): Q(E)k*! QE*(E_k) ! E*(E; o); (6:3)
where Lemma 4.7(c) provides the factorization Qoek* of oek* .
We thus have the factorization into E*module homomorphisms
oek* = Q(oe) Oqk: E*(E_k) ! Q(E)k*! E*(E; o); (6:4)
where we arranged for Q(oe) to have degree zero and qk to have degree k.
Definition 6.5 Given an additive operation r: k ! m, i. e. an element rA 2
P Em (E_k), we define the associated E*linear functional
k *
: Q(E)k*! QE*(E_k) ! E (6:6)
of degree mk (with no sign).
Now we can make the degree shift suggested by eq. (5.4). We have the strong
duality P E*(E_k) ~=DQE*(E_k) from Lemma 4.16(a). Given an object M of FMod ,
we use [8, Lemma 6.16(b)] and the freeness of QE*(E_k) to define the natural is*
*omor
phism of degree k
Mk k
FMod *(P E*(E_k); M) ~=M b QE*(E_k) ! M b Q(E)* : (6:7)
Lemma 6.8 Given an additive operation r: k ! m and an object M of FMod , the
composite (formed using (6.7))
M
FMod *(P E*(E_k); M) ~=M b Q(E)k*! M E* ~=M
coincides with the evaluation homomorphism er: FMod *(P E*(E_k); M) ! M defined
by erf = (1)m deg(f)frA.
JMB, DCJ, WSW  23  23 Feb 1995
Unstable cohomology operations
Proof We choose x 2 M, c 2 QE*(E_k), and evaluate. 
With Defn. 6.5 in hand, we extend Prop. 2.7 and identify:
(i)the additive operation r: Ek() ! Em ();
(ii)the cohomology class r = rA = rk 2 P Em (E_k);
(iii)the morphism of group objects r: E_k! E_m in Ho ; (6.9)
(iv)the E*linear functional = : Q(E)k*! E*, of degree
mk, defined by eq. (6.6).
(We drop the decorations A and Q on r except when we need to compare different
versions.) As Q(E)**is smaller than P E*(E_*), (iv) is the preferred choice. *
*We do
have to be careful with degrees, as (ii) has a different degree from (i) and (i*
*v), while
(iii) has no degree at all.
Scholium on signs We construct the duality diagram in FMod *
rS rA rU
oe*k
E*(E; o)_______P E*(E_k)_______E*(E_k)
  
~= (1)k ~= (1)k ~= (6:10)
  
? DQ(oe) ? Dq ?
DE*(E; o) ______D(Q(E)k*) ______DE*(E_k)k
whose center isomorphism is taken as
d D(k) k
P E*(E_k) ! DQE*(E_k) ! D(Q(E)*) :
Because D is contravariant, each square commutes up to the sign (1)k.
On restriction to spaces, a stable operation r of degree h yields an additi*
*ve unstable
operation r: k ! k + h, and we obtain elements rS, rA, and rU lying in the indi*
*cated
groups. From these, we get the linear functionals , , and by eq*
*. (6.6)
also . We note that rS and rQ have degree h, while rA and rU have degre*
*e k+h.
The algebra forces us to work with the element rA and the functional ; *
*we are
not really interested in the functional , which appears only in the defi*
*nition of
, and the element rQ will occur nowhere.
The complication is that these six elements do not all correspond in obvious
ways under the morphisms of diag. (6.10). The first surprise was [8, eq. (9.9)*
*], that
oe*krS = (1)khrA. Of course, rA and rU do correspond, because they are the sa*
*me
element regarded as being in different groups. The second surprise is that rA d*
*oes not
correspond to , because the definition [8, eq. (6.4)] of D(k) requires *
*the sign
(1)k(h+k), which is absent from Defn. 6.5. In fact, matters are simpler if we*
* work
with elements and refrain from turning everything into E*module homomorphisms.
Proposition 6.11 In diag. (6.10):
JMB, DCJ, WSW  24  23 Feb 1995
x6. Unstable comodules
(a) Given a stable operation r, the homomorphism DQ(oe) takes to ,
or in elements,
= for c 2 Q(E)k*, (6:12)
and also
= for c 2 E*(E_k); (6:13)
(b) Given an additive operation r: k ! m, the homomorphism Dqk takes
to (1)k(mk), or equivalently, in elements,
= for c 2 E*(E_k). (6:14)
Proof We just proved (a), except for eq. (6.13), which combines eqs. *
* (6.12)
and (6.14). In (b), is simply the restriction of , so that
= = :
But the definition of Dqk adds the unwanted sign (1)k(mk). 
Q(E)**as an algebra There is much structure on Q(E)**. First, it is by constru*
*ction
a left E*module.
Proposition 6.15 For any ring spectrum E, Q(E)**has the properties:
(a) Q(E)**is a bigraded E*algebra, with multiplication Q(OE) defined by the*
* com
mutative diagram (6.16)
x OEU*
E*(E_k) E*(E_m ) ______E*(E_kxE_m) _________E*(E_k+m )
 
 
qkqm qk+m
 
? Q(OE) ?
Q(E)k* Q(E)m* ______________________________Q(E)k+m* (6:16)
 
 
Q(oe)Q(oe) Q(oe)
 
? x OES ?
E*(E; o) E*(E; o) ______E*(E^E;o) ___________E*(E;o)*
and unit Q(j) defined by the commutative diagram
E*(T )________E*= _______E*(T+;=o)
  
jU* Q(j) jS* (6:17)
  
? q ? ?
0 Q(oe)
E*(E_0) ______Q(E)0*______E*(E; o)
(b) The stabilization Q(oe): Q(E)**! E*(E; o) is a homomorphism of E*algebr*
*as.
Proof Q(OE) is inherited, with a shift, from the multiplication on QE*(E_*) co*
*n
structed by Lemma 4.9. It thus fills in diag. (6.16), which is derived from [8,*
* eq. (9.15)]
by applying Ehomology and the factorization (6.4). We simply define Q(j) = q0O*
* jU*,
JMB, DCJ, WSW  25  23 Feb 1995
Unstable cohomology operations
to fill in diag. (6.17). This comes from diag. [8, eq. (9.4)] by taking x = 1T *
*2 E*(T ).
The algebraic properties of Q(OE) and Q(j) are inherited from the E*algebra ob*
*ject
n 7! E_nin Ho . Part (b) is clear from the diagrams. 
Q(E)**as a bimodule We also need the right E*action. By Lemma 4.5, the functor
QE*(): Gp(Ho ) ! Mod preserves finite products. We apply [8, Lemma 7.6(a)] to
the E*module object n 7! E_nin Gp(Ho ), to obtain, for each v 2 Eh, homomorphi*
*sms
Q(v) that fill in the commutative diagram
qk Q(oe)
E*(E_k) ________Q(E)k*_______E*(E; o)
  
  
(Uv)* Q(v) (Sv)* (6:18)
  
? q ? ?
k+h Q(oe)
E*(E_k+h) ______Q(E)k+h*______E*(E; o)
and make Q(E)**a module object in Mod *, i. e. an E*bimodule. This diagram came
from diag. [8, eq. (9.8)] by taking r = v.
We have the additive analogue of the stable right unit.
Definition 6.19 We define the right unit function jR: E* ! Q(E)**on v 2
Eh = Eh(T ) by jRv = qhv*1 2 Q(E)h0, using the homology homomorphism
v*: E* ~=E*(T ) ! E*(E_h) induced by the map v: T ! E_h.
It is clear from [8, eq. (9.4)] and the factorization (6.4) that compositio*
*n with Q(oe)
yields the stable right unit jR: E* ! E*(E; o) of [8, Defn. 11.2].
Proposition 6.20 For any ring spectrum E, the algebra Q(E)**has the properti*
*es:
(a) It is a bigraded E*bimodule, with components Q(E)ki= QEi(E_k) which are
assigned the degree ki;
(b) It has the welldefined unit element 1 = Q(j)1 = jR1 2 Q(E)00;
(c) The left action of v 2 Eh is left multiplication by v1 2 Q(E)0h;
(d) The right action of v 2 Eh is right multiplication by jRv 2 Q(E)h0;
(e) The stabilization Q(oe): Q(E)** ! E*(E; o) is a homomorphism of *
*E*
bimodules.
Remark Props. 6.15 and 6.20 are similar to [8, Prop. 11.3], except that Q(E)**
**is
bigraded and the conjugation O is conspicuous by its absence. The examples of x*
*16
show that O does not exist, at least, not in any obvious sense. (This is why we
eschewed O in [8].)
Proof Most of the proof is formally identical to the stable case [8, Prop. 11.*
*3]. For
(d), we apply Ehomology to the factorization [8, eq. (3.27)] of v. Part (e) is*
* clear
from diag. (6.18). 
We write the left and right E*actions as L: Eh Q(E)ki ! Q(E)kihand
R: Q(E)ki Eh ! Q(E)k+hi. Explicitly, the signs for R are
R(c v) = c . v = c(jRv) = (1)h deg(c)(jRv)c = (1)h deg(c)Q(v)c;(6:21)
JMB, DCJ, WSW  26  23 Feb 1995
x6. Unstable comodules
where v 2 Eh and c . v denotes the right action. For future use, we rewrite (d)*
* as the
commutative square
Q(OE)
Q(E)k* Q(E)m* ________Q(E)k+m*
 
 
Q(v)1 Q(v) (6:22)
 
? Q(OE) ?
Q(E)k+h* Q(E)m* ______Q(E)k+m+h*
The functor A0 Given an E*module M, we define (as promised in x5) the graded
group A0M as having the components
(A0M)k = Mi biQ(E)ki= (M b Q(E)k*)k (6:23)
(where the tensor product b iis formed using the two E*actions indexed by i. *
*We
have no use for the rest of M b Q(E)k*!) We use the isomorphism (6.7) to define*
* the
isomorphism AM ~=A0M as having the components
(AM)k = AkM = FMod (P E*(E_k); M) ~=Mi biQ(E)ki= (A0M)k: (6:24)
We use this isomorphism to transfer all the structure of x5 from A to A0and mak*
*e A0
a comonad, just as we did stably in [8]. (We generally drop the decorations 0ex*
*cept
when comparing different versions.)
In particular, we use (6.24) to convert modules to comodules. If M is an a*
*ddi
tively unstable module with coaction aeM : M ! AM (as in Defn. 5.9), we deduce *
*the
equivalent coaction ae0M: M ! A0M with components
ae0M: Mk ! (A0M)k = Mi biQ(E)ki M b Q(E)k*: (6:25)
In particular, for a space X, we convert the action aeX in (5.6) to
ae0X: Ek(X) ! Ei(X) biQ(E)ki E*(X) b Q(E)k*: (6:26)
Q(E)**as a coalgebra The stable discussion carries over, except that Q(E)**is
bigraded. The comonad structure ( ; ffl) on A translates into a comonad struct*
*ure
( 0; ffl0) on A0. By naturality and the case M = iE*, 0M: (A0M)k ! (A0A0M)k
must take the form M b for a certain comultiplication
= Q( ): Q(E)ki! Q(E)jij Q(E)kj (6:27)
(where we sum over j as in eq. (6.23)), and ffl0M: (A0M)k ! Mk must take the fo*
*rm
M b ffl for a certain counit
ffl = Q(ffl): Q(E)ki! Eki: (6:28)
By construction, these are both E*bimodule homomorphisms of degree zero.
Proposition 6.29 Assume that E*(E_k) and QE*(E_k) are free E*modules for all
k. Then:
(a) The homomorphisms = Q( ) and ffl = Q(ffl) in diags. (6.27) and (6.28) *
*make
Q(E)**a coalgebra over E*;
(b) If E*(E; o) is also free, the stabilization Q(oe): Q(E)**! E*(E; o) is a*
* morphism
of coalgebras (cf. [8, Lemma 11.8]).
JMB, DCJ, WSW  27  23 Feb 1995
Unstable cohomology operations
Proof By taking M = iE*, the comonad axioms [8, eq. (8.6)] for A0 yield the
coassociativity
Q( )
Q(E)kh _________________Q(E)jhj Q(E)kj
 
Q( ) 1Q( ) (6:30)
 
? Q( )1 ?
Q(E)ihiQ(E)ki ________Q(E)ihiQ(E)jij Q(E)kj
of Q( ) and the two counit axioms
Q( ) k Q( ) j k
Q(E)ki ______Q(E)jijQ(E)kj Q(E)i ______Q(E)ij Q(E)j
  
   
= Q(ffl)1 = 
   1Q(ffl) (6:31)
   
? ? ? ?
L k R j kj
Q(E)ki _______EjijoQ(E)kje Q(E)i ______oQ(E)iej E
Part (b) is the translation of Thm. 5.8(b). 
Comodules Now that we have the coalgebra Q(E)**, we can convert Defn. 5.9 and
Thm. 5.12.
Definition 6.32 An unstable (Ecohomology) comodule is an A0coalgebra*
* in
FMod .
In detail, given a complete Hausdorff filtered E*module M (i. e. object of*
* FMod ),
an unstable comodule structure on M consists of a coaction aeM : M ! A0M, with *
*com
ponents Mk ! Mi biQ(E)kias in diag. (6.25), that is a continuous homomorphism
of E*modules (i. e. morphism in FMod ) and satisfies the axioms
aeM
aeM M ______Mb Q(E)**
M ______________M b Q(E)** Q
Q ~= 
aeM MQ( ) Q MQ(ffl)
? ae ? Q 
M 1 Qs ? (6:33)
M b Q(E)** ______Mb Q(E)**bQ(E)** M E*
(i)
(ii)
This is a stronger structure than a stable comodule (assuming that E*(E; o)*
* is free,
so that stable comodules can be defined). Given a coaction aeM as above, Prop. *
*6.29(b)
shows that the coaction
aeM * MQ(oe)
M ! M b Q(E)* ! M b E*(E; o) (6:34)
makes M a stable comodule.
Remark We regard comodules as essentially additive constructs, as we find no a*
*na
logue in the fully unstable context. We therefore omit the adjective "additive"*
* from
comodules.
JMB, DCJ, WSW  28  23 Feb 1995
x6. Unstable comodules
Theorem 6.35 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k
(which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a).) Then
given a complete Hausdorff filtered E*module M (i. e. object of FMod ), an add*
*itively
unstable module structure on M in the sense of Defn. 5.9 is equivalent to an un*
*stable
comodule structure on M in the sense of Defn. 6.32.
Proof We have the isomorphism AM ~= A0M in eq. (6.24). The axioms (6.33) are
just the general coaction axioms [8, eq. (8.7)] interpreted for A0. 
Theorem 6.36 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k
(which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a).) Then:
(a) For any space X, there is a natural coaction
aeX : E*(X) ! E*(X) b Q(E)**
that makes E*(X)^ an unstable comodule, which corresponds by Thm. 6.35 to the
additive module structure given by Thm. 5.12;
(b) If also E*(E; o) is free, we recover the stable coaction [8, eq. (11.15)*
*] on E*(X)
from aeX as in diag. (6.34);
(c) ae is universal: given a discrete E*module N and an integer k, any addi*
*tive
natural transformation X: Ek(X) ! E*(X) b N (or bX: Ek(X)^ ! E*(X)^ bN)
that is defined for all spaces X is induced from aeX by a unique homomorphism
f: Q(E)k*! N of E*modules as
aeX * k 1f *
X: Ek(X) ! E (X) b Q(E)* ! E (X) b N :
Proof We deduce (a) from Thm. 5.12(a) and Thm. 6.35, just as we did stably in *
*[8,
Thm. 11.14]. In eq. (6.26), we defined the coaction ae0Xas corresponding to aeX*
* . In (b),
the stabilization Q(oe) clearly dualizes to oe*k: E*(E; o) ! P E*(E_k), which w*
*e used in
eq. (5.7) to define the stabilization oe: A ! S of comonads.
In (c), the natural transformation is classified by the element u = k 2
E*(E_k) b N. Additivity of for the universal example (2.6) states that
(*kN)u = (p*1N)u + (p*2N)u in E*(E_kxE_k) b N.
By [8, Lemma 6.16(a)], u corresponds to a homomorphism f: E*(E_k) ! N of E*
modules. The above property dualizes to
f Ok * = f Op1*+ f Op2*: E*(E_kxE_k) ! N;
which shows that f factors through Q(E)k*as required. 
Remark Just as stably, (c) allows us to use diags. (6.33) to define Q( ) and Q*
*(ffl) in
terms of ae. Three applications of the uniqueness in (c) show that Q( ) is coas*
*sociative
and has Q(ffl) as a twosided counit.
Linear functionals Theorem 6.35 establishes the equivalence between unstable
modules and comodules. For applications, we need the details. All our formulae *
*sta
bilize to the corresponding formulae of [8, x11] by applying Q(oe), which conve*
*niently
has degree zero.
JMB, DCJ, WSW  29  23 Feb 1995
Unstable cohomology operations
Given an unstable comodule M, we recover the action of the additive operati*
*on
r: k ! m on M from Lemma 6.8 as
aeM k M *
r: Mk ! M b Q(E)* ! M E ~=M: (6:37)
Because has degree mk, r takes values in Mm . To make this action expli*
*cit,
let us choose x 2 Mk and write
X
aeM x = (1)deg(xff) deg(cff)xff cff in M b Q(E)k*, (6:38)
ff
where the sum may be infinite, and of course deg(xff) = k  deg(cff). (As in [8*
*], we
insert signs here to keep the next formula simple.) Then
X
rx = xff in M, for all r: k ! m, (6:39)
ff
where the cffand xffdepend only on x, not on r. Because M is assumed complete, *
*this
sum converges if it is infinite. (Recall that Q(E)k*always has the discrete top*
*ology.)
Remark It is important for our applications not to require the cffto form a ba*
*sis
of Q(E)k*, or even be linearly independent; but if they do form a basis, the xf*
*fare
uniquely determined by eq. (6.39) as xff= c*ffx, where c*ffdenotes the operatio*
*n dual
to cff.
The fact that aeM is an E*module homomorphism is expressed by
X X
r(vx) = xff= (1)h deg(cff) xff in M,(6:40)
ff ff
for any v 2 Eh and all operations r: k + h ! m.
Because Q(ffl): Q(E)k*! E* corresponds to ffl in eq. (5.15), which is evalu*
*ation on
k, we have immediately
= Q(ffl): Q(E)k*! E*; (6:41)
as is obvious by comparing axiom (6.33)(ii) with eq. (6.37). In other words, i*
*n the
list (6.9), the identity operation k corresponds to the functional Q(ffl).
The cohomology of a point Our first test space is the onepoint space T .
Proposition 6.42 In the unstable comodule E*(T ) = E*:
(a) The action of the additive operation r: k ! m on v 2 Ek is given by
rv = in E*(T ) = E*; (6:43)
(b) The coaction aeT: E* ! E* Q(E)**~= Q(E)**coincides with the right unit
jR: E* ! Q(E)*0(see Defn. 6.19).
Proof We imitate [8, Prop. 11.22]. The map v: T ! E_kyields
rv = = = = = ;
by eq. (6.14) and Defn. 6.19 of jR. We compare eqs. (6.38) and (6.39) and rewr*
*ite
this as aeTv = 1 jRv, to give (b). 
Homology homomorphisms A class x 2 Ek(X) may be regarded as a map
x: X ! E_k. We need information about the induced homology homomorphism
x*: E*(X) ! E*(E_k).
JMB, DCJ, WSW  30  23 Feb 1995
x6. Unstable comodules
Proposition 6.44 Assume that E*(E_k) and QE*(E_k) are free E*modules for all
k. Given x 2 Ek(X), suppose that rx is given by eq. (6.39). Then the homomorphi*
*sm
qk Ox*: E*(X) ! Q(E)k*induced by the map x: X ! E_kis given on z 2 Eh(X) by
X X
qkx*z = (1)deg(cff)(deg(xff)+h) cff= cff in Q(E)k*.*
*(6:45)
ff ff
Proof For any additive r: k ! m, we have = by eq. (6.1*
*4). The
rest of the proof is formally identical to the stable analogue [8, Prop. 11.26]*
*. 
Conversely, we can recover aeX x from x* when X is well behaved, just as we*
* did
stably. If E*(X) is free, we have strong duality E*(X) ~=DE*(X) by Thm. 1.18(a),
and [8, Lemma 6.16(a)] supplies the isomorphism
E*(X) b Q(E)k*~=Mod *(E*(X); Q(E)k*) : (6:46)
Proposition 6.47 Assume that E*(X), E*(E_k), and QE*(E_k) are free E*modules
for all k. Take x 2 Ek(X). Then under the isomorphism (6.46), the element aeX*
* x
corresponds to the homomorphism qk Ox*: E*(X) ! E*(E_k) ! Q(E)k*.
Proof We apply the isomorphism to eq. (6.38) and compare with eq. (6.45). 
In particular, it is important to know the homomorphism of E*modules
Qr* m
Q(r): Q(E)k*~=QE*(E_k) ! QE*(E_m ) ~=Q(E)* (6:48)
induced by an additive operation r: k ! m (which by Prop. 2.7(c) is a morphism
of group objects in Ho ). It has degree mk. The Q(r) provide a convenient fait*
*h
ful representation of the additive operations. The translation of diag. (5.11)*
* is the
commutative square
r
Mk ________________Mm
aeM aeM (6:49)
? MQ(r) ?
Mi iQ(E)ki ________MiiQ(E)mi
which stabilizes to diag. [8, eq. (11.29)].
Just as stably, we easily recover the functional from Q(r) as
Q(r) Q(ffl)
: Q(E)k*! Q(E)m*! E*: (6:50)
Conversely, we have the additive analogue of [8, Lemma 11.31].
Lemma 6.51 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k. *
*If
r: k ! m is an additive operation, then the homology homomorphism Q(r): Q(E)k*!
Q(E)m*in diag. (6.48) has the properties:
(a) The diagram
Q(r)
Q(E)k* ________________Q(E)m*
 
Q( ) Q( ) (6:52)
? 1Q(r) ?
Q(E)** Q(E)k* ________Q(E)**Q(E)m*
JMB, DCJ, WSW  31  23 Feb 1995
Unstable cohomology operations
commutes; in other words, Q(r) is a morphism of left Q(E)**comodules;
(b) Q(r): Q(E)k*! Q(E)m*is the unique homomorphism of left E*modules that
satisfies eq. (6.50) and is a morphism of left Q(E)**comodules in the sense of*
* (a);
(c) Q(r) is given in terms of the functional as
Q( ) j 1 j
Q(r): Q(E)ki! Q(E)i j Q(E)kj ! Q(E)i j Emj
R m
! Q(E)i : 
We deduce from (c) that the composite sr: k ! n of the operations r: k ! m *
*and
s: m ! n corresponds to the functional
Q( ) j 1 j
: Q(E)ki! Q(E)i j Q(E)kj! Q(E)i j Emj
R m ni (6:53)
! Q(E)i ! E :
Remark From diags. (6.30) and (6.31)(ii) we observe that for fixed h, Q( ) mak*
*es the
graded group n 7! Qnhan additively unstable comodule, if we use the right E*mo*
*dule
action (6.21). Then by (c), the action of r: k ! m is just Q(r), and diag. (6.*
*52)
becomes a special case of diag. (6.49).
7 What is an additively unstable algebra?
In this section, we define an additively unstable algebra by enriching each*
* of the
four Answers in x5 with multiplicative structure. The treatment is closely par*
*allel
to the stable case [8, x12] and we give only the significant additions. The lo*
*gical
sequence is made slightly complicated by the fact that the monoidal structure i*
*s most
easily described in the context of the Second (or Third) Answer, while the como*
*nad
structure prefers the Fourth Answer.
In Defn. 7.13 we introduce the collapse operation, which detects the connec*
*tedness
of a space.
We assume throughout this section that E*(E_k) and QE*(E_k) are free E*mod*
*ules
for all k, which is true for our five examples by Lemma 4.17(a). Then by Cor. 2*
*.9,
P E*(E_k) is an object of FMod .
First Answer We have, for any space X, the additively unstable action (5.1)
O: P Em (E_k) x Ek(X) ! Em (X) :
Given x 2 Ek(X), y 2 Em (X), and r 2 P E*(E_k+m ), we would like to have a Cart*
*an
formula X
r(xy) = (r0ffx)(r00ffy) in E*(X), (7:1)
ff
for suitably chosen operations r0ffand r00ff(depending on k and m as well as r)*
*. For
the universal example
X = E_kx E_m; with x = k x 1, y = 1 x m , xy = OE = k x m , (7:2)
JMB, DCJ, WSW  32  23 Feb 1995
x7. What is an additively unstable algebra?
where OE: E_kxE_m ! E_k+m denotes the multiplication map of [8, Thm. 3.25], eq.*
* (7.1)
reduces to X
OE*r = r0ffx r00ff in E*(E_kxE_m ) .
ff
To ensure that OE*r is expressible in this form, we need to allow infinite sums*
* and use
the K"unneth homeomorphism E*(E_kxE_m ) ~=E*(E_k) b E*(E_m ) from Thm. 1.18(c).
We need to know more, that r0ff; r00ff2 P E*(E_*). We have enough duality i*
*somor
phisms to dualize the multiplication in Lemma 4.9 and define a comultiplication*
* P
by the commutative diagram
P
P E*(E_k+m )__________________________P E*(E_k) b P E*(E_m )
 
 
  (7:3)
 
? OE* ~ ?
E*(E_k+m )_________E*(E_kxE_m ) ______E*(E_k)obeE*(E_m=)
P 0 00
Then we write P r = ffrff rff, as required.
We must not forget the unit element 1X 2 E*(X). We define the counit
fflP : P E*(E_0) ! E* as the restriction of j*: E*(E_0) ! E*(T ) = E*, s*
*o that
r1X = (fflP r)1X in E*(X).
It is now clear what an additively unstable algebra should be. Given an E**
*algebra
M, we need actions P Em (E_k) x Mk ! Mm that compose correctly, are biadditive
and E*bilinear in the sense of diag. (5.2), satisfy the Cartan formula (7.1), *
*and respect
the unit in the sense that r1M = (fflP r)1M . In the classical case E = H(F p)*
*, there is
a good Cartan formula and this approach is useful. For more general E, such as *
*MU
and BP , this structure seems even more impractical than it was stably.
Second Answer We have the coaction (6.26),
aeX : Ek(X) ! Ei(X) biQ(E)ki:
In contrast to the Cartan formula of the First Answer, and just as stably in [8*
*], all we
have to do is observe that as k varies, aeX is a homomorphism of E*algebras, w*
*here
we use the bigraded algebra structure on Q**= Q(E)**from Prop. 6.15.
Explicitly, if for particular x; y 2 E*(X) we have, as in eq. (6.39),
X X
rx = xff; ry = yfi; for all r,
ff fi
the Cartan formula (7.1) becomes (cf. the stable analogue [8, eq. (12.5)])
X X
r(xy) = (1)deg(dfi) deg(xff) xffyfi in E*(X)^, for(all7*
*r.:4)
ff fi
Lemma 7.5 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k. Th*
*en
the homomorphisms Q( ) and Q(ffl) in (6.27) and (6.28) are multiplicative and r*
*espect
the unit element.
We defer the proofs until after Thm. 7.9, as the coalgebra structure on Q(E*
*)**is
not easily handled directly. The Lemma makes the following definition reasonabl*
*e.
JMB, DCJ, WSW  33  23 Feb 1995
Unstable cohomology operations
Definition 7.6 We call an unstable comodule M in the sense of Defn. 6.32 an
unstable (Ecohomology) comodule algebra if M is a filtered algebra (i. e. obje*
*ct of
FAlg) and its coaction aeM : M ! M b Q(E)**is a homomorphism of E*algebras.
In detail, M is a complete Hausdorff commutative filtered E*algebra, equip*
*ped
with a structure map aeM that is a continuous homomorphism of E*algebras and
makes diags. (6.33) commute.
Theorem 7.7 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k
(which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) For any space X, aeX makes E*(X)^ an unstable comodule algebra in the se*
*nse
of Defn. 7.6;
(b) ae is universal: given a (possibly bigraded) discrete E*algebra B, any *
*natural
transformation of rings X: E*(X) ! E*(X) b B (or bX: E*(X)^ ! E*(X)^ bB)
that is defined for all spaces X is induced from aeX by a unique homomorphism
f: Q(E)**! B of left E*algebras as
aeX * * 1f *
X: E*(X) ! E (X) b Q(E)* ! E (X) b B :
Proof This will follow from Thm. 7.9 in the same way that the stable result Th*
*m. 12.8
followed from Thm. 12.10 in [8]. 
Third Answer We use the multiplication Q(OE): Qk* Qm*! Qk+m*from Prop. 6.15
to make A0a symmetric monoidal functor (A0; iA0; zA0) in FMod , with
iA0(M; N): (A0M)k b(A0N)m ! (A0(M b N))k+m
given by
iA0(M; N): M b Qk*b N b Qm*~= M b N b(Qk* Qm*)
(7:8)
! M b N b Qk+m*
and zA0 = jR: Eh ! E* Qh*~= Qh*. Thus when M is an E*algebra, so is A0M.
We see that A0, equipped with natural transformations 0: A0! A0A0and ffl0: A0!*
* I
constructed from Q( ) and Q(ffl), becomes a symmetric monoidal comonad in FMod
and therefore a comonad in FAlg.
Fourth Answer For suitable E, we can make A a comonad in FAlg.
Theorem 7.9 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k
(which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) We can enrich A to make it a symmetric monoidal comonad in FMod and
therefore a comonad in FAlg;
(b) If also E*(E; o) is free, the stabilization oe: A ! S is a monoidal natu*
*ral trans
formation in FMod .
The relevant definition is now clear.
Definition 7.10 An additively unstable (Ecohomology) algebra is an Acoalgeb*
*ra
in FAlg, i. e. a complete Hausdorff commutative filtered E*algebra M equipped *
*with
a morphism aeM : M ! AM in FAlg that satisfies the coaction axioms [8, eq. (8.7*
*)].
JMB, DCJ, WSW  34  23 Feb 1995
x7. What is an additively unstable algebra?
If the closed ideal L M is invariant, the quotient algebra M=L inherits a *
*well
defined Acoalgebra structure.
Theorem 7.11 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k
(which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then
given a complete Hausdorff commutative filtered E*algebra M (i. e. object of F*
*Alg),
an unstable comodule algebra structure on M in the sense of Defn. 7.6 is equiva*
*lent
to an additively unstable algebra structure on M in the sense of Defn. 7.10.
Theorem 7.12 Assume that E*(E_k) and QE*(E_k) are free E*modules for all k
(which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) For any space X, the coaction aeX : E*(X) ! A(E*(X)^) in diag. (5.6) is a
homomorphism of E*algebras and makes E*(X)^ an additively unstable algebra;
(b) ae is universal: given a graded monoid object n 7! Cn in FMod op, so th*
*at (by
[8, Lemma 7.9]) n 7! Gn(X) = FMod (Cn; E*(X)^) is a graded ring, any natural
transformation of graded rings X: E*(X) ! G*(X) (or bX: E*(X)^ ! G*(X)), that
is defined for all spaces X, is induced from aeX by a unique morphism in FMod *
*opof
graded monoid objects with components fn: Cn ! P E*(E_n) in FMod , as
aeX * * Hom(fn;1) n *
X: En(X) ! FMod (P E (E_n); E (X)^) ! FMod (C ; E (X)^) :
Proof of Thms. 7.9 and 7.12 The main proof proceeds by the same five steps as *
*stably
for [8, Thms. 12.10, 12.13], except based on Thm. 5.8 instead of [8, Thm. 10.12*
*]. We
give only the major changes. We recall the universal class k 2 Ek(E_k), element
idk 2 AkP E*(E_k), and aek from the proof of Thm. 5.8.
Step 1. We construct the symmetric monoidal functor
(A; iA; zA): (FMod ; b; E*) ! (Mod ; ; E*) :
Then A will take monoid objects in FMod (i. e. objects of FAlg) to monoid obje*
*cts in
Mod (i. e. E*algebras).
By Lemma 4.16(a), we can construct the diagram (7.3) that defines P and ve*
*rify
its properties, which are dual to those of Q(OE) in Props. 6.15 and 6.20. The c*
*ounit
fflP : P E*(E_0) ! E* is the restriction of j*: E*(E_0) ! E*(T ) = E*. These m*
*ake
n 7! P E*(E_n) an E*algebra object in FMod *op, to which we apply [8, Lemma 7.*
*14].
The necessary compatibility axiom [8, eq. (7.13)] is the dual of diag. (6.22). *
*As stably,
we use [8, eq. (7.15)] to identify zA with aeT: E*(T ) ! AE*(T ).
If E*(E; o) is also free, we can dualize Prop. 6.15(b) to see that the dest*
*abi
lizations oe*n: E*(E; o) ! P E*(E_n) form a morphism of graded monoid objects in
(FMod *op; b; E*). Then [8, Lemma 7.9(b)] shows that oe: A ! S is monoidal.
Step 2. The proof that ae is monoidal is similar to the stable case. Here,*
* the
universal example is X = E_k and Y = E_m , with the element k m . The two
elements of Ak+m E*(E_kxE_m ) to be compared are
P * * * *
P E*(E_k+m ) ! P E (E_k) b P E (E_m )E (E_k) b E (E_m )
x *
! E (E_kxE_m )
JMB, DCJ, WSW  35  23 Feb 1995
Unstable cohomology operations
and
OE* *
P E*(E_k+m ) E*(E_k+m ) ! E (E_k x E_m):
These agree by diag. (7.3). The second condition needed is just zA = aeT.
Step 3. The analogue of diag. [8, eq. (12.17)] for this situation is fig. 1*
*. To
Figure 1: Additive operations and comultiplication
P
P E*(E_k+m )________P E*(E_k) b P E*(E_m )

 Pae Paem
  k
 ?
Paek+m AP E*(E_k) b AP E*(E_m )

 
 iA
? A ?
AP E*(E_k+m ) ______A(PE*(E_k)Pb P E*(E_m ))
establish this, we proceed as in [8, Thm. 12.10]. Because ae is monoidal and na*
*tural,
we have the commutative diagram fig. 2 (cf. diag. [8, eq. (12.16)]) which incl*
*udes an
Figure 2: The monoidality of ae
aekaem
E*(E_k) b E*(E_m )______AE*(E_k)b AE*(E_m )

 iA
 
 ?
~=x A(E*(E_k) b E*(E_m ))

 ~ 
 = Ax
? ae ?
E*(E_kxE_m ) ___________AE*(E_kxE_m )
6 * 6 *
OE AOE
 aek+m 
E*(E_k+m )______________AE*(E_k+m)
isomorphism from Thm. 1.18(c). Figure 1 is obtained from this by restriction, u*
*sing
the coaction (5.13) and diag. (7.3).
Step 4. The monoidality of follows formally from that of ae, just as stab*
*ly (cf.
diags. [8, eq. (12.18)]). The universal example is M = P E*(E_m ) and N = P E*(*
*E_n),
with element idm idn. We use fig. 1 instead of diag. [8, eq. (12.17)].
Step 5. The proof that ffl is monoidal is formally the same as stably, exce*
*pt for the
insertion of indices.
In Thm. 7.12(b), C has comultiplications C : Ck+m ! Ck b Cm and a counit
fflC : C0 ! E* which make n 7! FMod (Cn; E*(X)^) a graded ring. For each n,
Thm. 5.12(c) provides a morphism fn: Cn ! P E*(E_n) in FMod . For the universal
JMB, DCJ, WSW  36  23 Feb 1995
x7. What is an additively unstable algebra?
example (7.2), the multiplicativity (X)(xy) = ((X)x)((X)y) reduces to the com
mutativity of the outside of the diagram in fig. 3. The lower rectangle is dia*
*g. (7.3).
Figure 3: Comparison of comultiplications
C
Ck+m _______________CkbCm

fk+m fkfm
? ?
P E*(E_k+m )______PE*(E_k)PbP E*(E_m )

 
 ?

 E*(E_k) bE*(E_m )

 ~=
? OE* ?
E*(E_k+m )__________E*(E_kxE_m )
It follows that the upper square commutes, so that f preserves the comultiplica*
*tion.
Similarly, (T )1 = 1 yields fig. 4, which shows that f preserves the counit. 
Figure 4: Comparison of counits
________f0* ______*
C0 P E (E_0) E (E_0)
Q Q  j
QQfflC fflPj j
Q Qs ?jjj+j*
E*
Proof of Thm. 7.11 We use the isomorphism (6.7) to translate the monoidal stru*
*cture
of A to A0. From iA, which is given by [8, eq. (7.11)], we obtain eq. (7.8). We*
* have
identified both zA and zA0 with the coaction aeT. 
Proof of Lemma 7.5 Theorem 7.9(a) shows in particular that : A ! AA and ffl: *
*A !
I are monoidal natural transformations. By the isomorphism (6.24), so are 0: A*
*0!
A0A0and ffl0: A0! I. Evaluation of the relevant diagrams involving i for M = N *
*= E*
show precisely that Q( ) and Q(ffl) are multiplicative. Since zA0 = jR: Eh ! E*
Q(E)h*~= Q(E)h*, the two diagrams involving z show that 1 = 1 1 and ffl1 = 1,
simply because jR1 is the unit element of Q(E)**. 
Proof of Thm. 7.7 Part (a) follows from Thm. 7.12(a). In (b), Thm. 6.36(c) pr*
*ovides
for each n the E*module homomorphism fn: Q(E)n*! B that induces X: En(X) !
E*(X) b B. As in the proof of [8, Thm. 12.8(b)], the resulting f: Q(E)**! B is *
*an
E*algebra homomorphism. 
Connectedness There is a particular operation that is useful for expressing the
concept of connectedness in a cohomology algebra. It sees only the path compone*
*nts
of a space.
JMB, DCJ, WSW  37  23 Feb 1995
Unstable cohomology operations
Definition 7.13 For each n, we define the collapse operation n: n ! n as the *
*map
n: E_n! E_n(well defined up to homotopy) that sends each path component of E_n
to one point in that path component.
It is clearly additive, multiplicative ((xy) = (x)(y)), unital (01X = 1X ),*
* and
idempotent. It commutes with all operations in the sense that m Or = r Ok: k ! m
for all r: k ! m; in particular, is E*linear. It is zero in any degree n for*
* which
En = 0. In spite of being defined in all degrees, it is not at all stable, as n*
* = 0. All
these properties carry over to any additively unstable algebra M; in particular*
*, we
always have the E*module decomposition M = Im Ker, with (E*)1M Im .
For a connected space X with basepoint o, it is clear that the augmentation*
* ideal
E*(X; o) E*(X) is precisely Ker . In general, Ker = F 1E*(X) for any space X,
the first stage of the skeleton filtration. This suggests the following definit*
*ion.
Definition 7.14 We call the additively unstable algebra M connected if Im =
(E*)1M . We call M spacelike if it is a product (in FAlg) of connected algebras.
In particular, for a space X, E*(X)^ is always spacelike, and is connected *
*if and
only if X is connected.
8 What is an unstable object?
In this section, we interpret what it means to have an algebra over all the*
* un
stable operations on Ecohomology. Tensor products rapidly become unworkable for
nonadditive operations, with the effect that only the First and Fourth Answers *
*from
x5 survive intact.
We generally assume that E*(E_k) is a free E*module for all k. Then Thm. 1*
*.18
provides all the K"unneth and duality isomorphisms and homeomorphisms we need.
Of course, when we compare with the additive or stable theory, we impose the ap
propriate extra conditions.
As in (2.1), we identify:
(i)The cohomology operation r: Ek() ! Em ();
(ii)The class r = r(k) 2 Em (E_k);
(iii)The representing map r: E_k! E_m;
and write any of these as r: k ! m. (We shall retain the parentheses in r(x) wh*
*enever
r is nonadditive.)
We first deal with the constant operations r: k ! m, those of the form r(x)*
* =
v1X 2 Em (X) for all x 2 Ek(X) and all spaces X, where v 2 Em .
Lemma 8.1 Any operation r: k ! m decomposes uniquely as the sum of a based
operation s: k ! m and a constant operation.
Proof We set v = r(0) 2 Em (T ) = Em and define the operation s by s(x) =
r(x)  v1X in E*(X), to make s(0) = 0. 
JMB, DCJ, WSW  38  23 Feb 1995
x8. What is an unstable object?
First Answer Since Ek() is represented in Ho by E_k, we have as in (5.1) the
actions
O: Em (E_k) x Ek(X) ! Em (X); (8:2)
except that we cannot write them using tensor products. Instead, we need a Cart*
*an
formula for r(x+y) as well as for r(xy).
To find r(x+y), we consider the abelian group object E_k of Ho provided by *
*[8,
Cor. 7.8], which is equipped with the addition map k: E_kx E_k! E_kand zero map
!k: T ! E_k. By Lemma 8.1, we may restrict attention to based operations r. The
group axioms on E_klead (as in any Hopf algebra) to a formula of the form
X
*kr = rx1 + r0ffxr00ff+ 1xr in E*(E_kxE_k) ~=E*(E_k) b E*(E_k),
ff
where the r0ffand r00ffare also based. The only novelty is that the sum may be *
*infinite.
This translates into the desired Cartan formula
X
r(x + y) = r(x) + r0ff(x) r00ff(y) + r(y) in E*(X) (8:3)
ff
for any x; y 2 Ek(X).
There is a similar Cartan formula for multiplication, given x 2 Ek(X) and y*
* 2
Em (X), of the form
X
r(xy) = r0ff(x) r00ff(y) in E*(X), (8:4)
ff
for certain (other) based operations r0ffand r00ff(which depend on k and m).
This suggests that an unstable algebra should consist of an E*algebra M eq*
*uipped
with operations r that compose correctly and satisfy both Cartan formulae. This
requires knowing the operations r0ffand r00ffin eqs. (8.3) and (8.4) for all r.*
* In x10, we
shall in effect expand both Cartan formulae explicitly.
Second Answer We convert the First Answer to adjoint form, corresponding to
the Fourth Answer in x5. (We skip the Second and Third Answers.) Everything
becomes far cleaner, more evidence that this is the natural answer.
Any element x 2 Ek(X), regarded as a map x: X ! E_k, induces the continuous
homomorphism x*: E*(E_k) ! E*(X) of E*algebras. By Thm. 1.18(a), E*(E_k) is
Hausdorff and so in FAlg; we may therefore define, for any object M of FAlg,
UkM = FAlg(E*(E_k); M); (8:5)
the set of all continuous E*algebra homomorphisms E*(E_k) ! M. This encodes the
set of all possible actions on a typical element of degree k. We convert the a*
*ction
(8.2) to what we continue to call a coaction,
aeX : Ek(X) ! Uk(E*(X)^) = FAlg(E*(E_k); E*(X)^); (8:6)
by defining aeX x = x*, completing E*(X) if necessary to get it into FAlg. We a*
*ssemble
the sets UkM to form the graded set UM, which has the component (UM)k = UkM
in degree k, and obtain aeX : E*(X) ! U(E*(X)^).
JMB, DCJ, WSW  39  23 Feb 1995
Unstable cohomology operations
We compare UM with the stable and additive versions. Restriction to P E*(E_*
*k)
induces the natural transformation
(oM)k: UkM = FAlg(E*(E_k); M) ! FMod (P E*(E_k); M) = AkM : (8:7)
These form oM: UM ! AM. Composition with oeM: AM ! SM (see eq. (5.7))
yields
UkM = FAlg(E*(E_k); M) ! FMod k(E*(E; o); M) = (SM)k;
which is induced by the destabilization oe*k: E*(E; o) ! P E*(E_k) E*(E_k).
Apparently only a morphism of graded sets, aeX has far more structure, than*
*ks to
the rich structure on the spaces E_k.
Theorem 8.8 Assume that E*(E_k) is a free E*module for all k (which is true*
* for
E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) We can make the functor U, defined in eq. (8.5), a comonad in the catego*
*ry
FAlg of filtered E*algebras;
(b) If QE*(E_k) is a free E*module for all k, o: U ! A (see (8.7)) is a mor*
*phism
of comonads in FAlg;
(c) If E*(E; o) is a free E*module, oe Oo: U ! S (see (8.7) and (5.7)) is a*
* morphism
of comonads in FAlg.
Our main definition is now clear.
Definition 8.9 An unstable (Ecohomology) algebra is just a Ucoalgebra in FA*
*lg,
i. e. a complete Hausdorff filtered E*algebra M equipped with a continuous mor*
*phism
aeM : M ! UM of E*algebras that satisfies the coaction axioms [8, eq. (8.7)]. *
*We then
define the action of r 2 E*(E_k) on x 2 Mk by r(x) = aeM (x)r 2 M.
A closed ideal J M is called (unstably) invariant if the quotient algebra *
*M=J
inherits a welldefined unstable algebra structure from M.
It follows that the Cartan formulae (8.3) and (8.4) hold in M. The constant*
* op
erations behave correctly because aeM (x) is required to be a morphism of E*al*
*gebras.
We need to be able to recognize invariant ideals.
Lemma 8.10 Given an unstable algebra M, a closed ideal J M is unstably in
variant if and only if r(y) 2 J for all y 2 J and all based operations r.
Proof To make aeM=J well defined, we need r(x+y) r(x) mod J, for all x 2 M and
y 2 J. This is trivial for constant operations r, and so by Lemma 8.1, we need *
*only
check for based r. The stated condition is obviously necessary, by taking x = 0*
*. It is
also sufficient, by eq. (8.3). 
Theorem 8.11 Assume that E*(E_k) is a free E*module for all k (which is tru*
*e for
E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then:
(a) For any space X, the coaction (8.6) factors through E*(X)^ to make E*(X)^
an unstable Ecohomology algebra;
JMB, DCJ, WSW  40  23 Feb 1995
x8. What is an unstable object?
(b) We recover the additively unstable coaction (5.6) from aeX as
aeX * o *
E*(X) ! U(E (X)^) ! A(E (X)^);
(c) If E*(E; o) is Hausdorff, we recover the stable coaction [8, eq. (10.10)*
*] from aeX
as aeX o oe
E*(X) ! U(E*(X)^) ! A(E*(X)^) ! S(E*(X)^);
(d) ae is universal: given an object B of FAlgand an integer k, any natural *
*transfor
mation of sets X: Ek(X) ! FAlg(B; E*(X)^) (or bX: Ek(X)^ ! FAlg(B; E*(X)^),
that is defined for all spaces X, is induced from aeX by a unique morphism f: B*
* !
E*(E_k) in FAlg as
aeX * * *
X: Ek(X) ! U(E (X)^) = FAlg(E (E_k); E (X)^)
Mor(f;1)
! FAlg(B; E*(X)^)
Proof of Thms. 8.8 and 8.11 The proof breaks up into the same seven steps as a*
*ddi
tively (and stably), in Thms. 5.8 and 5.12. However, it is far simpler than Thm*
*s. 7.9
and 7.12 on algebras, because we are able to treat the multiplicative and module
structures together. At each step, we also discuss o and o Ooe, assuming the e*
*xtra
conditions hold.
Corollary 7.8 of [8] provides the E*algebra object n 7! E_nin Ho. We again*
* write
aek for aeX when X = E_k.
Step 1. We endow the functor U with an E*algebra structure. For each obje*
*ct
M of FAlg, we observe that according to [8, Lemma 6.9], the functor
E*()^ op Mor(;M)
FAlg(E*()^; M): Ho ! FAlg ! Set
preserves enough products that by [8, Lemmas 7.6(a), 7.7(a)] it takes the E*al*
*gebra
object n 7! E_n to the E*algebra object UM in Set; i. e. UM is an E*algebra. *
*It is
clear that UM is functorial in M. We shall filter it in Step 3.
To see that oM is a homomorphism of E*modules, we apply [8, Lemma 7.6(c)]
to the E*module object n 7! E_nin Gp(Ho ), using the natural transformation
FAlg(E*()^; M) ! FMod (P E*()^; M)
defined by restriction. To see that o is monoidal, we apply [8, Lemma 7.9(b)]. *
*The
monoidal structure of U is simply the multiplicative part of the algebra struct*
*ure,
and diag. (7.3) shows that the inclusions P E*(E_n) E*(E_n) form a morphism of
graded monoid objects in FMod op. The units are correct by definition. For o Oo*
*e, we
bypass P E*(E_n) and use the duals of diags. (6.16) and (6.17) instead.
Step 2. In order to define aeX (in (8.6)) as a morphism of E*algebras, we *
*consider
the Setvalued natural transformation
Ho (X; ) ! FAlg(E*()^; E*(X)^)
induced by E*()^: Hoop ! FAlg. We apply [8, Lemma 7.6(c)] to the E*algebra
object n 7! E_n, to obtain Thm. 8.11(a). Then Thm. 8.11(b) is clear by comparing
with the additive coaction (5.6), and for Thm. 8.11(c), we combine with Thm. 5.*
*12(b).
JMB, DCJ, WSW  41  23 Feb 1995
Unstable cohomology operations
Step 3. For U to take values in FAlg, we must filter UM. If M is filtered b*
*y the
ideals F aM, we filter UM by the ideals
" ! #
M
F a(UM) = Ker UM ! U _____
F aM
Just as stably, this filtration is complete Hausdorff and makes aeX continuous*
* by
naturality. This allows us to factor aeX through E*(X)^. Similarly, oM and oeM *
*OoM
are also filtered and therefore continuous.
Step 4. We convert the object E*(X)^ of FAlg to the corepresented functor
FX = FAlg(E*(X)^; ): FAlg ! Set. For example, when X = E_k, FX = Uk. As
suggested by [8, eq. (8.16)], we also convert the coaction aeX to the natural t*
*ransfor
mation aeX : FX ! FX U: FAlg! Set. Given M in FAlg, aeX M: FX M ! FX UM is thus
defined on f 2 FX M = FAlg(E*(X)^; M) as
(aeX M)f = Uf OaeX : E*(X)^ ! U(E*(X)^) ! UM; (8:12)
an element of FX UM.
Step 5. We define the natural transformation
M: UkM = FAlg(E*(E_k); M) ! FAlg(E*(E_k); UM) = UkUM (8:13)
by taking X = E_kin eq. (8.12). On the element f: E*(E_k) ! M of UkM, it is
aek * Uf
( M)f: E*(E_k) ! UE (E_k) ! UM :
(In terms of elements, this is r 7! [s 7! f(r*s) = f(sr)].) If we substitute t*
*he E*
algebra object n 7! E_n for X in eq. (8.12), [8, Lemma 7.6(c)] shows that M ta*
*kes
values in Alg. Naturality in M shows that M is filtered and so takes values in*
* FAlg
as required.
Step 6. The other required natural transformation,
fflM: UkM = FAlg(E*(E_k); M) ! M;
is defined simply as evaluation on k 2 E*(E_k). As before, naturality in M sho*
*ws
that fflM is filtered, but we have to calculate that ffl is an E*algebra homom*
*orphism.
Take any binary operation s(; ) in E*algebras (addition, multiplication,*
* or
any other), represented in Ho by the map s: E_kx E_m! E_q, which therefore indu*
*ces
s*q = s(p*1k; p*2m ). We need to show that the square
s
UkM x Um M ______UqM
fflxffl ffl
? ?
s
Mk x Mm ________Mq
commutes. We evaluate on f 2 UkM and g 2 Um M. Because E*(E_kxE_m ) is by [8,
Lemma 6.9] the coproduct in FAlg, there is a unique h: E*(E_kxE_m ) ! M in FAlg
such that h Op*1= f and h Op*2= g. Then by definition of the algebra structure*
* of
UM, s(f; g) = h Os*: E*(E_q) ! M. Since h is an algebra homomorphism,
ffls(f; g) = hs*n = hs(p*1k; p*2m ) = s(hp*1k; hp*2m ) = s(fk; gm ) = s(ffl*
*f; fflg):
JMB, DCJ, WSW  42  23 Feb 1995
x9. Unstable, additive, and stable objects
For unary and 0ary operations, we may adapt the above proof, or simply thr*
*ow
away any unwanted arguments. (For example, given v 2 E*, we could define the
constant binary operation s(x; y) = v1 in any E*algebra, to deduce that fflv =*
* v.)
Step 7. The proof that E*(X)^ is a Ucoalgebra and that U is a comonad is
formally identical to the stable case, except that we need versions of [8, Lemm*
*as 8.20,
8.22] for graded objects.
We use [8, Lemma 8.24] to show that o: U ! A is a natural transformation of
comonads. We take R as n 7! E*(E_n), R0 as n 7! P E*(E_n), 1R = 10Ras n 7!
n, and u: P E*(E_k) E*(E_k) as the inclusion. The first hypothesis on u is the
commutativity of the diagram
P E*(E_k) ______________________E*(E_k)
 
 aek
 ?

Paek FAlg(E*(E_k); E*(E_k))

 
 
? ?
FMod (P E*(E_k); P E*(E_k))_____FMod (P E*(E_k); E*(E_k))
which is obvious by construction, as r 2 P E*(E_k) yields r*P E*(E_k).
The proof of Thm. 8.11(d) is formally the same as stably. Since Ek() is re*
*pre
sented by k 2 Ek(E_k), is classified by f = (E_k)k 2 FAlg(B; E*(E_k)). 
9 Unstable, additive, and stable objects
In previous sections and [8], we constructed five different kinds of object*
*: stable
modules and algebras, additively unstable modules and algebras, and unstable al*
*ge
bras. In this section we compare them all. Unstable modules are conspicuous by *
*their
absence; Thm. 9.4 will show that they cannot be defined compatibly with our oth*
*er
objects.
Each kind of object is defined by a comonad. Theorems 8.8(b) and 7.9(b) pro*
*vide
natural transformations
o oe
U ! A ! S in FAlg (9:1)
between the comonads that define unstable, additively unstable, and stable alge*
*bras.
Theorem 5.8(b) provides the natural transformation
__ _oe__
A ! S in FMod (9:2)
between the comonads that define additively unstable and_stable_modules (where *
*we
temporarily rename the module versions of A and S toA andS ). They are related*
* __
to the algebra_versions by the forgetful functor V : FAlg! FMod , so that V A =*
*A V
and V S =S V .
We have the category, e. g. Ucoalgebras, of each kind of object. We consi*
*der
the diagram of categories and functors in fig. 5. For example, a stable alge*
*bra
B with coaction aeB :_B_! SB in FAlg yields the stable module V B with coaction
V aeB : V B ! V SB =S V B in FMod .
JMB, DCJ, WSW  43  23 Feb 1995
Unstable cohomology operations
Figure 5: Five kinds of object
unstable ______additivelyunstableo_stableoe
algebras algebras algebras
 
V V
? ?
additively _oe
unstable ______ stable
modules modules
Theorem 9.3 Assume that E*(E_k), QE*(E_k), and E*(E; o) are free E*modules
for all k (which is true for E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)
and [8, Lemma 9.21]). Then we have the diagram fig. 5 of categories and functor*
*s.
For any space X, E*(X)^ is an object in each of the five categories, relate*
*d by
these functors.
Proof The last assertion combines Thms. 5.12, 7.12, and 8.11 with Thms. 10.16
and 12.13 of [8]. 
There is a glaring gap: we have not defined unstable modules. We now show
that this gap cannot be filled, for rather silly reasons. In fact, the three mo*
*st natural
definitions are for stable modules, additively unstable modules, and unstable a*
*lgebras.
We can enrich the two kinds of module with multiplicative structure, but it is *
*not
possible to remove the multiplicative structure from the definition of unstable*
* algebra.
This is already strongly suggested by the appearance of multiplication in the C*
*artan
formula (8.3) for r(x+y).
We ignore most of the structure and the topology, fix k, and restrict atten*
*tion to
the two functors Uk: FAlg! Ab and Ak: FMod ! Ab and the natural transformation
ok: UkV ! Ak.
Theorem 9.4 Even in the classical case E = H(F p), unstable_modules do not e*
*xist
in the sense that we cannot insert a suitable comonadU into diag. (9.2). Speci*
*fically,
for fixed k > 0 there do not exist:
__k
(i)a functorU : FMod ! Ab;
__k
(ii)a natural isomorphismU V ~= Uk: FAlg! Ab;
__k
(iii)a natural transformation _ok:U ! Ak of functors FMod ! Ab;
__k
such that on FAlg, _okV :U V ! Ak agrees with ok: Uk ! Ak.
__k _
Proof We assume thatU and ok exist as stated and derive a contradiction. Giv*
*en
any (filtered) graded Fpmodule M, we construct the Fpalgebra M+ = Fp M with
the unit element 1 2 Fp and xy = 0 for all x; y 2 M. Then M is a retract in FMod
JMB, DCJ, WSW  44  23 Feb 1995
x9. Unstable, additive, and stable objects
of V M+ and we can compute _okM from the commutative diagram
FAlg(Ak; M+ )
=
~ ?
__k __________k + _______=k +
U M U V M U M
_okM _okV M+ okM+
? ? ?
__k ________o__ke + _______oek~+=
A M A V M A M
= =
? ?
FMod (P Ak; M) FMod (P Ak; M+ )
where Ak = H*(H_k). Because M+ has no decomposables, every homomorphism
P Ak ! M in the image of _okM kills the decomposable elements of P Ak (of which
there are many).
But for a general algebra B, okB: UkB ! AkB does not have this property, e.*
* g.
(okAk)idk 2 AkAk is the inclusion P Ak Ak. Taking M = V B shows that _okV B
does not agree with okB. 
Objects in ordinary cohomology Theorem 9.4 demands an immediate explana
tion of our terminology even in the case of ordinary cohomology. We give detail*
*s for
E = H(F 2); the case E = H(F p) for odd p is similar, with the usual changes.
The Steenrod algebra A = E*(E; o) is exactly as expected: it is the F2alg*
*ebra
generated by the Steenrod squares Sq ifor i > 0, subject to the standard Adem
relations. It is useful to write Sq0 = . We note that for E = H(F 2):
(i)oe*kmakes P E*(E_k) a quotient of E*(E; o);
(ii)E*(E_k) is a primitively generated Hopf algebra.
Below, M is to be an object of FMod (or FAlg), i. e. a complete Hausdorff *
*filtered
graded F2module (or commutative F2algebra). Topological conditions apply (whi*
*ch
we ignore for now). We list the five kinds of object we have defined, under our*
* names
for them:
(i)A stable module M is an Amodule.
(ii)A stable algebra M is both an F2algebra and an Amodule that satisfies
the Cartan formula
Xk
Sq k(xy) = (Sq ix)(Sq kiy) for k > 0.
i=0
It follows by induction that Sqk1M = 0 for all k > 0.
(iii)An additively unstable module M is an Amodule that satisfies the ext*
*ra
condition
Sqix = 0 for all x 2 M and all i > deg(x). (9:5)
Since Sq0x = x, it follows that Mn = 0 for all n < 0.
(iv)An additively unstable algebra is a stable algebra that satisfies (9.5*
*).
JMB, DCJ, WSW  45  23 Feb 1995
Unstable cohomology operations
(v) An unstable algebra M is a stable algebra that satisfies (9.5) as well*
* as the
extra condition
Sqk x = x2 for x 2 M and k = deg(x).
The objects normally known as unstable modules appear here as additively unstab*
*le
modules (although the word "additively" could well be omitted, there being no d*
*anger
of confusion with something that does not exist).
However, we do have two kinds of unstable algebra. We emphasize that in (iv*
*),
the squaring operation Mk ! M2k given by x 7! x2 (which looks additive but from
our point of view is not, because it is defined only when M is an algebra) is u*
*nrelated
to Sqk.
We have equivalent comodule descriptions in terms of E*(E; o) = F2[1; 2; 3;*
* : :]:
and the corresponding bigraded algebra Q(E)**= F2[0; 1; 2; : :]:, which has pol*
*yno
mial generators i2 Q(E)12i(as we shall see in Thm. 16.2):
(i)A stable comodule M has a coaction
aeM : M ! M b E*(E; o) = M b F2[1; 2; 3; : :]:
that satisfies the usual axioms [8, eq. (8.7)]. Then Sqk is dual to k1.
(ii)A stable comodule algebra M is both a stable comodule and a commutative
F2algebra, in such a way that aeM is an algebra homomorphism.
(iii)An unstable comodule M has coactions
aeM : Mk ! Mi biQ(E)ki M b F2[0; 1; 2; : :]:
that satisfy the coaction axioms (6.33). The unstable operation Sqi: *
*k !
k + i is now dual to ki0i1for i k, or is zero if i > k.
(iv)An unstable comodule algebra M is an unstable comodule that is also a *
*com
mutative F2algebra, in such a way that aeM is an algebra homomorphis*
*m.
The special features of H(F 2) allow us to handle unstable algebras too:
(v) For any x 2 Mk, aeM x contains the term x2 k1.
Remark There is one candidate for an unstable module, but it does not work. One
could try defining GkM = FMod (E*(E_k); M) for any object M of FMod , with
aeX : Ek(X) ! GkE*(X) defined as usual, by aeX x = x*. We would like aeX to be
at least additive, but the standard additive structure on FMod does not give t*
*his.
Indeed, it is easy to see that in general no abelian group structure on GkM*
* makes
aeX additive (not even for E = H(F p)). By [8, Lemma 7.7(d)], such a structure *
*would
have to be induced by some morphism : E*(E_k) ! E*(E_k) E*(E_k) in FMod .
Take any r 2 E*(E_k) and write r = (r0; r00). Then additivity of aeX translate*
*s into
r(x+y) = r0x + r00y for all x; y 2 Ek(X), which is absurd unless r happens to be
additive.
In fact, these objects appear to be particularly devoid of interest. In th*
*e case
E = H(F 2), for example, they are modules equipped not only with Steenrod squar*
*es
Sq ithat behave as expected, but also operations such as x 7! (Sq 2x)(Sq 3x), w*
*ithout
having cup products.
JMB, DCJ, WSW  46  23 Feb 1995
x10. Enriched Hopf rings
10 Enriched Hopf rings
In Defn. 8.9 we condensed all the structure of an unstable algebra down to *
*the
single word Ucoalgebra. In this section, we unpack the information again to gi*
*ve a
complete description of an unstable algebra in the language of Hopf rings, enri*
*ched
with certain additional structure. This description is summarized in Thm. 10.4*
*7,
which may be regarded as the unstable analogue of Thm. 11.14 of [8] and Thm. 6.*
*36.
Indeed, we find a whole new paradigm for handling unstable operations, making
computations with them reasonably practical and efficient. It serves as the tr*
*ue
successor to the Second Answer of x5 and [8, x10].
We assume in this section that E*(E_k) is a free E*module for all k, which*
* is
true for our five examples by Lemma 4.17(a). Thus all the results of x8 are ava*
*ilable,
and by [8, Lemma 6.16(c)], the topological dual FMod *(E*(E_k); E*) of E*(E_k)*
* is
E*(E_k).
We shall consistently identify (with some abuse of notation):
(i)the cohomology operation r: Ek() ! Em ();
(ii)the cohomology class r(k) 2 Em (E_k), which we often write simply
as r 2 Em (E_k); (10.1)
(iii)the representing map of spaces r: E_k! E_m;
(iv)the E*linear functional : E*(E_k) ! E* of degree m.
Remark In some situations, these identifications can obscure the correct signs*
* in
formulae. Considered as a cohomology class or functional, r has degree m, while*
* its
degree as an operation is mk, and as a map of spaces, r has no degree at all.
In any unstable algebra M, including E*(X)^ for any space X, Defn. 8.9 give*
*s,
for each x 2 Mk, the homomorphism aeM (x): E*(E_k) ! M. Then we defined r(x) =
aeM (x)r 2 M for any operation (i. e. class) r 2 E*(E_k). In practice, we find *
*it more
convenient to revert to the First Answer r(x) of x8, although the Second Answer*
*, in
terms of aeM , will continue to inform us as to what to do, even when only impl*
*icit.
Classically, one investigates cohomology operations by studying what happens to*
* r(x)
when r is fixed and x varies; but it is clear from x8 that what we should do is*
* fix x
and allow r to vary.
Linear functionals We need to develop a computational description of aeM in an
unstable algebra M. We start from the fact that aeM (x) is E*linear, i. e. r(*
*x) is
E*linear in r.
Definition 10.2 Let M be an unstable algebra, and fix an element x 2 Mk. We
say r(x) is written in standard form if
X
r(x) = xff in M (for all r); (10:3)
ff
for suitable choices cff2 E*(E_k) and xff2 M, where deg(xff) =  deg(cff). If *
*the
sum is infinite, we require each ideal F aM in the filtration of M to contain a*
*ll except
finitely many of the xff.
JMB, DCJ, WSW  47  23 Feb 1995
Unstable cohomology operations
This is the closest we will come to an unstable replacement for the tensor *
*products
and homomorphisms of x6 and [8, x11]. Our convention here and in all similar
formulae is that r runs through all unstable cohomology operations having the c*
*orrect
domain degree (different in nearly every formula, and rarely specified) but arb*
*itrary
target degree. The indexing set for ff is often left implicit.
It is easy to achieve eq. (10.3) in the universal form
X
r(x) = rff(x) in M (for all r); (10:4)
ff
by allowing cffto run through some basis of E*(E_k), which forces us to take xf*
*f=
rff(x), where rffdenotes the operation (linear functional) dual to cff. Contin*
*uity of
aeM (x): E*(E_k) ! M assures the finiteness condition in Defn. 10.2. We may the*
*refore
always assume that r(x) is written in standard form.
Where we depart from tradition is in not picking a definite basis of E*(E_k*
*) in
advance. We do not even insist on the cffbeing linearly independent. Nor do we
require the cffto span; we may obviously omit zero terms. This does not affect
the linearity of eq. (10.3) and allows the flexibility that our formulae requir*
*e. One
consequence is that most cohomology operations will never acquire names.
We have the analogue of Prop. 6.44.
Proposition 10.5 Given x 2 Ek(X), regarded as a map of spaces x: X ! E_k,
assume that r(x) is given by eq. (10.3). Then x*: E*(X) ! E*(E_k) is given by
X X
x*z = (1)deg(cff)(deg(xff)+deg(z)) cff= cff :*
* 
ff ff
The nonuniqueness in eq. (10.3) is really not a problem because we are usin*
*g it to
describe, not define the structure on M. The real definitions are all in x8; he*
*re, we
are only reinterpreting them. Nevertheless, it is easy to convert one standard*
* form
to another.
Lemma 10.6 Any standard form (10.3) can be transformed into the universal fo*
*rm
(10.4), and hence into any other standard form, by iterating three kinds of rep*
*lacement
(in either direction):
(i) =x0 x0+ x0;
(ii) x0= (1)deg(c) deg(v) vx0;
(iii) x0+ =x00 (x0+ x00):
(Infinitely many replacements may be needed; however, each F aM contains x0for *
*all
except finitely many of them.) 
Stabilization We need to record how eq. (10.3) behaves when we restrict the op
eration r to be additive or stable. We recall from [8, Defn. 9.3] the stabiliz*
*ation
homomorphism oek*: E*(E_k) ! E*(E; o) and from eq. (6.2) the algebraic homomor
phism qk: E*(E_k) ! Q(E)k*, both of which have degree k under our conventions.
JMB, DCJ, WSW  48  23 Feb 1995
x10. Enriched Hopf rings
Lemma 10.7 Let M be an unstable algebra, and assume that r(x) is expressed in
the standard form (10.3), where x 2 Mk. Then:
(a) The unstable comodule coaction aeM : Mk ! M b Q(E)k*is given by
X
aeM x = (1)deg(xff)(kdeg(xff))xff qkcff in M b Q(E)k*,
ff
provided QE*(E_k) is a free E*module;
(b) The stable comodule coaction aeM : M ! M b E*(E; o) is given by
X
aeM x = (1)deg(xff)(kdeg(xff))xff oek*cff in M b E*(E; o),
ff
provided E*(E; o) is a free E*module.
The signs are as expected, once we remember that if deg(xff) = i, then deg(*
*cff) =
i and deg(qkcff) = deg(oek*cff) = k  i.
P
Proof For additive r, Prop. 6.11 converts eq. (10.3) to rAx = ff*
*xff. We
deduce aeM x in (a) by comparing eqs. (6.38) and (6.39). Part (b) is similar, u*
*sing [8,
eq. (11.18), eq. (11.19)] instead. 
Unstable algebra structure Our task is to convert all the algebraic structure *
*of
an unstable algebra M in Defn. 8.9 into the current context. There are in effec*
*t four
pairs of axioms:
(a) Two axioms to make aeM (x): E*(E_k) ! M an E*algebra homomorphism,
rather than merely E*linear: (r ^ s)(x) = r(x)s(x) and 1(x) = 1M , wh*
*ich
will become eqs. (10.14) and (10.15);
(b) Two axioms to make aeM : M ! UM E*linear: aeM (x+y) = aeM (x) + aeM (*
*y)
and aeM (vx) = vaeM (x), which will become eqs. (10.20) and (10.16);
(c) Two axioms to make aeM : M ! UM multiplicative: aeM (1M ) = 1UM and
aeM (xy) = aeM (x)aeM (y), which will become eqs. (10.41) and (10.34);
(d) Two axioms to make M a Ucoalgebra: (sr)(x) = s(r(x)) and kx = x,
which will become eqs. (10.45) and (10.43).
The natural language for expressing the first three pairs is that of Hopf rings*
*, while
the last requires some additional structure.
Hopf rings We recall from [8, Lemma 6.12] that in Coalg, tensor products of co*
*al
gebras serve as products and E* is the terminal object. A commutative (graded) *
*ring
object in Coalg is called a Hopf ring over E*. (The terminology and some of the
notation were suggested by Milgram [17]; see [23, x1] for a detailed exposition*
*.)
We start from the E*algebra object n 7! E_nin Ho provided by [8, Cor. 7.8]*
*. We
apply [8, Lemma 7.6(a)], using the homology functor E*(), which takes values in
Coalg on the spaces we need and preserves enough products to make n 7! E*(E_n)
an E*algebra object in Coalg. In particular, this is an E*module object, and *
*each
E*(E_k) is an abelian group object in Coalg and thus a Hopf algebra.
There are seven parts to the Hopf ring structure of n 7! E*(E_n): two from *
*the
coalgebra, three from the abelian group object E_k, and two from the multiplica*
*tive
JMB, DCJ, WSW  49  23 Feb 1995
Unstable cohomology operations
monoid object, in addition to the underlying E*module structure on Ehomology.
They are as follows (for each k and m, where relevant):
(i) : E*(E_k) ! E*(E_k) E*(E_k), the comultiplication induced by the dia*
*go
nal map : E_k! E_kx E_k;
(ii)ffl: E*(E_k) ! E*, the counit for , induced by the map q: E_k! T ;
(iii)*: E*(E_k) E*(E_k) ! E*(E_k), a multiplication, induced by the addit*
*ion
map k: E_kx E_k! E_k;
(iv)1k = !k*1 2 E0(E_k), the *unit element, induced by the zero map !k: T*
* !
E_k;
(v) O: E*(E_k) ! E*(E_k), the canonical (anti)automorphism of the Hopf alg*
*ebra
E*(E_k), induced by the inversion map k: E_k! E_k;
(vi)O: E*(E_k) E*(E_m ) ! E*(E_k+m ), another multiplication, induced by *
*the
multiplication map OE: E_kx E_m ! E_k+m;
(vii)[1] = j*1 2 E0(E_0), the Ounit element, induced by the algebra unit m*
*ap
j: T ! E_0.
Because n 7! E*(E_n) is an E*algebra object rather than merely a ring obje*
*ct, we
have, for each v 2 Eh, the actions (v)*: E*(E_k) ! E*(E_k+h). As in x6, this re*
*duces
to a simpler structure.
Definition 10.8 We define the right unit function jR: E* ! E*(E_*). We regard
v 2 Eh = Eh(T ) as a map v: T ! E_h, and use the induced homomorphism v*: E* ~=
E*(T ) ! E*(E_h) to define [v] = v*1 2 E0(E_h) and jR(v) = [v].
In particular, this includes [1] = j*1 as in (vii), and [0k] = !k*1 = 1k as*
* in (iv). It
is clear from Defn. 6.19 and [8, Defn. 11.2] that qh[v] and oeh*[v] are the add*
*itive and
stable versions of jRv. The elements [v] determine the E*module object struct*
*ure
completely, because when we apply Ehomology to [8, eq. (7.5)], we obtain
(v)*c = [v] Oc for all c 2 E*(E_*). (10:9)
For the sake of completeness, we list all 33 laws that a Hopf ring satisfie*
*s, beyond
the usual axioms for an E*module. (Your count may vary.)PMost need no comment.
They are as follows, where in several we write c = ic0i c00i:
(i)The five operations are (bi)additive: (b+c) = b + c, ffl(b+c) = ffl*
*b + fflc,
(a+b) * c = a*c + b*c, O(b+c) = Ob + Oc, and (a+b) Oc = aOc + bOc;
P 0 00
(ii)The five operations are E*linear: (vc) = ivcici, ffl(vc) = vfflc, *
*(vb)*c =
v(b * c), O(vc) = vOc, and (vb) Oc = v(b Oc), for all v 2 E*;
(iii)Three coalgebra axioms: is coassociativePand cocommutative (with t*
*he
standard sign), and ffl is a counit: i(fflc0i)c00i= c;
(iv)The five parts of the ring object structure respect : (b * c) = ( b)*
* * ( c)
(where we give E*(E_k) E*(E_k) the obvious *multiplication,Pwith sig*
*ns),
(b Oc) = ( b) O( c) (similarly), 1k = 1k 1k, Oc = iOc0i Oc00i, a*
*nd
[1] = [1] [1];
JMB, DCJ, WSW  50  23 Feb 1995
x10. Enriched Hopf rings
(v) The five parts of the ring object structure respect ffl: ffl(b * c) =*
* (fflb)(fflc),
ffl1k = 1, fflOc = fflc, ffl(b Oc) = (fflb)(fflc), and ffl[1] = 1;
(vi)Four abelian group object axioms: associativity (a * b) * c = a * (b *
** c),
commutativity b * c = (1)ijcP* b (where i = deg(b), j = deg(c)), unit
1k * c = c, and inverse iOc0i* c00i= (fflc)1k;
(vii)Three axioms for a commutative monoid: associativity (a Ob) Oc = a O(b*
* Oc),
commutativity, which takes the somewhat complicated form (see *
*[23,
Lemma 1.12(c)(v)])
b Oc = (1)ijOkmc Ob (10:10)
for b 2 Ei(E_k) and c 2 Ej(E_m ) (where Okm = O if k and m are odd, an*
*d is
the identity otherwise, as in Prop. 10.12(b) below), and [1] Oc = c;
(viii)Three ring object axioms to state that  Oc respects the abelian group*
* object
structure: for addition, which yields the distributive law, in the com*
*plicated
form [ibid. (vi)]
X 0
(a * b) Oc = (1)deg(ci) deg(b)aOc0i* bOc00i;(10:11)
i
for the zero, 1m Oc = (fflc)1m+k [ibid. (ii)]; and for the inverse, O*
*(b Oc) =
(Ob) Oc.
Many standard laws follow from these axioms. In order to simplify notation in
eq. (10.11) and elsewhere, we give Omultiplication greater binding strength th*
*an *
multiplication, so that a * b Oc always means a * (b Oc), never (a * b) Oc. In *
*all our Hopf
rings, Prop. 11.2 will provide the laws relating the added elements [v] and ide*
*ntify
the useful element O[1] with [1].
Proposition 10.12 In any Hopf ring, the operation O has the following proper*
*ties:
(a) Oc = O[1] Oc, so that O[1] determines O;
(b) OOc = c;
(c) O(a * b) = Oa * Ob;
(d) O[1] OO[1] = [1].
Proof For (a), Oc = O([1] Oc) = O[1] Oc. Since [1] = [1] [1] and hence O[1*
*] =
O[1] O[1], the distributive law gives (c), by
O(a * b) = O[1] O(a * b) = O[1]Oa * O[1]Ob = Oa * Ob :
Also, we have O[1] * [1] = 10 and similarly OO[1] * O[1] = 10, which yield
OO[1] = OO[1] * 10 = OO[1] * O[1] * [1] = 10 * [1] = [1]:
But (a) gives OO[1] = O[1] OO[1], and hence (d) and the general case of (b). 
Generators We wish to use the laws to reduce any element of a Hopf ring to some
standard form. The distributive law (10.11) plays a key role. We shall describe*
* our
Hopf rings H by specifying two sets of elements:
(i)the Ogenerators of H;
JMB, DCJ, WSW  51  23 Feb 1995
Unstable cohomology operations
(ii)the *generators of H, each of which is a Oproduct of Ogenerators and
possibly O[1], where we allow the empty Oproduct [1].
We require every element of H to be an E*linear combination of *products of t*
*he
*generators of H; in other words, the *generators generate H as an E*algebra.
For each Ogenerator g, we need formulae for g (so we can expand eq. (10.11)),*
* fflg,
and Og. Although Hopf rings tend to be huge, each of our examples (see x17) has*
* a
conveniently small set of Ogenerators.
Hopf rings over Fp One can define the Frobenius operator F c = c*pin any algeb*
*ra
with multiplication *, and it is multiplicative if * is commutative. It is addi*
*tive if also
the ground ring has characteristic p. It is most useful when the ground ring i*
*s Fp,
because it is then automatically Fplinear. Commutativity of *multiplication i*
*mplies
that F c = 0 whenever c has odd degree (unless p = 2).
Moreover, in a Hopf ring (or cocommutative Hopf algebra) H over Fp, one has
dually the Verschiebung operator V : H ! H, defined so that DV = F : DH ! DH
in the dual Hopf algebra. It divides degrees by p. Then V c = 0 unless deg(c)*
* is
divisible by 2p (if p 6= 2). Both F and V preserve all the Hopf algebra struc*
*ture:
F (a * c) = F a * F c, F 1k = 1k, F c = (F F ) c, fflF c = fflc, and dually V*
* (a * c) =
V a * V c, V 1k = 1k, V c = (V V ) c, and fflV c = fflc. For Oproducts, we c*
*an iterate
eq. (10.11) and obtain the identity
a O(F c) = F (V a Oc); (10:13)
which is useful for reducing elements of the Hopf ring to standard form. (Norma*
*lly,
a and c both have even degree.)
Multiplication of operations The first pair of axioms on M we listed earlier, *
*that
for fixed x 2 M, aeM (x) is a homomorphism of E*algebras, is easily translated*
* into
Hopf rings. Because the diagonal map in E_kinduces both the cup product r ^ s a*
*nd
the comultiplication on E*(E_k), we can write down the cup product from eq. (*
*10.3)
as X X
(r ^ s)(x) = xfl= xfl in M :
fl fl
The product r(x)s(x) becomes, after some shuffling,
X X
r(x)s(x) = (1)deg(xff) deg(xfi) xffxfi:
ff fi
Since (r ^ s)(x) = r(x)s(x) has to hold for all r and s, we deduce the identity
X X X
cfl xfl= (1)deg(xff) deg(xfi)cff cfi xffxfi (10:14)
fl ff fi
in (E*(E_*) E*(E_*)) b M, where the tensor products are formed using only the
usual left E*actions.
The identity element 1k 2 E0(E_k) is the constant operation Ek(X) ! E0(X)
that sends everything to 1X ; regarded as a linear functional, it is simply ffl*
*. In terms
of eq. (10.3), the axiom 1k(x) = 1M becomes
X
(fflcff) xff= 1M in M : (10:15)
ff
JMB, DCJ, WSW  52  23 Feb 1995
x10. Enriched Hopf rings
Linear structure We next decode the statement that aeM : M ! UM is linear,
namely that aeM (x+y) = aeM (x) + aeM (y) and aeM (vx) = vaeM (x). Related to t*
*he first
is the formula for r*(b * c), which can be shown to be the translation of the s*
*tatement
that M: UM ! UUM is additive. We assume that r(x) is given by eq. (10.3), where
x 2 Mk.
The vaction UkM ! Uk+hM was given by composing with (v)*: E*(E_k+h) !
E*(E_k); dually, we use eq. (10.9) to translate aeM (vx) = vaeM (x) into
X
r(vx) = xff in M (for all r): (10:16)
ff
For addition, the idea is that k: E_kx E_k ! E_k induces both the additive
structure in UM and the *multiplication in E*(E_k). Of course, r*c is not addi*
*tive
in r. Given two operations r; s: k ! m, their sum may be constructed as
rxs m
r + s: E_k! E_kx E_k! E_m x E_m ! E_m ;
as we can check by composing with x: X ! E_k. When we apply Ehomology, we
find X
(r + s)*c = r*c0i* s*c00i in E*(E_m ), (10:17)
i
P 0 00
if we write c = ici ci for c 2 E*(E_k). (In other words, we add r* and s*
according to the group structure on Mod (E*(E_k); E*(E_m )) described by Milnor*
* and
Moore in [19, Defn. 8.1], which makes use of the coalgebra structure of E*(E_k)*
* and
the algebra structure of E*(E_m ).) To add more than two operations, we need it*
*erated
coproducts: given any finite indexing set , we write the iterated comultiplica*
*tion
: E*(E_k) ! ff2E*(E_k) in the form
X
c = ci;ff in ff2E*(E_k) (10:18)
i ff
for suitable elements ci;ff2 E*(E_k). We can of course replace E_k by any space*
* for
which we have the necessary K"unneth formulae.
Theorem 10.19 Let M be an unstable algebra and assume that E*(E_k) is a free
E*module for all k. Take x; y 2 Mk and assume that r(x) is in the standard form
(10.3). Then:
(a) We have the Cartan formula for addition
X
r(x + y) = xffr00ff(y) for all r: k ! m, (10:20)
ff
where for each ff, the operation r00ff: k ! m + deg(cff) is defined as having t*
*he func
tional
= (1)deg(cff)(m+deg(cff)) for all c 2 E*(E_k);*
*(10:21)
(b) If, similarly, r(y) has the standard form
X
r(y) = yfi; (10:22)
fi
JMB, DCJ, WSW  53  23 Feb 1995
Unstable cohomology operations
then we have the full Cartan formula for addition,
X X
r(x+y) = (1)deg(xff) deg(yfi) xffyfi(10:23)
ff fi
for all r: k ! m;
(c) Assume a; b 2 E*(E_k). Let cffrunPthrough a basis ofPE*(E_k), and denote*
* by
r0ffthe operation dual to cff. Let a = i ffai;ffand b = j ffbj;ffbe the i*
*terated
coproducts of a and b as in eq. (10.18), where in both cases, we ignore those f*
*f for
which
r0ff*ai;ff= (fflai;ff)1 for all i (10:24)
(see the Remark following). Then the homology homomorphism r*: E*(E_k) !
E*(E_m ) satisfies
X X
r*(a * b) = * r0ff*ai;ffOr00ff*bj;ff in E*(E_m ),(10:25)
i j ff
where r00ffis defined by eq. (10.21) and the only signs come from shuffling the*
* factors
to form (a x b).
Remark The formula (10.25) demands some explanation. The proof will show that
the relevant set of ff is in fact finite, so that the iterated coproducts a and*
* b are
defined.
If ff satisfies eq. (10.24), we have
r0ff*ai;ffOr00ff*bj;ff= (fflai;ff)1 Or00ff*bj;ff= (fflai;ff)fflr00ff*bj*
*;ff= (fflai;ff)(fflbj;ff)1m :
In the usual (and sufficient) case when ffla = fflb = 0, we can easily arrange *
*for each
ai;ffand bj;ffto be 1 or lie in Ker ffl, by breaking up terms and shuffling as *
*necessary.
Then the ijterm contributes nothing to r*(a * b) unless ai;ff= 1 and bj;ff= 1 *
*for all
ff 2 that satisfy eq. (10.24). Such an index ff may be omitted from the *prod*
*uct
in eq. (10.25) and the iterated coproducts a and b.
Proof We first assume that the cffform a basis of E*(E_k), so that xff= r0ff(x*
*) as in
eq. (10.4). By the K"unneth homeomorphism, we can write
X
*kr = r0ffx r00ff in E*(E_kxE_k), (10:26)
ff
for uniquely determined elements r00ff2 E*(E_k). In other words, in the diagram
u ______r0ffxr00ff
X _________E_kxE_k E_?x E_?
Q Q  
Q Qx+y k OE (10:27)
Q Q  
Qs ? r ?
E_k ___________E_m
the map r Ok is expressed as the sum of the maps gff= OE O(r0ffxr00ff), and is *
*the
universal example for computing r(x+y), where u: X ! E_k x E_k has coordinates
JMB, DCJ, WSW  54  23 Feb 1995
x10. Enriched Hopf rings
x: X ! E_kand y: X ! E_k. Evaluation on cffx c identifies r00ffas in eq. (10.21*
*), with
the help of
<*kr; cffxc> = = :
Then eq. (10.20) is induced from eq. (10.26). To deduce (b), we substitute eq. *
*(10.22)
in eq. (10.20) and watch the signs.
To remove the requirement that the cffform a basis, we note that by lineari*
*ty,
eq. (10.20) is preserved by each of the replacements listed in Lemma 10.6. (The
operation r0ffis no longer defined, but appears only in (c).)
For (c), we apply homology everywhere. We have to add the homomorphisms gff*
in the sense of eq. (10.17), using the iterated coproduct (a x b), which is obt*
*ained
from a x b by shuffling. We note that any a 2 E*(E_k) comes from some finite
subcomplex Y of E_k. All but finitely many of the r0ffvanish on Y , by the stro*
*ng duality
for E_k; these ff satisfy eq. (10.24), as we see by computing the iterated copr*
*oduct a
first in Y , since the zero operation 0: k ! m induces 0*c = (fflc)1m . 
Similarly, the zero map !k: T ! E_kand inversion map k: E_k! E_kof E_kyield
the useful formulae
r(0k) = 1M in M (for all r) (10:28)
and X
r(x) = xff in M (for all r): (10:29)
ff
For some applications, it is useful to cut out the finiteness argument in t*
*he proof
of Thm. 10.19(c) and work directly in a finite space Y .
Proposition 10.30 Let f: Y ! E_k be a map, where E*(Y ) is a free E*module
of finite rank, with basis elements zff. Let yff2 E*(Y ) be dual to zff. Then f*
*or any
a 2 E*(Y ), b 2 E*(E_k), and operation r: k ! m,
X X
r*(f*a * b) = * yff*ai;ffOr00ff*bj;ff in E*(E_m ),
i j ff
where r00ff: k ! m + deg(zff) denotes the operation having the functional
= (1)deg(zff)(m+deg(zff));
a and b are computed as in eq. (10.18), and we use yff*: E*(Y ) ! E*(E_?).
Proof By Thm. 1.18(a), E*(Y ) is dual to E*(Y ) and yffis defined. We modify t*
*he
proof of the Theorem by composing the square in diag. (10.27) with f x 1: Y x E*
*_k!
E_kPx E_k. We work in E*(Y xE_k) instead of E*(E_kxE_k) and write (f x1)**kr =
ffyffx r00ff. We evaluate this on zffx c to determine r00ff. 
Remark The commutativity of *multiplication ensures that r(x+y) = r(y+x).
Conversely, one could say that x + y = y + x in M requires *multiplication to
be commutative. The universal example has M = E*(E_kxE_k), x = k x 1, and
y = 1 x k, and cffand dfirun through bases of E*(E_k). Then r(x+y) = r(y+x)
for all r implies that cff* dfi= dfi* cfffor all ff and fi. The commutativity *
*of * in
general follows by linearity.
JMB, DCJ, WSW  55  23 Feb 1995
Unstable cohomology operations
Similar discussions hold for other laws in a ring. In particular, x + 0 = x*
* corre
sponds in this way to c * 1k = c, (x) = x to OOc = c, (x + y) = (x) + (y) *
*to
O(a * b) = Oa * Ob, and the associativity of + to the associativity of *.
Given a prime p, we can iterate eq. (10.23) to get
X
r(px) = r(x+x+ : :+:x) = xff1xff2: :x:ffp:
If the indices ffi are not all the same, we can permute them cyclically and obt*
*ain p
distinct terms which by commutativity are all equal, with the same sign. This l*
*eaves
only the terms with ffi= ff for all i, and we find
X
r(px) F xffmod p: (10:31)
ff
This is particularly useful when E* has characterisitic p, so that px = 0, beca*
*use
comparison with eq. (10.28) then yields
X
F xff= 1M in M (for all r):(10:32)
ff
Multiplicative structure The multiplication maps OE: E_kx E_m ! E_k+m induce
both the multiplication in UM and the Omultiplication in E*(E_*). This allows *
*us to
translate the axiom that aeM is multiplicative, aeM (xy) = aeM (x)aeM (y) in U*
*M.
Theorem 10.33 Let M be an unstable algebra, and assume that E*(E_k) is a fr*
*ee
E*module for all k. Take x 2 Mk and y 2 Mm and assume that r(x) is in the
standard form (10.3). Then:
(a) We have the Cartan formula for multiplication
X
r(xy) = xffr00ff(y) for all r: k + m ! h, (10:34)
ff
where for each ff, the operation r00ff: m ! h + deg(cff) is defined as having t*
*he func
tional
= (1)deg(cff)(h+deg(cff)) for all c 2 E*(E_m()*
*;10:35)
(b) If, similarly, r(y) is given by eq. (10.22), we have the full Cartan for*
*mula for
multiplication, X X
r(xy) = (1)deg(xff) deg(yfi) xffyfi(10:36)
ff fi
for all r: k + m ! h;
(c) Take a 2 E*(E_k) and b 2 E*(E_m ). Assume that cffruns throughPa basis
of E*(E_k),Pand denote by r0ffthe operation dual to cff. Let a = i ffai;ffa*
*nd
b = j ffbj;ffbe the iterated coproducts of a and b as in eq. (10.18), where *
*in both
cases, we ignore all ff that satisfy eq. (10.24). Then the homology homomorphi*
*sm
r*: E*(E_k+m ) ! E*(E_h) satisfies
X X
r*(a Ob) = * r0ff*ai;ffOr00ff*bj;ff in E*(E_m ),(10:37)
i j ff
where r00ffis defined by eq. (10.35) and the only signs come from shuffling the*
* factors
to form (a x b).
JMB, DCJ, WSW  56  23 Feb 1995
x10. Enriched Hopf rings
The Remark following Thm. 10.19 applies.
Proof This is formally identical to the proof of Thm. 10.19, with k: E_kx E_k!*
* E_k
replaced everywhere by OE: E_kx E_m ! E_k+m. 
By naturality, we can adapt eq. (10.36) to xproducts.
Corollary 10.38 Given spaces X and Y and elements x 2 Ek(X) and y 2 Em (Y ),
assume that r(x) and r(y) are given by eqs. (10.3) and (10.22). Then we have t*
*he
Cartan formula
X X
r(xxy) = (1)deg(xff) deg(yfi) xffxyfi(10:39)
ff fi
in E*(X xY ), for any operation r: k + m ! h. 
We have also the analogue of Prop. 10.30.
Proposition 10.40 Let f: Y ! E_k be a map as in Prop. 10.30. Then for any
a 2 E*(Y ), b 2 E*(E_k), and operation r: k + m ! h,
X X
r*(f*a Ob) = * yff*ai;ffOr00ff*bj;ff in E*(E_h),
i j ff
where r00ff: m ! h + deg(zff) denotes the operation having the functional
= (1)deg(zff)(m+deg(zff))
and a and b are computed as in eq. (10.18). 
Since the unit element of UM is jM Oj*: E*(E_0) ! E* ! M, the axiom aeM (1M*
* ) =
1UM translates easily into
r(1M ) = 1M = 1M = 1M in(M10:41)
for all r.
Just as with addition, certain laws in the Hopf ring correspond to laws in *
*an E*
algebra M. For example, associativity of Omultiplication corresponds to associ*
*ativity
of multiplication in M. Commutativity is slightly trickier: given x 2 Mk and y *
*2 Mm ,
r(yx) = r((1)kmxy) leads to the identity (10.10), thus explaining the signs an*
*d the
appearance of O.
The comonad structure Finally, we translate the two axioms which state that
aeM makes M a Ucoalgebra. Since we have in effect returned to the First Answe*
*r of
x8, these are the usual axioms for an action, (sr)(x) = s(r(x)) and kx = x.
The second is easily handled. From Prop. 6.11, we can use (6.41) to express*
* the
identity operation k as the functional
= Q(ffl) Oqk: E*(E_k) ! Q(E)k*! E*(E; o) ! E*: (10:42)
When we put r = k, eq. (10.3) expands easily to yield the axiom
X
(Q(ffl)qkcff) xff= x in M (10:43)
ff
JMB, DCJ, WSW  57  23 Feb 1995
Unstable cohomology operations
for x 2 Mk. We have thus interpreted the counit natural transformation fflM: UM*
* !
M of the comonad U, which was defined by (fflM)f = fk. The functional Q(ffl) Oq*
*k =
fflS Ooek*is not part of the Hopf ring structure as given so far, so we add it.*
* (It is
unrelated to the counit ffl: E*(E_k) ! E* of the Hopf algebra E*(E_k).)
It is easy to recover the functional from r*, as in eq. (6.50), in t*
*he form
r* oem* fflS *
: E*(E_k) ! E*(E_m ) ! E*(E; o) ! E ; (10:44)
by writing = = and using eq. (10.42). In the additiv*
*e context,
the reverse construction of r* from was neatly encoded in the comultipli*
*cation
Q( ) on Q(E)**. Here, we have no such map and must rely on the definition of M,
which explicitly uses r*. In effect, we dualize and use r* instead.
The first is the most complicated of all the axioms. When we substitute sr *
*and
r in eq. (10.3) and use = = , the axiom (sr)(x) *
*= s(r(x))
expands to
X X
xff= s(r(x)) = s xff in M, (10:45)
ff ff
for all r, s. The right side is to be expanded using eqs. (10.20) and (10.16), *
*and in
general is extremely complicated.
Our conclusion is that we need to know the induced homology homomorphism
r*: E*(E_k) ! E*(E_m ) for every operation r: Ek() ! Em (). This is the final*
* piece
of structure to add to the Hopf ring. To compute it successfully, we need r*c f*
*or each
Ogenerator c of E*(E_*), and then use formulae (10.25) and (10.37) for r*(a * *
*b) and
r*(a Ob).
Summary We collect the various formulae to form the main theorem of this secti*
*on.
In addition to the Hopf ring structure on E*(E_*), we need:
(i)The element [v] 2 E0(E_*) for each v 2 E* (see Defn. 10.8);
(ii)The augmentation (see eq. (10.42))
Q(ffl) Oqk: E*(E_k) ! Q(E)k*! E*(E; o) ! E* (10.46)
which may be written fflS Ooek*;
(iii)The homomorphism r*: E*(E_k) ! E*(E_m ) induced by each opera
tion r: k ! m.
These constitute what we mean by an enriched Hopf ring structure.
Theorem 10.47 Let M be an object of FAlg, i. e. a complete Hausdorff filte*
*red
E*algebra, and assume that E*(E_k) is a free E*module for all k (which is tru*
*e for
E = H(F p), BP , MU, KU, or K(n) by Lemma 4.17(a)). Then an unstable algebra
structure on M consists of a value r(x) 2 M for all x 2 M and all r 2 E*(E_k) (*
*where
k = deg(x) and r(x) 2 Mm if r 2 Em (E_k)), which is E*linear in r and therefo*
*re (for
fixed x) expressible in the standard form (10.3)
X
r(x) = xff in M (for all r):
ff
These values are subject to the following axioms:
JMB, DCJ, WSW  58  23 Feb 1995
x11. The Ecohomology of a point
(a) For fixed x 2 Mk, r(x) satisfies the three consistency conditions:
X X X
(i) cfl xfl = (1)deg(xff) deg(xfi)cff cfi xffxfi
fl ff fi
in (E*(E_k) E*(E_k)) b M;
X
(ii) (fflcff)=xff1M in M ;
ff
X
(iii) (fflSoek*cff)=xffx in M;
ff
(b) As x varies, r(x) satisfies the following identitiesPin M for all r, whe*
*re we
assume similarly (as in eq. (10.22)) that r(y) = fi yfi:
X X
(i) r(x + y)= (1)deg(xff) deg(yfi) xffyfi;
ff fi
X
(ii) r(vx) = xff;
ff
X X
(iii) r(xy) = (1)deg(xff) deg(yfi) xffyfi;
ff fi
(iv) r(1M )= 1M ;
(c) The composition law
X X
xff= s(r(x)) = s xff in M
ff ff
holds for all r, s, and all x 2 M;
(d) For each of the ideals F aM in the filtration of M:
(i)For fixed x 2 M, all except finitely many of the xfflie in F aM;
(ii)There exists F bM such that r(x) 2 F aM for all x 2 F bM and all based
operations r.
Proof The equations in (a) are (10.14), (10.15), and (10.43). Those in (b) are*
* (10.23),
(10.16), (10.36), and (10.41). The equation in (c) is (10.45). In (d), (i) stat*
*es that
aeM (x): E*(E_k) ! M is continuous for each x, while (ii) states that aeM : M !*
* UM is
continuous. 
Remark By (b), an unstable algebra structure on M is determined by the values
r(x) on a set of (topological) E*algebra generators x of M. Moreover, the Hopf*
* ring
laws imply that it is sufficient to verify axioms (a) and (d)(i) on these gener*
*ators. In
practice, the topological conditions (d) rarely cause us any distress.
11 The Ecohomology of a point
In this section, we study the first of our test spaces, the onepoint space*
* T , for
which E*(T ) is by definition the coefficient ring E*. Its unstable structure *
*is com
pletely determined by eqs. (10.41) and (10.16) as
r(v) = in E* = E*(T ) (for all r); (11:1)
JMB, DCJ, WSW  59  23 Feb 1995
Unstable cohomology operations
which may be taken as an alternate definition of the Hopf ring elements [v], in*
*stead
of Defn. 10.8.
It is easy to deduce how [v] interacts with each piece of the structure on *
*E*(E_*).
Much of this can be read off from the Hopf ring structure in x10. In particular*
*, jR is
still in some sense a ring homomorphism.
Proposition 11.2 The Hopf ring elements [v] 2 E0(E_h) for each v 2 Eh have t*
*he
properties:
(a) [v] = [v] [v];
(b) ffl[v] = 1;
(c) [v + v0] = [v] * [v0] for v02 Eh;
(d) [v] = O[v];
(e) [vv0] = [v] O[v0] for v02 Ek;
(f) 1m O[v] = 1m+h ;
(g) r*[v] = [] (for all r);
(h) r*1h = [];
(i) qh[v] = jRv in Q(E)h0;
(j) oeh*[v] = jRv in Eh(E; o), under stabilization.
Proof For (a) and (b) we substitute eq. (11.1) in eqs. (10.14) and (10.15). Fo*
*r (c) and
(e), we write down the Cartan formulae (10.23) and (10.36) for r(v+v0) and r(vv*
*0)
and compare with eq. (11.1). For (d), we write down r(v) from eq. (10.29) and
compare with eq. (11.1). For (g), we use eq. (11.1) to compute s(r(v)) = ]>;
by eq. (10.45), this must agree with for all s. Since [0n] = 1n, (f)*
* and (h) are
special cases of (e) and (g). For (i) and (j), we compare eq. (11.1) with eq. (*
*6.43) and
[8, eq. (11.23)], and use eqs. (6.14) and (6.13). 
Constant operations Constant operations were introduced briefly in x8. Although
they are of no real interest and contain no information, they are undeniably na*
*tural
and we have to be able to recognize them in their several disguises.
Proposition 11.3 Let r: k ! m be the constant operation defined by r(x) = v1X
for all x 2 Ek(X), where v 2 Em . Then:
(a) As a class, r = v1k 2 E*(E_k);
(b) As a map, r is the composite v Oq: E_k! T ! E_m;
(c) As a functional, = (fflc)v in E* for all c 2 E*(E_k);
(d) r*: E*(E_k) ! E*(E_m ) is given by r*c = (fflc)[v] for all c 2 E*(E_k). *
* 
Based operations Given a based space (X; o), we consider the naturality of an
operation r: k ! m with respect to the inclusion of the basepoint. We augment
Lemma 2.3.
Proposition 11.4 The following conditions on an operation r: k ! m are equiv*
*a
lent:
(a) r(0) = 0 in E*(T ) = E*, i. e. r is based in the sense of Defn. 2.2;
JMB, DCJ, WSW  60  23 Feb 1995
x12. Spheres, suspensions, and additive operations
(b) r(0) = 0 in E*(X) for all spaces X;
(c) The operation r induces r: Ek(X; o) ! Em (X; o) for all X;
(d) The class r lies in Em (E_k; o) Em (E_k);
(e) The map r: E_k! E_m is based (up to homotopy);
(f) The linear functional satisfies = 0;
(g) The homomorphism r*: E*(E_k) ! E*(E_m ) satisfies r*1k = 1m .
Proof Part (b) is equivalent to (a) by naturality. Because r(0) = 1X *
* by
eq. (10.28), (f) is equivalent to (b), and with the help of Prop. 11.2(h), to (*
*g). 
We can generalize (f).
Lemma 11.5 Let (X; o) be a based space. Then for any x 2 Ek(X; o) and any
operation r: k ! m, we have r(x) 1X mod E*(X; o).
Proof We use eq. (10.28) and the naturality of r in diag. [8, eq. (3.2)]. 
It is sometimes useful to be more specific. If we choose a basis of E*(E_k)*
* consisting
of 1k and elements cff2 Ker ffl, then for any x 2 Ek(X; o), eq. (10.4) takes th*
*e form
X
r(x) = 1X + xff in E*(X)^ (for all;r)(11:6)
ff
where the elements xff2 E*(X; o)^.
Formulae are often simpler for based operations, but the case of general r *
*can be
recovered easily enough by decomposing as in Lemma 8.1.
Lemma 11.7 If we write r(x) = s(x)+v1X , where s is a based operation and v *
*2 Em ,
the homology homomorphism r*: E*(E_k) ! E*(E_m ) is given by r*c = s*c * [v], w*
*here
we recognize v = r(0) = .
Proof We write r as the composite
1xq sxv m
E_k! E_k x E_k! E_k x T ! E_m x E_m ! E_m
and take Ehomology. The first two maps just give E_k~= E_kx T . 
12 Spheres, suspensions, and additive operations
So far, except for adding an extra grading, our additive results are formal*
*ly very
similar to the stable case discussed in [8]. What is new is that suspension is *
*no longer
an isomorphism, but defines a new element e. The stable results can all be reco*
*vered
by stabilizing, which consists merely of setting e = 1.
We assume throughout that E*(E_k), QE*(E_k), and E*(E; o) are free E*modul*
*es,
so that we have available the machinery of comodule algebras of xx6, 7 as well *
*as the
stable results of [8]. In particular, the coaction aeX : E*(X) ! E*(X) b Q(E)**
**is a
homomorphism of E*algebras for any X.
Spheres Our second test space, after the onepoint space T , is the circle S1.*
* Its
cohomology E*(S1; o) is a free E*module with the basis {1S; u1}, where the can*
*onical
generator u1 2 E1(S1; o) is provided by [8, Defn. 3.23]. Thus aeS: E*(S1) ! E*(*
*S1)
Q(E)**is determined by aeSu1.
JMB, DCJ, WSW  61  23 Feb 1995
Unstable cohomology operations
Definition 12.1 We define the suspension element e = eQ 2 Q(E)11by the identi*
*ty
aeSu1 = u1 e in E*(S1; o) Q(E)1*~=Q(E)1*. (12:2)
It has degree zero.
More generally, for the ksphere Sk, E*(Sk) is free on the basis {uk; 1S}, *
*where
uk 2 Ek(Sk; o).
Proposition 12.3 The suspension element e 2 Q(E)11has the following properti*
*es,
where k 0:
(a) aeSuk = uk ek in E*(Sk) Q(E)k*;
(b) ruk = uk in E*(Sk) for any additive operation r: k ! m;
(c) The class uk 2 Ek(Sk), regarded as a map uk: Sk ! E_k, induces qkuk*z =
ek 2 Q(E)kk, where z 2 Ek(Sk) is dual to uk;
(d) In the coalgebra structure on Q(E)**, Q( )e = e e and Q(ffl)e = 1;
(e) Q( )(vekw) = vek ekw in Q(E)** Q(E)**, for any v 2 E* and w 2 jRE*;
(f) Given v 2 E* and w 2 jREh, the homomorphism Q(r): Q(E)k+h*! Q(E)m*
induced on homology by any operation r: k + h ! m satisfies
Q(r)(vekw) = vekjR in Q(E)m*;
(g) Under stabilization, Q(oe)e = 1 in E*(E; o).
Proof We prove (a) for k > 0 by induction on k, starting from eq. (12.2). If i*
*t holds
for k and m, the multiplicativity of ae gives
ae(ukxum ) = (ukxum ) ek+m in E*(SkxSm ).
The projection map q: Sk x Sm ! Sk+m induces q*uk+m = uk x um , which gives (a)
for k+m. The result is true also for k = 0, if we make the obvious identificat*
*ion
e0 = 1. Then (b) follows by eq. (6.39) and (c) is an application of Prop. 6.44.
To prove (d), we evaluate both axioms (6.33) for M = E*(S1) on u1. Part (e)
follows immediately from (d) and the fact that Q( ) is a homomorphism of algebr*
*as
and of E*bimodules. Then (f) follows from (e) and Lemma 6.51(c). For (g), we
apply 1 Q(oe) to eq. (12.2) and compare with the stable coaction aeSu1 = u1 1*
* in
[8, eq. (11.24)]. 
Remark As v, k, and w vary, the elements vekw span Q(E)** Q as a Q module.
(In fact, Q(oe) induces Q(E)k* Q ~=E*(E; o) Q if E is (k1)connected.) Thus
in the important case when Q(E)k*has no torsion, the innocuous formulae in (e) *
*and
(f) are powerful enough to determine Q( ) and Q(r) completely.
Corollary 12.4 Let r: k ! m be an additive operation, regarded as a map of
Hspaces r: E_k! E_m. Then the induced homomorphism on homotopy groups
r* *
E* ~=ss*(E_k; o) ! ss*(E_m ; o) ~=E
is given on v 2 Eh by r*v = .
JMB, DCJ, WSW  62  23 Feb 1995
x13. Spheres, suspensions, and unstable operations
Proof We reinterpret r* as the action of the operation r on Ek(Sk+h; o). The e*
*lement
v corresponds to the class vuk+h. From Prop. 12.3(b) and eq. (6.40),
r(vuk+h) = uk+h in E*(Sk+h; o) .
Suspensions More generally, the action of the operations on the suspension X of
a based space X is easily deduced from the action on X.
Lemma 12.5 Given a based space (X; o) and x 2 Ek(X; o), the coaction aeX x *
*is
the image of aeX x under
e: E*(X; o) b Q(E)k*! E*(X; o) b Q(E)k+1*;
where e denotes multiplication by the element e 2 Q(E)1*.
Proof The projection map S1 x X ! X embeds E*(X; o) in E*(S1xX; S1xo).
Here, x corresponds to u1 x x, whose coaction is known. 
We can mimic this algebraically. We defined the formal suspension M of any
E*module M in [8, Defn. 6.6], merely by shifting all the degrees up one.
Definition 12.6 Given any unstable comodule M, we make the suspension M
of M an unstable comodule by equipping it with the coaction aeM defined by the
commutative square
aeM
Mk __________Mb Q(E)k*
~  
=  e
? ae ?
M
(M)k+1 pppppppMb Q(E)k+1*
The axioms on aeM are readily verified.
13 Spheres, suspensions, and unstable operations
In this section, we continue x12 by computing all the unstable operations on
E*(Sk) for the spheres Sk, which requires one new Hopf ring element, the suspen*
*sion
element e. This leads to the unstable structure of E*(X) in terms of E*(X).
We recall that E*(Sk) is a free E*module with basis {1S; uk}, where uk is *
*the
standard generator. The algebra structure is given by u2k= 0, except that of co*
*urse
u20= u0. By the Remark after Thm. 10.47, we have only to find r(uk). Lemma 11.5
gives partial information.
We assume that E*(E_k) is a free E*module for all k.
Definition 13.1 We define the suspension element e = eU 2 E1(E_1) by the iden
tity
r(u1) = 1S + u1 in E*(S1) (for all r):(13:2)
Here and in similar definitions, we use the freeness of E*(S1) and the dual*
*ity
FMod *(E*(E_k); E*) ~=E*(E_k) to ensure that e exists and is well defined. We *
*note
that ffle = 0 from eq. (10.15). Rather than develop all the properties of e no*
*w, we
include them below in Prop. 13.7 as the special case e1 = e.
JMB, DCJ, WSW  63  23 Feb 1995
Unstable cohomology operations
Suspensions We deduce from eq. (13.2) the behavior of unstable operations under
the suspension isomorphism : E*(X; o) ~= E*(X; o). We take an element x 2
Ek(X; o) Ek(X) and assume that r(x) is given by eq. (11.6), so that fflcff= 0.*
* The
quotient map q: S1 x X ! X embeds E*(X) in E*(S1xX) ~=E*(S1) E*(X);
under this embedding, x corresponds to u1 x x. We compute r(u1xx) from the
Cartan formula (10.39) and find
X
r(x) = 1X + (1)deg(xff) xff (13:3)
ff
for all r. The other terms drop out because 11O cff= fflcff= 0 and e O1k = ffle*
* = 0.
This suggests how the suspension of an unstable algebra should be defined. *
*The
treatment is slightly different from the additive version in x12. First, we ne*
*ed a
basepoint.
Definition 13.4 We call the unstable algebra M based_if we are given an augme*
*n
tation M ! E* of unstable algebras._Then the kernelM is an invariant ideal, a*
*nd
we have the splitting M = E* M as E*modules.
We define the unstable suspension U M of M as the subalgebra
__ * 1
U M = (1SE*) (u1M ) E (S ) M: (13:5)
__
The action of r is given on u1 M by eq. (13.3) and on 1S E* by eq. (11.1).
For example, if (X; o) is a based space, we have the augmentation E*(X) !
E*(o) = E*, with kernel E*(X; o) (as in [8, eq. (3.2)]). Inspection shows that *
*much of
the structure on M is not used. The multiplication on M is totally ignored. Ind*
*eed,
we do not need an unstable structure on M at all.
__ __
Theorem 13.6 Given an additively unstable moduleM , we can make E* M
an_unstable algebra, with_1 2 E* as the unit element and trivial multiplication*
* on
k P
M , as follows. If x 2M and r(x) = ffxfffor additive operations *
*r, where
cff2 Q(E)k*, we lift each cffto "cff2 E*(E_k) such that qk"cff= cff, and define*
* the action
of unstable operations r on x by
X
r(x) = 1 + (1)deg(xff) xff:
ff
Proof Because e O1 = 0 and e O(b*c) = 0 whenever fflb = 0 and fflc = 0, r(x) is
independent of the choices of the "cff. The definition_(with signs) has been_ch*
*osen so
that: (a) the additive unstable structure on E* M restricts to that on M giv*
*en
by Defn. 12.6, and (b) it includes eq. (13.5) for a based unstable algebra M. (*
*For (a),
we note that diag. (6.16) gives qk+1(eU O"cff) = (1)keQ cff.) Verification of *
*the axioms
of Thm. 10.47 is tedious but routine. 
The elements ek It is convenient to use eq. (13.3) to find the structure of E**
*(Sk).
We deduce the fundamental properties of the Hopf ring element e.
Proposition 13.7 We define the Hopf ring elements ek 2 Ek(E_k) for k 0 in t*
*erms
of e 2 E1(E_1) by ek = (1)k(k1)=2eOk for k > 0 (so that e1 = e) and e0 = [1] *
* 10.
They have the following properties:
JMB, DCJ, WSW  64  23 Feb 1995
x13. Spheres, suspensions, and unstable operations
(a) In E*(Sk) we have, for any k 0:
r(uk) = 1S + uk (for all r); (13:8)
(b) The class uk, regarded as a map uk: Sk ! E_k, induces uk*z = ek 2 Ek(E_k*
*) in
homology, where z 2 Ek(Sk) is dual to uk;
(c) ek Oem = (1)kmek+m if k > 0 or m > 0;
(d) ek = ek1 + 1ek for all k > 0;
(e) fflek = 0 2 E* for all k 0;
(f) Oek = ek for all k > 0;
(g) ek O[] = ek for all rational numbers 2 E0 and all k > 0;
(h) r*ek = [] * []Oek for all k 0 and all r: k ! m;
(i) qkek = ekQ= ek in Q(E)k*, for all k 0, for additive operations;
(j) oek*ek = 1 in E*(E; o), for all k 0, under stabilization.
Remark The results make it clear that the correct interpretation of eO0is [1] *
* 10 =
[1]  [00], as in [28] and elsewhere, rather than just the element [1].
Proof We give extensive details of this proof (only), as a good example of our*
* ma
chinery in action.
We establish eq. (13.8) for k > 0, and thus (a), by induction on k. It hol*
*ds for
k = 1 by definition. We recognize Sk as Sk+1 and uk as uk+1; then by eq. (13.3),
eq. (13.8) holds for k + 1 if it holds for k, provided that ek+1 = (1)ke Oek. *
* Our
definition of ek is designed to do exactly this. More generally, we have (c).
For k = 0, we write E*(S0) = E* E*. In Alg, this is a product of algebras,*
* with
the projections induced respectively by the inclusions of the basepoint and the*
* other
point. In this presentation, u0 = (0; 1), and of course 1S = (1; 1). By eq. (11*
*.1), the
action on u0 is
r(u0) = r((0; 1)) = (; ) = (1S  u0) + *
*u0;
which gives (a) if we define e0 = [1]  10.
Then (b) is an application of Prop. 10.5. When we substitute eq. (13.8) in*
*to
eq. (10.14), we find, for k > 0,
1k1S + ekuk = 1k1k1S + 1kekuk + ek1kuk;
since u2k= 0. This gives (d). (But e0 acquires the extra term e0e0, because u2*
*06= 0;
this is obvious anyway from Prop. 11.2. Also, (c), (d), and (g) are clearly fa*
*lse for
k = m = 0.) Similarly, eq. (10.15) yields 1S + (fflek)uk = 1S (even for k = 0),*
* which
gives (e).
For (g), which includes (f) as the special case = 1 (by Prop. 10.12(a) and
Prop. 11.2(d)), the distributive law (10.11) and (d) yield ek O[+] = ekO[] + ek*
*O[]
for all ; 2 E0. Since ek O[1] = ek, (g) follows. (We are in effect expanding r*
*(uk).)
For (h), we substitute eq. (13.8) into eq. (10.45). On the left, we have
(sr)(uk) = 1S + uk;
while on the right, iteration of eq. (13.8) yields, after simplification,
s(r(uk)) = ]> 1S + ] * []Oek> uk;
JMB, DCJ, WSW  65  23 Feb 1995
Unstable cohomology operations
with the help of eqs. (10.16) and (10.23). Comparison of these gives r*ek.
For k = 1 in (i) and (j), we stabilize eq. (13.2) by Lemma 10.7 and compare*
* with
Defn. 12.1 and [8, eq. (11.24)]. For general k, we use the multiplicative prope*
*rties in
diag. (6.16) of qk and oek*. 
We have the analogue of Cor. 12.4. By Lemma 2.3(d), a based operation r: k *
*! m
is represented by a based map r: (E_k; o) ! (E_m ; o). We need to know its effe*
*ct on
homotopy groups.
Lemma 13.9 Given a based operation r: k ! m, the induced homomorphism on
homotopy groups
r* mh
Ekh ~=ssh(E_k; o) ! ssh(E_m ; o) ~=E
is given on v 2 Ekh for any h 0 by
r*v = in Emh .
Proof Viewed cohomologically, the element v 2 Ekh or map v: Sh ! E_k corre
sponds to vuh 2 Ek(Sh; o). From eqs. (10.16) and (13.8), we compute r(vuh) =
uh, which simplified because r is based, so that = 0. 
14 Complex orientation and additive operations
In this section, we study the effect of a complex orientation on additive o*
*perations.
The relevant test space is C P 1, for which E*(C P 1) = E*[[x]] by [8, Lemma 5.*
*4],
where x = x() is the Chern class of the Hopf line bundle . All the stable resu*
*lts
carry over, almost without change, except that now b1 = e2 instead of 1.
We assume that E*(E_k), Q(E)k*, and E*(E; o) are free E*modules.
Definition 14.1 We define elements bi2 Q(E)22ifor all i 0 by the identity
1X
aex = b(x) = xi bi in E*(C P 1) b Q(E)2*~=Q(E)2*[[x]]; (14:2)
i=0
where b(x) is a convenient formal abbreviation that rapidly becomes essential.
We use eq. (6.39) to convert eq. (14.2) to the equivalent form
1X
rx = xi in E*(C P 1) = E*[[x]], for all r . (14:3)
i=0
Since the Hopf bundle is universal, eqs. (14.2) and (14.3) hold for the Chern c*
*lass x =
x() of any complex line bundle over any space X (after completion, if necessar*
*y).
Proposition 14.4 The elements bi2 Q(E)22ihave the following properties:
(a) b0 = 0 and b1 = e2, so that b(x) = xe2 + x2b2 + x3b3 + : :;:
(b) The Chern class x 2 E2(C P 1), regarded as a map of spaces x: CP 1 ! E_2,
induces q2x*fii= bi2 Q(E)22i, where fii2 E2i(C P 1) is dual to xi;
JMB, DCJ, WSW  66  23 Feb 1995
x14. Complex orientation and additive operations
(c) Q( )bk is given by
Xk
Q( )bk = B(i; k) bi in Q(E)** Q(E)2*,
i=1
where B(i; k) denotes the coefficient of xk in b(x)i, or formally,
1X
Q( )b(x) = b(x)i bi;
i=1
(d) Q(ffl)bk = 0 for k > 1, or formally, Q(ffl)b(x) = x;
(e) The stabilization Q(oe): Q(E)2*! E*(E; o) sends the element bi 2 Q(E)22i*
*to
the stable element bi2 E2i2(E; o) of [8, Defn. 13.1].
Proof For (a), we restrict eq. (14.2) to C P 1~=S2 and compare with eq. (12.2)*
*. For
(b), we apply Prop. 6.44. For (c) and (d), we substitute ae into diags. (6.33)*
* and
evaluate on x. For (e), we compare Defn. 14.1 with [8, Defn. 13.1]. 
Still following the stable strategy, we next apply ae to the multiplication*
* map
: CP 1 x CP 1 ! CP 1, to obtain the formal identity
X
b(F (x; y)) = FR(b(x); b(y)) = b(x) + b(y) + b(x)ib(y)jjRai;j(14:5)
i;j
in Q(E)**[[x; y]], which looks exactly like the stable version [8, eq. (13.6)].*
* Again,
FR(X; Y ) is a convenient abbreviation. The consequences are the same.
The plocal case
Lemma 14.6 Assume that E* is a plocal ring. Then the generator bk of Q(E)***
*is
redundant unless k is a power of p.
Proof The proof of [8, Lemma 13.7] applies without change. 
We therefore reindex the b's.
Definition 14.7 When E* is a plocal ring, we define b(i)= bpifor each i 0.
As in [8, x13], we obtain
X
b([p](x)) = [p]R(b(x)) = pb(x) + b(x)i+1jRgi (14:8)
i>0
in Q(E)2*[[x]], which looks exactly like the stable version [8, eq. (13.11)] bu*
*t is in a
k
different place. Again, we extract the coefficient of xp .
Definition 14.9 For each k > 0, we define the k th main (additively unstable)
relation as
(Rk) : L(k) = R(k) in Q(E)2*, (14:10)
k
where L(k) and R(k) denote the coefficient of xp in the left and right sides of*
* eq. (14.8)
respectively.
JMB, DCJ, WSW  67  23 Feb 1995
Unstable cohomology operations
15 Complex orientation and unstable operations
In this section, we extend our study of the test space C P 1 to all unstabl*
*e op
erations. Everything we did in x14 carries over, with a lot more complication *
*but
no essential difficulty. Again, it is enough to know r(x) for all operations r*
*, where
x = x() 2 E2(C P 1) is the Chern class.
We assume that E*(E_k) is a free E*module for all k.
Definition 15.1 We define elements bi2 E2i(E_2) for i 0 by the identity
1X
r(x) = xi= in E*(C P 1) = E*[[x]] (15:2)
i=0
P i
for all r, where we take xi inside the < ; > and write formally b(x) = ibix .
We first determine how the elements bk interact with the Hopf ring structur*
*e.
Proposition 15.3 The elements bk 2 E2k(E_2) of the Hopf ring E*(E_*) have the
properties:
_
(a) b0 = 12 and b1 = e2 = eO2, so that b(x) = 12 + b(x) if we define
_ 1X i
b(x) = bix in E*(E_2)[[x]]; (15:4)
i=1
(b) The universal Chern class x 2 E2(C P 1), regarded as a map x: CP 1 ! E_2,
induces x*fik = bk 2 E2k(E_2), where fik 2 E2k(C P 1) is dual to xk (as in [*
*8,
Lemma 5.4]);
P
(c) bk = i+j=kbi bj, or formally, b(x) = b(x) b(x);
(d) fflbk = 0 if k > 0, and fflb0 = 1, or formally, fflb(x) = 1;
_ _ _ _
(e) Ob(x) = (12 + b(x))*(1)= 12  b(x) + b(x)*2 b(x)*3+ : :;:
(f) For all rational numbers 2 E0,
_ * _ (1) _ *2
b(x) O[] = (12 + b(x)) = 12 + b(x) + _______b(x) + : :;: (15:5)
2
(g) For all r, r*bk is given as the coefficient of xk in the formal identity
1
r*b(x) = [] * * b(x)OjO[] in E*(E_*)[[x]];
j=1
(h) q2bk = bk 2 Q(E)22k, the additively unstable element in Defn. 14.1;
(i) oe2*bk = bk 2 E2k2(E; o), the stable element in [8, Defn. 13.1].
Remark The sign in (a) is absent from [23, Prop. 2.4]. The commutativity of
diag. (6.16) requires
Q(OE)(q1q1)(ee) = (q1e)(q1e) = q2b(0)= q2e2 = q2(eOe);
bearing in mind that deg(q1) = 1. The unexpected sign first appeared in Prop. 1*
*3.7(c).
JMB, DCJ, WSW  68  23 Feb 1995
x15. Complex orientation and unstable operations
Proof Naturality for the inclusion S2 ~=C P 1 CP 1 gives (a), by comparison wi*
*th
Prop. 13.7. Part (b) comes from Prop. 10.5. We read off (c) and (d) from eqs. (*
*10.14)
and (10.15). Part (e) is the special case = 1 of (f). For (f), eq. (10.11) an*
*d (c)
give b(x) O[+] = b(x)O[] * b(x)O[] for all ; 2 E*. Since b(x) O[1] = b(x) and *
*we
are working in the *multiplicative group of formal power series over E*(E_2) o*
*f the
form 1 + : :,:which has no ntorsion if 1=n 2 E*, the result follows. (We are i*
*n effect
expanding r(x); cf. eq. (10.16).) For (g), we apply eq. (10.45) to x 2 E2(C P 1*
*) and
expand. For (h) and (i), we stabilize eq. (15.2) by Prop. 6.11 and compare with*
* the
additive and stable versions, eq. (14.3) and [8, eq. (13.3)]. 
From (c) and eq. (10.11), we deduce the convenient distributive law
(a * c) Ob(x) = aOb(x) * cOb(x); (15:6)
where a and c are allowed to involve x. This formal device will prove extremely*
* useful
for computations in Hopf rings. We have one immediate application to the Froben*
*ius
operator F defined by F c = c*p.
Corollary 15.7 For any element c in the Hopf ring E*(E_*),
ae
F (c Obn) mod p;if k = pn;
(F c) Obk
0 mod p; if k is not divisible by p.
Proof By iterating eq. (15.6) we have (F c) Ob(x) = F (c Ob(x)). We pick out *
*the
coefficient of xk, working mod p. 
We next study the naturality of operations with respect to the multiplicati*
*on
: CP 1 x CP 1 ! C P 1. We expand *r(x) = r(*x) by the formal group law
[8, eq. (13.5)] and the Cartan formulae, to obtain the analogue of eq. (14.5). *
* The
complicated result is best expressed formally as
i j
b(F (x; y)) = FR(b(x); b(y)) = b(x) * b(y) * * b(x)OiOb(y)OjO[ai;j](15:*
*8)
i;j
as in [23, Thm. 3.8(i)], where FR(X; Y ) = X * Y * *i;jXOiOY OjO[ai;j], in the *
*sense that
the O and *multiplications apply only to Hopf ring elements, not to x and y.
The plocal case Lemma 14.6 carries over.
Lemma 15.9 Assume that E* is a plocal ring. Then the Ogenerator bk of the *
*Hopf
ring E*(E_*) is redundant unless k is a power of p.
Proof As before, we takeithejcoefficient of xiyj in eq. (15.8), where i + j = *
*k. On the
left, there is a term kibk, from bk(x+y)k, and this is the highest b that occu*
*rs; on the
right,inojb beyond bi or bj occurs. We choose i and j as in [8, Lemma 13.7], to*
* make
k
i not divisible by p and therefore invertible, which shows that bk is redunda*
*nt. 
We therefore reindex the b's as usual.
Definition 15.10 When E* is a plocal ring, we define b(i)= bpifor each i 0.
JMB, DCJ, WSW  69  23 Feb 1995
Unstable cohomology operations
We extend standard multiindex notation slightly by defining
bOI= bOi0(0)ObOi1(1)ObOi2(2)ObOi3(3)O: : : (15:11)
for any multiindex I = (i0; i1; i2; : :):. We also need a shift operation.
Definition 15.12 Given a multiindex I = (i0; i1; i2; : :):, we define the *
*shifted
multiindex s(I) = (0; i0; i1; i2; : :):. We iterate this process h times, for*
* any h
0, to form sh(I) = (0; : :;:0; i0; i1; i2; : :):. We even undo it, by defining*
* s1(I) =
(i1; i2; i3; : :):, provided i0 = 0; our convention is that this is undefined i*
*f i0 6= 0.
This notation allows us to iterate Cor. 15.7 neatly in the form
( O
s1(I)) mod pif i = 0;
(F c) ObOI F (c Ob 0 (15:13)
0 mod p if i0 6= 0.
We follow the stable plan and study instead of the much simpler pth power
map i: CP 1 ! CP 1. Naturality of the general operation r is expressed by i*r(x*
*) =
r(i*x). When we substitute the pseries [8, eq. (13.9)] and expand, we obtain, *
*as in
[23, Thm. 3.8(ii)],
X
b px + gixi+1 = b(x)*p* * b(x)Oi+1O[gi] (15:14)
i i
in E*(E_*)[[x]], or, in condensed notation, b([p](x)) = [p]R(b(x)).
Definition 15.15 For each k > 0, we define the kth main unstable relation as
(Rk) : L(k) = R(k) in E*(E_2), (15:16)
k
where L(k) and R(k) denote respectively the coefficient of xp in the left and *
*right
sides of eq. (15.14).
k
Thus L(k) is the coefficient of xp in b([p](x)), exactly as in Defn. 14.9.*
* However,
R(k) is vastly more complicated than before, and we study it in more detail in *
*x19 in
the case E = BP . The work of RavenelWilson [23], which we review in x17, impl*
*ies
that, despite appearances, the relations (Rn) contain all the information prese*
*nt in
eq. (15.8), with the understanding that we use eq. (15.8), by way of Lemma 15.9*
*, only
to express the redundant bj's (which still appear in b(k), b(k)O[], and r*b(k)*
*) in terms
of the b(i).
16 Examples for additive operations
In x5, we developed a comonad to express all the structure of additive unst*
*able
Ecohomology operations, for favorable E. In x6, we developed a bigraded algeb*
*ra
Q(E)**that contains equivalent information, where Q(E)kihas degree k  i. In th*
*is
section, we describe Q(E)**for each of our five cohomology theories E*(), name*
*ly
E = H(F p), MU, BP , KU, and K(n). (The first example splits into two, and we
break out the degenerate special case K(0) = H(Q ).) As stably in [8], our purp*
*ose
is to exhibit the structure of the results, not to derive them.
JMB, DCJ, WSW  70  23 Feb 1995
x16. Examples for additive operations
All the results here are formally very close to the stable results. By Prop*
*. 12.3(g),
Q(oe)e = 1. As E*(E; o) = colimkQ(E)k*by eq. (4.8), where the suspensions Q(E)k*
**!
Q(E)k+1*have been revealed in Lemma 12.5 as simply multiplication by e, we stab*
*ilize
everything merely by setting the suspension element e = 1. In this way, we reco*
*ver
all the corresponding stable results. Indeed, in the case E = KU, we have to ob*
*tain
the stable structure this way.
All four answers of x5 are of course available, but the Second Answer remai*
*ns the
most practical, consisting as in Thm. 7.7 of the coactions
aeX : Ek(X) ! E*(X) b Q(E)k*:
These coactions are automatically additive, multiplicative (for cup products an*
*d x
products), and unital (aeX 1X = 1X 1). (We simplify notation by suppressing re*
*dun
dant completions and suffixes.)
We use exactly the same test spaces and test maps as we did stably. The poi*
*nt
remains that complete knowledge of the behavior of E*() on these is sufficient*
* to
suggest the correct structure of Q(E)**(except that the case E = K(n) requires *
*some
extra work). By Prop. 6.42(b), the onepoint space T in effect defines the righ*
*t unit
jR, and the circle S1 defines e 2 Q(E)11by eq. (12.2). As all our examples have
complex orientation, we have available the elements bi of Defn. 14.1.
In each case, we list the generators and relations for the bigraded E*alge*
*bra
Q(E)**, describe the right unit jR, and give the values of the algebra homomorp*
*hisms
= Q( ): Q(E)**! Q(E)** Q(E)**and ffl = Q(ffl): Q(E)**! E* on each generator.
In some cases, we can express the universal property of Q(E)**very simply. The
stabilization Q(oe) maps each generator to its stable namesake, except that of *
*course
Q(oe)e = 1.
Example: H(F 2) We take E = H = H(F 2), the EilenbergMacLane spectrum.
Our test space is R P 1, for which H*(R P 1) = F2[t], a polynomial algebra on t*
*he
generator t 2 H1(R P 1). We define elements ci2 Q(H)1*by the identity
1X
aet = ti ci in H*(R P 1) b Q(H)1*~=Q(H)1*[[t]].
i=0
Restriction to S1 = RP 1shows that c0 = 0 and c1 = e. As stably, the multiplica*
*tion
: RP 1 x RP 1 ! R P 1 implies that ci = 0 unless i is a power of 2. We therefore
write i= c2i2 Q(H)12ifor each i 0, so that
1X i
aet = t2 i in H*(R P 1) b Q(H)1*~=Q(H)1*[[t]]; (16:1)
i=0
which looks just like the stable version [8, eq. (14.1)], except that now 0 = e.
Theorem 16.2 For the EilenbergMacLane ring spectrum H = H(F 2):
(a) Q(H)**= F2[0; 1; 2; 3; : :]:, a polynomial algebra over F2 on generators*
* i 2
Q(H)12ifor i 0, where 0 = e;
(b) In the complex orientation for H(F 2), b(i)= 2ifor all i 0, and bj = 0 *
*if j is
not a power of 2;
JMB, DCJ, WSW  71  23 Feb 1995
Unstable cohomology operations
(c) is given by
Xn i
n = 2ni i in Q(H)** Q(H)1*;
i=0
(d) ffl is given by ffln = 0 for n > 0 and ffl0 = 1.
Proof Part (a) is of course a reformulation of classical results. For fixed k*
*, the
stabilization Q(oe): Q(H)k*! H*(H; o) is the monomorphism that is dual (with a
shift in degree) to the wellknown epimorphism oe*k: H*(H; o) ! P H*(H_k) that *
*tells
which Steenrod operations can act nontrivially on Hk(). The proof of (b) is t*
*he
same as stably. We prove (c) and (d) by taking M = H*(R P 1) in diags. (6.33) a*
*nd
evaluating on t. 
As stably in [8, x14], we combine the universal property of the polynomial *
*algebra
F 2[0; 1; 2; : :]:with Thm. 7.7(b).
Corollary 16.3 Let B be a discrete commutative graded F2algebra. Assume that
the ring homomorphism : H*(X) ! H*(X) b B is natural for spaces X. Then on
t 2 H1(R P 1), has the form
1X i
t = t2 0i in H*(R P 1) b B ~=B[[t]],
i=0
i1)
where the elements 0i2 B(2 determine uniquely for all X and may be chosen
arbitrarily. 
Example: H(F p) (for p odd) We write H = H(F p), the EilenbergMacLane spec
trum. The complex orientation defines elements i= b(i)for i 0, and, just as st*
*ably,
bj = 0 whenever j is not a power of p. The only difference now is that 0 = b1 =*
* e2
instead of 1.
The other test space is the lens space L = K(F p; 1), for which H*(L) = Fp[*
*x]
(u). As x is a Chern class, aeLx is given by eq. (14.2). This leaves only aeLu,*
* which
reduces (as stably) to
1X i
aeLu = u e + xp oi in H*(L) b Q(H)1*, (16:4)
i=0
for certain elements oi that it defines.
Theorem 16.5 For the EilenbergMacLane ring spectrum H = H(F p), with p odd:
(a) Q(H)**is the commutative algebra over Fp with generators:
e 2 Q(H)11, a polynomial generator;
i2 Q(H)22pifor all i 0, a polynomial generator for i > 0;
oi2 Q(H)12pifor all i 0, an exterior generator;
JMB, DCJ, WSW  72  23 Feb 1995
x16. Examples for additive operations
subject to the relation 0 = e2;
(b) is given by e = e e,
Xk i
k = pki i in Q(H)** Q(H)2*, (16:6)
i=0
and
Xk i
ok = ok e + pki oi in Q(H)**bQ(H)1*;
i=0
(c) ffl is given by ffle = 1, ffli= 0 for i > 0, and ffloi= 0 for all i.
Proof Part (a) is again a reformulation of classical results, which may be rec*
*overed
in this form from [27, Thm. 8.5], in somewhat different notation, by taking the*
* inde
composables. We obtain (b) and (c) by substituting aeL in diags. (6.33) and eva*
*luating
on x and u. 
We have the analogue of Cor. 16.3.
Corollary 16.7 Let B be a discrete commutative graded Fpalgebra. Assume that
the ring homomorphism : H*(X) ! H*(X) b B is natural for spaces X. Then on
H*(L) = Fp[x] (u), has the form
1X i
x = xe02+ xp 0i
i=1
1X i
u = ue0+ xp o0i
i=0
i1) 0 (2pi1)
where the elements e02 B0, 0i2 B2(p , and oi 2 B determine uniquely
for all X and may be chosen arbitrarily. 
Example: H(Q ) We write E = H = H(Q ), the EilenbergMacLane spectrum. As
always, there is the suspension element e 2 Q(H(Q ))11, whose properties we know
from Prop. 12.3. There is nothing else.
Theorem 16.8 For the ring spectrum H = H(Q ):
(a) Q(H)**= Q[e], a polynomial algebra on e 2 Q(H)11;
(b) The coalgebra structure is given by e = e e and ffle = 1. 
Example: MU The coefficient ring is MU* = Z[x1; x2; x3; : :]:, with a pol*
*y
nomial generator xi in degree 2i for each i. These give rise to the elements
jRxi2 Q(MU)2i0. We have complex orientation, almost by definition, and therefo*
*re
the elements bi 2 Q(MU)22i, with b0 = 0 and b1 = e2. We have the relations (14.*
*5)
between the b's and the jRv, but unlike the stable case, because e is no longer*
* invert
ible, they do not render the generators jRxi redundant. Implicit in [23, Cor. 4*
*.6(a)]
is that this is the whole story.
Theorem 16.9 (RavenelWilson) For the unitary Thom ring spectrum MU:
(a) Q(MU)**is the commutative algebra over MU* with generators:
JMB, DCJ, WSW  73  23 Feb 1995
Unstable cohomology operations
jRxi2 Q(MU)2i0(for i > 0);
e 2 Q(MU)11;
bi2 Q(MU)22i(for i 1);
all of even degree, subject to the relations (14.5) and b1 = e2;
(b) is given by e = e e and
Xk
bk = B(i; k) bi in Q(MU)** Q(MU)2*,
i=1
where B(i; k) denotes the coefficient of xk in b(x)i;
(c) ffl is given by ffle = 1 and fflbk = 0 for k > 1. 
Although we no longer have a polynomial algebra, part of Cor. 16.3 carries *
*over.
It applies equally well to the two following cases, which we include here.
Corollary 16.10 Let B be a discrete commutative E*algebra, where E = MU,
BP , or KU. Then a ring homomorphism : E*(X) ! E*(X) b B that is natural for
spaces X is uniquely determined by its values on E*(S1) and E*(C P 1). 
Example: BP The coefficient ring is now BP *= Z(p)[v1; v2; v3; :::], with pol*
*ynomial
generators vn in degree 2(pn1). We have complex orientation, but because BP *
is plocal, we need only the generators b(i)2 Q(BP )22pi, where b(0)= e2. Again*
*, [23,
Cor. 4.6(b)] implies that this is all there is; in particular, eq. (14.5) is re*
*dundant,
except to express the other bj in terms of the b(i)and the elements vi and wi= *
*jRvi.
Theorem 16.11 (RavenelWilson) For the BrownPeterson ring spectrum BP :
(a) Q(BP )**is the commutative algebra over BP *with generators:
i1)
wi= jRvi2 Q(BP )2(p0 (for i > 0);
e 2 Q(BP )11;
b(i)2 Q(BP )22pi(for i 0);
subject to the main relations (Rk) (from eq. (14.10)) for k > 0 and b(0)= e2;
(b) is given by e = e e and
pkX
b(k)= B(i; pk) bi in Q(BP )** Q(BP )2*,
i=1
k i
where B(i; pk) denotes the coefficient of xp in b(x) ;
(c) ffl is given by ffle = 1 and fflb(k)= 0 (for k > 0). 
We discuss the structure of Q(BP )**in more detail in x18.
Remark Alternatively, we could use the generator hiinstead of b(i)as in [6]; h*
*owever,
Quillen's element ti (see [21] or Adams [1, II.16]) does not exist in this cont*
*ext for
i > 1, for lack of conjugation in Q(BP )**.
JMB, DCJ, WSW  74  23 Feb 1995
x16. Examples for additive operations
Example: KU We take E = KU, the complex Bott spectrum, with the coefficient
ring KU* = Z[u; u1] (where u 2 KU2), right unit jR: KU* ! Q(KU)**given by
jRu = v, and Chern class x given by [8, eq. (5.2)]. The simple form [8, eq. (5.*
*16)] of
the formal group law reduces eq. (14.5) to
b(x + y + uxy) = b(x) + b(y) + b(x)b(y)v; (16:12)
which looks like the stable version [8, eq. (14.13)], with b(x) = b1x + b2x2+ b*
*3x3+ : :,:
except that now b1 = e2 6= 1. The coefficient of xiyj yields the relation
min(i;j)Xi+jk i
bibj = ukbi+jkv1; (16:13)
k=0 i k
like [8, eq. (14.15)], except that the case i = 1 now gives the reduction formu*
*la
b1bi= (i+1)bi+1v1 + iubiv1 for i > 0. (16:14)
The results here are much clearer than in the stable case, and there is some ov*
*erlap
with the work of tom Dieck [10].
Theorem 16.15 For the complex Bott spectrum KU:
(a) Q(KU)**is generated as an algebra over KU* = Z[u; u1] by the elements:
v = jRu 2 Q(KU)20;
v1 = jRu1 2 Q(KU)20;
e 2 Q(KU)11, the suspension element;
bi2 Q(KU)22ifor i > 0;
subject to the relations b1 = e2 and (16.13) for i > 0, j > 0;
(b) Q(KU)**is a free KU*module, with a basis consisting of all monomials of*
* the
forms vn, bivn, evn, and ebivn, for i > 0 and n 2 Z;
(c) is given by e = e e and
Xk
bk = B(i; k) bi in Q(KU)** Q(KU)2*,
i=1
where B(i; k) denotes the coefficient of xk in b(x)i;
(d) ffl is given by ffle = 1 and fflbk = 0 for all k > 1.
Proof We start with (b). We take the Hopf line bundle over CP 1 and regard the
element u1[] 2 KU2(C P 1) as a map f: CP 1 ! KU__2= Z x BU. By Lemma 4.6,
f induces an isomorphism of KU*modules
KU*(C P 1) ! QKU*(Z xBU) ~=KU* QKU*(BU);
which we compute. By the definition [8, eq. (5.2)] of the Chern class x, u1[] *
*= u1+x
in KU2(C P 1); geometrically, the components of f are the map CP 1 ! Z with ima*
*ge
1, and x: CP 1 ! BU.
Thus q2f*fi0 = v1 and q2f*fii = q2x*fii = bi for i > 0, with the help*
* of
Prop. 14.4(b); we have the desired basis of Q(KU)2*. For Q(KU)2n*, we multiply
by vn+1, an isomorphism.
JMB, DCJ, WSW  75  23 Feb 1995
Unstable cohomology operations
For the odd case, the description of KU*(U) in [8, Cor. 5.12] in terms of t*
*he
Bott map b: (Z xBU) ! U shows that multiplication by e induces an isomorphism
Q(KU)2n*~=Q(KU)2n+1*.
We have specified enough relations to reduce any monomial in the b's, e, v,*
* and
v1 to a linear combination of the elements in (b), which proves (a). Parts (c)*
* and
(d) are included in Props. 14.4 and 12.3. 
Now that we know the additive situation, we return to finish off the stable*
* case.
We may discard the odd spaces in eq. (4.8) and write
KU*(KU; o) = colimnQ(KU)2n*:
Corollary 16.16 In the stable algebra KU*(KU; o):
(a) Every element of KU*(KU; o) of even degree can be written in the form
c = uq(1u1 + 2u2b2 + : :+:nunbn)vm
for some integers q, m, n, and i;
(b) This element c = 0 if and only if i= 0 for all i.
Proof By Thm. 16.15(b), we can write the general element of Q(KU)2m+22quniquely
in the form
Xn
c = uq 0v1 + iuibi vm
i=1
with integer coefficients. Since e2 = b1, eq. (16.14) yields
Xn Xn
e2c = uq+1 0u1b1 + (i + 1)iui1bi+1+ iiuibi vm1
i=1 i=1
in Q(KU)2m+42q+2, which gives (a). Further, e2c = 0 only if c = 0, which implie*
*s (b). 
Example: K(n) The coefficient ring is K(n)* = Fp[vn; v1n], where vn*
* 2
n1)
K(n)2(p . We write wn = jRvn, as we did for BP . Obviously, wn and vn are
no longer equal as they were stably, because they lie in different groups.
We have a complex orientation, and therefore the usual elements bj. Because
K(n)* is plocal, we need only the b(i)for i 0. (In fact, bj = 0 if j is not a*
* power
of p and j < pn, for dimensional reasons, but not in general if j > pn.) When *
*we
n
apply ae to the pth powernmap i: CP 1 ! CP 1 , which induces i*x = vnxp as in*
* [8,
eq. (14.26)], we obtain bpjwn = vjnbj, and therefore
n pi 1 2pn
bp(i)= vn b(i)wn in Q(K(n))* (16:17)
for i 0. This stabilizesnto [8, eq. (14.27)].
In particular, bp(0)= vnb(0)w1n. As always, b(0)= e2. A more sophisticat*
*ed
analysis, involving other cohomology theories as in [28, Prop. 1.1(j)], shows t*
*hat this
relation can be desuspended once to give
n1 1 2pn1
ebp(0) = vnewn in Q(K(n))* . (16:18)
JMB, DCJ, WSW  76  23 Feb 1995
x16. Examples for additive operations
n
The other test space is the skeleton Y = L2p 1 of the lens space L, for w*
*hich
n
K(n)*(Y ) = (u) K(n)*[x: xp = 0]. We know aeY x, because x is inherited from
C P 1. As stably, we define elements ai; ci2 Q(K(n))1*by the coaction
pn1X pn1X
aeY u = xi ai+ uxi ci : (16:19)
i=0 i=0
By restriction to S1 Y , we see that a0 = 0 and c0 = e. Then eq. (16.18) is eq*
*uivalent
n1 *
to the statement aeY y = y e, where y = vnuxp 2 K(n) (Y ); in other words, y
behaves like u1 2 K(n)1(S1). The same partial multiplications : L2k+1x L2m ! Y
as in [8, x14] show that ci= 0 for all i > 0 and that ai= 0 for i not a power o*
*f p. We
therefore reindex, as usual.
Definition 16.20 We define a(i)= api2 Q(K)12pi, for 0 i < n.
In the new notation,
n1X i
aeY u = ue + xp a(i) in K(n)*(Y ) Q(K(n))1*. (16:21)
i=0
Having odd degree, the a(i)are exterior generators of Q(K(n))**. This is not al*
*l; we
again appeal to [28, Prop. 1.1(i)] to find that one more factor e can be squeez*
*ed out
of eq. (16.18) if we first multiply by a(0), to give the relation
n1 1 2pn1
a(0)bp(0) = vna(0)wn in Q(K(n))* . (16:22)
Theorem 16.23 For the Morava Ktheory ring spectrum K(n):
(a) Q(K(n))**is the commutative bigraded algebra over K(n)* = Fp[vn; v1n], *
*where
n1)
vn 2 K(n)2(p , with generators:
n1)
wn = jRvn 2 Q(K(n))2(p0 ;
w1n= jRv1n;
e 2 Q(K(n))11;
a(i)2 Q(K(n))12pi(for 0 i < n);
b(i)2 Q(K(n))22pi(for i 0);
subject to the relations b(0)= e2, (16.17), (16.18), and (16.22);
(b) is given by e = e e,
Xk i
a(k)= a(k)e + bp(ki)a(i) in Q(K(n))** Q(K(n))1*, (16:24)
i=0
and
pkX
b(k)= B(i; pk) bi in Q(K(n))** Q(K(n))2*, (16:25)
i=1
k i
where B(i; pk) denotes the coefficient of xp in b(x) (and Lemma 14.6 is used *
*to
express b(x) in terms of the b(j), vn, and wn);
(c) ffl is given by ffle = 1, ffla(k)= 0 (for k 0), and fflb(k)= 0 (for k >*
* 0).
JMB, DCJ, WSW  77  23 Feb 1995
Unstable cohomology operations
Proof The algebra structure (a) is implicit in the main theorem of [28], by ta*
*king
indecomposables. As always, we obtain a(i)and ffla(i)by evaluating the coacti*
*on
axioms (6.33) on u 2 K(n)*(Y ). The rest of (b) and (c) can be obtained similar*
*ly, or
by appealing to Props. 12.3 and 14.4. 
Corollary 16.26 Let B be a discrete commutative K(n)*algebra. Then a ring
homomorphism : K(n)*(X) ! K(n)*(X) b B that is natural for spaces X is uniquely
determined by its values on K(n)*(C P 1) and K(n)*(Y ). 
Remark If k n, eq. (16.25) simplifies just as in [8, Thm. 14.32] to
Xk i
b(k)= bp(ki) b(i) in Q(K(n))** Q(K(n))2*,
i=1
which resembles eq. (16.6).
17 Examples for unstable operations
In this section, we discuss the enriched Hopf ring for each of our five coh*
*omology
theories E*(), namely for E = H(F p), MU, BP , KU, and K(n). According to x10,
this is what we need to handle general unstable operations. As in x16, we divid*
*e the
case H(F p) in two and treat K(0) = H(Q ) separately. Even more than before, our
intent is to exhibit the structure of the results, not to reestablish them.
Our strategy is the same as in the stable and additive contexts, using exac*
*tly the
same test spaces and test maps. Each E has a complex orientation, which provides
by Defn. 15.1 the elements bi of the Hopf ring, in addition to e and the [v]. W*
*e have
O[1] = [1] by Prop. 11.2(d), and its properties were listed in Prop. 10.12.
As pointed out in (10.46), we need more than just the Hopf ring and the ele*
*ments
[v]. The elements Q(ffl)qkc = fflSoek*c are given by x16. We also need r*c fo*
*r each
operation r; by Thms. 10.19(c) and 10.33(c), it is in principle enough to know *
*these
for each Ogenerator c.
Our presentation changes somewhat from x16. Each family of Ogenerators has*
* its
own Proposition, which lists all the pertinent information. It is therefore suf*
*ficient to
describe each Hopf ring by listing its Ogenerators and the defining relations,*
* and to
refer to these propositions for further details. We recover all the results for*
* additive
operations merely by taking the indecomposables.
Example: MU We recall that MU* = Z [x1; x2; x3; : :]:, where deg(xi) = 2i, *
*is
better described as generated by the elements ai;j, as in [8, x14]. We have the*
* elements
bi, as well as e and [v] = jR(v). Stably, [8, eq. (13.6)] gave an inductive for*
*mula for
jRai;jin terms of MU* and the bi. Unstably, eq. (15.8) is only a relation betw*
*een
these elements. Corollary 4.6(a) of [23] says in effect that this is all there *
*is.
Theorem 17.1 (RavenelWilson) For the unitary cobordism ring spectrum MU,
MU*(MU__*) is the Hopf ring over MU* = Z[x1; x2; x3; :::] with Ogenerators:
[xi] 2 MU0(MU__2i) for each i > 0 (see Prop. 11.2);
e 2 MU1(MU__1) (see Prop. 13.7);
JMB, DCJ, WSW  78  23 Feb 1995
x17. Examples for unstable operations
bi2 MU2i(MU__2) for i 1 (see Prop. 15.3);
subject to the relations eO2= b1 and eq. (15.8). 
Example: BP The main reference is still [23]. As BP *is plocal, Lemma 15.9 a*
*nd
Defn. 15.10 apply, to define the elements b(i)of the Hopf ring. We have as alwa*
*ys e
and the elements [v] for each v 2 BP *.
Theorem 17.2 (RavenelWilson) For the BrownPeterson ring spectrum BP ,
BP*(BP__*) is the Hopf ring over BP *= Z(p)[v1; v2; v3; : :]:with the Ogenerat*
*ors:
[] 2 BP0(BP__0), for each 2 Z(p)(see Prop. 11.2);
[vi] 2 BP0(BP__2(pn1)), for i > 0 (see Prop. 11.2);
e 2 BP1(BP__1) (see Prop. 13.7);
b(i)2 BP2pi(BP__2) for i 0 (see Prop. 15.3);
subject to the relations [] O[0] = [0], [] * [0] = [ + 0], e O[] = e, b(i)O[] =*
* : : :
(see Prop. 15.3(f)), eO2= b(0), and the main relations (Rn) for n > 0 as in eq*
*. (15.16).
We implicitly use eq. (15.8), but only to express inductively the bj, for j*
* not a
power of p, in terms of the b(i), v, and [v]; this is needed for computing b(i*
*), Ob(i),
b(i)O[], and r*b(i).
Proof This is the content of [23, Cor. 4.6(b)]. By Prop. 11.2, each [v] for v *
*2 BP *
can be expressed in terms of the [] and [vi]; we have enough generators. The li*
*sted
relations come from Props. 11.2, 13.7, and 15.3, and eq. (15.16). This reduce*
*s the
*generators (see x10) to three types:
(i)bOIO[vJ];
(ii)e ObOIO[vJ]; (17.3)
(iii)[vJ];
in terms of the multiindex notation bOIintroduced in eq. (15.11).
For each k, the *generators that lie in BP*(BP__k) generate it as a BP *a*
*lgebra.
Assume first that k is even, so that we have only types (i) and (iii). We write
BP__k= BP kx BP__0kas in Lemma 4.17; then
BP*(BP__k) ~=BP*(BP k) BP*(BP__0k); (17:4)
where we recognize the first factor as the group ring over BP *of the abelian g*
*roup
BP kwith basis elements [v] for v 2 BP k. The type (i) generators lie in BP*(BP*
*__0k) and
the type (iii) in BP*(BP k), which is described by Lemma 4.4. Because [vJ]*[0vJ*
*] =
[(+0)vJ], we have enough relations for the type (iii) generators. The work of [*
*23]
reduces the type (i) generators to certain allowable generators bOIO[vJ], which*
* form a
system of polynomial generators BP*(BP__0k). Since this reduction (see x19) use*
*s only
the relations (Rn), we have enough relations.
If k is odd, only generators of type (ii) occur. These reduce similarly to*
* the
allowable generators of type (ii), which are exterior generators of BP*(BP__k).*
* 
Example: H(Q ) This example is of course classical.
JMB, DCJ, WSW  79  23 Feb 1995
Unstable cohomology operations
Theorem 17.5 For the ring spectrum H = H(Q ), H*(H_*) is the Hopf ring over Q
with generators:
[] 2 H0(H_0) for each 2 Q (see Prop. 11.2);
e 2 H1(H_1) (see Prop. 13.7);
subject to the relations [] O[0] = [0], [] * [0] = [+0], and e O[] = e.
Proof For k < 0, H_k = T , and we have only the Q basis element 1k.
For k = 0, H_0 = Q, regarded as a discrete group, and the grou*
*p ring
H*(H_0) = Q [Q ] has a basis consisting of the elements []. The first two rela*
*tions,
from Prop. 11.2, show how these multiply.
For k > 0, the third relation, from Prop. 13.7(g), reduces us to the single*
* *
generator eOk 2 Hk(H_k) of H*(H_k). We have the polynomial algebra Q [eOk] if *
*k is
even, or the exterior algebra (eOk) if k is odd. 
Example: H(F 2) We write H = H(F 2). As H*(H_*) is a Hopf ring over F2, we
have the Frobenius operator F and the Verschiebung V .
We imitate Defns. 15.1 and 15.10 in a mod 2 version, using the same test sp*
*ace
R P 1 = K(F 2; 1) as before, for which H*(R P 1) = F2[t]. We define ci 2 Hi(H_1*
*) =
Hi(R P 1) for i 0 by the identity
1X
r(t) = ti= in H*(R P 1) (for all r);(17:6)
i=0
P i
where we write formally c(t) = icit as in Defn. 15.1. In other words, ci is *
*dual to
ti and the elements ci form an F2basis of H*(H_1).
We are primarily interested in the accelerated elements c(i)= c2i. As befor*
*e, we
have the suspension element e. The complex orientation provides elements bi whi*
*ch
are redundant, as in x16.
Proposition 17.7 The Hopf ring elements ci2 Hi(H_1) (for i 0) and c(i)= c2i2
H2i(H_1) (for i 0) have the following properties:
(a) c0 = 11 and c(0)= c1 = e;
P
(b) ck = i+j=kci cj, or formally, c(t) = c(t) c(t);
(c) V c(i)= c(i1)for i > 0, and V c(0)= 0;
(d) fflck = 0 if k > 0, and fflc0 = 1, or formally, fflc(t) = 1;
(e) Oc(t) = c(t)*(1), expanded as in Prop. 15.3(e);
i j
(f) ci* cj = i+jici+j;
(g) F c(i)= c(i)* c(i)= 0;
(h) bi= ciOci in H2i(H_2);
(i) For all r, r*ck is the coefficient of tk in the formal identity
1
r*c(t) = * c(t)OjO[] in H*(H_*)[[t]];
j=0
(j) q1c(i)= i in Q(H)1*, and q1cj = 0 if j is not a power of 2;
(k) oe1*c(i)= i in H*(H; o), and oe1*cj = 0 if j is not a power of 2.
JMB, DCJ, WSW  80  23 Feb 1995
x17. Examples for unstable operations
Proof The naturality of r for the multiplication : RP 1 x RP 1 ! R P 1, which
induces *t = tx1 + 1xt, yields the identity
X X X
(tx1 + 1xt)k = tixtj
k i j
in H*(R P 1xR P 1) = F2[tx1; 1xt], with the help of the Cartan formula (10.23).*
* The
coefficient of tix tj gives (f). The special case (g) of (f) also follows from *
*eq. (10.32).
We expand r(t2) for the Chern class t2 by eqs. (17.6) and (10.36) and compare w*
*ith
eq. (15.2); most terms cancel, to give (h).
The other parts are formally as in Prop. 15.3, with all degrees halved, exc*
*ept that
(c) is immediate from (b). 
Just as in Lemma 15.9, except that everything is now explicit in (f), cj is*
* redundant
unless j is a power of 2. This leads to the following elegant description of th*
*e Hopf
ring, which is a reformulation of classical results.
Theorem 17.8 For the EilenbergMacLane ring spectrum H = H(F 2), H*(H_*) is
the Hopf ring over F2 with generators c(i)2 H2i(H_1) for i 0 (see Prop. 17.7),
subject to the relation [1]*2= 10.
Proof By Prop. 17.7(c), we can write cOI = V cOs(I)for any multiindex I =
(i0; i1; i2; : :):. Then F cOI = F ([1]OcOI) = F [1] OcOs(I)= 0 by eq. (10*
*.13), as in
eq.P(15.13), and H*(H_k) is an exterior algebra on those generators cOI for whi*
*ch
OI i1;i2;:::
tit = k. Here, c is dual to the primitive element Sq k in cohomology (*
*in
terms of the Milnor basis [18] of H*(H; o)). (The index i0 serves only as paddi*
*ng, to
ensure that i1 + i2 + i3 + : : :k.) 
Example: H(F p) (for p odd) We write H = H(F p). We have, as always, the
suspension element e. The complex orientation defines elements bi for all i 0;*
* but
Lemma 15.9 shows that only the b(i)= bpifor i 0 are needed. Also, b0 = 12 and
b1 = e2 =ieO2.j However, the bj for j not a power of p do not vanish, but sati*
*sfy
bi* bj = i+jibi+j, which is all that survives from eq. (15.8). In particular, *
*b*p(i)= 0 for
all i > 0, as is also clear from eq. (10.32) applied to x.
For the other test space L = K(F p; 1), we have H*(L) = Fp[x] (u). We only
need to know r(u). We define elements ai2 H2i(H_1) and ci2 H2i+1(H_1) by
1X 1X
r(u) = xi+ uxi in H*(L),
i=0 i=0
P i
which wePcondense formally to + u by writing a(x) = iaix *
* and
c(x) = icixi. Thus ai is dual to xi, ci is dual to uxi, and the ai and ci for*
*m a basis
of H*(H_1).
Again, we accelerate the indexing by defining a(i)= apifor i 0.
Proposition 17.9 The Hopf ring elements ai 2 H2i(H_1), a(i)= api2 H2pi(H_1),
and ci2 H2i+1(H_1), (for i 0), have the following properties:
(a) a0 = 11 and c0 = e;
JMB, DCJ, WSW  81  23 Feb 1995
Unstable cohomology operations
P
(b) ak = i+j=kai aj;
(c) V a(i)= a(i1)for i > 0, and V a(0)= 0;
(d) fflak = 0 for all k > 0;
(e) Oa(x) = a(x)*(1), expanded as in Prop. 15.3(e);
i j
(f) ai* aj = i+jiai+j;
(g) F a(i)= a*p(i)= 0;
(h) ci= e * ai;
(i) For all r, r*ak is the coefficient of xk in the formal identity
1 1
r*a(x) = * b(x)OiO[] * * a(x)Ob(x)OiO[] in H*(H_*)[[x*
*]];
i=0 i=0
(j) q1a(i)= oi in Q(H)1*, and q1aj = 0 if j is not a power of p;
(k) oe1*a(i)= oi in H*(H; o), and oe1*aj = 0 if j is not a power of p.
Proof We consider naturality of operations with respect to the multiplication *
*: L x
L ! L, for which *u = ux1 + 1xu. In condensed notation, we compare
*r(u) = + (ux1 + 1xu)
with r(*u), which we expand by eq. (10.23) as
r(*u) = + ux1
+ 1xu + uxu:
The coefficient of xix xj gives (f), which implies (g). (Alternatively, (g) fol*
*lows from
eq. (10.32) applied to u.) The coefficient of u x xi gives (h). The other parts*
* require
no new ideas. 
In particular, all the ci and most of the ai are redundant. We trivially ha*
*ve the
relation [1]*p = [p] = [0] = 1, from which it follows, as in the previous examp*
*le,
that (aOI)*p = 0 and (bOI)*p = 0 for all I. Once again, this is the whole stor*
*y. A
detailed exposition by Ravenel and Wilson from this point of view is presented *
*in [27,
Thm. 8.5] (with slightly different notation: a(i)is written ff(i), and b(i)is w*
*ritten fi(i)).
Theorem 17.10 (RavenelWilson) For the EilenbergMacLane ring spectrum H =
H(F p), H*(H_*) is the Hopf ring over Fp with the Ogenerators:
e 2 H1(H_1) (see Prop. 13.7);
a(i)2 H2pi(H_1), for i 0 (see Prop. 17.9);
b(i)2 H2pi(H_2), for i 0 (see Prop. 15.3);
subject to the relations [1]*p= 10 and eO2= b(0). 
Example: KU We recall that KU* = Z[u; u1]. The complex orientation defines
elements bi for i > 0. As before, these, along with elements [un] = [] O[un] an*
*d e,
are all we need.
In view of the formal group law F (x; y) = x+y+uxy, the relation (15.8) bec*
*omes
_ _ _ _ _
1 + b(x+y+uxy) = (1 + b(x)) * (1 + b(y)) * (1 + b(x)Ob(y)O[u]) (17:11)
JMB, DCJ, WSW  82  23 Feb 1995
x17. Examples for unstable operations
which is more complicated than the additive analogue (16.12), but still managea*
*ble.
Again, we take the coefficient of xiyj. The left side is the same as before. *
*On the
right, we may choose xsyt with s > 0 and t > 0 from the third factor, which for*
*ces
us to take xisfrom the first factor and yjt from the second; or we can take a*
*ll of
xiyj from the first two factors. The result, after some rearranging, is
min(i;j)Xi+jk i
biObj = ukbi+jkO[u1]
k=0 i k
i1Xj1X
 bisO[u1] * bjtO[u1] * bsObt
s=1t=1 (17:12)
i1X j1X
 bisO[u1] * bsObj  bjtO[u1] * biObt
s=1 t=1
 biO[u1] * bjO[u1]
This serves as an inductive reduction formula for biObj, for any i > 0 and j > *
*0. In
particular, the suspension formula becomes
b1O bj =(j+1)bj+1O[u1] + jubjO[u1]
j1X (17:13)
 bjkO[u1] * b1Obk  b1O[u1] * bjO[u1]
k=1
Theorem 17.14 For the complex Ktheory ring spectrum KU, KU*(KU__*) is the
Hopf ring over KU* = Z[u; u1] with the Ogenerators:
[u] 2 KU0(KU__2) (see Prop. 11.2);
[u1] 2 KU0(KU__2) (see Prop. 11.2);
e 2 KU1(KU__1) (see Prop. 13.7);
bi2 KU2i(KU__2) for i > 0 (see Prop. 15.3);
subject to the relations [u] O[u1] = [1], Oe = e, Obi = : : :(see Prop. 15.3(*
*e)),
eO2= b1, and eq. (17.12).
Explicitly, for the even spaces we have
M
KU*(KU__2n) = [mun] * KU*[b1O[un+1]; b2O[un+1]; b3O[un+1]; : :]:;
m2Z
a direct sum (over m) of polynomial algebras, and for the odd spaces
KU*(KU__2n+1) = (eO[un]; eOb1O[un+1]; eOb2O[un+1]; : :):;
an exterior algebra over KU* (where we use [mun] = [m] O[un], [un] = [u]On, [*
*un] =
[u1]On, [u0] = [1], [n] = [1]*n, and [n] = [1]*n = (O[1])*n).
Proof We computed KU*(BU) in [8, Lemma 5.6]. By Prop. 15.3(b), the Chern
class x: CP 1 ! KU__2 induces x*fii = bi, so that we may write KU*(0 x BU) =
KU*[b1; b2; : :]:. For the copy KU*(m x BU), we *multiply this by [m]. This gi*
*ves
KU*(KU__2). For other even spaces, we apply the *isomorphism  O[un].
JMB, DCJ, WSW  83  23 Feb 1995
Unstable cohomology operations
For the odd spaces, we quote [8, Cor. 5.12].
To see that we have specified enough relations, we note that every *genera*
*tor
reduces to e O[un] or e ObiO[un] on the odd spaces, or biO[un] or [] O[un] on t*
*he even
spaces, where 2 Z . We allow n = 0 and = 1 and use [um ] O[un] = [um+n ] and
[] O[0] = [0]. In the even case, we need at most one *factor of the form [] O[*
*un],
and we may always insert the redundant factor [0] O[un] = 1. Thus we can reduce*
* any
expression in the generators to standard form. 
Example: K(n) We use the same test spaces as before, C P 1 and the finite lens
n1
space L2p , and follow the same strategy. The main reference is [28]. Some of*
* the
algebra resembles the case E = H(F p).
As usual, the complex orientation determines Hopf ring elements bi, where b*
*0 = 12
and b1 = e2 = eO2. As K(n) is plocal, Lemma 15.9 shows that the bj other than
the b(i)= bpiarePredundant. If we apply eq. (10.32) to the Chern class x, we ob*
*tain
the identity j xpj = 1. This shows that F bj = 0 for all j *
*> 0; in
particular, b*p(i)= 0.
n
Next, we apply the general operation r to i*x = vnxp by eq. (15.2) to obta*
*in
n Opn pn+i
b(vnxp ) = b(x) O[vn]. Equating coefficients of x yields the relation
n pi 1
bOp(i)= vn b(i)O[vn ]; (17:15)
the obvious analogue of eq. (16.17).
n1 * * pn
For the other test space Y = L2p , we have K(n) (Y ) = (u) K(n) [x: x =
0]. The class x is a Chern class, which we know all about. Parallel to eq. (16.*
*19), we
use u 2 K(n)1(Y ) to define elements ai; ci 2 K(n)*(K(n)_1) for 0 i < pn by the
identity
pn1X pn1X
r(u) = xi+ uxi in K(n)*(Y ) (for all r):
i=0 i=0
Proposition 17.16 The Hopf ring elements ai 2 K(n)2i(K(n)_1) (for 0 i < pn),
a(i)= api2 K(n)2pi(K(n)_1) (for 0 i < n), and ci 2 K(n)2i+1(K(n)_1) (for 0 i <
pn) have the following properties:
(a) a0 = 11 and c0 = e;
P
(b) ak = i+j=kai aj;
(c) V a(i)= a(i1)for 0 < i < n, and V a(0)= 0;
(d) fflak = 0 for all k > 0;
(e) Oak is the coefficient of xk in a(x)*(1), expanded as in Prop. 15.3(e);
i j
(f) ai* aj = i+jiai+jif i + j < pn;
(g) F a(i)= a*p(i)= 0 for 0 i < n  1;
(h) ci= e * ai;
(i) For all r, r*ak is the coefficient of xk in the formal identity
pn1 pn1
r*a(x) = * b(x)OiO[] * * a(x)Ob(x)OiO[]
i=0 i=0
JMB, DCJ, WSW  84  23 Feb 1995
x17. Examples for unstable operations
n
in K(n)*(K(n)_*)[x : xp = 0];
(j) q1a(i)= a(i)2 Q(K(n))1*, and q1aj = 0 if j is not a power of p;
(k) oe1*a(i)= a(i)2 K(n)*(K(n); o), and oe1*aj = 0 if j is not a power of p.
Proof All the proofs are formally identical to those of Prop. 17.9, except tha*
*t we use
the space Y instead of L. As in x16, the partial multiplications : L2k+1x L2m !*
* Y
yield (f) and (h).
P pi
For (g), we apply eq. (10.32) to u and obtain i>0 x = 0. But be*
*cause
n *p n1
xp = 0 already, we are able to deduce that ai = 0 only for 0 < i < p . (We s*
*hall
see in a moment that a*p(n1)6= 0.) 
We have to rely on [28, Prop. 1.1] for two facts, just as in x16. The first*
* is that
when i = 0, eq. (17.15) desuspends once, exactly as eq. (16.18) suggests, to
n1 1
e ObOp(0)= vne O[vn ]: (17:17)
n1 1 *
*1 1
In other words, the class y = vnuxp 2 K(n) (Y ) still behaves like u1 2 K(n)*
* (S )
and satisfies eq. (13.2). The second is that when we take account of decomposab*
*les,
eq. (16.22) acquires an extra term,
n1
a*p(n1)= vna(0) a(0)ObOp(0)O[vn]: (17:18)
This complements (g). We have the material for the main theorem of [28].
Theorem 17.19 For the Morava Ktheory ring spectrum K(n), K(n)*(K(n)_*) is
the Hopf ring over K(n)* = Fp[vn; v1n] with the Ogenerators:
[vn] 2 K(n)0(K(n)_2(pn1)) (see Prop. 11.2);
[v1n] 2 K(n)0(K(n)_2(pn1)) (see Prop. 11.2);
e 2 K(n)1(K(n)_1) (see Prop. 13.7);
a(i)2 K(n)2pi(K(n)_1), for 0 i < n (see Prop. 17.16);
b(i)2 K(n)2pi(K(n)_2), for i 0 (see Prop. 15.3);
subject to the relations [1]*p = 10, [vn] O[v1n] = [1], eO2 = b(1), (17.15), *
*(17.17),
and (17.18). 
Thus we have the *generators:
(i)aOIObOJO[vkn] in even degrees;
(ii)e OaOIObOJO[vkn] in odd degrees;
where I = (i0; i1; : :;:in1), with each ir = 0 or 1, and J = (j0; j1; j2; : *
*:):, with
0 j < pn, and k 2 Z . In (ii), we may assume j0 < pn  1 by eq. (17.17). The
relations a*p(i)= 0 (for i < n  1) and b*p(i)= 0 (for all i) follow from [1]*p*
* = 10 by
eq. (10.13), as in Thm. 17.10.
JMB, DCJ, WSW  85  23 Feb 1995
Unstable cohomology operations
18 Relations for additive BP operations
In this section, we discuss relations in the bigraded algebra Q**= Q(BP )***
*, follow
ing [23], in preparation for discussing additive unstable operations in BP coh*
*omology.
In view of Thm. 16.11(a), Q**is spanned as a BP *module by the monomials
efflbIwJ = efflbi0(0)bi1(1):w:j:11wj22: :;: (18:1)
where ffl 1 and we use standard notation with multiindicesPI = (i0; i1; i2; :*
* :):andP
J = (j1; j2; : :):. We define the length of I as I = tit, and similarly J*
* = tjt.
We also need the special multiindex 0 = (1; 0; 0; : :):.
The main relations For E = BP , we easily compute the first main relation from
Defn. 14.9 and [8, eq. (15.4)] (or equivalently from eqs. (14.5) and [8, eq. (1*
*5.3)]) as
(R1) : v1b(0)= pb(1)+ bp(0)w1 in Q(BP )2*. (18:2)
(Indeed, this is the only candidate that stabilizes correctly to [8, eq. (15.6)*
*].) We
still have bi = 0 whenever i1 is not a multiple of p1. We can use the pserie*
*s [8,
eq. (15.5)], just as stably, to simplify the higher relations (Rk) by neglectin*
*g enough.
Denote by V and W respectively the ideals (p; v1; v2; : :):and (p; w1; w2; : *
*:):in Q**,
which correspond to the left and right actions of the ideal I1 . We also need t*
*he ideal
M = (e; b(0); b(1); b(2); : :): Q**, so that M + V is the obvious augmentati*
*on ideal
consisting of all the Qkifor i > 0. In particular, bi2 M + V for all i. From De*
*fn. 14.9
and [8, eq. (15.7)], the right side of (Rk) has the form
k1X i k
R(k) bp(ki)wi+ bp(0)wk mod V + M W 2, (18:3)
i=1
while the left side L(k) 2 V and will not much concern us here. The new feature*
* is
k 2
that because wk appears in the form bp(0)wk, where b(0)= e is no longer 1, (Rk*
*) fails
to express wk in terms of the other generators, and W 6= V ; this made it nece*
*ssary
to add wk as a new generator of Q**in Thm. 16.11.
The RavenelWilson basis The relations (Rk) show that many of the monomials
(18.1) are redundant. In defining the basis, it is easier to specify which mon*
*omials
are not wanted.
Definition 18.4 We disallow all monomials of the form
2 pn
bp(i1)bp(i2):b:(:in)wnc (i1 i2 : : :in, n > 0), (18:5)
where c stands for any monomial in the b(i), wi, and e (c = 1 is permitted). A*
*ll
monomials (18.1) not of this form are declared to be allowable.
Nevertheless, we need a positive construction of the allowable monomials, a*
*nd we
need to know how they behave under suspension. Given any indices
0 = k0 k1 k2 : : :kn; where n 0, (18:6)
we define the monomial
2 pn L
bL = b(k0)bp(k1)bp(k2):b:(:kn)= b(0)b 0 : (18:7)
JMB, DCJ, WSW  86  23 Feb 1995
x18. Relations for additive BP operations
It is easy to see that every allowable monomial can be written uniquely in the *
*canonical
form 2 n
c = efflbL0 bM wJ = efflbp(k1)bp(k2):b:p:(kn)bM wJ;(18:8)
where ffl = 0 or 1 and M and J satisfy the conditions:
(i)t < ku implies mt< pu, for 0 < u n;
(ii)t kn implies mt< pn+1; (18.9)
(iii)jt= 0 for all t n;
as well as (18.6). In detail, we choose, by induction on u, the smallest ku suc*
*h that
2 pu
bp(k1)bp(k2):b:(:ku)divides c, to make (i) hold for u. If no such ku exists, we*
* set n = u1
and have (ii). Since c is allowable, it can have no factor wu, which gives (iii*
*) for t = u.
(In case n = 0, we have merely c = efflbM wJ, (i) and (iii) are vacuous, and (i*
*i) says
only that mt< p for all t.)
The main technical result is that there is only one way the suspension ec o*
*f c can
fail to be allowable. (This is in effect equivalent to the discussion in [23, *
*x5].) We
recall from Defn. 15.12 the shifted multiindex s(I).
2 pn+1 L M
Lemma 18.10 Assume that the monomial bH = bp(i0)bp(i1):b:(:in)divides b b ,*
* where
bL (with the same n) is as in eq. (18.7), i0 i1 : : :in, and M satisfies cond*
*itions
(i) and (ii) of (18.9). Then:
(a) iu = ku for 0 u n, so that H = pL;
(b) We can write M = (p1)L + s(M0), where M0 again satisfies (i) and (ii).
Proof We show first that iu ku for all u. For any t < ku, we have mt< pu by (*
*i).
Then the exponent of b(t)in bLbM is at most
(1 + p + p2 + : :+:pu1) + (pu  1) < pu+1;
which shows that t 6= iu.
We proceed by induction on n. For n = 0, bp(i0)divides b(0)bM , where mt < *
*p for
all t. We must have i0 = 0 and m0 = p  1, which gives M the required form.
For n > 0 we must have in = kn, since in > kn is forbidden by (ii). Let ff *
* 0 be
the smallest index such that kff= kn; then we must have iff= iff+1= : :=:in = k*
*n.
From lkn = pff+ pff+1+ : :+:pn and mkn < pn+1 we deduce
mkn(p1)lkn < pn+1(p1)(pff+ : :+:pn) = pn+1(pn+1pff) = pff: (18:11)
If kn = 0 we clearly have ff = 0 and hence m0 = (p1)l0, and can write M =
u+1
(p1)L + s(M0). If kn > 0, we have ff > 0. We delete the factors bp(iu)for ff *
*u n
from both sides of our hypothesis, as well as any factors b(t)for t > kn, to de*
*duce
2 pff p pff1 M00 00
that bp(i0)bp(i1):b:(:iff1)divides b(k0)b(k1): :b:(kff1)b , where M satisf*
*ies (i) and (ii)
for the sequence (k0; k1; : :;:kff1). By induction, we deduce that H = pL and *
*that
M has the form (p1)L + s(M0).
If t kn, we have m0t= mt+1 < pn+1, which gives (ii) for M0. To establish
(i), assume that t < ku. If also t + 1 < ku, we have m0t mt+1 < pu, as desired.
Otherwise, ku = t + 1. Let fi be the smallest index such that kfi> t + 1, so t*
*hat
JMB, DCJ, WSW  87  23 Feb 1995
Unstable cohomology operations
ku = ku+1 = : :=:kfi1= t + 1. Then mt+1< pfiand lt+1 pu + pu+1 + : :+:pfi1.
As in eq. (18.11), we find m0t= mt+1 (p1)lt+1< pu. 
Lemma 18.12 In the bigraded algebra Q**= Q(BP )**:
(a) Every allowable monomial can be written uniquely in the form (18.8), sub*
*ject
to the conditions (18.6) and (18.9), and conversely, every monomial of this for*
*m is
allowable;
(b) The suspended monomial from eq. (18.8)
n M J
b(0)c = efflbLbM wJ = efflb(0)bp(k1):b:p:(kn)b w
is disallowed if and only if jn+1 > 0 and b(p1)Ldivides bM , in which case we *
*can write
0 M (p1)Ls(M0) L M0 J0
wJ = wn+1wJ and b = b b , with b 0 b w allowable;
(c) Every allowable monomial can be written uniquely in the extended canonic*
*al
form
2(L)+:::+sh1(L))sh(M)hJ
c = efflbL0 b(p1)(L+s(L)+s b wn+1w (18:13)
with L as in eq. (18.7), where h 0, either b(p1)Ldoes not divide bM or jn+1 *
*= 0 (or
both), and conditions (18.6) and (18.9) hold;
(d) In (c), the monomial bLbM wJ is allowable.
Proof In (a), we need to establish the converse. If c is disallowed, so is b(*
*0)c. By
Lemma 18.10(a), b(0)c can be disallowed only if H = pL; but by Lemma 18.10(b), *
*the
necessary factors b(0)are not present in c.
Moreover, b(0)c is disallowed if and only if it contains bpLwn+1 as a facto*
*r (using
0) L M0 J0
the same n). If so, we write bM = b(p1)Lbs(M , where b 0 b w is allowab*
*le by
(a). This proves (b).
Parts (c) and (d) follow by induction on h. We take h maximal. 
Lemma 18.14 In the stable range defined by i pk, every allowable monomial in
Qki= Q(BP )kihas the form efflbi0(0)bi1(1):,:w:ith no factors of the form wjt.
Proof For each monomial c 2 Q**, we define g(c) = i  pk, where c 2 Qki. We
compute g from g(b(n)) 0 if n > 0, g(wn) = 2p(pn1), g(e) = (p1), and
g(b(0)) = 2(p1), using g(ac) = g(a) + g(c). Thus if c contains wn as a facto*
*r,
g(c) > 0 unless c contains at least {2p(pn1)  (p1)}=2(p1) factors b(0), whi*
*ch
disallows it. 
Theorem 18.15 (RavenelWilson) The allowable monomials (18.1) form a basis of
the free BP *module Q**= Q(BP )**.
This is proved in [23, Thm. 5.3, Prop. 5.1]. We content ourselves with show*
*ing,
as part of Thm. 18.16, that the allowable monomials span Q**, assuming that it *
*is
spanned by all the monomials (18.1). We shall obtain for each disallowed monomi*
*al
(18.5) a reduction formula that expresses it in terms of other monomials. A fin*
*iteness
argument then implies that the allowable monomials must span. A counting argume*
*nt
is needed to show they in fact form a basis. As only the relations (Rk) and e2 *
*= b(0)
are used in the reduction, they constitute sufficient relations in Thm. 16.11.
JMB, DCJ, WSW  88  23 Feb 1995
x18. Relations for additive BP operations
Knowing that the allowable monomials form a basis of Q**is not enough. In o*
*rder
to work with this basis, we need to know how the ideal W looks in terms of the*
* basis.
We therefore define Am for any m 0 as the BP *submodule of Q**spanned by all *
*the
allowable monomials efflbIwJ that have I 6= 0 and J m. Although A m is not an
ideal for m > 0, it is convenient for computation, because when an element c 2 *
*Q**
is expressed in terms of the basis, it is obvious whether or not it lies in A m*
*. We
shall prove the following parts of the structure of Q**, after developing the n*
*ecessary
reduction formula.
Theorem 18.16 In the bigraded algebra Q**= Q(BP )**:
(a) Am + V = M W m + V for any m > 0 (or_A0 + V = M + V if m = 0), so that
*
the image of Am in the quotient algebraQ *(see eq. (18.17)) is an ideal;
(b) The allowable monomials span Q**= Q(BP )**as a BP *module.
Lemma 15.2 of [8] allows us to work mod V , in the quotient Fpalgebra
__* *
Q *= Q*=V ~=QH*(BP__*; Fp) : (18:17)
__even
Better yet, we may ignore e and work in the subalgebraQ * .
Higher order relations As they stand, the relations (Rk) are not very practica*
*l.
j
We derive a more useful relation by eliminating the terms that come from b(x)p *
*wj
for j < n in eq. (18.3) from the n relations (Rk1), (Rk2), . . . , (Rkn), as in*
* [23, Lemma
5.13]. The result is of course a determinant. For ulterior purposes, we make *
*the
elimination totally explicit.
Definition 18.18 Given any positive integers i1; i2; : :;:in, where n 1, we *
*define
i1pi2 pin*
*1pin
L(i1; i2; : :;:in) and R(i1; i2; : :;:in) as the coefficient of xp1x2 : :x:n1*
*xn in
2 pn1
b(x1)pb(x2)p : :b:(xn1) b([p](xn))
and
2 pn1
b(x1)pb(x2)p : :b:(xn1) [p]R(b(xn))
respectively. By eq. (14.8), these are equal in Q**.
Then given any integers 0 < k1 < k2 < : :<:kn, where n > 1, we deduce the n*
*th
order derived relation
X X
(Rk1;k2;:::;kn) : fflssL(i1; i2; : :;:in) = fflssR(i1;(i2;1:8*
*:;:in):19)
ss ss
in Q**by summing over all permutations ss 2 n, where fflssdenotes the sign of s*
*s and
we permute the n entries in (i1; i2; : :;:in) = ss(k1; k2; : :;:kn). (For n = 1*
*, it reduces
to L(k1) = R(k1), which is just (Rk1).)
We note that this relation lies in Qf(n)*, where the numerical function
2(pn  1)  deg(vn)
f(n) = 2(1 + p + p2 + : :+:pn1) = _________= _________
p  1 p  1
was introduced in eq. (1.4).
JMB, DCJ, WSW  89  23 Feb 1995
Unstable cohomology operations
The left side of (Rk1;k2;:::;kn) lies in V and will be of little interest h*
*ere. By (18.3),
the right side reduces to
X p p2 pn1 pj 2
fflssb(i11)b(i22): :b:(in1n+1)b(inj)wj mod V + M W ,(18:20)
ss;j
where we sum over all permutations ss and all j > 0, and adopt the convention t*
*hat
b(i)= 0 for i < 0. However, we have arranged matters so that no (explicit) term*
*s in
wj with j < n survive; when we interchange ij and in, we find identical terms h*
*aving
opposite signs. The term of most interest is the leading term with ss = id,
2 pn1 pn
bLwn = bp(k11)bp(k22):b:(:kn1n+1)b(knn)wn ; (18:21)
which is thereby expressed in terms of other monomials and hence redundant. (The
multiindex L serves only as a convenient abbreviation, unrelated to eq. (18.7)*
*. The
indices ku are different, too.)
To make this more precise, we note that all terms bIwj in the sum (18.20) h*
*ave
I = L = p + p2 + : :+:pn if j = n, or I > L if j > n. We order the
terms thatPcontain wn by defining the weight of any multiindex I = (i0; i1; i2*
*; : :):as
wt (I) = ttit (which is not the weight used in [23]). This makes bLwn the hea*
*viest
term with its length, because if we improve the ordering of the indices of any *
*other
term in (18.20) by interchanging ir and is, where r < s and ir > is, we increas*
*e its
weight by
(isr)pr + (irs)ps  (irr)pr  (iss)ps = (iris)(pspr) > 0 :
Thus (Rk1;k2;:::;kn) provides a reduction formula
2 pn X I
bLwn = bp(k11)bp(k22):b:(:knn)wn b wj (18:22)
I;j
__* 2
inQ *mod M W , where the sum is taken over certain pairs (I; j) with j n, f*
*or
which I > L, or I = L and wt(I) < wt(L).
The first nth order relation (R1;2;:::;n) is particularly important, as onl*
*y one term
of the sum (18.20) is meaningful, namely bpm(0)wn, where m = f(n)=2. We observe*
* that
this monomial lies just inside the stable range of Lemma 18.14. In this simple *
*case,
we can do better with a little more attention to detail, to obtain the direct a*
*nalogue
of [8, Lemma 15.8].
Lemma 18.23 In Qf(n)*= Q(BP )f(n)*we have the relation
bpm(0)wn vnbm(0) mod InQ(BP )f(n)*
for each n > 0, where m = f(n)=2 = 1 + p + p2 + : :+:pn1.
Proof We proceed by induction on n, starting from eq. (18.2), and work through*
*out
n
mod InQ**. On the left side of eq. (14.8) we have b([p](x)) b(vnxp + : :):, *
*by [8,
eq. (15.5)]. Then R(j) = L(j) 0 for all j < n, and the only surviving terms in
(R1;2;:::;n) are bh(0)L(n) bh(0)R(n), where h = p + p2 + : :+:pn1. On the le*
*ft, we
clearly have L(n) vnb(0). On the right, bh(0)wj 0 for all j < n, by the induc*
*tion
JMB, DCJ, WSW  90  23 Feb 1995
x18. Relations for additive BP operations
hypothesis;nby eq. (18.3) and dimensional reasons, the only surviving term in R*
*(n) is
bp(0)wn. 
__*
Proof of Thm. 18.16 We work entirely in the quotient algebra Q * defined by
eq. (18.17). We first generalize (18.22) to show that
M W m Am + M W m+1 (18:24)
for any m 1. As an Fpmodule, M W m is generated by those monomials efflbIwJ
that have J m. These lie in Am or M W m+1 except for the disallowed monomi*
*als
that have J = m. On comparing the monomial (18.5) with eq. (18.21), we see
that each such monomial has the form bLwnc, where L is given by eq. (18.21) and
c = efflbLwN , with N = m1. When we multiply eq. (18.22) by c, both ordering*
*s are
preserved, and we express the general disallowed monomial bLwnc as a signed sum*
* of
monomials with greater length, or the same length and lower weight, mod M W m+*
*1.
Because there are only finitely many monomials in each bidegree, eq. (18.24) fo*
*llows
by induction.
For any i > m, eq. (18.24) gives
Am + M W i Am + Ai+ M W i+1= Am + M W i+1:
Then by induction on i, starting from eq. (18.24),
M W m Am + M W i
for all i > m. In any fixed bigrading, M W iis zero for large i. Thus M W m *
*A m
and we have (a) for m > 0. For m = 0, we note that every monomial in M either *
*lies
in M W A1 or is automatically allowable and so lies in A0.
On reinstating the monomials_of the form wJ, which are all allowable, we se*
*e that
*
the allowable monomials spanQ *. Then (b) follows by Nakayama's Lemma in the
form [8, Lemma 15.2(d)]. 
The ideals Jn Just as the ideal I1 BP *led to the introduction of the ideal
W Q**, the ideal Jn, needed for our splitting theorems, leads to an ideal in *
*Q**.
Definition 18.25 We define the ideal Jn = (wn+1; wn+2; wn+3; : :): Q**.
We need to know how Jn sits inside Q**. The answer is remarkably clean, in*
* a
certain range.
Lemma 18.26 Assume n 0. Then:
(a) If k < f(n+1), Qk*\ Jn is the left BP *submodule of Qk*spanned by all t*
*he
allowable monomials efflbIwJ 2 Qk*that contain an explicit factor wt for some t*
* > n;
(b) If k = f(n+1), Qk*\ Jn is the left BP *submodule of Qk*spanned by all t*
*he
allowable monomials as in (a), together with all disallowed monomials of the fo*
*rm
2 pn+1
bp(i1)bp(i2):b:(:in+1)wn+1, where 0 i1 i2 : : :in+1.
Remark The first disallowed monomial in (b) is bpm(0)wn+1, where m = f(n+1)=2.
Lemma 18.23 shows it definitely does not lie in the submodule described in (a).
Proof The stated elements obviously lie in Jn. To show the converse, we fix k *
*and a
large integer m, and prove by downward induction on h that all elements in Qkio*
*f the
JMB, DCJ, WSW  91  23 Feb 1995
Unstable cohomology operations
form cwh lie in the indicated submodule whenever i < m. This statement is vacuo*
*us
for sufficiently large h (depending on m and k). We therefore fix t > n, assume*
* the
statement for all h > t, and prove it for h = t. We ignore efflthroughout and a*
*ssume
k is even.
Case 1: c = bI. The number I of b factors in c is k=2 + pt  1. In (a)*
*, as
k < f(n+1) and t > n, this is always less than p + p2 + : :+:pt, which makes
cwt = bIwt automatically allowable. The same holds in (b), except in the extre*
*me
case bIwn+1, which may be allowable or disallowed; either way, it is in.
Case 2: c = bIwhwJ allowable, with h t. Then cwt= bIwhwtwJ remains allow
able, by the form of Defn. 18.4.
Case 3: c = awh, with h > t, any a. Then cwt = (awt)wh is in by induction,
provided i < m.
t1)
By Thm. 18.15, these c generate Qk+2(p* as a BP *module. 
19 Relations in the Hopf ring for BP
In this section, we develop the unstable analogues of the results of x18, w*
*orking
in the Hopf ring BP*(BP__*) for BP . By taking account of *decomposable elemen*
*ts,
we can improve many of these results by one. The structure of the Hopf ring was
described briefly in x17. Before we can even state some of our results precisel*
*y, it is
necessary to clarify the concept of ideal in a Hopf ring.
Hopf ring ideals As it is obviously impractical to retain everything in typica*
*l Hopf
ring calculations (the preceding sections should convince), we need to control *
*carefully
what is thrown away. There is an obvious relevant concept, valid in any Hopf ri*
*ng H.
We concentrate on the structure of H as a *algebra, treating Omultiplication *
*chiefly
as a means of creating new *generators from old.
Definition 19.1 We call a bigraded Rsubmodule I of any Hopf ring H over R a
Hopf ring ideal if the quotient H=I inherits a welldefined Hopf ring structure*
* from
H (over the possibly smaller ground ring R=fflI).
If we ignore the Omultiplication and coalgebra structure, I must obviously*
* be a
*ideal in the ordinary sense, i. e. an Rsubmodule for which b * c 2 I wheneve*
*r b 2 H
and c 2 I.
Lemma 19.2 Let H be a Hopf ring over R and I R an ideal. Let I be the *ide*
*al
in H generated by the elements cff. Then I is a Hopf ring ideal, with quotient *
*a Hopf
ring over R=I, if and only if:
(i) cff2 IH + H I for all ff;
(ii)fflcff2 I for all ff;
(iii)a Ocff2 I for all a 2 H and all ff;
(iv)IH I.
Proof The conditions are evidently necessary. Conditions (i) and (ii) ensure *
*that
H=I inherits a comultiplication and counit ffl. Condition (iv) shows that H=*
*I is
JMB, DCJ, WSW  92  23 Feb 1995
x19. Relations in the Hopf ring for BP
defined over R=I. For any a; b 2 H, eq. (10.11) and (iii) show that a O(b * cff*
*) 2 I; this
is enough to furnish H=I with a Omultiplication. All the necessary identities *
*in H=I
(see x10) are inherited from H. 
Remark It is clear from the Lemma that the sum I + J of two Hopf ring ideals is
another Hopf ring ideal. However, their *product ideal I * J (defined as the u*
*sual
product of ideals) need not be a Hopf ring ideal, as (i) can fail. We note that*
* (ii) and
(iii) nevertheless continue to hold for I * J, with the help of eq. (10.11).
When R = Fp, we can define a rather more useful ideal.
Definition 19.3 Given an ideal I in a Hopf ring over F p, we define F I as the
*ideal generated by {F x : x 2 I}.
The ideal F Iis far smaller than I*p, and clearly is a Hopf ring ideal by L*
*emma 19.2
whenever I is. (We use eq. (10.13) to verify (iii).)
The redundant generators We proved in Lemma 15.9 that the generator bi is
redundant unless i is a power of p. As in (17.3), this implies that BP*(BP__*)*
* is
*generated as a BP *algebra by Omonomials of the forms (cf. eq. (18.1))
bOIO[vJ]= bOi0(0)ObOi1(1)ObOi2(2)O:O:[:vj11vj22: :]:
(i) Oi Oi Oi Oj Oj
= b(00)Ob(11)Ob(22)O: :O:[v1] 1O[v2] 2O: :;:
(19:4)
(ii) e ObOIO[vJ];
(iii)[vJ] = [] O[v1]Oj1O[v2]Oj2O: :;:
in the notation of eq. (15.11). To carry out computations, we need to express *
*the
redundant bi in terms of these *generators.
In_order to make the finiteness of_our computations apparent, we write b(x)*
* =
12 + b(x) as in eq. (15.4) and use 1 Ob(x) = 0. Then eq. (15.8) expands to
_ X i j
12 + b x + y + ai;jx y
i;j (19:5)
_ _ n _ Oi _ Oj o
= (12 + b(x)) * (12 + b(y)) * * 12 + b(x) Ob(y) O[ai;j]
i;j
As in Lemma 15.9, if n is not a power of p, we take s as the largest power of p*
* less
than n, and the coefficient of xsyns then yields a reduction formula for bn. F*
*or the
low bn's we can be explicit; they are no longer trivially zero, as in x18.
Lemma 19.6 For 1 i < p we have bi= b*i(0)=i! .
Proof All that is left of eq. (19.5) in this range is b(x+y) = b(x) * b(y). He*
*nce b(x)
must be the exponential series exp(b1x), expanded using *multiplication. 
Beyond this range, we must settle for inductive formulae in terms of Omono*
*mials
of the form
bi1Obi2O: :O:birO[vJ] : (19:7)
JMB, DCJ, WSW  93  23 Feb 1995
Unstable cohomology operations
We expand the formal group law F (x; y) fully, in the form
X
F (x; y) = x + y + vIxiyj;
;I;i;j
summing over appropriate quadruples (; I; i; j) consisting of a coefficient 2 *
*Z(p), a
multiindex I, and exponents i and j. The right side of eq. (19.5) becomes
i _ j i _ j n _ _ * o
12 + b(x) * 12 + b(y)* * 12 + b(x)OiOb(y)OjO[vI];
;I;i;j
where {1 + : :}:* is expanded by the binomial series as in eq. (15.5). Every el*
*ement
of the Hopf ring that appears here is a *product of elements of the form (19.7*
*).
This is still not enough! To make the induction succeed, we really need a r*
*eduction
formula for every Omonomial (19.7) that contains a Ofactor bn with n not a po*
*wer
of p, without relying on iterated appeals to the distributive law (10.11). A re*
*duction
formula for bn Obh1Obh2O: :O:bhq, whenever n is not a power of p and the hi are*
* any
positive integers, will suffice, as  O[vI] is a *homomorphism and [vI] O[vJ] *
*= [vI+J].
We therefore Omultiply eq. (15.8) by b(z1) Ob(z2) O: :O:b(zq) (and thus wo*
*rk in the
(q+2)fold product (C P 1)q+2). On the right, we use the distributive law (15.*
*6) to
move all the b()'s inside the *factors, to obtain
_ X I i j _ _
12q+2+ b x + y + v x y Ob(z1) O: :O:b(zq)
n ;I;i;j
_ _ _ o
= 12q+2+ b(x)Ob(z1)O : :O:b(zq)
n _ _ _ o (19:8)
* 12q+2+ b(y)Ob(z1)O : :O:b(zq)
n _ _ _ _ * o
* * 12q+2+ b(x)OiOb(y)OjOb(z1)O : :O:b(zq)O[vI]
;I;i;j
The coefficient of xsynszh11: :z:hqqyields the desired reduction formula. Insp*
*ection of
the Omonomials that appear on the right shows that they are all simpler, so th*
*at the
induction makes progress. (In detail, they all have lower height, or thePsame h*
*eight
but more b factors, if we define the height of the monomial (19.7) as rir.)
None of this is necessary for the other generators, (19.4)(ii). For these *
*it is far
simpler to start from Lemma 14.6, work in Q(BP )**, and suspend by applying e O*
*.
The main relations As given in Defn. 15.15, the main relations are particularly
opaque. We make eq. (15.14) more useful in our situation by first expanding the
pseries [8, eq. (13.9)] for BP in full as
X
[p](x) = px + vIxm ; (19:9)
;I;m
much as we just did for F (x; y), and summing over appropriate combinations of
coefficient 2 Z(p), multiindex I, and exponent m. Then eq. (15.14) becomes
_ X I m n _ o*p n _ Om I o*
12 + b px + v x = 12 + b(x) * * 12 + b(x) O[v ] (19:10)
;I;m ;I;m
JMB, DCJ, WSW  94  23 Feb 1995
x19. Relations in the Hopf ring for BP
where we again expand {12 + : :}:* by the binomial series as in eq. (15.5).
The first main relation, the coefficient of xp, simplifies (with the help o*
*f [8,
eq. (15.4)] and Lemma 19.6) to
b*p(0)
(R1) : v1b(0)= pb(1)+ bOp(0)O[v1]  _______ in BP*(BP__2) (19:11)
(p1)!
(although it is far easier to extract this as the coefficient of xp1y in eq. (*
*15.8), using
[8, eq. (15.3)]). Subsequent relations rapidly become extremely complicated an*
*d can
be handled only by neglecting terms wholesale. We need some ideals.
Let V be the ideal (p; v1; v2; : :):in BP*(BP__*) (more accurately, genera*
*ted as a
graded *ideal by all the elements p1k and vn1k for each k). We need the unsta*
*ble
analogue of the ideals M W m of x18, coming from the right action of I1 on Q**
**. It
is obvious how to handle the generators vi of I1 . For the generator p, eq. (1*
*0.13)
shows that in the quotient Hopf ring BP*(BP__*)=V over Fp = BP *=I1 , we may w*
*rite
c O[p] = c O(F [1]) = F (V cO[1]) = F V c. Indeed, it is even more convenient t*
*o ignore e
and work in the Hopf subring
__
H = BP*(BP__even)=V ~=H*(BP__even; Fp); (19:12)
using only those elements that do not involve the Ogenerator e (though of cour*
*se we
keep b(0)= eO2).
__
Definition 19.13 We define M 0 as the *ideal inH generated by all the elem*
*ents
bOIO[vJ] with I 6= 0, whether allowable or not. For m > 0, we define M m induct*
*ively
as the *ideal generated by F M m1 and all elements bOIO[vJ] with I 6= 0, whet*
*her
allowable or not, that have J m.
Equivalently, M m is the *ideal generated by all elements F h(bOIO[vJ]) wi*
*th I 6= 0
and h + J m. (Thus M m is roughly, but not quite, the Hopf ring analogue of *
*the
right BP *action of the ideal Im1.) We thus have the decreasing sequence of id*
*eals
__
H M 0 M 1 M 2 : :::
__
We note that M 0 is just the obvious augmentation ideal inH consisting of all*
* the
Hi(BP__even; Fp) with i > 0.
Lemma 19.14 For all m 0:
__
(a) M m is a Hopf ring ideal in the Hopf ringH = BP*(BP__even)=V ;
(b) M m O[vn] M m+1 for all n > 0;
(c) M m O[vJ] M m+J;
(d) M m O[p] M m+1.
Proof We first prove (b), from which (c) follows by induction. As  O[vn] is *
*a *
homomorphism, it is enough to check that c O[vn] 2 M m+1 for the generators c *
*of
M m. For c = bOIO[vJ], we use [vJ] O[vn] = [vJvn]. For c = F a = a*p, where a *
*2 M m1,
we have c O[vn] = F (a O[vn]) by eq. (10.13). This lies in F M m, by induction *
*on m.
We next apply Lemma 19.2 to prove (a). Clearly, fflM m = 0. For a genera*
*tor
of the form c = bOIO[vJ], with J m, we have a Oc = (a ObOI) O[vJ] 2 M m by
JMB, DCJ, WSW  95  23 Feb 1995
Unstable cohomology operations
P
(c), sincePa ObOI 2 M 0. Similarly, if we write bOI = iB0i B00i, we find t*
*hat
c = iB0iO[vJ] B00iO[vJ] has the required form, because for each i, either B*
*0i2 M 0
or B00i2 M 0 for reasons of degree.
For a generator F c with c 2 M m1, we use induction on m. By eq. (10.13),
a O(F c) = F (V a Oc) 2 F M m1. Also, F c = (F F ) c has the required form.
Because M m is now known to be a Hopf ring ideal, we have V a 2 M m for a*
*ny
a 2 M m. Then (d) is immediate from eq. (10.13), using [p] = F [1]. 
__
We now have the tools to handle eq. (19.10). We work entirely_in H , so th*
*at
by [8, eq. (15.5)], the left side is trivial. By Lemma 19.14, b(x)Om O[vI] 2 M *
*I. Most
*factors on the right side of eq. (19.10) are trivial mod M 2and we are left w*
*ith only
n _ o n _ j o __
12 + F b(x) * * 12 + b(x)Op O[vj] inH [[x]] mod M 2.
j>0
k
When we pick out the coefficient of xp and neglect also certain products, we o*
*btain
Xk j __
R(k) F b(k1)+ bOp(kj)O[vj] inH mod M 2 + M 1*M 1, (19:15)
j=1
analogous to eq. (18.3). Although the ideal here is not a Hopf ring ideal, (ii)*
* and (iii)
of Lemma 19.2 still hold, according to the Remark following that lemma.
The RavenelWilson generators We lift the allowable monomials of x18 via the
canonical projections qk: BP*(BP__k) ! Q(BP )k*, so that multiplication is now *
*to be
interpreted as Omultiplication.
Definition 19.16 We disallow all Omonomials of the form
2 Opn
bOp(i1)ObOp(i2)O:O:b:(in)O[vn] Oc (i1 i2 : : :in, n > 0),(19:17)
where c stands for any Omonomial in the b(i), [vj], and e (c = [1] is permitte*
*d). All
Omonomials (19.4)(i) and (ii) not of this form are declared to be allowable.
It follows from Thm. 18.15 (and local finiteness) that the allowable Omono*
*mials
generate BP*(BP__*), but far more is true, by [23, Thm. 5.3, Rk. 4.9].
Theorem 19.18 (RavenelWilson) In the Hopf ring for BP :
(a) If k is even, denote by BP__0kthe zero component of the space BP__k(so t*
*hat
BP__0k= BP__kif k > 0). Then BP*(BP__0k) is a polynomial algebra over BP *on th*
*ose
allowable Omonomials bOIO[vJ] with I 6= 0 that lie in it. If k 0, BP*(BP__k)*
* =
BP*(BP k) BP*(BP__0k) as in eq. (17.4).
(b) If k is odd, BP*(BP__k) is an exterior algebra over BP *on those allowab*
*le
Omonomials e ObOIO[vJ] that lie in it. 
As in x18, we need information on where the disallowed monomials lie. The
difficulty with eq. (19.15) is that it is hard to tell whether a given element *
*lies in
M 2. We therefore define analogous ideals in terms of the polynomial generato*
*rs in
Thm. 19.18 for which this problem_does not exist. Again, we ignore e and neglec*
*t V
by working in the Hopf ringH over Fp (see eq. (19.12)).
JMB, DCJ, WSW  96  23 Feb 1995
x19. Relations in the Hopf ring for BP
__
Definition 19.19 We define A 0as the *ideal inH generated by all the allow*
*able
Omonomials bOIO[vJ] that have I 6= 0. For m > 0, we define Am inductively as *
*the *
ideal generated by F Am1 and all the allowable Omonomials bOIO[vJ] for which *
*I 6= 0
and J m.
In other words, Am is the *ideal generated by all the elements F h(bOIO[vJ*
*]), where
bOIO[vJ] is allowable, I 6= 0, and h + J m.
Theorem_19.20 For all m 0, A m = M m and is therefore a Hopf ring ideal in
H = BP*(BP__even)=V ~=H*(BP__even; Fp).
This result we shall prove in full. For m = 0, it is part of Thm. 19.18.
Higher order relations As in x18, we derive a more useful relation by eliminat*
*ion
from the n relations (Rk1), (Rk2), . . . (Rkn), with multiplication now interpr*
*eted as
Omultiplication. We find it simpler to return to eq. (19.10) rather than try *
*to deal
directly with eq. (19.15).
Definition 19.21 Given any positive integers i1, i2, . . . , in, where n 1, *
*we define
i1pi2 pin
L(i1; i2; : :;:in) and R(i1; i2; : :;:in) as the coefficient of xp1x2 : :x:n in
2 Opn1
b(x1)OpOb(x2)Op O: :O:b(xn1) Ob([p](xn))
and
2 Opn1
b(x1)OpOb(x2)Op O: :O:b(xn1) OP (xn) (19:22)
respectively, where P (x) denotes the right side of eq. (19.10).
Then given any integers 0 < k1 < k2 < : :<:kn, where n > 1, we define the n*
*th
order derived relation
X X
(Rk1;k2;:::;kn) : fflssL(i1; i2; : :;:in) = fflssR(i1; i2; *
*: :;:in)
ss ss
by summing over all permutations ss 2 n, where (i1; i2; : :;:in) = ss(k1; k2; :*
* :;:kn).
(For n = 1, we recover (Rk1).)
This relation lies in BP*(BP__f(n)),_where f(n) denotes the usual numerical*
* function
(1.4). To study it, we work inH . The left side of (Rk1;k2;:::;kn) vanishes,*
* as before.
To handle the right side, we first rewrite (19.22) just as we did eq. (19.8), b*
*y using
eq. (15.6) to move all the Ofactors b() inside the *factors. The term px of*
* [p](x)
produces the *factor
n _ _ 2 _ n1_ *p o
1 + b(x1)OpOb(x2)Op O: :O:b(xn1)Op Ob(xn); (19:23)
and the general term vIxm produces the *factor
n _ _ 2 _ n1_ * o
1 + b(x1)OpOb(x2)Op O: :O:b(xn1)Op Ob(xn)OmO[vI];
to be expanded as in eq. (15.5). By the form [8, eq. (15.5)] of the pseries, t*
*he only
*factors of the latter kind that are not trivial mod M 2 are
_ Op _ Op2 _ Opn1_ Opj
1 + b(x1) Ob(x2) O: :O:b(xn1) Ob(xn) O[vj] (19:24)
JMB, DCJ, WSW  97  23 Feb 1995
Unstable cohomology operations
for j > 0. We can now efficiently extract the coefficient R(i1; i2;*
* : :;:in) of
i1pi2 pin
xp1x2 : :x:n . From the factor (19.23) we have the term
i O O 2 O n1 j
F b(in1)Ob(pi12)Ob(pi23)O:O:b:p(in1n);
after some shuffling, while the factor (19.24) yields
2 Opn1 Opj
bOp(i11)ObOp(i22)O:O:b:(in1n+1)Ob(inj)O[vj]:
(We continue the convention of x18 that meaningless terms, those involving any *
*b(i)
with i < 0, are treated as zero.) We now sum over ss and j, taking the opportun*
*ity to
permute the ir in the terms with F (which introduces a sign), to obtain (Rk1;k2*
*;:::;kn)
in the desired form
X i Op Opn1j
(1)n1 fflssF b(i11)Ob(i22)O: :O:b(inn)
ssX
2 Opn1 Opj (19:25)
+ fflssbOp(i11)ObOp(i22)O:O:b:(in1n+1)Ob(inj)O[vj] 0
ss;j
__
inH mod M 2 + M 1*M 1. As before, the terms involving [vj] for j < n cancel:*
* when
we interchange ij and in, we obtain two identical terms having opposite signs. *
* We
therefore sum only over j n. The terms of most interest are the two leading te*
*rms
with ss = id:
i j i O O n1j
(1)n1F bOL = (1)n1F b(k11)Ob(pk22)O:O:b:p(knn) (19:26)
and 2 n
bOpLO[vn] = bOp(k11)ObOp(k22)O:O:b:Op(knn)O[vn]; (19:27)
for a certain multiindex L (different from x18).
The reduction formula We obtain a reduction formula for the general disallowed
Omonomial (19.17) in BP*(BP__k). First, we assume k is even. For any n > 0,
0 < k1 < k2 < : :<:kn, and multiindices M and J, the desired formula is:
2 Opn OM J
bOp(k11)ObOp(k22)O:O:b:(knn)Ob O[vnv ]
X Op Opn O
 fflssb(i11)O: :O:b(inn)Ob M O[vnvJ]
ss6=id i
n1 Os1(M) J j
+ (1)nF b(k11)ObOp(k22)O:O:b:Op(knn)Ob O[v ] (19:28)
X i Op Opn1 O 1 j
+ (1)n fflssF b(i11)Ob(i22)O: :O:b(inn)Ob s (M)O[vJ]
__ss6=id
inH mod M h+2 + M h+1*M h+1,
where we sum over permutations ss 2 n, (i1; i2; : :;:in) = ss(k1; k2; : :;:kn),*
* and
h = J. (Terms involving s1(M) with m0 6= 0 are to be omitted.) To obtain thi*
*s,
we first apply  ObOM to eq. (19.25), using eq. (15.13) to rewrite the terms in*
*volving
F . The suppressed terms lie in M 2O bOM M 2 and (M 1 * M 1) ObOM M 1* M 1, as
we know from Lemma 19.14(a) that M 2 and M 1 are Hopf ring ideals. Then we apply
the *homomorphism  O[vJ] and use Lemma 19.14(c).
JMB, DCJ, WSW  98  23 Feb 1995
x19. Relations in the Hopf ring for BP
Remark Strictly speaking, this is only a reduction formula mod V , but it meet*
*s our
present needs. One can work modulo the slightly smaller ideal (v1; v2; : :):in*
*stead
and extract a more complicated reduction formula that is valid in BP*(BP__*) it*
*self,
without recourse to Nakayama's Lemma.
For odd k, the reduction formula takes the far simpler form
2 Opn OM J
e ObOp(k11)ObOp(k22)O:O:b:(knn)Ob O[vnv ]
X Op Op2 Opn O
 fflsse Ob(i11)Ob(i22)O: :O:b(inn)Ob M O[vnvJ]
ss6=id
mod M h+2. To see this, one can suspend eq. (19.28) by applying e O, which kil*
*ls all
*products, including F c; but it is far simpler to suspend eq. (18.22) instead.
Proof of Thm. 19.20 For m > 0, it follows from eq. (19.28) that
__
M m Am + M m+1 + M m*M m+ F M m1 inH , (19:29)
by using exactly the same orderings of monomials (reinterpreted) as in the proo*
*f of
Thm. 18.16. For m = 0, we clearly have M 0 = A 0+ M 1 because the generators of
M 0that are not in M 1 are all allowable.
We show by induction on m that the term F M m1 is not needed, that
M m Am + M m+1 + M m*M m (19:30)
for all m 1. This is clear for m = 0. If it holds for m1, applying F yields
F M m1 F Am1 + F M m + F M m1*F M m1 :
Each term on the right is already included in the other terms of eq. (19.29) an*
*d may
be omitted.
Next, we dispose of M m * M m. On *multiplying eq. (19.30) by M *i1we have
M m*M *i1 Am *M i1+ M m+1*M i1+ M m*M m*M i1
Am + M m+1 + M m*M i+11:
It follows by induction on i that
M m Am + M m+1 + M m*M *i1
for all i. Since M *i1is zero in each bigrading for large enough i, we must h*
*ave
M m Am + M m+1. As in the proof of Thm. 18.16, this implies M m = Am . 
The suspension We can use eq. (19.28) to extract detailed information about the
suspension homomorphism e O: Qk*! P BP*(BP__k+1) when k is odd. (When k is
even, there is nothing to discuss: the allowable monomial bIwJ 2 Qk*suspends to*
* the
allowable Omonomial e ObOIO[vJ] 2 P BP*(BP__k+1).)
By Lemma 18.12(c), we can write every allowable monomial in Qk*uniquely in *
*the
extended canonical form
h1(L))sh(M)hJ
c = e bL(0) b(p1)(L+s(L)+:::+s b wn+1w ;
n
where 0 = k0 k1 k2 : : :kn, n 0, bL = b(k0)bp(k1):b:p:(kn), M and J satisfy
the conditions (18.9), and h 0 is maximal. What happens to e Oc is that if h >*
* 0,
JMB, DCJ, WSW  99  23 Feb 1995
Unstable cohomology operations
it is disallowed, as the derived relation (Rk0+1;k1+2;:::;kn+n+1) applies, and *
*we pick out
the leading term (19.26) mod V . If h > 1, we can repeat this cycle h times (al*
*ways
with the same indices ku). In all cases, e Oc has the leading term
F h(bOLO bOM O[vJ]); (19:31)
__ *
* O
where bOLO bOM O[vJ] is allowable by Lemma 18.12(d) and primitive inH because*
* b L
contains the factor b(0).
In fact, one can show that every primitive allowable Omonomial in BP*(BP__*
*k+1)
can be written uniquely in the form bOLO bOM O[vJ], subject to the conditions (*
*18.9).
We have a computational verification mod V of the isomorphism Qk*~=P BP*(BP__k+*
*1)
induced by suspension.
The first nth order relation The relation (R1;2;:::;n) is particularly importa*
*nt,
as only the two leading terms are meaningful. Bendersky has pointed out (during
the proof of [3, Thm. 6.2]) that with a little more attention to detail, one ob*
*tains a
sharper version, the unstable analogue of Lemma 18.23.
Lemma 19.32 (Bendersky) In BP*(BP__f(n)) we have the relation
bOpm(0)O[vn] vnbOm(0)+ (1)n(bOm(0))*p mod InBP*(BP__f(n)), (19:33)
for each n > 0, where m = f(n)=2 = 1 + p + p2 + : :+:pn1.
Proof Although this result can be extracted from (R1;2;:::;n) by detailed exam*
*ination,
it is far simpler to return to (Rn). We proceed by induction on n, starting fr*
*om
eq. (19.11) for n = 1. For n > 1, we assume the result for all smaller n, and o*
*btain it
for n by evaluating bOph(0)O(Rn) mod In, where h = f(n1)=2 = 1 + p + p2+ : :+:*
*pn2.
n
We recall that (Rn) is defined as the coefficient of xp in eq. (19.10). On*
* the left,
n *
* Oph+1
we have bOph(0)Ob(vnxp + : :):by [8, eq. (15.5)], which provides only the term *
*vnb(0) .
The right side simplifies enormously, because h > 0 and b(0)O kills *decompos*
*ables;
we obtain _ X _
bOph(0)OP (x) = pbOph(0)Ob(x) + bOph(0)Ob(x)OmO[vI] :
;I;m
By induction, bOph(0)O[vj] 0 mod In for all j < n  1, since h = f(n1)=2 > f(*
*j)=2.
n
Thus the only terms of interest in [p](x) in our range of degrees are vnxp *
* and
n1
vn1xp , as it follows from [8, eq. (14.26)] and the map BP ! K(n1) of ring
spectra that any terms in eq. (19.9) of the form vin1xm with i > 1 have divis*
*ible
n Oph Opn
by p. The term vnxp yields b(0)Ob(0)O[vn], which is the leading term (19.27).*
* By
n1
induction and eq. (15.13), vn1xp yields
n1 n1 Oh Opn1 n1 Oh Opn1
bOph(0)ObOp(1)O[vn1] (1) F (b(0)) Ob(1) (1) F (b(0)Ob(0) );
which is the other leading term, (19.26). 
The ideals Jn For the unstable version of our splitting theorems we need the u*
*n
stable analogue of the ideal Jn of Defn. 18.25.
Definition 19.34 For n 0, we define Jn BP*(BP__*) as the *ideal generated
by all elements of the form c O([vj]1), where j > n.
JMB, DCJ, WSW  100  23 Feb 1995
x19. Relations in the Hopf ring for BP
Lemma 19.35 Jn is a Hopf ring ideal in BP*(BP__*).
Proof We apply Lemma 19.2; only (i) requires any comment. It holds for [vj]  *
*1,
by the identity
([v]  1) = ([v]1)[v] + 1([v]1); (19:36)
P *
*0 00
which is valid for any v 2 BP *by Prop. 11.2(a). We combine this with c = ic*
*ici
to obtain
X X
(cO([v]1)) = c0iO([v]1) c00iO[v] + c0iO1 c00iO([v]1);(19:37)
i i
which shows that (i) holds for the typical *generator of Jn. 
Lemma 19.38 [v] 1 mod J nfor all v 2 Jn.
Proof Suppose v = v0+ vjvK with j > n. As Jn is a Hopf ring ideal, we have
[v] = [v0] * [vK ]O[vj] [v0] * [vK ]O1 = [v0] mod Jn .
The result follows by induction on the number of terms in v. 
The unstable analogue of Lemma 18.26 requires more detail but no new ideas.
Lemma 19.39 For k f(n+1), Jn\BP*(BP__k) is the *ideal in BP*(BP__k) genera*
*ted
by all elements that lie in BP*(BP__k) and have any of the following forms, whe*
*re vJ
contains a factor vj with j > n:
(i)(if k is even) an allowable monomial bOIO[vJ];
(ii)(if k is odd) an allowable monomial e ObOIO[vJ];
(iii)(if k 0 and is even) [vJ]  1k, with 2 Z(p);
(iv)(if k = f(n+1)) a disallowed monomial
2 Opn+1
bOp(k11)ObOp(k22)O:O:b:(kn+1n1)O[vn+1]
with 0 < k1 < k2 < : :<:kn+1.
Remark To make (i) correct for I = 0, it is necessary to define bO0= eO0= [1] *
* 1 as
in Prop. 13.7, so that bO0O[vJ] = [vJ]  1.
Proof Denote by I the *ideal in BP*(BP__k) generated by the stated elements. *
*It is
clear from Lemma 19.38 that I Jn.
To show the converse, we fix k and a large m, and prove by downward inducti*
*on on
h that all elements in BPi(BP__k) of the form c O([vh]1) lie in I whenever i <*
* m. This
statement is vacuous for sufficiently large h (depending on m and k). We theref*
*ore
fix t > n and assume the statement holds for all h > t.
Case 1: c = [vJ]. (This includes the degenerate cases [1] and 1k = [0k].) T*
*hen
c O([vt]1) = [vJvt]  1 is listed in (iii).
Case 2: c = efflObOI. As in Lemma 18.26, c O([vt]1) = efflObOIO[vt] has t*
*o be
allowable, except in the extreme case when k = f(n+1) and j = n + 1; either way*
*, it
is a listed generator of I.
Case 3: c = efflObOIO[vhvJ] allowable, where h t. From the form of Defn. 1*
*9.16,
c O([vt]1) = efflObOIO[vhvtvJ] remains allowable and is thus a listed generato*
*r of I.
JMB, DCJ, WSW  101  23 Feb 1995
Unstable cohomology operations
Case 4: c = efflObOIO[vhvJ], with h > t. We can write c O([vt]*
*1) =
efflObOIO[vtvJ] O([vh]1), which lies in I by induction, provided i < m.
By Thm. 19.18, we have enough *generators c. If c = a*d, eqs. (10.11) and *
*(19.36)
give
c * ([vt]1) = aO([vt]1) * dO[vt] + aO1 * dO([vt]1);
which shows that the statement holds for c = a * d whenever it holds for a and *
*d. 
20 Additively unstable BP objects
In this section, we discuss the additively unstable structures developed in*
* xx5, 7
in the case E = BP , with particular attention to what becomes of the stable re*
*sults
of [8, x15]. We easily recover Quillen's theorem, that for any space X, the ge*
*nera
tors of BP *(X) all lie in nonnegative degrees. Our main result Thm. 20.11 say*
*s in
effect that there are no relations there either; more precisely, all relations *
*follow from
relations in nonnegative degrees. We apply the theory to Landweber filtration*
*s of
an additively unstable module or algebra M, and find that the presence of addit*
*ive
unstable operations implies severe constraints on the degrees of the generators*
* of M;
this may be viewed as a better version of Quillen's theorem.
By Thms. 6.35 and 7.11, module and comodule structures are equivalent, with
or without multiplication. The most convenient context remains the Second Answer
of x5, that an additively unstable BP cohomology module (algebra) consists of a
BP *module (BP *algebra) M equipped with coactions
aeM : Mk ! M b Q(BP )k* (20:1)
that (as k varies) form a homomorphism of BP *modules (BP *algebras) and sati*
*sfy
the usual coaction axioms (6.33). We continue to abbreviate Q(BP )**to Q**. T*
*he
bigraded algebra Q**was discussed in detail in x18.
Connectedness The principle is that nothing interesting ever happens in negati*
*ve
degrees. The first result in this direction is due to Quillen [22, Thm. 5.1].
Theorem 20.2 (Quillen) For any space X, BP *(X)^is generated, as a BP *modu*
*le,
(topologically if X is infinite) by elements of positive degree and exactly one*
* element
of degree 0 for each component of X.
This will be an immediate consequence of Lemmas 4.10 of [8] and 20.5 (below*
*).
Quillen's proof is geometric; in contrast, x6 provides a global algebraic proof*
* of the
weak form of Quillen's theorem.
Theorem 20.3 Given any integer k < 0, there exist for n 1:
(i)additive unstable BP operations rn defined on BP k(), with deg(rn) !*
* 1
and deg(rn) k for all n;
(ii)elements v(n) 2 BP *;
such that in any additively unstable BP cohomology module M (e. g. BPP*(X)^for*
* any
space X), any x 2 Mk decomposes as the (topological infinite) sum x = nv(n)rn*
*x,
with deg(rnx) 0 for all n.
In particular, M is generated (topologically) by elements of degree 0.
JMB, DCJ, WSW  102  23 Feb 1995
x20. Additively unstable BP objects
Proof Let {c1; c2; c3; : :}:be the RavenelWilson (or any other) basis of the *
*free BP *
module Qk*. By eq. (6.39) and the following Remark, we can write
X
x = kx = xn (20:4)
n
with xn = rnx, where rn denotes the operation dual to cn. If cn 2 Qkj, we must *
*have
j 0; then deg(rn) =  deg(cn) = j  k k gives (i). We put v(n) = and
note that deg(xn) = deg(rn) + deg(x) = j 0. 
Remark The coefficients in eq. (20.4) are readily computed from eq. (6.41) as *
*v(n) =
Q(ffl)cn. Thus v(n) = vJ if cn = efflbm(0)wJ, and vanishes for monomials cn not*
* of this
form, so that many terms in eq. (20.4) are zero.
If M is bounded above or X is finitedimensional, the sum is finite and no *
*topology
on M is needed.
To handle the generators in degree 0, we need a stronger hypothesis.
Lemma 20.5 Let M be a connected (see Defn. 7.14) additively unstable algebra
(e. g. BP *(X)^ for any connected space X). Then as a topological BP *module, *
*M is
generated by 1M 2 M0 and elements of strictly positive degree. The generator 1*
*M is
never redundant.
Remark Again, we may ignore the topology on M if M is bounded above or X is
finitedimensional.
Proof We choose a basis {c1; c2; c3; : :}:of Q0*with c1 = 1; then given x 2 M0*
*, we
have eq. (20.4) with deg(xn) =  deg(cn) > 0 for all n > 1. Thus x <0; 1> x1
mod L, where L denotes the BP *submodule of M generated (topologically) by the
elements of positive degree.
For the collapse operation 0 introduced in Defn. 7.13, we similarly have 0x
<0; 1> mod L. But <0; 1> = <0; 1> = 1. As M is connected, 0x = 1M for some
2 Z(p), by Defn. 7.14. We deduce from Thm. 20.3 that M = L + (BP *)1M . Since
L = 0 and (v1M ) = v1M for any v 2 BP *, this is a direct sum decomposition. 
Primitive elements We generalize the theory of Landweber filtrations to the ad
ditive unstable context by following the same strategy as stably. We explore a *
*general
unstable comodule M by looking for morphisms f: BP *(Sk; o) ! M, for any k 0.
As a BP *module, BP *(Sk; o) is free on the canonical generator uk. Thus f is *
*deter
mined, as a homomorphism of BP *modules, by the element x = fuk 2 M. Since
aeSuk = uk ek by Prop. 12.3(a), the condition we need is clear.
Definition 20.6 Let M be any unstable comodule. If k 0, we call x 2 Mk
additively unstably primitive if aeM x = x ek in M b Qk*.
This obviously stabilizes to [8, Defn. 15.9], so that the additively unstab*
*le primi
tives of M form a subgroup of the stable primitives of M. We do not define prim*
*itives
in negative degrees, for lack of a space Sk, and because ek is meaningless. In *
*fact, for
k < 0, x 1 does not in general lie in the image of the stabilization
M Q(oe): M b Qk*! M b BP*(BP; o) :
JMB, DCJ, WSW  103  23 Feb 1995
Unstable cohomology operations
(Perhaps it never does?)
Remark One might object that we have abolished primitives in negative degrees *
*by
simply defining them away, while some alternate definition might work. However,*
* no
such definition can be satisfactory.
It is obvious from Defn. 12.6 that if x 2 M is primitive, so is x 2 M. On
the other hand, we shall find (nontrivially) in Cor. 20.12 that the only primit*
*ive in
M of degree zero is 0 (at least, for the kind of comodule we discuss). It foll*
*ows,
by suspending enough, that no definition of primitive can have both these prope*
*rties
and produce anything interesting in negative degrees.
It is immediate from the definition that if x 2 Mk is primitive,
aeM (vx) = x ekjRv in M b Q** (for v 2 BP *). (20:7)
We again recall from eq. (1.4) the numerical function
2(pn  1) n1 n2
f(n) = _________= 2(p + p + : :+:p + 1)
p  1
and remind that deg(vn) = (p1)f(n) for n > 0.
Lemma 20.8 Let x 2 Mk be a nonzero primitive element of the unstable BP 
cohomology comodule M, and take n > 0.
(a) If k < pf(n), then vinx 6= 0 for all i > 0 and is not additively unstabl*
*y primitive;
(b) If k pf(n) and Inx = 0, then vnx is additively unstably primitive.
Corollary 20.9 If the additively unstably primitive element x 2 M satisfies *
*Inx =
0 and is a vntorsion element, then:
(a) deg(vinx) pf(n) whenever vinx 6= 0;
(b) vinx is additively unstably primitive or zero for all i.
Proof We apply the Lemma to vinx by induction on i. Part (a) never applies (u*
*nless
vinx = 0); hence (b) must apply, to show that vi+1nx is primitive. 
All this follows easily from Lemma 18.23.
Proof of Lemma 20.8 From eq. (20.7) we have
aeM (vinx) = x ekwin:
In case (a), we note that by Defn. 18.4, ekwinis a basis element of Q**, so tha*
*t aeM (vinx)
is clearly nonzero. Even if k 2(pn1)i, vinx is not primitive because aeM (vi*
*nx) is
different from n n
vinx ek2(p 1)i= x vinek2(p 1)i:
In case (b), we use the same formulae, with i = 1. The difference is that *
*by
Lemma 18.23, they now coincide, since e2 = b(0)and Inx = 0. 
Remark For any x 2 Mk, where k 0, the coaction axiom (ii) of [8, eq. (8.7)] f*
*orces
aeM x to have the form X
aeM x = x ek + xff cff;
ff
JMB, DCJ, WSW  104  23 Feb 1995
x20. Additively unstable BP objects
where the cffare other RavenelWilson basis elements and deg(xff) > k. Assuming
that k < pf(n), so that ekwn is a basis element, let r be the operation (or fun*
*ctional)
dual to it. Proceeding as in the proof of the Lemma, we obtain
X
r(vnx) = x + xff;
ff
which shows that vnx 6= 0 if (for example) x is a module generator of M.
Landweber filtrations The preceding results allow us to sharpen Thms. 15.10
and 15.11 of [8].
Theorem 20.10 Let M be the BP *module with the single generator x 2 Mk and
Ann (x) = In, so that M ~=k(BP *=In).
(a) If n > 0, M admits an unstable comodule structure if and only if k f(n)*
*2,
and it is unique. The additively unstably primitive elements are those of the *
*form
vinx, where 2 Fp, and k + deg(vin) f(n) if i > 0.
(b) If n = 0, M ~= kBP *admits an unstable comodule structure if and only if
k 0, and it is unique. The additively unstably primitive elements are those of*
* the
form x, with 2 Z(p).
Remark Unlike the stable case, there are only finitely many primitives for n >*
* 0.
Of course, our definition forces this by requiring the degree of a primitive el*
*ement to
be nonnegative. However, the theorem gives a much stronger condition.
Proof By Thm. 20.3, we must have k 0, the canonical generator x is necessarily
primitive, and ae must be given by eq. (20.7). Thus in (a), ae will be well de*
*fined if
and only if ek(jRv) 2 InQ**whenever v 2 In. Lemma 18.23 shows that this holds
for v = vi for all i < n, since k f(n)  2 pf(i); this is sufficient. On the *
*other
hand, if k < f(n)  2 = pf(n1), Lemma 20.8(a) (with n replaced by n1) would
contradict vn1x = 0.
Because ae is a BP *module homomorphism (when it exists), the coaction axi*
*oms
[8, eq. (8.7)] need only be checked on x, where they are obvious. (Alternative*
*ly,
k(BP *=In) is a quotient of the geometric comodule BP *(Sk; o).)
Since any additively unstably primitive element is also by design stably pr*
*imitive,
[8, Thm. 15.10] restricts the candidates for primitives to vinx. Lemma 20.8 sho*
*ws,
by induction on i 0, that vi+1nx is additively unstably primitive if and only *
*if
deg (vinx) pf(n). This is what we want, since  deg(vn) = (p1)f(n).
The proof of (b) is similar, but far simpler. 
With this restriction on the basic building blocks for an unstable module, *
*we
obtain the expected improvement in [8, Thm. 15.11].
Theorem 20.11 Let M be an unstable BP cohomology comodule that is finitely
presented as a BP *module and has the discrete topology. Then there exists a f*
*iltration
by subcomodules
0 = M0 M1 : : :Mm = M;
JMB, DCJ, WSW  105  23 Feb 1995
Unstable cohomology operations
where each Mi=Mi1 is generated, as a BP *module, by a single element xi, whose
annihilator ideal Ann (xi) = Ini for some ni, and deg(xi) f(ni)  2 (if ni > 0*
*), or
deg (xi) 0 (if ni= 0).
If, further, M is a spacelike BP *algebra (see Defn. 7.14), for example BP*
* *(X) for
any finite complex X, we can take each Mito be an invariant ideal in M. At the *
*last
stage, we may take xm = 1 and nm = 0 or 1.
Unfortunately, although the statement of the Theorem is exactly as expected,
Landweber's method fails; Lemma 2.3 of [16] does not appear to be available her*
*e.
(The BP *submodule 0: In = {y 2 M: Iny = 0} of M is defined but does not appea*
*r to
be unstably invariant, owing to the dimensional restriction in Lemma 18.23.) I*
*nstead,
we are forced to construct a suitable primitive x1 2 M directly. We would have
preferred Landweber's construction because it guarantees that Ann (x1) is maxim*
*al,
which is useful in applications.
Proof We start with a nonzero element x 2 Mk of top degree; by Thm. 20.2, k
0 and x is automatically primitive. We construct a sequence of nonzero primiti*
*ve
elements ys 2 M such that Isys = 0, starting with y0 = x. (Here, it is conveni*
*ent
to write v0 = p.) We stop when we reach an element yn that is vntorsionfree
(vinyn 6= 0 for all i > 0) and put x1 = yn and n1 = n; this must occur eventual*
*ly,
by Lemma 20.8(a) (e. g. when 2pn > k). Assume we have ys, where s 0. If it is
vstorsionfree, we stop; this is yn. Otherwise, take the smallest exponent q s*
*uch that
vqsys = 0 and put ys+1 = vq1sys, to get Is+1ys+1 = 0. By Cor. 20.9 (with s in *
*place of
n), ys+1 is primitive and deg(ys+1) pf(s).
We have found a primitive x1 such that Inx1 = 0, x1 is vntorsionfree, and
deg (x1) pf(n1) = f(n)  2. (If n = 0, there was no induction, and deg(y0) =
k 0.) As Ann (x1) is an invariant ideal (in the stable sense), its radical ide*
*al must
be a finite intersection of invariant prime ideals in BP *, therefore be Im for*
* some m.
That is, q ________
In Ann (x1) Ann (x1)= Im :
q ________
Since vn 62 Ann (x1), we conclude that m = n and Ann (x1) = In.
We finish as in the stable case, by setting M1 = (BP *)x1, observing that t*
*his
submodule is invariant by eq. (20.7), and replacing M by M=M1. The induction
continues until M = 0, and must terminate (easily, unlike the stable case), bec*
*ause
each Mk is a finitely generated module over the Noetherian ring Z (p)and we need
consider only k 0.
Now assume that M is a spacelike algebra, i. e. a product of connected alge*
*bras.
This product is evidently finite, otherwise M would be uncountable. We easily r*
*educe
to the case when M is connected, which includes the case when M = BP *(X) for a
connected finite complex X. By Lemma 20.5, the module (BP *)x is automatically *
*an
ideal in M; by induction, so is (BP *)ys for each s, in particular M1. At the l*
*ast step,
the module M=Mm1 is also an algebra; we therefore have 1 = vxm and x2m= v0xm
for some v,v0 2 BP *. Then xm = 1xm = vx2m = vv0xm = v01, which shows that
Ann (1) = Ann (xm ) = Inm , and we may replace the generator xm by 1. This impl*
*ies
nm 1, since f(n) > 2 for n 2. 
JMB, DCJ, WSW  106  23 Feb 1995
x21. Unstable BP algebras
Corollary 20.12 For M as in Thm. 20.11, the suspension M contains no nonzero
additively unstably primitive elements in degree zero.
Proof We observe that
0 = M0 M1 : : :Mm = M
is a Landweber filtration of M. By Thm. 20.10, the only unstable comodule of the
form k(BP *=In) that has a nonzero primitive in degree zero is BP *, which does*
* not
occur as a Landweber factor Mi=Mi1of M. 
21 Unstable BP algebras
In this section, we apply the theory of xx10, 19 to an unstable BP cohomol*
*ogy
algebra M. Our main application is Thm. 21.12 on Landweber filtrations of M,
which contains Thm. 1.5 and improves on Thm. 20.11 by one degree.
Of course, we can always recover an additively unstable algebra from an uns*
*table
algebra simply by discarding the nonadditive operations. As a general rule, we *
*can
improve our results by one degree (but never more than one, in view of Thm. 13.*
*6)
by retaining all operations, at the cost of working in a far more complicated a*
*nd
unfamiliar environment. We developed the necessary machinery in x10.
Primitive elements It is clear from x20 that the way to study a general unstab*
*le
algebra M is to look for unstable morphisms f: BP *(Sk) ! M from the (relativel*
*y)
well understood object BP *(Sk). Since BP *(Sk) is a free BP *module with bas*
*is
{1S; uk}, f is uniquely determined, as a homomorphism of BP *modules, by f1S =*
* 1M
and the element x = fuk 2 Mk. We extend the concept of primitive element to the
unstable context, using Prop. 13.7 as a guide.
Definition 21.1 We call x 2 Mk (where k 0) unstably primitive if
r(x) = 1M + x for all r, (21:2)
where we interpret e0 = [1]  10 (as in Prop. 13.7).
This is a necessary and sufficient condition for f to be a morphism of unst*
*able
algebras, by eqs. (10.41), (10.16), and the Cartan formula (10.23). Among the u*
*nsta
ble operations is the squaring operation, defined by r(y) = y2 for all y, which*
* implies
that f is a homomorphism of BP *algebras (even if k = 0). When we restrict to
additive operations, x is automatically additively primitive, and we have avail*
*able all
the results of x20.
Many elementary properties of primitives follow directly from the definitio*
*n.
Proposition 21.3 Let M be an unstable algebra. Then:
(a) Unstable primitives are natural: if x 2 M is unstably primitive and f: M*
* ! N
is a morphism of unstable algebras, then fx 2 N is also unstably primitive;
(b) The elements 0 2 Mk (for any k 0) and 1M are unstably primitive;
(c) If x 2 Mk is unstably primitive, where k > 0, then x2 = 0;
JMB, DCJ, WSW  107  23 Feb 1995
Unstable cohomology operations
(d) If x 2 Mk is unstably primitive, where k > 0, then x is unstably primiti*
*ve
for any 2 Z(p);
(e) If k > 0 is odd, the unstable primitives in Mk form a Z(p)submodule;
(f) If k > 0 is even and x; y 2 Mk are unstably primitive, then x + y is uns*
*tably
primitive if and only if xy = 0;
(g) The only nonzero unstable primitive in BP *= BP *(T ) is 1;
(h) Any unstable primitive x 2 M0 is idempotent, x2 = x;
(i) If x 2 M0 is unstably primitive (and therefore idempotent), then the con*
*jugate
idempotent 1M x is also unstably primitive, but x is never unstably primitive
(unless x = x).
Proof Part (a) is trivial. Part (b) is clear from eqs. (10.41) and (10.28). As*
* noted
above, f is an algebra homomorphism, which gives (c) and (h). Then (g) follows *
*from
(b) and (h).
In (d), eq. (10.16) gives
r(x) = 1M + x :
Since k > 0, Prop. 13.7(g) gives [] Oek = ek, which shows that x is primitive.
We prove (e) and (f) together. If x; y 2 Mk are primitive, the Cartan form*
*ula
(10.23) yields
r(x+y) = 1M + x + y + (1)k xy;
which is to be compared with eq. (21.2). The unwanted last term vanishes if k i*
*s odd,
because ek is then an exterior generator; but if k is even, ek * ek is a basis *
*element of
BP*(BP__k). For (e), we combine this with (d).
For (i), we first use eq. (10.29) to compute r(x) = 1M + x,
which shows that x is not primitive. We then use eqs. (10.23) and (10.41) to c*
*ompute
r(1M x) = 1M + x, which shows that 1M  x is primitive.*
* 
We deduce that the Remark following Defn. 20.6 extends to show that unstable
primitives cannot usefully be defined in negative degrees, even though the unst*
*able
suspension (see Defn. 13.4) had to be defined somewhat differently.
__
Corollary 21.4 Let M = BP *M be a based unstable BP algebra.
__ __
(a) If x 2M is unstably primitive, so is x 2 BP * M ;
(b) If M is the kind of algebra considered_in Thm. 20.11, there are no unsta*
*ble
primitives of degree zero in BP * M other than 0 and 1 2 BP *.
*
* __
Proof Part (a) is clear from eq. (13.3). For (b), take any primitive y 2 BP * M
in degree 0. By Prop. 21.3(g), its augmentation in BP *must be_0 or 1; if_1,_we*
* use
Prop. 21.3(i) to replace y by 1  y. Then y = x for some x 2M . As y 2 M is
also additively primitive, Cor. 20.12 shows that y = 0. 
` * *
If X is the disjoint union X1 X2 of two spaces, we have BP (X) = BP (X1)
BP *(X2), a product of unstable algebras. By Prop. 21.3, the elements (1; 0) an*
*d (0; 1)
are primitive idempotents in BP *(X). The converse is also true, algebraically.
JMB, DCJ, WSW  108  23 Feb 1995
x21. Unstable BP algebras
Theorem 21.5 If x 2 M0 is an unstably primitive element in the unstable alge*
*bra
M, other than 0 and 1M , so that x and 1M  x are idempotents, we have the spli*
*tting
M ~=xM (1M x)M of M as a product of unstable algebras.
Proof By Prop. 21.3(i), both x and 1M  x are primitive and idempotent. We def*
*ine
the first projection pK : M ! K = xM by pK y = xy; since x is idempotent, pK is
a homomorphism of BP *algebras. We define pL: M ! L = (1M x)M similarly, by
pLy = (1M x)y. These will give the desired splitting of M.
Given y 2 M, we assume that rM (y) is in the standard form (10.22), where rM
denotes the operation of r on M. By the Cartan formula (10.36),
X X
rM (xy)= yfi+ xyfi
fi fi
X
= xrM (y) + (1M x)yfi:
fi
Hence xrM (xy) = xrM (y), which shows that pK is an unstable morphism, provided
we define the action rK : K ! K of r on K by rK (z) = xrM (z) for z 2 K M. All
the necessary laws are inherited from M. We treat pL similarly. 
Landweber filtrations We repeat the theory of x20, with an improvement of one
in degree. If x 2 Mk is primitive in the unstable algebra M, where k > 0, we co*
*mpute
from eq. (10.16) that
r(vx) = 1M + x (21:6)
for any v 2 BP h.
Lemma 21.7 Let M be an unstable algebra, and x 2 Mk an unstably primitive
element, where k > 0. Then the BP *submodule (BP *)x generated by x is an unst*
*ably
invariant ideal in M, provided it is an ideal.
Proof We apply Lemma 8.10, with the help of eq. (21.6). 
It is still true that an element of positive top degree in M is automatical*
*ly primi
tive, for lack of any other possible terms in r(x).
We now use the additional structure of the unstable operations to sharpen
Lemma 20.8. We recall once more from eq. (1.4) the numerical function
2(pn  1) n1 n2
f(n) = _________= 2(p + p + : :+:1) :
p  1
Lemma 21.8 Let x 2 Mk be a nonzero unstably primitive element of the unstable
algebra M, and n > 0.
(a) If k pf(n), then vinx 6= 0 for all i > 0 and is not unstably primitive;
(b) If k > pf(n) and Inx = 0, then vnx is unstably primitive.
Corollary 21.9 If the unstably primitive element x 2 M satisfies Inx = 0 and*
* is
a vntorsion element, where n > 0, then:
(a) deg(vinx) > pf(n) whenever vinx 6= 0.
(b) vinx is unstably primitive or zero for all i.
JMB, DCJ, WSW  109  23 Feb 1995
Unstable cohomology operations
Proof This is formally the same as for Cor. 20.9. 
Proof of Lemma Part (a) adds nothing to Lemma 20.8(a) unless k = pf(n), in
which case we must take i = 1 if we are to have deg(vinx) 0.
To test whether or not vnx is primitive, we have to compare
r(vnx) = 1M + x
from eq. (21.6) with
1M + vnx = 1M +