Unspecified Journal
Volume 00, Number 0, Pages 000-000
S ????-????(XX)0000-0
THE UNITS OF A RING SPECTRUM AND A LOGARITHMIC
COHOMOLOGY OPERATION
CHARLES REZK
Abstract.We construct a "logarithmic" cohomology operation on Morava E-
theory, which is a homomorphism defined on the multiplicative group of i*
*nvertible
elements in the ring E0(K) of a space K. We obtain a formula for this ma*
*p in terms
of the action of Hecke operators on Morava E-theory. Our formula is clos*
*ely related
to that for an Euler factor of the Hecke L-function of an automorphic fo*
*rm.
1. Introduction
Recall that if R is a commutative ring, then the set Rx R of invertible el*
*ements of
R is naturally an abelian group under multiplication. This construction is a fu*
*nctor from
commutative rings to abelian groups. In general, there is no obvious relation b*
*etween
the additive group of a ring R and the multiplicative group of units Rx . Howev*
*er, under
certain circumstances one can define a homomorphism from (a subgroup of) Rx to a
suitable completion of R, e.g., the natural logarithm Qx>0! R, or the p-adic lo*
*garithm
(1 + pZp)x ! Zp.
The "logarithmic cohomology operation" is a homotopy theoretic analogue of t*
*he
above, where R is a commutative S-algebra and "completion" is Bousfield localiz*
*ation
with respect to a Morava K-theory. The purpose of this paper is to give a formu*
*la for the
logarithmic operation (in certain contexts) in terms of power operations. Befor*
*e giving
our results we briefly explain some of the concepts involved.
1.1. Commutative S-algebra. A spectrum is a topological object which represents
a generalized homology and cohomology theory. A commutative S-algebra is a spec-
trum equipped with a commutative multiplication; such a spectrum gives rise to *
*a coho-
mology theory which has a commutative product, as well as power operations. Jus*
*t as
any ordinary commutative ring is an algebra over the ring Z of integers, so a c*
*ommutative
S-algebra is an algebra over the sphere spectrum S.
The definition of commutative S-algebra is rather technical; it is the resul*
*t of more
than twenty years of effort, by many people. There are in fact several differen*
*t models of
commutative ring spectra; the commutative S-algebra in the sense of [EKMM97 ],*
* or a
symmetric commutative ring spectrum in the sense of [HSS00 ], or some other equ*
*ivalent
model. These models are equivalent, in the sense that they have equivalent homo*
*topy
theories; for the purpose of stating results it does not matter which model we *
*use.
____________
Received by the editors June 9, 2005.
1991 Mathematics Subject Classification. 55N22; 55P43, 55S05, 55S25, 55P47, *
*55P60, 55N34, 11F25.
The author was partially supported by the National Science Foundation.
cO0000 (copyright hold*
*er)
1
2 CHARLES REZK
1.2. Power operations. A spectrum R which admits the structure of a commutative
ring up to homotopy gives rise to a cohomology theory X 7! R*(X) taking values
in graded commutative rings. The structure of commutative S-algebra on R is mu*
*ch
stronger than this; it provides not just a ring structure on homotopy groups, b*
*ut also
power operations, which encode "higher commutativity". Let m denote the symmet-
ric group on m letters, and let B m denote its classifying space. If R is a com*
*mutative
S-algebra, there are natural maps
Pm :R0(X) ! R0(B m x X),
with the property that the composite of Pm with restriction along an inclusion *
*{*}xX !
B m x X is the mth power map ff 7! ffm on R0(X). In other words, for a commutat*
*ive
S-algebra, the mth power map is just one of a family of maps parameterized (in *
*some
sense) by the space B m . An exposition of power operations and their propertie*
*s can
be found in [BMMS86 ].
For suitable R, one can construct natural homomorphisms of the form
ae: R0(B m x X) ! D R0 R0(X),
where D is an R0-algebra, and so get a cohomology operation of the form
op :R0(X) Pm--!R0(B m x X) ae-!D R0 R0(X).
Such functions op are what are usually called power operations.
For instance, the Eilenberg-MacLane spectrum HR associated to an ordinary co*
*m-
mutative ring R is the spectrum which represents ordinary cohomology; when R = *
*Fp,
power operations are the Steenrod operations. Topological K-theory spectra, bot*
*h real
and complex, admit a number of power operations, including the exterior power o*
*pera-
tions ~k and the Adams operations _k. Other theories of interest include some e*
*lliptic
cohomology theories, including the spectrum of topological modular forms [Hop02*
* ]; bor-
dism theories, including the spectrum MU of complex bordism.
1.3. Formal groups and isogenies. Recall that a multiplicative cohomology theory
is complex orientable if R*(CP 1) is the ring of functions on a one-dimensional*
* com-
mutative formal group. (In this paper, all formal groups are commutative and o*
*ne-
dimensional.)
If R is a commutative S-algebra whose associated cohomology theory is complex
orientable with formal group G, and op is a power operation on R which is a ring
homomorphism as above, then
R0(CP 1) ! D R0 R0(CP 1)
is an homomorphism i*G ! G of formal groups; here i: R0 ! D is a map of rings, *
*and
i*G is the formal group obtained by extension of scalars along i. For example, *
*complex
K-theory is complex orientable, and K0(CP 1) is the ring of functions on the fo*
*rmal
multiplicative group ^Gm; the Adams operation _k corresponds to the k-th power *
*map
G^m ! ^Gm.
The philosophy is that power operations (of degree m) on a complex orientabl*
*e com-
mutative S-algebra R should be parameterized by a suitable family of isogenies *
*(of degree
m) to the associated formal group. This philosophy is best understood in the c*
*ase of
Morava E-theories, which we now turn to.
LOGARITHMIC COHOMOLOGY OPERATION 3
1.4. Power operations on Morava E-theory. Fix 1 n < 1 and a prime p. Let
k be a perfect field of characteristic p, and 0 a height n formal group over k*
*. Such a
formal group admits a Lubin-Tate universal deformation [LT66 ], which is a form*
*al group
defined over a ring
O Wk[[u1, . .,.un-1]];
W k is the ring of p-typical Witt vectors on k. Morava E-theory is a 2-periodic
complex orientable cohomology theory with as its associated formal group; thu*
*s ss*E
O[u, u-1] with O in degree 0 and u in degree 2. The Hopkins-Miller theorem (see*
* [GH ],
[RR04 ]) states that Morava E-theories admit a canonical structure of commutati*
*ve S-
algebra.
Power operations for Morava E-theories were constructed by Ando [And95 ]; se*
*e also
[AHS04 ]. These operations are parameterized by level structures on the associa*
*ted Lubin-
Tate universal deformation. To each finite subgroup A of the infinite torsion *
*group
* (Qp=Zp)n is associated a natural ring homomorphism _A :E0X ! D E0 E0X,
where D is the E0-algebra representing a full Drinfel'd level structure f : * !*
* i* ;
the ring D was introduced into homotopy theory in [HKR00 ]. The associated isog*
*eny
i*G ! G has as its kernel the subgroup generated by the divisor of the image of*
* f|A in
i*G.
An expression in the _A's which is invariant under the action of the automor*
*phism
group of * descends to a maprE0X ! E0X (see x1.12 below). When n = 1 and E is
p-adic K-theory, then _A = _p for A Z=pr * Qp=Zp. A precise definition *
*of
the operations _A is given in x11.7.
1.5. Units of a commutative ring spectrum. To a commutative S-algebra R is
associated a spectrum gl1(R), which is analogous to the units of a commutative *
*ring (see
x2). The 0-space of the spectrum gl1(R) is denoted GL1(R), and it is equivalent*
* up to
weak equivalence with subspace of 1 R.
Write Hq(X; E) def=Eq(X) for a space X and a spectrum E. Then gl1(R) gives a
generalized cohomology theory which, in degree q = 0 is given by
H0(X; gl1(R)) (R0(X))x .
The higher homotopy groups for the spectrum gl1(R) are given by
ssqgl1(R) = eH0(Sq; gl1(R)) (1 + eR0(Sq))x (R0(Sq))x , q > 0.
In particular, there is an isomorphism of groups ssqgl1(R) ssqR for q > 0, de*
*fined
by "1 + x 7! x". This isomorphism of homotopy groups is induced by the inclusi*
*on
GL1(R) ! 1 R of spaces, but not in general by a map of spectra.
The main interest in the cohomology theory based on gl1(R) is that the group
H1(X; gl1(R)) contains the obstruction to the R-orientability of vector bundles*
* over
X, according to the theory of [May77 ]. The present paper was motivated by one *
*partic-
ular application: the construction of a MO<8>-orientation for the topological m*
*odular
forms spectrum. This application will appear in joint work with Matt Ando and M*
*ike
Hopkins.
1.6. K(n) localization. Let F be a homology theory. Bousfield F -localization
consists of a functor LF on the homotopy category of spectra, and a natural map
'X :X ! LF X for each spectrum X, such that 'X is the initial example of a map
of spectra out of X which is an F*-homology isomorphism [Bou79 ]. Distinct homo*
*logy
4 CHARLES REZK
theories may give rise to isomorphic Bousfield localizations, in which case the*
*y are called
Bousfield equivalent.
Given a Morava E-theory spectrum, there is a an associated "residue field" F*
* , a
spectrum formed by killing the sequence of generators of the ideal m = (p, u1, *
*. .,.un-1)
in ss0E, so that ss*F k[u, u-1]. The spectrum F is not a commutative S-algeb*
*ra,
although it is a ring spectrum up to homotopy.
The Bousfield class of F depends only on the prime p and the height n of the*
* formal
group of E, and this is the same as the Bousfield class of the closely related *
*Morava
K-theory spectrum K(n). (The spectrum F is isomorphic to a finite direct sum of
suspensions of K(n).) As is standard, we will write LK(n) for the localization *
*functor
associated to any of these Bousfield equivalent theories.
In many respects, K(n)-localization behaves like completion with respect to *
*the ideal
m O. In particular, K(n)-localization allows us to define a modification of t*
*he homol-
ogy functor E* associated to a Morava-E theory, called the completed E-homology
E^*and defined by
E^*(X) def=ss*(LK(n)(X ^ E)).
This functor takes values in complete E*-modules; if E*X is a finitely generate*
*d E*-
module, then E^*(X) (E*(X))^m. See [HS99 , x8] for a discussion of K(n)-local*
*ization
and completed homology.
1.7. The logarithmic cohomology operation. For each commutative S-algebra R,
there is a natural family of "logarithm" maps from gl1(R) to various "completio*
*ns" of
R. For each prime p and n 1, there exists a natural map
`n,p:gl1(R) ! LK(n)R.
This map is defined using the construction due to Bousfield and Kuhn [Bou87 ], *
*[Kuh89 ],
which is a functor n from spaces to spectra, with the property that n 1 (X)
LK(n)X for any spectrum X. If R is a commutative S-algebra, the spaces 1 gl1(*
*R)
and 1 R have weakly equivalent basepoint components, and so the Bousfield-Kuhn
construction gives an equivalence LK(n)gl1(R) LK(n)R of spectra. The map `n,p*
*is
the composite
'gl1(R)
gl1(R) ----! LK(n)gl1(R) LK(n)R.
The construction of `n,pis described in detail in x3.
The map `n,pgives a natural transformation of cohomology theories, and thus *
*for any
space X a group homomorphism
`n,p:(R0X)x ! (LK(n)R)0(X),
natural as X varies over spaces and R varies over commutative S-algebras.
The purpose of this paper is the computation of this "logarithmic" map in te*
*rms
of power operations, when R is a reasonable K(n)-local commutative S-algebra. *
*It is
convenient to consider the cases n = 1 and n > 1 separately, though the proof f*
*or n = 1
is really a corollary of the general case.
1.8. The logarithm for K(1)-local spectra. Let p be a prime, and let R be a K(1*
*)-
local commutative S-algebra, satisfying the following technical condition: the *
*kernel of
ss0LK(1)S ! ss0R contains the torsion subgroup of ss0LK(1)S. This condition is *
*always
satisfied if p > 2 (since ss0LK(1)S is torsion free for odd p), and is satisfie*
*d at all primes
when R is the p-completion of the periodic complex or real K-theory spectra, or*
* if R is
the K(1)-localization of the spectrum of topological modular forms.
LOGARITHMIC COHOMOLOGY OPERATION 5
Such a ring R admits canonical cohomology operations _ and `, such that (in *
*partic-
ular) _ is a ring homomorphism, and for x 2 R0X,
_(x) = xp + p`(x).
(See x13.) When R is the p-completion of real or complex K-theory, then _ is t*
*he
classical pth Adams operation _p.
1.9. Theorem. Let p be any prime, and let R be a K(1)-local commutative S-
algebra, satisfying the technical condition above. For a finite complex X, the *
*logarithm
`1,p:(R0X)x ! R0X is given by the infinite series
1X pk-1 ``(x)' k
`1,p(x) = (-1)k____ ____p ,
k=1 k x
which converges p-adically for any invertible x, and so is a well-defined expre*
*ssion.
Note that the series can be formally rewritten as
` ' ` ' p
`1,p(x) = 1_plog_____1_____1 +=p1`(x)=xp_plogx__(x),
and that this new expression is still meaningful, up to p-torsion. Since xp=_(*
*x) 1
mod p for invertible x, this can be written
`1,p(x) = (id-1_p_)(log(x)),
which is meaningful (up to p-torsion) when x - 1 is nilpotent. If x = 1 + ffl w*
*ith ffl2 = 0,
then the formula of (1.9)becomes
`1,p(1 + ffl) = ffl - `(ffl) = ffl - 1_p_(ffl).
The proof of (1.9)is given in x13, as a corollary of (1.11)below.
1.10. The logarithm for Morava E-theory. For general n 1, we give a result for
Morava E-theory, in terms of power operations. We have
1.11. Theorem. Let p be any prime, n 1, and let E be a Morava E-theory associ*
*ated
to a height n formal group law over a perfect field of characteristic p. Then t*
*he logarithm
`n,p:(E0X)x ! E0X is given by
1X pk-1 1
`n,p(x) = (-1)k-1____M(x)k = _ log(1 + p . M(x)),
k=1 k p
where M :E0X ! E0X is the unique cohomology operation such that
Yn` Y ' (-1)jp(j-1)(j-2)=2
1 + p . M(x) = _A(x) .
j=0 A *[p]
|A|=pj
Here *[p] * denotes the kernel of multiplication by p on *.
In the case when n = 1, then 1 + p . M(x) = xp=_Z=p(x), and thus we recover *
*the
result for K-theory.
6 CHARLES REZK
1.12. Interpretation in terms of Hecke operators. Define formal expressions Tj,p
for 0 j n and T (pk) for k 0 by
X 1 X
Tj,pdef=1_pj _A and T (pk) = __k _A;
A *[p] p A *
|A|=pj |A|=pk
these give rise to well defined additive operations E0X ! p-1E0X, which we call*
* Hecke
operators. The T (pk) operators can be expressed as polynomials in the Tj,poper*
*ators,
and there is an action of the commutative ring Hn,p= Z[T1,p, . .,.Tn,p] on p-1E*
*0X. (An
account of this action is sketched in x14.)
Formally, we can rewrite the expression of (1.11)as
`n,p(x) = F1(logx)
(using the fact that logtakes products to sums, and that the _A's are ring homo*
*mor-
phisms), where
Xn
FX = (-1)jpj(j-1)=2Tj,p. Xj 2 Hn,p[X].
j=0
In particular, if x = 1+ffl 2 (E0X)x with ffl2 = 0, then `n,p(1+ffl) = F1(ffl) *
*(up to torsion).
The formal operator inverse of FX is
1X
FX-1= T (pk) . Xk 2 Hn,p[[X]],
k=0
and both these expressions appear in the theory of automorphic forms. Namely, i*
*f f is
an eigenvector of the action of the algebraPHn,pon a space of automorphic forms*
*, then
Fp-1-sf = Lp(s; f)f, where Lp(s; f) = akp-ks is the p-th Euler factor of the *
*Hecke
L-function of f, and Fp-sf = (1=Lp(s; f))f, so that 1=Lp(s; f) is a polynomial *
*of degree
n in p-s. (See [Shi71, 3.21].)
It is notable that this expression from the theory of L-functions arises nat*
*urally from
a purely topological construction; it came as a suprise to the author, and he s*
*till has no
good explanation for it. It is also significant for the application to elliptic*
* cohomology; in
the presence of an elliptic curve, these Hecke operators coincide with the clas*
*sical action
of Hecke operators on modular forms.
1.13. Structure of the proof. The proof of (1.11)falls naturally into two parts.
First part. The logarithm is equal to a certain power operation in E-cohomo*
*logy,
corresponding to a particular element v 2 E^0 1 S, the completed E-homology of *
*the
0-th space of the sphere spectrum (5.8). Furthermore, the element v (the "logar*
*ithmic
element") is completely characterized by certain algebraic properties (9.3). Th*
*e proof of
the first part comprises xx4-9.
Second part. An element v 2 E^0 1 Sis constructed, and shown to be a logarit*
*hmic
element (12.3). The explicit form of v gives the formula of (1.11). The proof*
* of the
second part comprises xx10-12.
The proof of (1.11)is completed in x12.4.
1.14. Conventions on spaces and spectra. We write Spaces for a category of
"spaces" (such as topological spaces or simplicial sets), and Spaces*for based *
*spaces.
We write Spectrafor any suitable category of spectra. Most of this paper takes *
*place in
suitable homotopy categories of spaces or spectra; therefore, we will usually n*
*ot specify
LOGARITHMIC COHOMOLOGY OPERATION 7
a particular model. There are exceptions, namely x2 (where we refer to the mod*
*el of
[LMSM86 ]) and x6 (where we use the model of [BF78 ]).
By commutative S-algebra, we mean any suitable category of commutitive ring *
*objects
in spectra (e.g., the model of [EKMM97 ], or any equivalent model, see [MM02 *
*]). In x2,
we will use the particular model of algebras over the linear isometries operad,*
* in the
sense of [LMSM86 ].
There are pairs of adjoint functors with units and counits
(- )+ :Ho SpacesAE HoSpaces*:(- )- , pK :K ! (K+ )- , qK :(K- )+ ! K,
where K+ K q pt, and K- K with the basepoint forgotten,
1 : HoSpaces*AE HoSpectra: 1 , jK :K ! 1 1 K, fflX : 1 1 X ! X,
and
1+: HoSpacesAE HoSpectra: 1 , j+K:K ! 1 1+K, ffl+X: 1+ 1 X ! X,
so that 1+K 1 (K+ ) and 1 X ( 1 X)- . Also, note that 1 (X ^ Y ) 1 X*
* ^
1 Y , while 1+(X x Y ) 1+X ^ 1+Y ; we will use this identifications often.
1.15. Conventions on localization. When n 1 and the prime p are fixed, we wri*
*te
L = LK(n) for the Bousfield localization functor, and write 'X :X ! LX for its *
*coaug-
mentation.
We make the following convention for the sake of legibility: in general we *
*do not
specify the augmentation 'X . Thus, if f :X ! Y is a map of spectra, the nota*
*tion
"Lf :X ! LY " is understood to denote the composite of Lf :LX ! LY with the
coaugmentation 'X :X ! LX. Note that little information is lost, since Lf :LX !*
* LY
is in fact the unique factorization of X ! LY through 'X up to homotopy.
Likewise, if f :X ^ Y ! Z is a map, the notation Lf :X ^ LY ! LZ denotes
the unique extension of X ^ Y ! LZ along the map X ^ Y ! X ^ LY (which is a
K(n)-homology equivalence).
If R is a K(n)-local spectrum, we write
R^qX def=ssqL(X ^ R), RqX def=[X, qR].
Both functors Rq and R^qtake K(n)-homology isomorphisms to isomorphisms.
1.16. Acknowledgements. This work began as a joint project with Paul Goerss and
Mike Hopkins. In particular, the first proof of (1.9)when R is p-completed K-th*
*eory
was proved jointly with them. The original proof of this was somewhat different*
* than
the one offered here; it involved an explicit analysis of the Bousfield-Kuhn fu*
*nctor in the
K(1)-local case.
I would like to acknowledge both Paul and Mike for their assistance at vario*
*us points
in this project. I would also like to thank Matt Ando for many tutorials on po*
*wer
operations and level structures. I would also like to thank Nick Kuhn and Nora *
*Ganter
for various comments which improved the paper.
2.The units of a commutative ring spectrum
In this section, we describe the units spectrum of a structured commutative *
*ring spec-
trum. The notion of the units of a commutative S-algebra has a long history, pa*
*ralleling
the long history of constructions of structured ring spectra. The notion seems *
*to have
arisen from work of Segal (as in [Seg75]) and Waldhausen. Our discussion of the*
* units
spectrum is based on the construction of [May77 ], as corrected in [May82 ]. T*
*here is
8 CHARLES REZK
another approach for constructing the units spectrum due to Woolfson [Woo79 ], *
*based
on Segal's theory of -spaces.
2.1. Definition of the units spectrum. Let R be a commutative ring spectrum in *
*the
sense of [LMSM86 ], i.e., a spectrum defined on a universe, and equipped with *
*an action
of the linear isometries operad L. Then R(0) = 1 R, the 0-space of the spectru*
*m R, is
itself an algebra over L. Let GL1(R) 1 R denote the subspace of 1 R defined*
* by
the pullback square
GL1(R) _j__//_ 1 R
| |
| |
fflffl| fflffl|
(ss0R)x_____//ss0R
We write ` :GL1(R) ! 1 R for the inclusion. Then GL1(R) is a grouplike E1 -spa*
*ce,
and so by infinite loop space theory is the 0-space of a (-1)-connective spectr*
*um, which
we denote gl1R. Note that the identity element of GL1(R) is not the usual basep*
*oint of
1 R.
The construction which associates R 7! gl1R defines a functor HoS-alg! HoSpe*
*ctra.
(It can be lifted to an honest zig-zag of functors between underlying model cat*
*egories;
we don't need this here.)
2.2. Example. Let R = S. Then GL1(S) G, the monoid of stable self-homotopy
equivalences of the sphere.
2.3. Example. Let R = HA, the Eilenberg-Mac Lane spectrum associated to a commu-
tative ring A. Then gl1HA HAx , the Eilenberg-Mac Lane spectrum on the group *
*of
units in A.
The spectrum gl1R defines a cohomology theory on spaces; it is convenient to*
* write
X 7! Hq(X; gl1R) for the group represented by homotopy classes of stable maps f*
*rom
1+X to qgl1R. In general, there seems to be no convenient description of thes*
*e groups
in terms of the cohomology theory R, except when q = 0.
2.4. Proposition. There is a natural isomorphism of groups
H0(X; gl1R) (R0X)x .
Furthermore, if X is a pointed and connected space, then this isomorphism ident*
*ifies
eH0(X; gl1R) (1 + eR0X)x .
In particular, if we take X = Sk for k 1, we obtain isomorphisms of groups
ssk(gl1R) eH0(Sk; gl1R) (1 + eR0Sk)x eR0(Sk) sskR;
this uses the isomorphism (1 + eR0Sk)x eR0Sk defined by 1 + ffl 7! ffl, and i*
*s realized by
a map of spaces GL1(R) ! 1 R.
A main motivation for studying the units is their role in the obstruction to*
* orientations.
For instance, if V ! X is a spherical fibration, the obstruction to the existen*
*ce of an
orientation class in the R-cohomology of the Thom space is a certain class w(V *
*) 2
H1(X; gl1(R)); see [May77 , IV,x3].
LOGARITHMIC COHOMOLOGY OPERATION 9
2.5. A rational logarithm. In x3 we will construct a "logarithm" in the K(n)-lo*
*cal
setting, for n 1. This construction does not extend to the case of n = 0, whe*
*re "K(0)-
local" means "rational". For completeness, notice an ad hoc logarithm in the ra*
*tional
setting; it will not be used elsewhere in the paper.
Let R be a commutative S-algebra, and let RQ denote the rationalization of R*
*; its
homotopy groups are ssnRQ (ssnR) Q. Let (gl1R)1 denote the 0-connected cove*
*r of
the spectrum gl1R.
The group H0(X; (gl1R)1) is equal to the subgroup of (R0X)x consisting of cl*
*asses
ff which restrict to 1 2 R0({x}) for each point x 2 X.
2.6. Proposition. There exists a map `0: (gl1R)1 ! RQ of spectra, unique up to *
*ho-
motopy, which when evaluated at a space X is a map
H0(X; (gl1R)1) ! H0(X; RQ)
given by the formula
1X (ff - 1)k
ff 7! log(ff) = (-1)k-1_______ .
k=1 k
Proof.The indicated formula is in fact well-defined; convergence of the series*
* fol-
lows because H0(X, RQ) limH0(X(k), RQ), where X(k)is the k-skeleton of a CW -
approximation to X, and because ff - 1 2 H0(X, R) is nilpotent when restricted *
*to any
finite dimensional complex.
Therefore, the indicated formula gives rise to a natural transformation of f*
*unctors to
abelian groups, and therefore is represented by a map 1 (gl1R)1 ! 1 RQ of H-s*
*paces.
Since RQ is a rational spectrum, it is straightforward to show that this map de*
*loops to
a map of spectra, unique up to homotopy.
When X = Sk, k 1, and ff 2 eR0Sk, this gives `0(1 + ff) = ff; that is, `0 *
*is the
"identity" on homotopy groups in dimensions 1.
3. The Bousfield-Kuhn functor and the construction of the logarithm
Fix a prime p and an integer n 1. Write L = LK(n)for localization of spect*
*ra with
respect to the nth Morava K-theory, as in x1.15.
3.1. The Bousfield-Kuhn functor.
3.2. Proposition (Bousfield [Bou87 ], Kuhn [Kuh89 ], Bousfield [Bou01 ]). There*
* exists a
functor = n: Spaces*! Spectraand a natural weak equivalence of functors o : *
* O
1 ~-!L. Furthermore, (f) is a weak equivalence whenever f :X ! Y induces an
isomorphism on ssn for all sufficiently large n.
That is, L: Spectra! Spectrafactors through 1 :Spectra! Spaces*, up to homo-
topy.
3.3. Remark. In fact, a stronger result applies. There is a functor fnand a n*
*atural
equivalence LfK(n)= fnO 1 , where LfK(n)is Bousfield localization with respec*
*t to a
vn-telescope of a type n finite complex. In fact, the functor constructed in [B*
*ou01 ] is
fn, in which case n = LK(n) Fn.
In fact, everywhere in this paper where LK(n) and n appear, they may be rep*
*laced
by LfK(n)and fn, including in the key results (5.8)and (9.3).
10 CHARLES REZK
3.4. The basepoint shift. Let (K, k0) be a pointed space and X spectrum, and let
f :K ! 1 X be an unbased map. Write j(f): K ! 1 X for the based map defined
by
k 7! (jf)(k) = ~(f(k), '(f(k0))),
where ~: 1 Xx 1 X ! 1 X and ': 1 X ! 1 X are the addition and inverse maps
associated to the infinite loop space structure. Colloquially, (jf)(k) = f(k) -*
* f(k0). In
terms of the cohomology theory represented by X, this induces the evident proje*
*ction
X0(K) ! eX0(K) X0K to reduced cohomology summand.
3.5. Construction of the logarithm. For a commutative S-algebra R, we write
` :GL1(R) ! 1 R for the standard inclusion, as in x2. It is not a basepoint pr*
*eserving
map; however, j` is.
3.6. Definition. Define ` = `n,p:gl1(R) ! LR to be the composite
gl1(R) ! L gl1(R) o-!~ (GL1(R)) -(j`)--! ( 1 R) LR.
Note that the map (j`) is an equivalence, since j` is a weak equivalence on*
* basepoint
components.
This construction (for fixed n and p) gives rise for each space K and each c*
*ommutative
S-algebra R a map
`: (R0K)x ! (LR)0K,
which is natural in the variables K and R. It factors through the composite
(R0K)x ! ((LR)0K)x -`!(LLR)0K (LR)0K;
thus, when attempting to calculate the effect of ` on R, it will suffice to ass*
*ume that R
is already L-local.
4.A formula for the Bousfield-Kuhn idempotent operator
The Bousfield-Kuhn functor produces an idempotent operator which turns unsta*
*ble
maps between infinite loop spaces into infinite loop maps; the logarithm of x3 *
*is the
result of applying the Bousfield-Kuhn idempotent to the inclusion ` :GL1(R) ! *
*1 R.
To derive a formula for the logarithm, we will first give a formula for the Bou*
*sfield-Kuhn
idempotent. In this section, we do this for a version of the Bousfield-Kuhn op*
*erator
'ewhich acts on basepoint preserving maps. In the next section, we extend this *
*to an
operator ' which acts on arbitrary maps; that form will apply to the logarithm.
For based spaces K, L, we write [K, L]* for the set of basepoint preserving *
*maps up
to homotopy. For spectra X, Y , we write {X, Y } for the the set of maps in the*
* stable
homotopy category.
In what follows, we assume that X and Y are spectra, and that Y is an L-lo*
*cal
spectrum, so that 'Y :Y ! LY is a weak equivalence.
4.1. The Bousfield-Kuhn operator. We define an operator 'e:[ 1 X, 1 Y ]* !
[ 1 X, 1 Y ]* which sends f : 1 X ! 1 Y to the map obtained by applying 1
to the composite
X 'X--!LX 1 X -f-! 1 Y LY Y.
The operator e'has the following properties, where f, f0 2 [ 1 X, 1 Y ]*:
LOGARITHMIC COHOMOLOGY OPERATION 11
(a)e'is a natural with respect to maps of spectra g :X ! X0, and to maps of*
* L-local
spectra h: Y ! Y 0, in the sense that
'e( 1 h O f O 1 g) = 1 h O e'f O 1 g.
(b) e'is additive: e'(f + f0) = e'f + e'f0, where addition is defined using *
*the infinite
loop structure of 1 Y .
(c)If f = 1 g, then e'f = f.
(d) e'f is an infinite loop map.
In particular, e'2= e', and thus the group of stable maps {X, Y } can be identi*
*fied with
a summand of the group of unstable maps [ 1 X, 1 Y ]*.
Our approach to this operation relies on two facts. First, the fact that all*
* unstable
maps f : 1 X ! 1 Y between infinite loop spaces factor as f = 1 1 f O j 1 X,
where 1 1 f is an infinite loop map and j 1 X is the unit of the 1 - 1 adjun*
*ction.
Therefore, we really only need to understand the effect of the Bousfield-Kuhn f*
*unctor
on the "universal example" of an unstable map out of an infinite loop space, wh*
*ich is
j 1 X. Second, if 1 X 1 1 K where K is a based space, then we can translate
the problem of understanding the effect of the Bousfield-Kuhn functor on j 1 1*
* Kto
that of understanding its effect on j 1 S; this is a consequence of the fact th*
*at the
Bousfield-Kuhn functor can be modelled as a simpliciial functor.
4.2. The natural transformation ~. Let ~X :X ! L 1 1 X be the map defined
by
X 'X--!LX 1 X -j-1-X--! 1 1 1 X L 1 1 X;
~ is a natural transformation between functors on the homotopy category of spec*
*tra. It
is the same natural transformation considered in [Kuh ].
4.3. Proposition. Let f : 1 X ! 1 Y be a based map, and let ef: 1 1 X ! Y
denote its stable adjoint. There is a commutative diagram
~X 1 1
X _______//_L X
ppp
fO'X|| ppppp
fflffl|Lfewwppp
LY Y
so that e'f = 1 (LfeO ~X ).
Proof.Observe that f is equal to the composite
1 ef
1 X j-1-X--! 1 1 1 X ---! 1 Y.
Apply to this diagram.
4.4. Corollary. The transformation ~ is a section of Lffl. That is, LfflX O ~X *
*= 'X :X !
LX.
Proof.Set f = 1 'X in (4.3).
12 CHARLES REZK
4.5. Formula for e'. Let K denote an arbitrary based space. A map f 2 [ 1 X, 1*
* Y ]*
gives rise to a cohomology operation f*: eX0(K) ! eY(0K) byf*(ff) = f O ff. We *
*remind
the reader that f* is not necessarily a homomorphism of abelian groups, althoug*
*h it is
the case that f*(0) = 0. Our goal is to calculate the cohomology operation indu*
*ced by
'ef in terms of that induced by f. The formula we give is in the form of a comp*
*osite
Xe0(K) Pe-!eX0( 1 S ^ K) f*-!eY(0 1 S ^ K) Qe-!eY(0K),
where ePand eQare certain functions which we define now.
Given ff 2 eX0(K), represented by a map a: 1 K ! X, write ePff 2 eX0( 1 S ^*
* K)
for the class represented by
1 ( 1 S ^ K) 1 1 S ^ 1 K fflS^a---!S ^ X X.
If Y is an L-local spectrum, then we define a natural map
eQ:eY(0 1 S ^ K) ! eY(0K)
as follows: represent a class ff 2 eY(0 1 S ^ K) by a map a: 1 ( 1 S ^ K) ! Y *
*, and
let eQ(ff) be the class represented by
1 K S ^ 1 K ~S^1---!L 1 1 S ^ 1 K ! L 1 ( 1 S ^ K) La--!LY Y.
The main result of this section is.
4.6. Proposition. We have an identity ('ef)* = eQO f* O ePof operations Xe0(K) !
Ye0(K).
4.7. The natural transformation ffi. For a based space K and a spectrum Z, let
ffi = ffiZ,K: 1 Z ^ K ! 1 (Z ^ 1 K) be the map adjoint to
1 ( 1 Z ^ K) 1 1 Z ^ 1 K fflZ^1-1-K----!Z ^ 1 K.
4.8. Lemma. Given ff 2 eX0(K) represented by a stable map a: 1 K ! X, the elem*
*ent
Pe(ff) is represented by the stable map 1 a O ffiS,K: 1 S ^ K ! 1 X.
Proof.The composite
1 a
1 S ^ K ffi-! 1 (S ^ 1 K) ---! 1 X
is adjoint to
1 ( 1 S ^ K) 1 1 S ^ 1 K fflS^1-1-K----!S ^ 1 K a-!X,
which represents eP(ff).
4.9. The proof of (4.6).
Proof of (4.6).Let ff 2 Xe0(K), represented by a spectrum map a: 1 K ! X. Let
b: 1 S ^ K ! 1 X be the map of based spaces which represents ePff 2 eX0( 1 S *
*^ K);
it is adjoit to the stable map fflS^a: 1 1 S^ 1 K ! S^X X. Let ef: 1 1 X !*
* Y
LOGARITHMIC COHOMOLOGY OPERATION 13
be the map adjoint to f : 1 X ! 1 Y . We will refer to the following diagram.
~S^' 1_K____L010(_1_S_^_K)______________________________*
*_____________________________________________________________________________
_________________________________________________________*
*_____________________________________________________________________________*
*________________________________________________________
_______________________|L|1_ffi_____________________________*
*_____________________________________________________________________________*
*_________________________
________________________fflffl|______________________________*
*__________________
S ^ 1 K __________//_L 1 1 (S ^ 1 K) _L_1_b__________________*
*___________________
~S^ 1 K __________________________*
*___________________
| | 1 1 ____________________________*
*_________________
a| |L ______________________________*
*_______________a
fflffl| fflftt_______________________________*
*_________________________________fl|
X _______~X________//_Lh1 1 X
hhhh
fO'X|| hhhhhhhhh
fflffl|tthhhLfeh
LY Y
We are going to prove that this diagram commutes. Given this, the proposition i*
*s derived
as follows. Note that the composite f O'X Oa is precisely the map representing*
* the class
('ef)*(ff). We claim that the long composite S ^ 1 K ! Y around the outer edge *
*of the
diagram is a map representing (QeOf*OPe)(ff). The composite efO 1 b: 1 ( 1 S ^*
*K) !
Y is adjoint to f O b, which represents f*(Peff), and so it is clear from the d*
*efinition of eQ
that the long composite in fact the desired class.
To show that the diagram commutes, we need to check the commutativity of eac*
*h of
four subdiagrams. The central square commutes because ~ defines a natural trans*
*for-
mation 1 ! L 1 1 . The bottom triangle commutes by (4.3).
The commutativity of the right-hand triangle of the diagram follows from (4.*
*8).
We defer the proof of the commutativity of the upper-left triangle(6.1)to x6.
5.An unbased Bousfield-Kuhn operator
In this section, we define a Bousfield-Kuhn operator on unbased maps, using *
*the stable
basepoint splitting, and derive a formula for it similar to (4.6); from this we*
* will produce
the formuala (9.3)for the logarithm.
For unbased spaces K and L, we write [K, L] for the set of unbased homotopy *
*classes
of (not necessarily basepoint preserving) maps.
5.1. The stable basepoint splitting. For a spectrum Y and a based space K, we
consider functions
[K, 1 Y ]* i-![K, 1 Y ] j-![K, 1 Y ]*.
The function i is the evident inclusion, while j is the basepoint shift operato*
*r of x3.4.
These operations give rise to the direct sum decomposition
[K, 1 Y ] [K, 1 Y ]* ss0Y.
In particular, ji = id. Note this direct sum decomposition arises from the stab*
*le splitting
1 K+ 1 K _ S, which is realized by maps
_1+zK//_ _1_qK//_
1+(pt)oo___ 1+K oo___ 1 K ,
1+ssK flK
where zK :pt ! K, ssK :K ! pt, and qK :K+ ! K are the evident maps of spaces,
and flK = 1 1+K is the stable map with 1 qK O flK = 1 1 K and 1+ssK O flK = 0*
*. With
this notation, the operators i and j are induced by qK and flK respectively. We*
* record
the relation between j and fl in the following
14 CHARLES REZK
5.2. Lemma. Let K be a based space, and f :K ! 1 Y an unbased map. Then the
fe
based map jf is adjoint to 1 K flK--! 1+K -! Y , where efis adjoint to f. In p*
*articular,
ffl+YO fl 1 Y = fflY , since ffl+Yis adjoint to the identity map of 1 Y , whic*
*h preserves the
basepoint.
Parameterized Version: Let K be a based space and L be an unbased space, and*
* let
f :K x L ! 1 Y be an unbased map, with ef: 1+K ^ 1+L ! Y its adjoint. Then
feO (flK ^ 1 1+L) is adjoint to f - f O (zK ssK x 1L) 2 [K x L, 1 Y ].
In particular, if f is such that f O (zK x 1): L ! 1 Y is homotopic to the *
*null map,
then efO (flK ^ 1 1+L) is adjoint to jf.
5.3. The natural transformation ~+ . Let ~+X:X ! L 1+ 1 X be the map defined
by
X ~X--!L 1 1 X Lfl-1-X---!L 1+ 1 X.
5.4. The operator '. Now suppose that X and Y are spectra, and that Y is L-loca*
*l.
We define an operator ': [ 1 X, 1 Y ] ! [ 1 X, 1 Y ] on the set of unbased ma*
*ps,
by 'f def=(i O e'O j)f. The operator ' is idempotent, and has as its image the*
* set of
infinite-loop maps; it coincides with e'on the summand of basepoint-preserving *
*maps.
We are going to prove a formula for ' analogous to the one proved for e'.
For an unbased space K we define natural functions
P :X0(K) ! X0( 1 S x K) and Q: Y 0( 1 S x K) ! Y 0(K),
as follows.
Given ff 2 X0(K), represented by a map a: 1+K ! X, write P ff 2 X0( 1 S x K)
for the class represented by
ffl+S^a
1+( 1 S x K) 1+ 1 S ^ 1+K ---! S ^ X X.
If Y is an L-local spectrum, then we define a natural map
Q: Y 0( 1 S x K) ! Y 0(K)
as follows: represent a class ff 2 Y 0( 1 S x K) by a map a: 1+( 1 S x K) ! Y *
*, and
let eQ(ff) be the class represented by
~+S^1 1 1 1 1 1 La
1+K S ^ 1+K ---! L + S ^ + K ! L + ( S x K) --!LY Y.
5.5. Proposition. We have an identity ('f)* = Q O f* O P of operations X0(K) !
Y 0(K).
Proof.Let K be an unbased space, and let ff 2 X0(K) eX0(K+ ). We will show th*
*at
Q(f*(P ff)) = eQ((jf)*(Pe(ff))) = ('ejf)*(ff); the result follows when we note *
*that there
is an identity of cohomology operations (ig)* = g* when g 2 [ 1 X, 1 Y ]*, so *
*that
('ejf)* = (i'ejf)* = ('f)*.
We have a direct sum decomposition
X0( 1 S x K) X0(K) eX0( 1 S ^ K+ ) X0(K) X0( 1 S x K, ptx K),
which is produced by smashing K+ with the maps
(pt)+ -z!( 1 S)+ -q! 1 S and ss :( 1 S)+ ! (pt)+ .
LOGARITHMIC COHOMOLOGY OPERATION 15
The remaining projection of the splitting comes from smashing 1+K with the sta*
*ble
map fl = fl 1 S: 1 1 S ! 1 ( 1 S)+ . We claim that with respect to this split*
*ting,
the maps in
Xe0(K+ ) P-!eX0(K+ ) eX0( 1 S ^ K+ ) f*-!eY(0K+ ) eY(0 1 S ^ K+ ) Q-!eY(0*
*K+ )
satisfy
P (ff) = (0, eP(ff)), f*(0, fi) = (f*(0), (jf)*(fi)), Q(ff, fi) = eQ*
*(fi).
The proposition will follow immediately.
The formula for P follows, using a standard adjunction argument, from the fa*
*cts that
ffl+SO fl = fflS, by (5.2), and ffl+SO 1 z = 0.
The formula for Q follows from the facts that L 1 q O ~+S= L 1 q O Lfl O ~S *
*= ~S
and L 1 ss O ~+S= L 1 ss O Lfl O ~S = 0.
To prove the formula f*, let b: 1 S x K ! 1 X be the map representing fi 2
Xe0( 1 S ^ K+ ) X0( 1 S x K, ptx K); in particular, b O (z x 1): ptx K ! 1 X
is the null map, and b can be taken to be basepoint preserving. Set f*(0, fi) =*
* (x, y).
It is clear that x is represented by f O b O (z x 1) = f O 0: ptx K ! 1 Y , an*
*d so
x = f*(0) 2 X0(K) Xe0(K+ ). We have that y is represented by the stable map
feO 1+b O fl ^ 1, which by the parameterized version of (5.2)is adjoint to j(f*
* O b). Since
b preserves basepoints, j(f O b) = (jf) O b, which represents (jf)*(fi).
5.6. Application to ring spectra. We now introduce the additional hypothesis th*
*at
Y is a commutative ring spectrum; here we only need that Y be a commutative rin*
*g up
to homotopy, not a structured ring spectrum. We write Y0^(X) def=ss0L(X ^ Y ) f*
*or the
completed homology of a spectrum X with respect to Y . If K is a based space, w*
*e write
Y0^(K) for Y0^( 1+K).
Let v 2 Y0^( 1 S) denote the Hurewicz image of ~+S2 ss0L 1 S; i.e., the homo*
*logy
class represented by
~+S^1Y 1 1 1 1
S 1S^~---!S ^ Y ----! L + S ^ Y ! L( + S ^ Y ),
where ~: S ! Y represents the unit of the ring spectrum.
Since Y is a ring spectrum, there is a slant product operation
ff oe 7! ff=oe :Y 0(X1 ^ X2) Y0^(X2) ! Y 0(X1)
defined by
X1 1^oe--!X1 ^ L(X2 ^ Y ) ! L(X1 ^ X2 ^ Y ) L(ff^1)-----!L(Y ^ Y ) ! LY *
* Y.
5.7. Proposition. For an L-local ring spectrum Y , and ff 2 Y 0(K), we have
Q(ff) = ff=v.
In particular, for ff 2 X0(K), we have ('f)*(ff) = f*(P ff)=v.
Proof.The second statement is immediate from the first. The first statement is *
*straight-
forward from the definitions.
Now suppose that Y = R is an K(n)-local commutative S-algebra, and that X =
gl1(R). Let ` : 1 gl1(R) ! 1 R be the standard inclusion; it corresponds to t*
*he
"cohomology operation" R0(K)x ! R0(K) which is the inclusion of the units into *
*the
ring. The logarithm is ` = ('`)*, and thus we have proved
16 CHARLES REZK
5.8. Theorem. Let v 2 R^0 1+ 1 S denote the Hurewicz image of ~+S2 ss0L 1+ 1 S.
Then `(ff) = `*(P ff)=v.
We will often abuse notation by taking `* to be understood, so that we write*
* `(ff) =
(P ff)=v.
6.Simplicial functors
For a spectrum X and a space K, recall (from x4.7) that ffiX,K : 1 X ^K+ ! *
*1 (X ^
1+K) is the map adjoint to fflX ^ id: 1 1 X ^ 1+K ! X ^ 1+K, where ffl is t*
*he
counit of the adjunction ( 1 , 1 ). The purpose of this section is to prove th*
*e following.
6.1. Proposition. For every spectrum X and space K, the diagram
LX ^ 1+K ___~X^id___//_L( 1 1 X) ^ 1+K
| | 1
| |L( ffiX,K)
fflffl| fflffl|
L(X ^ 1+K) __~_______//_L 1 1 (X ^ 1+K)
X^ 1+K
commutes in the homotopy category of spectra; the vertical maps involve the loc*
*alization
augmentation, as described in x1.15.
The idea of the proof is easy to describe: the functors in question come fro*
*m simplicial
functors on an underlying simplicial model category of spectra, and the natural*
* transfor-
mation ~ on the homotopy category comes from a natural transformation between t*
*hese
simplicial functors; the vertical maps in the square are certain natural transf*
*ormations
associated a simplicial functor. To carry out the proof, we will choose explici*
*t models
for these functors.
6.2. Simplicial functors. Recall that if C is a simplicial model category, then*
* for any
X, Y 2 C and K 2 sSetthere are objects
mapC(X, Y ) 2 sSet, X K 2 C, Y K 2 C,
which come with isomorphisms
map C(X K, Y ) mapC (X, Y K) mapsSet(K, mapC(X, Y )).
We will usually write TK X def=X K and MK Y def=Y K; these are functors TK , MK*
* :C !
C. It is standard that these give rise to derived functors LTK , RMK :Ho C ! Ho*
* C,
and that furthermore LTK and RMK are adjoint to each other. See [GJ99 , IX] for*
* a full
treatment.
A simplicial functor F :C ! D is a functor which is enriched over simplicial*
* sets,
in the sense that for every pair of objects there is an induced map map C(X, Y *
*) !
map D(F X, F Y ) which on 0-simplicies coincides with F . For any such functor,*
* there is
a natural transformation
aeFX,K:F (X) K ! F (X K),
which is adjoint to the map
K ! mapC (X pt, X K) ! mapD (F (X pt), F (X K)).
LOGARITHMIC COHOMOLOGY OPERATION 17
A simplicial natural transformation is a natural transformation ff: F ! G su*
*ch
that the two evident maps map C(X, Y ) ! map D(F X, GY ) sending f to ffY O F f*
* and
Gf O ffX coincide. Such transformations give rise to commutative square
aeFX,K
F X K ____//_F (X K)
ffX 1K|| |ffX|K
fflffl| fflffl|
GX K aeG_//_G(X K)
X,K
6.3. Bousfield-Friedlander spectra. Let BFSpectradenote the Bousfield-Friedland*
*er
model category of spectra [BF78 ], [HSS00 ], [GJ99 , X.4]. This category has as*
* objects
X = {Xn 2 sSet*, fXn:S1 ^ Xn ! Xn+1}n 0, where where S1 = [1]=@ [1], with
morphisms g :X ! Y being sequences {gn: Xn ! Yn}n 0 commuting with the structure
maps f. It simplicial model category.
We note the following functors; all of them are simplicial functors.
(1) 1 :sSet*! BFSpectra, defined on objects by ( 1 K)n = (S1)^n^ K, with
the evident structure maps.
(2) 1 :BFSpectra, defined on objects by 1 X = X0.
(3)F :BFSpectra! BFSpectra, defined on objects by
(FX)n = colimm m Sing|Xm+n |,
where Sing:sSet*AE Top* :|- | are geometric realization and singular com*
*plex,
and K = map*(S1, K). (The functors | |, Sing, and are simplicial func*
*tors,
and thus so is F.)
(4) : sSet*! BFSpectra, the functor defined in [Bou01 ]; there it is constr*
*ucted as
a simplicial functor.
(5)TK :sSet*! sSet, where K is an unpointed simplicial set, defined on obje*
*cts by
TK (L) = L ^ K+ .
(6)TK :BFSpectra ! BFSpectra, where K is an unpointed simplicial set, defin*
*ed
on objects by TK (X)n = Xn ^ K+ .
A spectrum X 2 BFSpectrais cofibrant if and only if each structure map fXn:X*
*n !
S1 ^ Xn+1 is an inclusion of simplicial sets. For all K 2 sSet*, the spectrum *
*1 K is
cofibrant.
A map f :X ! Y 2 BFSpectrabetween cofibrant objects is a weak equivalence if*
* and
only Ff is a weak equivalence of simplicial sets in each degree. The functor F *
*comes
equipped with a natural transformation 1 ! F, which on degree n of a spectrum X*
* is
the map adjoint to the evident map |Xn| ! colimm 1 |Xm+n |.
The functors 1 , , and both versions of TK are homotopy functors: they pre*
*serve
weak equivalences. The functor F is a homotopy functor on the full subcategory*
* of
cofibrant objects. The functor 1 is not a homotopy functors; however, the com*
*posite
1 F is a homotopy functor on the full subcategory of cofibrant spectra.
There is a simplicial natural transformation j :id! 1 F 1 of functors sSet*
**!
sSet*; it is the evident map
X ! colimm m Sing|Sm ^ X|.
Define
~: 1 F = id 1 F ! 1 F 1 1 F
to be the natural simplicial transformation j 1 F.
18 CHARLES REZK
By the above remarks, both functors are homotopy functors on cofibrant spect*
*ra.
Both functors are simplicial functors. Therefore, the square
TK 1 FX TK~X//_TK 1 F 1 1 FX
ae|| ae0||
fflffl| fflffl|
1 FTK X ~T__//_ 1 F 1 1 FTK X
KX
commutes for all X. Applied to cofibrant X, it gives rise to a commutative diag*
*ram in
the homotopy category of spectra. It is clear that the horizontal arrows are p*
*recisely
those of (6.1). It remains to identify the arrows labelled ae and ae0. They can*
* be factored
1 F 0 1 1 1 F 1 1 1 1 F 1 1
ae = ae and ae = ( F)( )ae O( F)ae ( F)Oae ( )( F*
*).
6.4. Proposition.
(a)The natural transformation ae 1 F: TK 1 F ! 1 FTK , on the homotopy
category of spectra, induces the map
LX ^ 1+K ! L(X ^ 1+K),
which is the unique map up to homotopy compatible with the augmentations
'X ^ id:X ^ 1+K ! LX ^ 1+K and1': X ^ 1+K ! L(X ^ 1+K).
(b) The natural transformation ae :TK 1 ! 1 TK , on the homotopy catego*
*ry
of spectra, induces the canonical equivalence
1 X ^ 1+K ~-! 1 (X ^ 1+K).
(c)The natural transformation ae 1 F:TK 1 F ! 1 FTK , on the homotopy cat*
*e-
gory of spectra, induces the map
ffiX,K : 1 X ^ 1+K ! 1 (X ^ 1+K)
adjoint to fflX ^ id: 1 1 X ^ 1+K ! X ^ 1+K.
Proof. (a) The composite 1 F is a model for the Bousfield localization func*
*tor
on spectra.1
(b) The map ae 1 is easily seen to be an isomorphism on objects in BFSpectra.
(c)The map ae F is adjoint to the arrow ff in the square
TK 1 1 F_TKfflF//_TK F
ae 1 1~F|| ~ aeF||
fflffl| fflffl|
1 TK 1 F__ff_//FTK
which is seen to commute by appeal to the definitions. The vertical arro*
*w on the
left is an isomorphism, and the vertical arrow on the right is a weak eq*
*uivalence
when applied to cofibrant spectra. This gives the desired result.
LOGARITHMIC COHOMOLOGY OPERATION 19
7.Power operations and the units spectrum
Power operations on a representable functor on the homotopy category of spac*
*es are
defined via the action of some E1 -operad on the representing space. We need to*
* consider
two flavors of E1 -spaces, and their associated power operations: the "additive*
*" E1 -
structure on the 0-space of any spectrum (power operations here amount to the t*
*heory
of transfers), and the "multiplicative" E1 -structure on the 0-space of any com*
*mutative
S-algebra. The purpose of this section is to show that both flavors are united *
*in the E1 -
space GL1(R), which can be viewed as "multiplicative" by the inclusion GL1(R) *
* 1 R,
or as "additive" by GL1(R) 1 gl1(R).
There are some standard references for some of this material, notably [BMMS8*
*6 ], who
frame their power operations in terms of extended power of spectra. (Their form*
*ulation
works naturaly not just for commutative S-algebras (a.k.a., E1 -ring spectra), *
*but with
the weaker notion of an H1 -ring spectra, and much of what we say in this secti*
*on applies
in that context, too.)
7.1. Power operations associated to E1 -spaces. Let Z be an E1 -space, as in
[May72 ]. That is, for a suitable E1 -operad E, there are structure maps Em x m*
* Zm ! Z,
satisfying certain axioms. Each space Em of the operad is a free contractible *
*m -space; we
will not refer explicitly to the operad E, but rather to a structure map E m x *
*m Zm ! Z.
Given a map f :K ! Z, we let bPm(f): E m x m Km ! Z denote the composite
E m x m Km ! E m x m Zm ! Z. This gives rise to functions bPm and Pm on
homotopy classes of maps
Pbm
[K, Z]____//[E m x m Km , Z]
PPP
PPPP | *m
Pm PPPP''Pfflffl||
[B m x K, Z]
Here m :B m x K E m x m K ! E m x m Km denotes the diagonal map, and
Pm def= *mO bPm.
The set [K, Z] is a commutative monoid via the E1 -structure of Z. With res*
*pect
to this, bPmand Pm are homomorphisms of commutative monoids. Furthemore, these
operations are natural with respect to maps of spaces in the K-variable, and ma*
*ps of
E1 -spaces in the Z-variable.
7.2. Power operations associated to a commutative S-algebra. Let R be a com-
mutative S-algebra, and Z = 1 R, viewed as an E1 -space using the "multiplicat*
*ive"
structure (which, we emphasize, is not the same as the "additive" structure com*
*ing from
Z being an infinite loop space). Then the constructions of the previous section*
* specialize
to natural power maps bPmand Pm :
bPm 0 m
R0K _____//OOOR (E m x m K )
OOOO |*
Pm OOOO''O fmflffl||
R0(B m x K)
These operations are multiplicative: bPm(fffi) = Pbm(ff)Pbm(fi) and Pbm(1) = 1*
*, and
similarly for Pm . Hence if ff 2 R0K is multiplicatively invertible, then Pbm(*
*ff) and
20 CHARLES REZK
Pm (ff) are also invertible, and so these operations restrict to functions (R0K*
*)x !
(R0E m x m Kn)x ! (R0B m x K)x .
These operations are not in general additive. Instead, we have
7.3. Proposition. We have
X
bPm(ff + fi) = eTij[Pi(ff)Pj(ff)]
i+j=m
and X
Pm (ff + fi) = Tij[Pi(ff)Pj(fi)],
i+j=m
where eTijand Tijdenote the transfers associated to the covering maps E m x ix j
Km ! E m x m Km and B ix B j ! B m respectively.
Proof.This is [BMMS86 , Lemma 2.1].
7.4. Power operations associated to an infinite loop space, and transfers. Let
Y be a spectrum, and let Z = 1 Y , viewed as an E1 -space using the "additive"
structure. Then we again have natural power maps, which we denote bPm+and Pm+:
bP+m
Y 0K_____//OY 0(E m x m Km )
OOO |
OOOO |*m
P+m OOO''O fflffl|
Y 0(B m x K)
These operations are additive: Pb+m(ff + fi) = bPm+(ff) + bPm+(fi) and bPm+(0) *
*= 0, and so
likewise for Pm+.
We recall the well-known relation between such "power operations" and the th*
*eory of
transfers.
7.5. Proposition. The map Pb+m(j+K): E m x m Km ! 1 1+K is adjoint to the
composite
1+(proj)1
1+(E m x m Km ) transfer-----! 1+(E m x m (m_x Km )) ------! + K,
where m_ is a fixed set of size m permuted by m . In particular, Pb+m(j+*): B*
* m !
1 1+S is adjoint to the composite
1+(proj)1
1+B m -transfer----! 1+B m-1 ------! + (*) S,
(take B -1 = ?).
7.6. The natural transformation ffi+ . Let Y be a spectrum and K an unbased spa*
*ce.
Let ffi+Y,K: 1 Y x K ! 1 (Y ^ 1+K) denote the map adjoint to
ffl+Y^ id: 1+ 1 Y ^ 1+K ! Y ^ 1+K.
This is a variant of the map ffi defined in x4.7.
LOGARITHMIC COHOMOLOGY OPERATION 21
7.7. A total power operation for infinite loop spaces. Let K be a space and Y a
spectrum. We define operations bP:Y 0K ! Y 0 1 1+K, and P :Y 0K ! Y 0( 1 1+Sx
K), as follows. Given ff: K ! 1 Y , let bP(ff) def= 1 eff, where eff: 1+K ! Y*
* is the
adjoint to ff. Let P (ff) def=bP(ff) O ffi+S,K.
Since 1 Y is an infinite loop space, it admits "additive" power operations *
*of the type
described in x7.4. The following lemma says that the two kinds of operations co*
*incide,
via the standard maps
eimdef=bPm+(j+K): E m x m Km ! 1 1+K and im def=Pm+(j+*): B m ! 1 S.
7.8. Lemma. Let ff: K ! 1 Y be a map. The diagram
Pb+m(ff)
B m x K _____//E m x m Km _____//818Yp
ppp
im xidK|| eim|| ppppp
fflffl| 1 fflffl|1bP(ff)pp
1 S x K __ffi+_// + K
S,K
commutes up to homotopy.
Proof.The adjoint pair ( 1+, 1 ) gives a commutative diagram
K ____ff__//919Ys
sss
j+K|| ssss1
fflffl|ss "ff
1 1+K
The two maps ff and j+K from K give rise to maps Pb+m(ff) and Pb+m(j+K) out of
E m x m Km , and the resulting triangle which appears in the statement of the p*
*ropo-
sition commutes, by the naturality of bPm+with respect to E1 -maps.
The commutativity of the left-hand square comes from the fact that the map
Pb+m(j+X): E m x m Xm ! 1 1+X is a natural transformation of functors, and ca*
*n in
fact be realized as a toplological natural transformation, by taking E m to be *
*the m-th
space of the little cubes operad.
The example we are interested in is Y = gl1(R). In this case, "classical" *
*power
operations are just the standard power operations in R-cohomology, restricted t*
*o units.
7.9. The universal example of the "infinite-loop" total power operation. The
"infinite-loop" power operation bP:[K, 1 Y ] ! [ 1 1+K, 1 Y ] of the previou*
*s sec-
tion is completely determined by its restriction to a "canonical" class, namely*
* the map
j+K:K ! 1 1+K. This leads to the following.
7.10. Proposition. Given a map ff: K ! 1 Y , we have that
ffl+ 1*
*+K
(a)bP(ff): 1 1+K ! 1 Y is adjoint to the composite 1+ 1 1+K ----!
1+K eff-!Y .
(b) P (ff): 1 S x K ! 1 Y is adjoint to the composite
ffl+S^id 1+K eff
1+ 1 S ^ 1+K -------! S ^ 1+K 1+K -! Y.
22 CHARLES REZK
Here effdenotes the adjoint to ff. In particular, the operator P coincides with*
* the one
called P in x5.4.
Proof.Consider the diagram
1+ffi+S,K 1 1 1 1+ 1 eff 1 1
1+ 1 S ^ 1+K __________//_ + + K_________//_ + Y
UUUUU | +
UUUUU |ffl 1+K ffl+Y|
ffl+S^idU1+KUU**UUUUUfflffl| fflffl||
1+K _______eff______//_Y
It is clear that the square commutes (up to homotopy), because ffl+ is a natura*
*l trans-
formation, while the triangle commutes because ffi+S,Kis adjoint to ffl+S^ 1 1+*
*K.
bP(ff) = 1 effis adjoint to ffl+YO 1+ 1 eff, which equals effO ffl+ 1 by *
*commutativity
+ K
of the diagram. Thus P (ff) = bP(ff) O ffi+S,Kis adjoint to effO ffl+ 1+KO 1+f*
*fi+S,K, which the
diagram shows is equal to effO ffl+S^ id 1+K.
7.11. Power operations associated to homology classes. We now assume that R
is a K(n)-local commutative S-algebra. From now on, we will be interested only*
* in
two related kinds of power operations: the operations Pbnand Pn associated to *
*the
mulplicative struture on 1 R (x7.2), and the total operations bPand P associat*
*ed to
the infinite loop space GL1(R) (x7.4).
Given u 2 R^0B m , define opu:R0K ! R0K by opu(x) def=Pm (x)=u, using the sl*
*ant
product map -=u: R0(B m x K) ! R0(K).
Similarly, for a class u 2 R^0 1 S we define opu:R0(K)x ! R0(K) by opu(ff) d*
*ef=
P (ff)=u, where we implicitly regard P (ff) as an element of R0( 1 S x K) by th*
*e usual
inclusion R0( 1 S xK)x R0( 1 S xK). Such operations defined using 1 S coinci*
*de
with those defined using B m , via the map in = Pbm(j+*) used in (7.8). That i*
*s, if
u 2 R0B n, and u0= im*(u) 2 R0 1 S, then opu= opu0.
With these definitions, (5.8)becomes
`(ff) = opv(ff).
More generally, if D is a flat extension of R^0(pt), then we obtain operat*
*ions
opu: R0K ! D R R0K parameterized by elements u 2 D R R^0B m (respectively,
opu: (R0K)x ! D R R0K parameterized by elements u 2 D R R^0 1 S).
8.The structure of the spaces parameterizing power operations
We summarize here some structure which is relevant to power operations, and *
*which
is used in xx9-12. Most of what we say in this section is well-known; the summa*
*ry is
provided mainly to fix notation. For another summary of this sort of structure*
*, in a
similar context, the reader is directed to [ST97 ].
Let R be a K(n)-local commutative S-algebra, and D a flat extention of R^0(p*
*t). Set
h(X) def=D R^0(pt)R^0(X); this is a multiplicative, homological functor. Set h*
*0(X) def=
D R^0(pt)R0X. We write h for h(pt). The functor h admits a K"unneth map x: h(X)*
* h
h(Y ) ! h(X x Y ). Let the symbol M denote either B or 1 S.
We give below a combined list of structure maps (numbered items) involving t*
*he
group h(M), and a list of properties they satisfy (lettered items). After the l*
*ist, we will
give the definitions of each of the structure maps, and sketch the proofs of ea*
*ch of the
LOGARITHMIC COHOMOLOGY OPERATION 23
properties. We will also show that in every case in which the structure maps de*
*fined for
both M = B and M = 1 S, they commute with the standard map B ! 1 S. It
may be helpful to point out here that structures (1) through (7) make h( 1 S) i*
*nto a
Hopf ring.
8.1. Summary list of structure maps for B and 1 S. The letters u, v denote
elements of h(M), while ff denotes an element of R0(K), for an arbitrary space *
*K.
(1)h(M) is a module over h = h(pt), such that
(a)opu+v(ff) = opu(ff) + opv(ff), and opcu(ff) = c . opu(ff), for c 2 h*
*(pt).
(2)There is a distinguished class 1 2 h(M), and
(3)a product u v 7! u . v :h(M) h h(M) ! h(M), such that
(b) "." is associative and commutative, with unit 1, and
(c)op1(ff) = 1 and opu.v(ff) = opu(ff)opv(ff).
Structures (1), (2), and (3) can be summarized: h(M) is a commutative h-algebra*
*, and
for ff 2 h0(K), the map u 7! opu(ff): h0(M) ! h0(K) is a map of h-algebras.
(4)There is a distinguished class [1] 2 h(M), and
(5)a product u v 7! u O v :h(M) h h(M) ! h(M), such that
(d) "O" is associative and commutative, with unit [1], and
(e)op[1]= id.
(6)There is an h-module map ss :h(M) ! h, such that
(g)ss(1) = 1 and ss(u . v) = ss(u)ss(v), and
(h) ss(u) = opu(1).
(7)There is an h-module map x :h(M) ! h(M x M).
Say that an element u 2 h(M) is grouplike if (i) ss(u) = 1, and (ii) *
*x (u) =
u x u. We write h(M)grpfor the set of grouplike elements.
(i)If u 2 h(M) is grouplike, then opu is multiplicative: opu(1) = 1 and
opu(fffi) = opu(ff)opu(fi).
(j)If u is grouplike, then we have the identities u O 1 = 1 and u O (v *
*. w) =
(u O v) . (u O w). That is, x 7! u O x is a ring homomorphism if u i*
*s grouplike.
(k) 1 2 h(M)grp, and u, v 2 h(M)grpimplies u.v 2 h(M)grpand uOv 2 h(M)gr*
*p.
Thus, the set of grouplike elements h(M)grpforms a commutative semi-*
*ring,
in which "addition" is given by u . v, "multiplication" is given by *
*u O v, the
"zero" element is 1, and the "one" element is [1]. Furthermore, the *
*grouplike
elements of h( 1 S) admit "additive" inverses, so that h( 1 S)grpis *
*not just
a semi-ring but a ring.
(8)There is an h-module map o :h(M) ! h, such that
(l)o([1]) = 1 and o(u O v) = o(u)o(v), and
(m) o(1) = 0 and o(u . v) = o(u)ss(v) + ss(u)o(v).
In particular, o defines a semi-ring homomorphism h(M)grp! h.
The following structures are only defined for M = B .
(9)There is an h-module map i :h(B ) ! h(pt), such that
(n) i(1) = 1 and i(u . v) = i(u)i(v),
(o)i([1]) = 0 and i(u O v) = i(u)ss(v) + ss(u)i(v), and
(p) opu(0) = i(u).
(10)There is an h-module map + :h(B ) ! h(B x B ).
Say that u 2 h(B ) is primitive if (i) i(u) = 0, and (ii) + (u) = ux1*
*+1xu.
(q) If u 2 h(B ) is primitive, then opuis additive: opu(0) = 0 and opu(f*
*f+fi) =
opu(ff) + opu(fi).
24 CHARLES REZK
8.2. Definition of the structure maps. Recall that 1+B DS, the free commu-
tative S-algebra on the 0-sphere.
(2)The class 1 2 h(M) is the image of the canonical class under the maps pt
B 0 B i-! 1 S. Equivalently, it is induced by the unit map S 1-!DS of*
* the
ring DS.
(3)The product "." is induced on B and 1 S by maps
Bq: B x B ! B and Hq : 1 S x 1 S ! 1 S.
Here q: x ! is the coproduct functor on finite sets; 1+Bq corresp*
*onds
to the ring product DS ^ DS ! DS. Hq is the "additive" H-space product on
1 S, obtained by applying 1 to the fold map S x S S _ S ! S.
(4)The class [1] 2 h(M) is the image of the canonical class under the maps *
*pt
B 1 B -i! 1 S. Equivalently, it is induced by the "cannonical" map
S ! DS.
(5)The product "O" induced on B and 1 S by maps
Bx: B x B ! B and Hx :B x B ! B .
Here x: x ! is the cartesian product functor on finite sets, and Hx *
*is the
"multiplicative" H-space product on 1 S, and is adjoint to ffl+ ^ ffl+ *
*: 1+ 1 S ^
1+ 1 S ! S ^ S S.
(6)The map ss is induced by the projection map M ! pt.
(7)The map x is induced by the diagonal map M ! M x M.
(8)The map o is induced on 1+B by the composites
1+(proj)1
1+B m -transfer----! 1+B m-1 ------! + (pt) = S,
(let B -1 = ?). On 1+ 1 S the map o is induced by the counit map
ffl+ : 1+ 1 S ! S.
(9)The map i is induced on 1+B by maps jk: 1+B n ! 1+B 0 S, which
is the identity if k = 0, and null homotopic if k > 0. Alternately, i is*
* induced
by the ring map DS ! S free on the null map S ! * ! S.
(10)The map + is induced on 1+B by the transfer maps
1+B k ! 1+(B ix B j), i + j = k,
associated to the inclusion ix j n. Alternately, + is induced by th*
*e ring
map DS ! D(S _ S) DS ^ DS which is free on the pinch map S ! S _ S.
8.3. Compatibility of the structure maps. We need to know that in each of the c*
*ases
(1) through (8), the two structure maps are compatable with respect to B ! 1 *
*S.
For (1), (2), (4), (6), and (7), compatibility is clear.
For (3) and (5), compatibility amounts to saying that B ! 1 S is a map of *
*"semi-
ring spaces", which is well-known.
For (8), this is (7.5).
8.4. Proof of properties.
(a)The slant product h0(M x K) h h(M) ! h0(K) is h-linear.
(b) Bq (resp. Hq) make M into a commutative and associative H-space.
(c)For M = B , this follows from the fact that r*ijbPm(ff) = bPi(ff) x bPj(*
*ff), where
i + j = m and rij:(E ix i Ki) x (E j x j Kj) E m x ix j Km !
E m x n Km . Restricting along the diagonal maps iand j gives the resu*
*lt.
LOGARITHMIC COHOMOLOGY OPERATION 25
For M = 1 S, this follows from the fact that H*qbP(ff) = bP(ff)xPb(ff*
*), viewed
as a class in h0( 1 1+K x 1 1+K)x .
(d) Bx (resp. Hx ) make M into a commutative and associative H-space.
(e)P1:h0(K) ! h0(E 1 x 1 K) h0(K) is the identity map.
(g)M ! ptis a map of H-spaces.
(h) P :h0(pt) ! h0(M x pt) sends 1 to 1, and ss(u) equals the slant product *
*of 1
with u.
(i)If u is grouplike, then the slant product map -=u: h0(M x K) ! h0K is a *
*map
of rings. Since P :h0(K) ! h0(M x K) is mulitplicative, the result follo*
*ws.
(j)M is a (semi-)ring space.
(k) The two products are defined via space-level maps M x M ! M, and so are
compatible with diagonal.
(l)For M = B , this is a consequence of the "double-coset formula" applied *
*to the
homotopy pullback square
B k-1 x B `-1 _____//B k`-1
| |
| |
fflffl| Bx fflffl|
B k x B ` _______//_B k`.
For M = 1 S, it follows from the fact that the composite
1+Hx 1 1 ffl+
1+ 1 S ^ 1+ 1 S ----! + S -! S,
which induces u v 7! o(u O v), equals ffl+ ^ ffl+ .
(m) For M = B , this is a consequence of the "double-coset formula" applied *
*to the
homotopy pullback square
(B k-1 x B `) q (B k x B `-1)_____//B k+`-1
| |
| |
fflffl| Bq fflffl|
B k x B ` ______________//_B k+`.
To prove it for M = 1 S, note that for any spectrum X there is a diagram
( 1+ 1 ss1, 1+ 1 ss2)1 1 1 1 ffl+_ffl+
1+ 1 (X x X) ________________//_ + X _ + X___//_X _ X
WWWWW |
WWWWWWW |fold |fold
1+ 1 (fold)WWWWWW++WWWWWWfflffl|| fflffl||
1+ 1 X ___________//_X
ffl+
where ssi:XxX ! X denote the projection maps. This diagram is commutative
up to homotopy. Evaluating at X = S gives the desired result.
(n) This is clear from the observation that i arises from a ring map DS ! S.
(o)Bx maps B k x B ` to BQ0 only if either k or ` equals 0.
(p) P :h0(pt) ! h0(B ) kh0(B k) sends 0 to (1, 0, 0, . .)..
(q) This follows from (7.3).
9. The logarithmic element
In this section, we characterize the element v 2 R^0( 1 S) which appears in *
*the state-
ment of (5.8). The main result is (9.3), which states that v is satisfies certa*
*in algebraic
identities, which characterize it uniquely; i.e., it is the unique logarithmic *
*element. These
26 CHARLES REZK
algebraic identities in some sense encode the properties of the idempotent oper*
*ator ':
namely, that ' is the identity on infinite loop maps, and is idempotent.
We note that all the results in this section still hold if we take L to be t*
*he telescopic
localization functor LfK(n), rather than K(n)-localization; see (3.3).
9.1. Logarithmic elements. Recall from x8.1(8) the homomorphism o :R^0 1 S !
R^0S induced by the counit map ffl+S: 1+ 1 S ! 1+(pt) = S, and from x8.1(5) the
product O: R^0 1 S R^0(pt)R^0 1 S ! R^0 1 S.
A logarithmic element for R is an element v 2 R^0 1 S with the following two
properties:
(La) o(v) = 1 in R^0S.
(Lb) x O v = o(x)v for all x 2 R^0 1 S.
9.2. Proposition. There is at most one logarithmic element in R^0 1 S. A map R *
*! R0
of K(n)-local commutative S-algebras carries the logarithmic element for R (if *
*it exists)
to the logarithmic element for R0.
Proof.The second statement is a consequence of the uniqueness of the logarithmi*
*c ele-
ment. If both v and v0are logarithmic elements in R^0 1 S, then
v = 1 . v = o(v0) . v = v0O v = v O v0= o(v) . v0= 1 . v0= v0,
using x8.1(b) and (d).
9.3. Theorem. The element v of (5.8)is a logarithmic element in R^0 1 S, and he*
*nce
is the unique logarithmic element.
In particular, there is a logarithmic element for R = LS, the K(n)-localizat*
*ion of
the sphere. By (9.2)all logarithmic elements for K(n)-local commutative S-algeb*
*ras are
determined by the logarithmic element for the case R = LS.
To give a proof of (9.3), we need some results involving the natural transfo*
*rmations
~+ (x5.3) and ffi+ (x7.6).
Recall the structure map fl of x5.1.
9.4. Proposition. The diagram
fl 1 Y^1 1+K
1 1 Y ^ 1+K ____________// 1+ 1 Y ^ 1+K
1 ffiY,K|| ||1+ffi+Y,K
fflffl| fflffl|
1 1 (Y ^ 1+K) fl_1_(Y_^_1//_ 1+ 1 (Y ^ 1+K)
+ K)
commutes in the homotopy category of spectra.
Proof.This is immediate from (9.5), and the stable basepoint splitting of x5.1.
9.5. Lemma. For every unbased space K and spectrum Y , the diagram
1+ss^1 1 q^1 1 1 1
1+(pt) ^ 1+Koo___ 1+ 1 Y ^ 1+K ______//_ Y ^ + K
1+ss|| 1+ffi+Y,K|| ||1 ffiY,K
fflffl| fflffl| fflffl|
1+(pt)oo_1_____1+ 1 (Y ^ 1+K) __1_//_ 1 1 (Y ^ 1+K)
+ ss q
commutes in the homomotpy category of spectra. (The map q was defined in x5.1.)
LOGARITHMIC COHOMOLOGY OPERATION 27
Proof.The commutativity of the left-hand square follows from the fact that ffi+*
* is a
natural transformation, and so commutes with the map induced by the projection *
*Y !
pt.
The right-hand square is equal to 1 applied to the square
( 1 Y x K)+ ( 1 Y )+ ^ K+____q^1____// 1 Y ^ K+
(ffi+Y,K)+|| ffiY,K||
fflffl| fflffl|
( 1 (Y ^ 1+K))+_______q________// 1 (Y ^ 1+K)
so it suffices to show that this square commutes in the homotopy category of po*
*inted
spaces. In fact, formal properties of adjunction show that
q O (ffi+Y,K)+ = ffi+Y,K= ffiY,K O (q ^ 1).
9.6. Proposition. Let K be a space and Y a spectrum. The diagram
~+Y^id 1 1 1
LY ^ 1+K ____________//L + Y ^ + K
| | 1 +
| + |L( + ffiY,K)
fflffl| ~Y ^ 1+K fflffl|
L(Y ^ 1+K) __________//_L 1+ 1 (Y ^ 1+K)
commutes. (We use the conventions for localization described in x1.15.)
In particular, taking Y = S gives ~+ 1+K= L( 1+ffiS,K) O (~+S^ id).
Proof.This square breaks up into two squares:
LY ^ 1+K ___~Y^id___//_L 1 1 Y ^ 1+K_Lfl_1_Y^id//_L 1+ 1 Y ^ 1+K
| 1 | | 1 +
| L ffiY,K| |L + ffiY,K
fflffl| fflffl| fflffl|
L(Y ^ 1+K)__~________//L 1 1 (Y ^ 1+K)_________//_L 1+ 1 (Y ^ 1+K)
Y ^ 1+K Lfl 1 (Y ^ 1+K)
The commutativity of the left-hand square is (6.1), while commutativity of the *
*right-hand
square is proved by applying L to the square of (9.4).
9.7. Lemma. The composite
ffi+X, 1 Y 1 (idX^ffl+Y)
1 X x 1 Y -----! 1 (X ^ 1+ 1 Y ) ---------! 1 (X ^ Y )
is the K"unneth map, i.e., the map representing the external product map X0K x *
*Y 0L !
(X ^Y )0(K xL) in generalized cohomology. In particular, for X = Y = S, the com*
*posite
ffi+S, 1 S 1 ffl+S
1 S x 1 S -----! 1 1+ 1 S ----! 1 S
is precisely the K"unneth product for S.
Proof.The K"unneth map 1 X x 1 Y ! 1 (X ^ Y ) is characterized as the adjoint
to
ffl+X^ ffl+Y: 1+ 1 X ^ 1+ 1 Y ! X ^ Y.
The result follows by factoring ffl+X^ ffl+Y= (1 ^ ffl+Y) O (ffl+X^ 1) and taki*
*ng adjoints.
28 CHARLES REZK
As a consequence, we have
9.8. Lemma. The diagram
~+S^id 1 1 1 1
LS ^ 1+ 1 S __________//_L + S ^ + S
id^ffl+S|| |L|1+m
fflffl| fflffl|11
LS __________~+________//L + S
S
commutes, where m denotes the K"unneth product map 1 S x 1 S ! 1 S. (Recall
the conventions described in x1.15 for localization.)
Proof.Consider the diagram
~+S^id 1 1 1 1
LS ^ 1+ 1 S ____//_L + S ^ + S
| | 1 +
| + L(|+ ffiS, 1 S)
fflffl|~ 1+ 1 S fflffl|
L( 1+ 1 S) ______//_L 1+ 1 ( 1+ 1 S)
Lffl+S|| L|1+|1 ffl+S
fflffl| ~+S 1fflffl|1
LS ______________//L + S
The top square is just (9.6)specialized to K = 1 S and Y = S, and so commutes.*
* The
bottom square commutes because ~+ :L ! L 1+ 1 is a natural transformation. The
composite of the right-hand vertical maps is L 1+m, by (9.7), and therefore the*
* outer
rectangle is the desired square.
9.9. Proof of (5.8). We must verify for v the identities (La) and (Lb) of the d*
*efinition
of logarithmic element.
Proof of (9.3), property (La).We have that ffl+ = ffl O q, ~+ = Lfl O ~, and q *
*O fl = id; thus
Lffl+SO ~+S= idS.
Proof of (9.3), property (Lb).Apply R-homology to the commutative square of (9.*
*8)to
get
R^0( 1 S)_vx__//R^0( 1 S x 1 S)
o|| O||
fflffl| fflffl|
R^0(pt)___v.___//_R^0( 1 S)
On the bottom and left, the composite is x 7! o(x) . v. On the top and right, *
*the
composite is x 7! x O v.
10.Level structures and the cocharacter map
10.1. Universal deformations. In this section, we fix a prime p and a height n *
* 1,
a perfect field k of characteristic p, and a height n-formal group 0 over k. W*
*e let E
denote the Morava E-theory associated to the universal deformation of 0.
LOGARITHMIC COHOMOLOGY OPERATION 29
10.2. Level structures. For a profinite abelian p-group M, we write
M* def=homcts(M, Qp=Zp),
where Qp=Zp is given the discrete topology, and we write M[pr] for the subgroup*
* of
pr-torsion elements of M.
If F is a formal group over a complete local ring R with maximal ideal mR, t*
*hen
F (mR) denotes the additive group with underlying set mR and group law given by*
* F .
For a discrete abelian group A, a homomorphism f :A ! F means a homomorphism
A ! F (mR) of abelian groups. The set of such homomorphisms is denoted hom(A, F*
* ).
10.3. Proposition. Let A be a finite abelian group.
(a)The O-module E0BA* is free and finitely generated over O. There are natu*
*ral
isomorphisms
E0(BA* x X) E0BA* O E0X
and
R O E0BA* homctsO-mod(E0BA*, R).
(b) Let i: O ! R be a local homomorphism to a complete local ring R, classif*
*ying a
deformation F . Then there is a natural isomorphism
hom ctsO-alg(E0BA*, R) hom(A, F ).
Proof.Part (a) is [HKR00 , 5.10 and 5.11], and part (b) is [HKR00 , 5.12].
We set O(hom (A, )) def=E0BA*; it carries the universal homomorphism from A*
* to a
deformation of 0.
A homomorphism f :A ! F is called a level structure if on the formal scheme F
over R one has the inequality of divisiors
X
[f(a)] F [p],
a2A[p]
where the left-hand side is over the elements of the p-torsion subgroup of A, a*
*nd the right-
hand side denotes the divisor of the p-torsion subgroup of F . In terms of a co*
*ordinate
T on F , this amounts to the condition that
Y
(T + T (f(a))) divides [p]F (T ) in R[[T ]].
a2A[p]
Write level(A, F ) for the set of level structures; note that by definition it *
*is a subset of
hom (A, F ).
10.4. Proposition. Fix a deformation F of 0, classified by the homomorphism i:*
* O !
R.
(a)For each finite abelian group A, there exists a complete local ring O(le*
*vel(A, ))
over O and natural bijections
homO-alg(O(level(A, )), R) level(A, F ).
The ring O(level(A, )) is a quotient of O(hom (A, )).
(b) If f :A ! B is an inclusion of finite abelian groups, then there is an e*
*vident
natural transformation level(B, F ) ! level(A, F ). The map O(level(A, *
* )) !
O(level(B, )) classifying the universal example of this transformation *
*is finite
and flat.
30 CHARLES REZK
(c)The invariant subring of the evident action of the ring Aut ((Z=prZ)n) on
O(level((Z=prN)n), ) is exactly O.
Proof.Part (a) is [Str97, Proposition 22] or [AHS04 , 10.14]. The existence of *
*the trans-
formation of part (b) is clear. That the map classifying it is finite and flat*
* is [Str97,
Theorem 34(ii)], while (c) is [Str97, Theorem 34(iii)].
Level structures enter topology in the statement of the character theorem of*
* [HKR00 ],
though this point of view is not made explicit there. The most useful reference*
*s for level
structures in the context of algebraic topology are [Str97, x7] and [AHS04 , x1*
*0].
10.5. The cocharacter map. Fix Znp, so that * (Qp=Zp)n. Write Dr def=
S
O(level( *[pr], )), and let D def=Dr. The group GL( ) acts in a natural way on*
* each
Dr on the left, through the finite quotient GL( =pr ), in such a way that DGL(r*
*) O.
Let G denote a finite group (not necessarily abelian). If M is any profinite*
* abelian
p-group, we let GM def=homcts(M, G)=G, where G acts by conjugation. In the spe*
*cial
case M = it is called the set of of generalized p-conjugacy classes. There is*
* an
evident right action of GL( ) on G .
In [HKR00 ], the authors define a character map, which is a ring homomorphism
OG :E0BG ! map(G , D)GL( ).
Their theorem [HKR00 , Thm. C] states that this becomes an isomorphism after in*
*verting
p. It is more convenient for us to use a dual construction, which we call a coc*
*haracter
map. The cocharacter map !G :G ! D O E0BG is defined as follows: An element x 2
G is represented by some homomorphism f : =pr ! G for sufficiently large r. T*
*here
is a homomorphism E0B =pr i Dr D classifying the underlying homomorphism
of the universal =pr -level structure. Write !r 2 D O E^0B =pr for the homol*
*ogy
class corresponding to this homomorphism, by (10.3). Then we define
!G (x) def=f*(!r) 2 D O E^0BG.
One checks that this definition does not depend on the choice of r or f, and th*
*at the
HKR character map is derived from the cocharacter map: the evaluation of OG at a
given x 2 G is given by Kronecker pairing with !G (x).
Recall that an element u 2 D O E^0X is grouplike if ss*(u) = 1 where ss :X *
*! *,
and if *(u) = u x u, where "x" denotes the external product, and : X ! X x X *
*is
the diagonal.
10.6. Proposition.
(a)The image of !G is contained in the grouplike elements of D O E^0BG.
(b) Under the evident bijection (GxH) G xH , we have !GxH (x, y) = !G (x)x
!H (y).
(c)The cocharacter map !G is equivariant with respect to the actions of GL(*
* ) on
G and D.
(d) If H is a subgroup of G, and T :E^0BG ! E^0BH denotes transfer, we have
X
T (!G (x)) = !H (xg),
gH2(G=H)x( )
where x: ! G is a fixed representative of the generalized conjugacy cl*
*ass,
xg(~) def=g-1x(~)g, and (G=H)x( )is the subset of G=H fixed by the image
x( ) G.
LOGARITHMIC COHOMOLOGY OPERATION 31
Proof.For (a), it suffices to see that !r 2 D E^0B =pr is grouplike, which i*
*s a conse-
quence of (10.3)(a) and the fact that it is dual to a ring homomorphism E0B =pr*
* ! D.
The proofs of (b) and (c) are straightforward.
The equation of part (d) is proved in the same way as the transfer formula f*
*or charac-
ters [HKR00 , Theorem D]. (The statement of the HKR transfer formula directly i*
*mplies
(d) modulo torsion, which is enough for our purposes.)
10.7. A congruence formula. Let i: O ! O(level(A, )) be the standard inclu-
sion, and let f :A ! i* be the tautological level structure. Let I(level(A, *
*))
O(level(A, )) denote the ideal generated by the elements {T (f(a))}a2A, where *
*T is any
coordinate for ; the ideal does not depend on the choice of T .
10.8. Proposition. Suppose A is a finite abelian p-group of rank 1 r n. The*
*n there
is an isomorphism
O(level(A, ))=I(level(A, )) O=(p, . .,.ur-1),
and so I(level(A, )) \ O (p, . .,.ur-1).
Proof.By construction, the O-algebra O(level(A, ))=I(level(A, )) is universal*
* for level
structures which are trivial homomorphisms. A deformationrF of 0 admits at mos*
*t one
such level structure, and one exists if and only if T p divides [p]F (T ). Thus*
* F admits
such a trivial level structure if and only if i(uk) = 0 for k = 0, . .,.r - 1, *
*and we conclude
that O(level(A, ))=I(level(A, )) O=(p, . .,.ur-1).
10.9. Remark. As a special case of (10.8), we see that any deformation of 0 to*
* a ring
with p = 0 admits a unique Z=p-level structure. I am indebted to Mike Hopkins *
*for
pointing out this fact to me, which led to the proof of the congruence (12.2).
Consider a finite subgroup V *. Let S denote the kernel of the projec*
*tion
dual to this inclusion:
0 ! S ! ! V *! 0.
Thus S is an open subgroup of . The inclusion V * determines a homomorphism
O(level(V, )) ! D, classifying the restriction of level structures.
10.10. Proposition. Let x, y 2 G , such that x|S = y|S in GS. Then
!(x) !(y) mod I(level(V, )) . D O E^0BG.
Proof.Let I def=I(level(V, )) . D. Let r be chosen sufficiently large such tha*
*t pr S,
and such that x and y are represented by maps f, g : =pr ! G. It suffices to s*
*how, in
j
D O E^0B =pr __ss//_D=I O E^0B =pr oo___D=I O E^0BS=pr ,
jjjj
f*||g*|| jjjjjjjj
fflffl|fflffl|ttjjjj
D=I E^0BG
that ss(!r) is in the image of j; given this, the result follows, because x|S =*
* y|S implies
that f|S=pr and g|S=pr are conjugate by an element of G, and so induce identica*
*l maps
E^0BS=pr ! E^0BG. Dualizing, we are asking that a dotted arrow exist making the
32 CHARLES REZK
following a commutative square of rings:
!r
E0B =pr ______//D
| |
| |
fflffl| fflffl|
E0BS=pr ` ` `//D=I
The existence of such a dotted arrow is a tautology; it amounts to a factorizat*
*ion
0_____//V_______//__*_____//_S*___//_v0
| v
| v v
fflffl|zzv
(mD=I)
of a diagram of abelian groups.
11.Burnside ring elements
11.1. The Morava E-theory of B and 1 S.
11.2. Proposition. For all k, E^0B k is a finitely generated, free O-module.
Proof.This is [Str98, Theorem 3.3].
11.3. Proposition. E^0 1 S is the completion of an infinitely generated free O-*
*module.
It is flat over O, and thus in particular is p-torsion free. The union of the i*
*mages of the
maps E^0{k} x B ` ! E^0 1 S are dense.
Proof.Let f :B ! B denote the map given by fk: B k-1 ! B k. It is well known
that 1 S is stably equivalent to hocolim(B f-!B f-!. .).. Furthermore, fk ad*
*mits
a stable retraction 1+B k ! 1+B k-1 (the stable homotopy version of a theorem*
* of
Dold [Dol62]). From this and (11.2), it is clear that ssk(E ^ f-1 B ) is a free*
* O-module
for even k, and 0 for odd k. Thus E^0 1 S is the m-adic completion of this free*
* module.
The flatness result follows from (11.4)below.
11.4. Lemma. Let A be a Noetherian commutative ring, I A an ideal. Then the
I-adic completion of any free A-module is flat over A.
Proof.For a free module on one generator, this is well-known; the usual proof (*
*e.g.,
[AM69 , Prop. 10.14]) using the Artin-Rees lemma generalizes to give the lemma,*
* as we
show below. L
Let S be a set, and define a functor on A-modules by F (M) def= s2SM ^I. *
*We
claim that
(i)F is exact on the full subcategory of finitely generated A-modules, and
(ii)the evident map F (A) A M ! F (M) is an isomorphism when M is finitely
generated.
Then F (A) A - is exact on the full subcategory of finitely generated modules,*
* and thus
F (A) is flat.
Recall the Artin-Rees lemma [AM69 , Thm. 10.11]: given a finitely generated *
*module
M and a submodule M0, there exists c 0 such that for all k 0,LIk+cM0 LIkM *
*\M0
Ik-cM0. Therefore the same is true when M and M0 are replaced by sM and sM0.
This implies (i), by [AM69 , Cor. 10.3]. To prove (ii), note that the map is an*
* isomorphism
if M is free and finitely generated, and therefore surjective for all finitely *
*generated
LOGARITHMIC COHOMOLOGY OPERATION 33
M, using the exactness result (i). Since A is Noetherian, (ii) follows by a 5-*
*lemma
argument.
11.5. The cocharacter map for B and 1 S. As in x10, define the profinite abel*
*ian
group def=Znp. Let A+k( ) denote the set of isomorphism classes of discrete c*
*ontinuous
-sets which have exactly k elements. We identify A+k( ) ( k) : to a general*
*ized
conjugacy class x: ! k associate X = k= k-1, regarded as a -set via x.
` +
Let A+ ( ) def= kAk ( ). The set A+ ( ) admits the structure of a semi-ring,*
* with
addition and multiplication corresponding to coproduct and product of sets. Let*
* A( )
be the ring obtained from A+ ( ) by adjoining additive inverses; it is the Burn*
*side ring
of , isomorphic to the direct limit A( =S) where S ranges over open subgroups *
*of .
For the following proposition we need the notation introduced in x8.1.
11.6. Proposition. The cocharacter maps (10.5)for symmetric groups fit together*
* to
give a map
!+ :A+ ( ) ! D O E^0B .
It is a homomorphism into the semi-ring of grouplike elements. That is, for x, *
*y 2 A+ ( ),
!+ (0) = 1, !+ (1) = [1], !+ (x + y) = !+ (x) . !+ (y), !+ (xy) = !+ (x*
*) O !+ (y).
If X is a transitive -set, then !+ ([X]) is also primitive.
Furthermore, this map extends uniquely to a map
! :A( ) ! D O E^0 1 S
which is a homomorphism into the ring of grouplike elements in D O E^0 1 S.
We have that for x 2 A( ),
o(!(x)) = d(x),
where d: A( ) ! Z is the ring homomorphism defined by d([X]) = #(X ).
The maps !+ and ! are GL( )-equivariant, and so ! induces a map A( )GL( )!
E^0 1 S.
Proof.That !+ lands in the grouplike elements follows from (10.6)(a). That it *
*is a
homomomorphism of semi-rings follows from the fact that the operations of sum a*
*nd
product on A+ ( ) are derived, using (10.6)(b), from the maps
( k) x ( `) ( k x `) q-!( k+`)
and
( k) x ( `) ( k x `) x-!( k`) ,
which are also the origin of the product maps "." and "O", as defined in x8.1. *
*Similarly,
the additive and multiplicative units of A+ ( ) arise as the unique elements of*
* ( 0) and
( 1) , respectively.
The primitivity of !+ ([X]) when X is transitive follows from [ST97 , 4.3].
The map o is induced by the stable map B k transfer-----!B k-1 proj--!pt. Th*
*e transfer
formula (10.6)(d) gives
X
transfer(! k(x)) = ! k-1(xg).
x2( k= k-1)x( )
The element !G (y) is always a grouplike element (10.6)(a), and so goes to 1 un*
*der the
projection BG ! pt. Under this projection, the element on the right-hand side o*
*f the
equation becomes an integer, equal to the size of ( k= k-1)x( )= X .
34 CHARLES REZK
To extend !+ to !, note that the grouplike elements of D E^0 1 S are inver*
*tible in
the "." product, so we may set
!([X] - [Y ]) def=!([X]) . !([Y ])-1.
The equivariance property follows from (10.6)(c).
11.7. Power operations. Let A * be a finite subgroup of order pr. Dualizing
gives a surjective homomorphism f : ! A*. Using this map, we can regard A* as
a set with a transitive -action, and hence an element in the Burnside ring, de*
*noted
s(A) 2 A+pr( ). We define _A :E0(X) ! D O E0(X) by _A = op!+(s(A)). According
to the remarks of the previous section, s(A) is both grouplike and primitive, a*
*nd thus _A
is a ring homomorphism (though not an O-algebra homomorphism). These operations
coincide with the ones constructed by Ando [And95 ], though the construction is*
* not
identical, since Ando did not have available to him the fact that the Morava E-*
*theories
are commutative S-algebras. Some dicussion of these operations is given in [AHS*
*04 ].
12. Construction of the logarithmic element
12.1. The element defined. We define a certain element e 2 A( )GL( )as follows:
nX
e = p (-1)jpj(j-1)=2ej,
j=0
where X
ej = 1_pj [ =S].
p S
=S (Z=p)j
This element e really lives in A( ) and not just A( ) Q, since 1 + j(j - 1)=2*
* - j =
(j - 1)(j - 2)=2 0 when j 0.
In this section, we will prove the following.
12.2. Proposition. The element !(e) is congruent to 1 modulo p in E^0 1 S.
12.3. Proposition. Let m 2 E^0 1 S such that 1 + p . m = !(e). The resulting el*
*ement
X1 pk-1 1
v def= (-1)k-1____mk = _ log!(e)
k=1 k p
is the logarithmic element for E.
12.4. Proof of the main theorem. We can now complete the proof of (1.11). By (5*
*.8)
and (9.3), we have that `(ff) = opv(ff), where v is the logarithmic element for*
* E. From
x8.1(a) and (c) we have that
opu+u0(ff) = opu(ff) + opu0(ff) and opuu0(ff) = opu(ff)opu0(ff),
and so 1
X pk-1
`(ff) = (-1)k-1____opm (ff)k.
k=1 k
The operation M of (1.11)is simply opm.
By construction, the operation M satisfies the formula for 1 + pM given in t*
*he state-
ment of (1.11). We claim it is the unique such operation. It is clear that the *
*formula
characterizes M up to p-torsion. Any operation E0 ! D O E0 corresponds to an e*
*le-
ment of D O E0 1 E. By, e.g., [BH04 , Thm. 1.4], E^0 1 E is a free E*-module i*
*n even
LOGARITHMIC COHOMOLOGY OPERATION 35
degress, whence D O E0 1 E is torsion free, and thus M must be the unique oper*
*ation
with this property.
12.5. Congruence for !(e). We use the notation of x11.7. In these terms, using *
*(11.6),
we have
Yn` Y ' (-1)jp(j-1)(j-2)=2
!(e) = !(s(U)) .
j=0 U *[p]
|U|=pj
Recall that *[p] = (Z=p)n (Q=Z(p))n = *.
12.6. Proposition. In D E E^0 1 S we have the congruence
!(e) 1 mod I(level(V, )) . D O E^0 1 S,
where V *[p] is any subgroup which is isomorphic to Z=p.
Reduction of (12.2)to (12.6).Let I = I(level(V, )). By (10.8)we have an inclus*
*ion of
short exact sequences
0 ____//_pO____//fflfflO//_fflfflO=pO//_fflffl0
| | |
| | |
fflffl| fflffl| fflffl|
0 ____//_ID____//D____//_D=ID____//_0.
Tensoring with the flat module E^0 1 S (11.3)preserves exact sequences and mono*
*mor-
phisms. The element 1 - !(e) 2 D O E^0 1 S lives in E^0 1 S by (11.6), and liv*
*es in
ID O E^0 1 S by (12.6), and so must be an element of pE^0 1 S.
12.7. Lemma. Consider a decomposition V V ? __ *[p] where V Z=p; let T =
Ker( ! V *). Given a subgroup U *[p], let U V ? *[p] denote the image *
*of
the projection of U to V ?. Then in A+ (T ),
( __
s(U)|T = s(U )|T_ if V 6 U,
p . s(U )|Tif V U.
As a consequence, we obtain the congruences
( __
!(s(U)) !(s(U ))__ if V 6 U,
!(p . s(U ))if V U,
modulo the ideal I(level(V, )) . D O E^0 1 S.
Proof.Let i: U ! * denote the given_inclusion,_and j :U ! * denote the map
factoring through the projection to U. By definiton, i j mod V , and so both*
* i and j
define the same composite U ! * ! T *. Dualizing, we see that i*|T = j*|T, vie*
*wed as
maps T ! ! U*. __
If V 6 U, then j*|T is surjective, whence s(U)|T_= s(U )|T. If V U, then *
*j*|T has
cokernel isomorphic to Z=p, whence s(U)|T = p . s(U )|T.
The congruences follow immediately from (10.10).
36 CHARLES REZK
Proof of (12.6).Choose any decomposition V V ? *[p] with V Z=p, as in the
lemma. Let d(j) = (-1)jp(j-1)(j-2)=2; note that d(j + 1) = -d(j)pj-1. We have
Yn Y n-1YY Yn Y
!(e) = !(s(U))d(j)= !(s(U))d(j). !(s(U))d(j)
j=0U *[p] j=0 V 6 U j=1V U
|U|=pj |U|=pj |U|=pj
which by (12.7)is congruent mod I to
n-1YY __ Yn Y __
I !(s(U ))d(j). !(p . s(U ))d(j)
j=0 V 6 U j=1V U
|U|=pj |U|=pj
which we reindex according to subgroups of V ?, to get
n-1YY j Yn Y
= !(s(W ))d(j)p. !(s(W ))d(j)p
j=0W V ? j=1W V ?
|W|=pj |W|=pj-1
n-1YY j
= !(s(W ))d(j)p +d(j+1)p.
j=0W V ?
|W|=pj
Since the exponents are always 0, the expression reduces to 1.
12.8. M"obius functions and the logarithmic element property. It remains to
show that the element v of (12.6)is in fact a logarithmic element. To do this, *
*we first
show that e=p 2 A( ) Q is the idempotent associated to the augmentation d: A( )*
* ! Z
sending d([X]) = #X .
We recall the theory of idempotents in a Burnside ring [Glu81], in the speci*
*al case
when the group is finite abelian. Thus, let G be a finite abelian group and A(*
*G) its
Burnside ring. The M"obius function of G is the unique function ~G defined on *
*pairs
C B of subgroups of G, characterized by the property that
(
X 1 if A = B,
~G (C, B) =
A C B 0 if A 6= B,
where the sum is over all subgroups C contained in B and containing A. Then the
elements X
eA def= ~G_(B,_A)_[G=B] 2 A(G) Q
B A #(G=B)
as A ranges over the subgroups of G are the primitive idempotents of A(G) Q [*
*Glu81,
p. 65], and furthermore,
yeA = dA(y)eA,
where dA :A(G) ! Z is given by dA([X]) = #(XA ).
Set ~G = ~G (0, G). Since the value of ~G (C, B) really only depends on the*
* poset
of subgroups of G between C and B, we see that ~G (C, B) = ~B=C , and that ~G o*
*nly
depends on the isomorphism class of G. The following lemma calculates ~A for al*
*l abelian
A.
12.9. Lemma. Let A be a finite abelian group.
LOGARITHMIC COHOMOLOGY OPERATION 37
Q Q
(1)If A Ap where the Ap are p-groups for distinct primes, then ~A = ~*
*Ap.
(2)If A is elementary p-abelian of rank j 0, then ~A = (-1)jpj(j-1)=2.
(3)If A is a p-group but not elementary p-abelian, then ~A = 0.
Proof.Part (1) is straightforward.
To prove part (2), note that it amounts to the identity [Shi71, Lemma 3.23]
Xn ~n ~ (1 if n = 0,
(-1)jpj(j-1)=2 =
j=0 j p 0 if n > 0,
n Q j-1(pn-pi)
where jp = i=0______(pj-pi)is the Gaussian binomial coefficient, which is th*
*e number of
elementary abelian subgroups of rank j inside (Z=p)n.
PWe prove part (3) by induction on the size of A. We have for A nontrivial ~*
*A =
- A)B ~B , the sum taken over proper subgroups of A. A proper subgroup B ( A
is one of two types: (a) it is an elementary abelian p-group, or (b) it isn't. *
* For (a),
suchPB are exactly the subgroups of A[p] ( A, the subgroup of p-torsion element*
*s, and
A[p] B~B = 0, since A[p] 6= 0. For (b), we have ~B = 0 by induction.
For r 1, the elements er def=e =pr 2 A( =pr ) Q are idempotents, and the
homomorphisms A( =pr ) Q ! A( =pr+1 ) Q carry er to er+1, as can be seen from
the explicit formula for these elements together with (12.9). Thus the limiting*
* element
e1 2 A( ) Q of this sequence is an idempotent in this ring, with [X]e1 = d([X*
*])e1 ,
where d([X]) = #(X ). By (12.9), we see that the element e defined in x12.1 is*
* equal
to pe1 , and thus we obtain
12.10. Proposition. In A( ) we have
(a)d(e) = p, and
(b) for all y 2 A( ), ye = d(y)e.
12.11. Proof of the logarithmic element property.
12.12. Lemma. Let ff 2 D O E^0 1 S be an element of the form ff = 1 + pfi, and*
* let
P pk
log(ff) def=k 1(-1)k-1__kfik-1.
(a)We have that
o(log(ff)) = o(ff)_ss(ff).
(b) If w 2 D O E^0 1 S is a grouplike element, then
w O log(ff) = log(w O ff).
Proof.First, note that the operations o, ss, and O on D O E^0(- ) are continuo*
*us with
respect to the maximal ideal topology, since they are induced by maps of spectr*
*a.
To prove (a), recall that o is a derivation (8.1)(m) with respect to the "."*
* product, so
that o(fik) = ko(fi)ss(fi)k-1. Thus
0 1
X pk X pk
o(log(ff))= o @ (-1)k-1__fikA = (-1)k-1__o(fik)
k 1 k k 1 k
X
= (-1)k-1pkss(fi)k-1o(fi) = o(fi)(1 + pfi)-1.
k 1
38 CHARLES REZK
To prove (b), recall that if w is grouplike, then w O (- ) is a homomorphism*
* of D-
algebras (8.1)(j). Thus,
0 1
X pk
w O log(ff)= w O @ (-1)k-1__fikA
k 1 k
X pk
= (-1)k-1__(w O fi)k = log(1 + p(w O fi)) = log(w O ff).
k 1 k
Proof of (12.3).We are going to prove that v = (1=p) log!(e) is a logarithmic e*
*lement.
We have that
o(1_plog!(e)) = 1_po(!(e))_ss(!(e))= 1_pd(e)_1= p_p= 1,
using (12.12)(a), (11.6), and (12.10)(a). This proves (La) of the logarithmic *
*element
property, that o(v) = 1.
Now we need to prove (Lb); that x O v = o(x)v for all x 2 E^0 1 S. The union*
* of the
images of E^0{k}xB ` ! E^0 1 S is dense, with respect to the maximal ideal topo*
*logy,
by (11.3). Thus it suffices to prove (Lb) for those x which are in the image of*
* one of
these maps. It is enough to do this after faithfully flat base change to D. Now*
*, E^0 1 S
is p-torsion free, and by the HKR theorem, p-1D E^0B m is spanned by elements*
* in
the image of the cocharacter map. Thus, it suffices to check (Lb) when x = !(y)*
* for any
y 2 A( ).
So let y 2 A( ). We have that
!(y) O 1_plog!(e)= 1_plog(!(y) O !(e)) by (12.12)(b),
= 1_plog!(ye) = 1_plog!(d(y)e) by (12.10)(b),
= 1_plog!(e)d(y)= d(y)_plog!(e)
= o(!(y))1_plog!(e) by (11.6).
Thus !(y) O v = o(!(y))v, as desired.
13.The logarithm for K(1)-local ring spectra
In this section, we describe the structure of ss0LK(1) 1+ 1 S (completely at*
* an odd
prime, and modulo torsion at the prime 2), outline its relation to power operat*
*ions on
K(1)-local commutative ring spectra, and give a proof of (1.9).
13.1. The p-adic K-theory of some spaces. We recall results on the p-completed
K-homology of 1 S, due to [Hod72 ]. We have that
K^0(B m ; Zp) homcts(R m , Zp),
where R m is the complex representation ring, topologized with respect to the i*
*deal of
representation of virtual dimension 0. Furthermore
K^0(B ; Zp) (Z[ 0, 1, 2, . .].)^p.
LOGARITHMIC COHOMOLOGY OPERATION 39
The elements k, k 0, are characterized implicitly by Witt polynomials
X pj
Wk = pi i ,
i+j=k
where Wk 2 K0(B pk; Zp) is the element corresponding to the continuous homomor-
phism R pk! Zp defined by evaluation of characters on an element g 2 pk which *
*is a
cycle of length pk.
Thus
K^0( 1 S; Zp) (Z[ 0 , 1, 2, . .].)^p.
k
Since Wk W0p mod pK^0B , the elements Wk become invertible in K^0 1 S.
The cocharacter map !+ :A(Zp) ! K^0(B ; Zp)grpsends [Zp=pk] to Wk. The opera-
tion corresponding to Wk is the Adams operation _pk.
According to (12.3), the logarithmic element for K^pis
1_log!(p[*] - [Z=p]) = 1_logW0p_= 1_log____1_____= X (-1)k-1pk-1__k1_.
p p W1 p 1 + p 1= p0 k 1 k pk0
13.2. The K(1)-local homotopy of B and 1 S. Recall that if X is a spectrum,
then there are cofibration sequences
KO ^ X ! KO ^ X ! K ^ X
and ~
LX ! L(K ^ X) (_--1)^id------!L(K ^ X) if p > 2,
and ~
LX ! L(KO ^ X) (_--1)^id------!L(KO ^ X) if p = 2,
where ~ 2 Zxp(respectively, ~ 2 Zx2={ 1}) is a topological generator. In partic*
*ular, we
have that (
ss0LK(1)S = Zp if p > 2,
Z2[x]=(x2, 2x)if p = 2,
where x comes from the non-trivial element of ss1KO Z=2.
13.3. Proposition. Let f denote either of the Hurewicz maps ss0L 1+B k ! K^0B k
or ss0L 1+ 1 S ! K^0 1 S. The map f is an isomorphism if p is odd, while if p =*
* 2 it
is surjective, and its kernel is the ideal generated by x 2 ss0LS.
Proof.This follows from the cofibration sequences mentioned above, together wit*
*h the
fact that the Adams operations _~, for ~ 2 Zxp, act as the identity map on K0(B*
* k; Zp).
13.4. The proof of (1.9). Let R be any K(1)-local commutative S-algebra satisfy*
*ing
the technical condition described in x1.8. By the above proposition, there is a*
* natural
factorization i: K^0 1 S ! R^0 1 S of the map ss0L 1+ 1 S ! R^0 1 S. Define pow*
*er
operations in R-theory by
_ def=opi(W1)= opi( p0+p 1)and ` def=opi(.1)
There is an identity _(x) = xp + p`(x), and _ is a ring homomorphism.
The derivation of (1.9)is now straightforward, since the logarithmic element*
* for R
must be the image under i of the logarithmic element for K, since both are the *
*image of
the logarithmic element for LS.
40 CHARLES REZK
13.5. Exponential maps for K-theory and KO-theory. The logarithm maps
gl1(K^p) ! K^pand gl1(KO^p) ! KO^pare seen to be weak equivalences on 3-connect*
*ed
covers in the first case, and 1-connected covers in the second. In other words,*
* the loga-
rithm admits inverse "exponential" maps
e: KSU(X; Zp) ! (1 + KSU(X; Zp))x and e: KSO(X; Zp) ! (1 + KSO(X; Zp))x ,
where KSU(- ; Zp) and KSO(- ; Zp) denote the cohomology theories defined by the*
*se
connective covers. Let `k: K0(X; Zp) ! K0(X; Zp) denote the operation correspon*
*ding
to the element k 2 K^0B pk described above, so that we have
k X i pj
_p (ff) = p `i(ff) .
i+j=k
13.6. Proposition. The exponential maps in K and KO theory are both given by the
formula 0 1
1Y X1 ` (ff)pj
e(ff) = exp@ _i____jA,
i=0 j=0 p
` P pk '
which converges p-adically. Formally this equals exp 1k=0___(ff)_pk.
Proof.We give the proof for K-theory; at the end, we indicate the changes neede*
*d for
KO-theory.
P 1 Tpj
Let f(T ) def=exp( j=0 ___pj) 2 Z(p)[[T ]] be the Artin-Hasse exponential. *
*We will define
Q1
a map e by e(ff) def=i=0f(`i(ff)). We will show below that this expression conv*
*erges
for ff 2 KSU(X; Zp) when X is a finite complex. It is then straightforward to c*
*heck that
`(e(ff)) = ff for any ff 2 KSU(X; Zp) where X is a finite complex. The represe*
*nting
space for KSU(- ; Zp) is BSU^p; the set KSU(BSU^p; Zp) = limKSU(X; Zp) as X
ranges over finite subcomplexes, and thus we can verify the identity ` O e = id*
*on the
universal example, which proves that e is the desired inverse.
S def
Suppose given a CW-model Xk = X. Let Ik = ker[K(X; Zp) ! K(Xk-1; Zp)];
the Ik's give a filtration of K(X; Zp) by ideals such that IkIk0 Ik+k0. We hav*
*e that
KSU(X; Zp) = I4. Since the `i are natural operations, and `i(0) = 0, they prese*
*rve the
ideals Ik; in particular, `j(I4) I4. Since X is finite, Ik = 0 for k sufficie*
*ntly large, and
so for ff 2 I4 each expression f(`i(ff)) is actually a finite sum, contained in*
* 1 + I4.
We now show that `i(ff) ! 0 as i ! 1 in the p-adic topology. Since I4 has a *
*finite
filtration by the Ik's, it suffices to do this one filtration quotient at a tim*
*e. By the Atiyah-
Hirzebruch spectral sequence, this amounts to a calculation on the reduced K-th*
*eory of
spheres. Thus, Ik=Ik-1 = 0 if k is odd, and `i(ff) = pi(k=2-1)ff for ff 2 Ik=2=*
*Ik=2-1if
k is even and k > 0. In particular, for ff 2 Ik, k 4, we see that the sequenc*
*e `i(ff)
mod Ik+1 approaches 0 p-adically.
It follows that f(`i(ff)) ! 1 as i ! 1, and therefore the infinite product c*
*onverges.
The argument for KO-theory is almost the same, except for the calculations o*
*n fil-
tration quotients. Here the additional observation is that `i(ff) = 0 for ff 2 *
*Ik=Ik-1, for
all k such that k 2 and k 1, 2 mod 8 (but not when k = 1).
13.7. Exponential operations of Atiyah-Segal. In [AS71 ], the authors construct
explicit exponential maps on K-theory and KO-theory completed at some prime p. *
*Their
construction startsPwith the observation that on any ~-ring R, the operator t:*
*R ! R[[t]]
given by t(x) = i 0~i(x)ti is exponential. By setting t to particular values*
* ff 2 Zp,
LOGARITHMIC COHOMOLOGY OPERATION 41
one can sometimes obtain series which converge p-adically, on some subsets of s*
*uitable p-
adic ~-rings R. In this way, Atiyah and Segal can piece together exponential op*
*erations,
and construct an exponential isomorphism for KO-theory (though not for K-theory*
*).
Their construction involves arbitrary choices, and leads to an operation which *
*is not
infinite-loop.
To compare our construction with theirs, we note thatPin a ~-ring we can set
St(x) = ( -t(x))-1; the operators si defined by St(x) = i 0si(x)ti correspond*
* to
taking symmetric powers of bundles. Adams operations are related to St by the e*
*qua-
tion 2 3
X _m (x)
St(x) = exp4 ______tm 5.
m 1 m
Thus, our exponential operator is a kind of "p-typicalization" of the symmetric*
* powers,
evaluated at t = 1.
14.The action of Hecke operators on Morava E-theory
We give a quick and dirty exposition of a fact which does not seem to be pro*
*ved in
the literature, but should be well-known; namely, that the Morava E-theory of a*
* space
carries an action by an algebra of Hecke operators.
14.1. Hecke operators. Let be a monoid, and a subgroup. Define
H = homZ[ ](Z[ = ], Z[ = ])
where Z[ ] denotes the monoid ring of , and Z[ = ] is the left-Z[ ]-module spa*
*nned
by cosets. If M is a left Z[ ]-module, then the -invariants M are naturally *
*a left
H-module.
Consider two examples:
(a)The algebra H = Hn, associated to = End(Zn) \ GLn(Q) and = GLn(Z).
(b) The algebra H = Hn,p, associated to = End(Znp)\GLn(Qp) and = GLn(Zp).
In either case, H has a basis which is in one-to-one correspondence with double*
* cosets
\ = ; a double cosetP x corresponds to the unique endomorphism "T x of Z[ = ]
which sends 1 7! [y ], where y ranges over representatives of the finite set *
* x = .
In these terms, Hn is the same as the Hecke ring for GLn as described for insta*
*nce in
[Shi71, Ch. 3]. One sees also that
O
Hn,p Z[T"1,p, . .,."Tn,p] and Hn Hn,p,
p
where "Tj,pcorresponds to the double coset of the diagonal matrix which has p i*
*n j entries
and 1 in the other n - j entries.
14.2. Morava E-theory is a module over Hn,p. We want to show that the algebra
Hn,pacts on the functor X 7! E0(X), where E is a Morava E-theory of height n.
(Warning: this only agrees up to scalar with the action described in x1.12; see*
* x14.5
below.) Let = Znp. The right cosets = are in one-to-one correspondence with*
* open
subgroups of , by x 7! x . The sum
X
!+ ([ =y ]) 2 D E^0B ,
42 CHARLES REZK
where y ranges over representatives of x = , is invariant under the action of *
*GL( ) on
A+ ( ), and so lives in E^0B , by (11.6). We define
_ x :E0X ! E0X
to be the operationPassociated to this class. In terms of the notation used in*
* x11.7,
we have i O _ x = _A, where i: E0X ! D O E0X is the evident inclusion (D
is faithfully flat over O), and the sum is over all finite subgroups A * suc*
*h that
ker( ! A*) = y for some y 2 x .
14.3. Proposition. The assignment "T x7! _ x makes E0X into an Hn,p-module.
This is a statement about compositions of the additive cohomology operations*
* _ x .
We will prove it by reducing to results about the composition of certain ring o*
*perations,
proved in [AHS04 , App. B].
14.4. Lemma. For each x 2 there is a ring homomorphism
_x: D O E0X ! D O E0X
natural in X, with the following properties:
(a)If x 2 = GL( ), then _x acts on D O E0X purely through the D-factor, *
*via
the the action of GL( ) on D described in x10.5.
(b) Under the inclusion i: E0X ! D O E0X, we have _x O i = _A, where A *
is the kernel of the adjoint x*: * ! * to x.
(c)If X is a finite product of copies of CP 1, then x 7! _x gives an action*
* of the
monoid on D O E0X.
Proof.Given x 2 , consider the following diagram of formal groups and level st*
*ructures
over D: *
A _____// *_x___//_ *
`|| |`0|
fflffl| fflffl|
i* __f_//_j*
Here is the universal deformation formal group over O, i: O ! D is the usual *
*inclusion,
A = kerx*, f is an isogeny with kernel `(A), such that modulo the maximal ideal*
* of D,
f reduces to a power of frobenius. Therefore the codomain of f is a deformation*
* of 0
to D, classified by a map j :O ! D. There is a commutative diagram
D MMoo____O = E0(pt)_________//_E0(X)
MM
MMM _| _|
O`0MMM&&MMAfflffl|| fAflffl||
D = D O E0(pt) _____//D O E0X
where _A is the operation associated to A as in x11.7. On the cohomology of a p*
*oint,
the map _A :O ! D equals j. The map labelled O`0is the map classifying the pair
consisting of the formal group j* over D, and the level structure `0. We define
_x: D O E0X ! D O E0X by x y 7! O`0(x)_A(y).
Properties (a) and (b) are immediate. Property (c) is proved by the arguments o*
*f [AHS04 ,
App. B] when X = ptor X = CP1 ; since the _x act as ring homomorphisms, they are
compatible with K"unneth isomorphisms, and so property (c) holds for finite pro*
*ducts of
projective spaces.
LOGARITHMIC COHOMOLOGY OPERATION 43
Proof of (14.3).We first show that the cohomology operations _ x make E0X into*
* a
Hn,p-module when X is a finite product of complex projective spaces, and hence *
*when X
is a finite product of CP1 's. Since E is a Landweber exact theory, the result *
*of [Kas94,
Thm. 4.2] applies to show the desired result.
Parts (a) and (c) of (14.4)show that the action of on D O E0X descends to
an action of Hn,pon (D O E0X) . Then (b) shows that the action of an operator
_ x on E0X as defined above coincides with this action of Hn,punder the inclus*
*ion
i: E0X ! (D O E0X) .
14.5. Renormalized operators. Setting Tj,pdef=(1=pj)T"j,p, we obtain the operat*
*ors
described in x1.12. Because of the denominators, this only gives an action of H*
*n,pon
p-1E0X. We introduce this apparently awkward renormalization because it coincid*
*es
with the usual normalization of Hecke operators acting on classical modular for*
*ms.
References
[AHS04] Matthew Ando, Michael J. Hopkins, and Neil P. Strickland, The sigma or*
*ientation is an
H1 map, Amer. J. Math. 126 (2004), no. 2, 247-334. MR MR2045503
[AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra,*
* Addison-Wesley
Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 39 #41*
*29
[And95] Matthew Ando, Isogenies of formal group laws and power operations in t*
*he cohomology
theories En, Duke Math. J. 79 (1995), no. 2, 423-485. MR MR1344767 (97*
*a:55006)
[AS71] M. F. Atiyah and G. B. Segal, Exponential isomorphisms for ~-rings, Qu*
*art. J. Math.
Oxford Ser. (2) 22 (1971), 371-378. MR 45 #344
[BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of fl-spaces, s*
*pectra, and bisim-
plicial sets, Geometric applications of homotopy theory (Proc. Conf., *
*Evanston, Ill., 1977),
II, Springer, Berlin, 1978, pp. 80-130. MR 80e:55021
[BH04] Martin Bendersky and John R. Hunton, On the coalgebraic ring and Bousf*
*ield-Kan spectral
sequence for a Landweber exact spectrum, Proc. Edinb. Math. Soc. (2) 4*
*7 (2004), no. 3,
513-532. MR MR2096616
[BMMS86] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, H1 ring sp*
*ectra and their
applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag*
*, Berlin, 1986. MR
88e:55001
[Bou79] A. K. Bousfield, The localization of spectra with respect to homology,*
* Topology 18 (1979),
no. 4, 257-281. MR MR551009 (80m:55006)
[Bou87] _____, Uniqueness of infinite deloopings for K-theoretic spaces, Pacif*
*ic J. Math. 129
(1987), no. 1, 1-31. MR 89g:55017
[Bou01] _____, On the telescopic homotopy theory of spaces, Trans. Amer. Math.*
* Soc. 353 (2001),
no. 6, 2391-2426 (electronic). MR 2001k:55030
[Dol62] Albrecht Dold, Decomposition theorems for S(n)-complexes, Ann. of Math*
*. (2) 75 (1962),
8-16. MR 25 #569
[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules*
*, and algebras in
stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, *
*American Math-
ematical Society, Providence, RI, 1997, With an appendix by M. Cole. M*
*R MR1417719
(97h:55006)
[GH] P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spec*
*tra, preprint,
http://www.math.nwu.edu/"pgoerss/.
[GJ99] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progre*
*ss in Mathematics,
vol. 174, Birkh"auser Verlag, Basel, 1999. MR MR1711612 (2001d:55012)
[Glu81] David Gluck, Idempotent formula for the Burnside algebra with applicat*
*ions to the p-
subgroup simplicial complex, Illinois J. Math. 25 (1981), no. 1, 63-67*
*. MR 82c:20005
[HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generali*
*zed group char-
acters and complex oriented cohomology theories, J. Amer. Math. Soc. 1*
*3 (2000), no. 3,
553-594 (electronic). MR 1 758 754
[Hod72] Luke Hodgkin, The K-theory of some wellknown spaces. I. QS0, Topology *
*11 (1972), 371-
375. MR 48 #9701
44 CHARLES REZK
[Hop02] M. J. Hopkins, Algebraic topology and modular forms, Proceedings of th*
*e International Con-
gress of Mathematicians, Vol. I (Beijing, 2002) (Beijing), Higher Ed. *
*Press, 2002, pp. 291-
317. MR MR1989190 (2004g:11032)
[HS99] Mark Hovey and Neil P. Strickland, Morava K-theories and localisation,*
* Mem. Amer. Math.
Soc. 139 (1999), no. 666, viii+100. MR MR1601906 (99b:55017)
[HSS00] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer*
*. Math. Soc. 13
(2000), no. 1, 149-208. MR 2000h:55016
[Kas94] Takuji Kashiwabara, Hopf rings and unstable operations, J. Pure Appl. *
*Algebra 94 (1994),
no. 2, 183-193. MR 95h:55005
[Kuh] Nicholas J. Kuhn, Localization of Andr'e-Quillen-Goodwillie towers, an*
*d the periodic ho-
mology of infinite loopspaces, preprint.
[Kuh89] Nicholas J. Kuhn, Morava K-theories and infinite loop spaces, Algebrai*
*c topology (Arcata,
CA, 1986) (Berlin), Lecture Notes in Math., vol. 1370, Springer, 1989,*
* pp. 243-257. MR
MR1000381 (90d:55014)
[LMSM86] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure, Equiva*
*riant stable homo-
topy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag,*
* Berlin, 1986, With
contributions by J. E. McClure. MR MR866482 (88e:55002)
[LT66] Jonathan Lubin and John Tate, Formal moduli for one-parameter formal L*
*ie groups, Bull.
Soc. Math. France 94 (1966), 49-59. MR MR0238854 (39 #214)
[May72] J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berl*
*in, 1972, Lectures
Notes in Mathematics, Vol. 271. MR MR0420610 (54 #8623b)
[May77] J. Peter May, E1 ring spaces and E1 ring spectra, Springer-Verlag, Ber*
*lin, 1977, With
contributions by Frank Quinn, Nigel Ray, and Jorgen Tornehave, Lecture*
* Notes in Mathe-
matics, Vol. 577. MR MR0494077 (58 #13008)
[May82] J. P. May, Multiplicative infinite loop space theory, J. Pure Appl. Al*
*gebra 26 (1982), no. 1,
1-69. MR MR669843 (84c:55013)
[MM02] M. A. Mandell and J. P. May, Equivariant orthogonal spectra and S-modu*
*les, Mem. Amer.
Math. Soc. 159 (2002), no. 755, x+108. MR 2003i:55012
[RR04] Birgit Richter and Alan Robinson, Gamma homology of group algebras and*
* of polyno-
mial algebras, Homotopy theory: relations with algebraic geometry, gro*
*up cohomology, and
algebraic K-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Provid*
*ence, RI, 2004,
pp. 453-461. MR MR2066509
[Seg75] Graeme Segal, The multiplicative group of classical cohomology, Quart.*
* J. Math. Oxford
Ser. (2) 26 (1975), no. 103, 289-293. MR MR0380770 (52 #1667)
[Shi71] Goro Shimura, Introduction to the arithmetic theory of automorphic fun*
*ctions, Publications
of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishe*
*rs, Tokyo, 1971,
Kan^o Memorial Lectures, No. 1. MR 47 #3318
[ST97] Neil P. Strickland and Paul R. Turner, Rational Morava E-theory and DS*
*0, Topology 36
(1997), no. 1, 137-151. MR 97g:55005
[Str97] Neil P. Strickland, Finite subgroups of formal groups, J. Pure Appl. A*
*lgebra 121 (1997),
no. 2, 161-208. MR 98k:14065
[Str98] N. P. Strickland, Morava E-theory of symmetric groups, Topology 37 (19*
*98), no. 4, 757-779.
MR 99e:55008
[Woo79] Richard Woolfson, Hyper- -spaces and hyperspectra, Quart. J. Math. Oxf*
*ord Ser. (2) 30
(1979), no. 118, 229-255. MR MR534835 (81b:55026)
Department of Mathematics, University of Illinois at Urbana-Champaign, Urban*
*a IL, 61820
E-mail address: rezk@math.uiuc.edu