EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA
J. P. MAY
Abstract. Let G be a compact Lie group, let RG be a commutative algebra o*
*ver
the sphere Gspectrum SG, and let R be its underlying nonequivariant alge*
*bra over
the sphere spectrum S. When RG is split as an algebra, as holds for examp*
*le for
RG = MUG, we show how to "extend scalars" to construct a split RGmodule
RG ^R M from an Rmodule M. This allows the wholesale construction of hig*
*hly
structured equivariant module spectra from highly structured nonequivaria*
*nt mod
ule spectra. In particular, it applies to construct MUGmodules from MUm*
*odules
and therefore gives conceptual constructions of equivariant BrownPeterso*
*n and
Morava Ktheory spectra.
We enrich the theory of highly structured modules over highly structured ring
spectra that was developed in [3] by showing how to construct highly structured
equivariant modules from highly structured nonequivariant modules. Throughout,
we let G be a compact Lie group. As pointed out in its introduction, although
[3] is written nonequivariantly, all of its theory applies verbatim equivariant*
*ly. The
equivariant ring spectra we are interested in are the algebras over the equivar*
*iant
sphere spectrum SG . These are essentially equivalent to the earlier E1 ring G*
*spectra
of [8, VIIx2]. The prime examples are SG itself and the spectra MUG of stabiliz*
*ed
equivariant complex cobordism. The latter are studied in [7]. In [4], Elmendorf*
* and I
showed how to construct examples from nonequivariant Salgebras. For finite gro*
*ups
G, a great many other examples are known.
In view of the completion theorem of [7], it is important to be able to const*
*ruct split
MUG modules from MUmodules. As we shall refine below, [7, 6.5] gives that the*
* un
derlying Gspectrum MUG is split (in the sense discussed in [6, x0] and [5, x3]*
*). This
means that there is a map from MU to the fixed point spectrum (MUG )G whose com
posite with the inclusion (MUG )G ! MU is the identity. Therefore MU* = ss*(MU)
is a direct summand of MUG*= ss*((MUG )G ). Starting from MUG , we can follow [*
*3,
Ch. V] and construct MUG module spectra by killing off any sequence of element*
*s of
ss*(MU) and inverting any other sequence. This allows us to construct equivaria*
*nt
versions of the BrownPeterson spectrum BP , the Morava Ktheory spectra k(n) a*
*nd
K(n), and all of the other spectra that are usually constructed from MU by means
of the BaasSullivan theory of manifolds with singularities.
1
2 J. P. MAY
However, we shall give a far more direct and elegant construction that will a*
*pply to
any MUmodule M. This allows us to start from, rather than repeat, the nonequiv
ariant constructions of [3]. One advantage of the more conceptual construction*
* is
that it allows us to bootstrap up the homotopical analysis of MUring spectra i*
*n [3,
Vx4] to a homotopical analysis of MUG ring Gspectra. We would not have suffic*
*ient
calculational control to do this if we used the construction of the previous pa*
*ragraph.
We let MR denote the category of modules over an Salgebra R and let GMRG
denote the category of modules over an SG algebra RG . We let DR and GDRG deno*
*te
the respective derived categories; these are obtained from the respective homot*
*opy
categories by formally inverting the weak equivalences or, equivalently, by pas*
*sing to
approximations by weakly equivalent cell modules.
Theorem 0.1. There is a monoidal functor MUG ^MU (?) : MMU ! GMMUG .
If M is a cell MUmodule, then MUG ^MU M is split as a module with underly
ing nonequivariant MUmodule M. The functor MUG ^MU (?) induces a derived
monoidal functor DMU ! GDMUG . Therefore, if M is an MUring spectrum (in
the homotopical sense), then MUG ^MU M is an MUG ring Gspectrum.
The term "split as a module" will be defined below. It means that MG is split*
* as
a Gspectrum in a fashion suitably compatible with its module structure.
A special case partially answers a problem raised by Carlsson [1]:
"Define and compute equivariant Morava Ktheory spectra."
The answer is only partial because we do not offer a calculation of the coeffic*
*ient
ring. As observed in the introduction of [7], this must await a better understa*
*nding
of MUG*. We should point out an anomaly in one of the few familiar cases: the
equivariant form MUG ^MU k of the MUmodule k that represents connective K
theory cannot represent equivariant connective Ktheory since Greenlees has obs*
*erved
that the latter theory does not take values in modules over MUG*. We do not know
whether or not the equivariant form MUG ^MU K of periodic Ktheory that we
construct represents equivariant Ktheory.
The theorem is a special case of one that applies to all SG algebras that ar*
*e "split
as algebras", in a sense that we shall make precise below.
Theorem 0.2. Let RG be a commutative SG algebra and assume that RG is split as
an algebra with underlying nonequivariant Salgebra R . Then there is a monoidal
functor RG ^R (?) : MR ! GMRG . If M is a cell Rmodule, then RG ^R M is split
as a module with underlying nonequivariant Rmodule M. The functor RG ^R (?)
induces a derived monoidal functor DR  ! GDRG . Therefore, if M is an Rring
spectrum (in the homotopical sense), then RG ^R M is an RG ring Gspectrum.
The application to MUG is a special case of a generic criterion for RG to be *
*split
as an algebra. The notion of a global I*FSP was defined in [7], and it was sho*
*wn
there that the sphere and cobordism functors provide examples.
EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA 3
Theorem 0.3. If T is a global I*FSP, then its associated commutative SG alge*
*bra
is split as an algebra for every compact Lie group G.
Although the basic idea and construction predate the writing of [4], this pap*
*er is
best understood as a sequel to that one, and we shall freely use its notations *
*and
results. The reader is referred to [8, 5, 9] for the relevant background on equ*
*ivariant
stable homotopy theory.
1. Change of universe and operadic smash products
The functor RG ^R M that we shall construct depends on an extension of the
operadic smash product
^L : GS U[L ] x GS U[L ] ! GS U[L ]
of [3, Ch I] that incorporates the change of universe functors
IUU0: GS U0[L 0] ! GS U[L ]
of [4, 1.1]. Recall from [4, 1.5] that the functors IUU0are monoidal equivalen*
*ces of
categories.
Definition 1.1.Let U, U0, and U00be Guniverses. For an L0spectrum M and an
L00spectrum N, define an Lspectrum M ^L N by
M ^L N = IUU0M ^L IUU00N:
Obviously, the formal properties of this product can be deduced from those of*
* the
functors IUU0together with those of the operadic smash product for the fixed un*
*iverse
U. In particular, since the functor IUU0takes SU0modules to SUmodules and the
smash product over SU is the restriction to SUmodules of the smash product over
L , we have the following observation. Here SU denotes the sphere Gspectrum
indexed on U.
Lemma 1.2. The functor ^L : GS U0[L 0] x GS U00[L 00] ! GS U[L ] restricts *
*to a
functor
^SU : GMSU0x GMSU00! GMSU :
There is an alternative description of this product that makes its structure *
*more
apparent. It depends on the following generalization of [3, I.5.4], which in f*
*act is
implied by that result; compare [4, 1.2].
Lemma 1.3. Assume given universes U, U0, U00, U0rfor 1 r i, and U00sfor
1 s j, where i 1 and j 1. Then the following diagram is a split coequalizer
4 J. P. MAY
of spaces and therefore a coequalizer of Gspaces; the maps fl are given by sum*
*s and
compositions of linear isometries.
I (U0 U00; U) x I (U0; U0) x I (U00; U00) x I (ir=1U0r; U0) x I (js=1U00s; U*
*00)
flxididxfl
fflfflfflffl
I (U0 U00; U) x I (ir=1U0r; U0) x I (js=1U00s; U00)
fl
fflffl
I ((ir=1U0r) (js=1U00s); U):
Lemma 1.4. There is a natural isomorphism
M ^L N ! I (U0 U00; U) nI (U0;U0)xI (U00;U00)M ^ N;
where ^ on the right is the external smash product GS U0xGS U00! GS (U0U00).
Proof.Expanding definitions, we see that M ^L N is
I (U U; U) nI (U;U)xI (U;U)[(I (U0; U) nI (U0;U0)M) ^ (I (U00; U) nI (U00;U00*
*)N)]:
Formal properties of the twisted halfsmash product allow us to rewrite this as
[I (U U; U) xI (U;U)xI (U;U)I (U0; U) x I (U00; U)] nI (U0;U0)xI (U00;U00)M *
*^ N:
The previous lemma gives a homeomorphism
I (U U; U) xI (U;U)xI (U;U)I (U0; U) x I (U00; U) ! I (U0 U00; U)
of Gspaces over I (U0 U00; U), and the conclusion follows. ___ 
Similarly, as in the proof of [3, I.5.5 and I.5.6], Lemma 1.3 implies the fol*
*lowing
associativity property of our generalized operadic smash products and therefore*
*, upon
restriction, of our generalized smash products over sphere Gspectra.
Lemma 1.5. Let M 2 GS U0[L 0], P 2 GS U00[L 00], and N 2 GS U000[L 000]. Then
both (M ^L 0P ) ^L N and M ^L (P ^L 000N) are canonically isomorphic to
I (U0 U00 U000; U) nI (U0;U0)xI (U00;U00)xI (U000;U000)M ^ P ^ N;
which in turn is canonically isomorphic to
IUU0M ^L IUU00P ^L IUU000N:
Using change of universe explicitly or, via the previous lemmas, implicitly, *
*we can
define modules indexed on one universe over algebras indexed on another.
Definition 1.6.Let R 2 GMSU00be an SU00algebra0and let M 2 GMSU0. Say that
M is a right Rmodule if it is a right IUU00Rmodule, and similarly for left mo*
*dules.
It is quite clear how one must define smash products over R in this context.
EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA 5
Definition 1.7.Let R 2 GMSU00be an SU00algebra, let M 2 GMSU0 be a right
Rmodule and let N 2 GMU000be a left Rmodule. Define
M ^R N = IUU0M ^IUU00RIUU000N:
0 U U
Here we have used that IUU00~=IUU0IUU00and that IU0M is therefore an IU00Rmo*
*dule,
and similarly for N. Expanding definitions and using our associativity isomorph*
*isms,
we obtain the following more explicit description.
Lemma 1.8. M ^R N is the coequalizer displayed in the diagram
I (U0 U00 U000; U) nI (U0;U0)xI (U00;U00)xI (U000;U000)M ^ R ^ N


fflfflfflffl
I (U0 U000; U) nI (U0;U0)xI (U000;U000)M ^ N


fflffl
M ^R N;
where the parallel arrows are induced by the actions of R on M and on N.
Evidently these smash products inherit good formal properties from those of t*
*he
smash products of Rmodules studied in [3]. Similarly, their homotopical proper*
*ties
can be deduced from the homotopical properties of the smash product of Rmodules
and the homotopical properties of the IUU0, which were studied in [4].
2.SG algebras and their underlying Salgebras
We are concerned with genuine Gspectra and their comparison with naive G
spectra. Recall that these are indexed respectively on a complete Guniverse U *
*and
its Gfixed point universe UG . We write SG for the sphere Gspectrum indexed o*
*n U
and S for the sphere spectrum indexed on UG . We regard nonequivariant spectra *
*such
as S as Gspectra with trivial Gaction. We have the forgetful change of unive*
*rse
functor i* : GS U  ! GS UG obtained by forgetting those indexing spaces of U
that are not contained in UG . The underlying nonequivariant spectrum E of a G
spectrum EG is defined to be i*EG , with its action by G ignored. Said another *
*way,
let U# denote U with its action by G ignored and let E#Gdenote the nonequivaria*
*nt
spectrum indexed on U# that is obtained from EG by forgetting the action of G.
Then E = i*E#G.
The Gfixed point spectrum of EG is obtained by taking the spacewise fixed po*
*ints
of i*EG . We say that EG is split if there is a map E ! (EG )G of spectra inde*
*xed on
UG whose composite with the inclusion of (EG )G in E is an equivalence. As obse*
*rved
in [6, 0.4], EG is split if and only if there is a map of Gspectra i*E  ! EG *
*that
is a nonequivariant equivalence, where i* : GS UG ! GS U is the left adjoint *
*of
6 J. P. MAY
i*. In either form, the notion of a split Gspectrum is essentially a homotopic*
*al one.
More precisely, it is a derived category notion: its purpose is to allow the co*
*mparison
of equivariant and nonequivariant homology and cohomology theories, which are
defined on derived categories. Thus we could have used weak equivalences in the
definitions just given, and we shall use the term equivalence to mean weak equi*
*valence
of underlying spectra or Gspectra in what follows; we will add the adjective "*
*weak"
in cases where we would not expect to have a homotopy equivalence in general.
We must modify these definitions in the context of highly structured ring and
module spectra. This point was left obscure in both [5] and [7], where referen*
*ce
was made to "the underlying Salgebra R of an SG algebra RG ": in fact there i*
*s no
obvious way to give R = i*R#Ga structure of Salgebra. The point becomes clear
when one thinks back to the underlying E1 ring structures. We are given Gmaps
(2.1) I (Uj; U) n RjG! RG ;
and there is no obvious way to obtain induced nonequivariant maps
(2.2) I ((UG )j; UG ) n Rj ! R:
Observe however, that we can forget the Gactions and regard the maps (2.1) as
maps of nonequivariant spectra. That is, before restricting to indexing spaces *
*in UG ,
we obtain a perfectly good nonequivariant SU# algebra R#Gfrom our equivariant *
*SU
algebra RG . We may choose a nonequivariant linear isomorphism f : UG ! U# .
By conjugation of linear isometries by f, we obtain an isomorphism between the
nonequivariant linear isometries operads of UG and of U# , and we find immediat*
*ely
that if R is an SU# algebra, then f*R is an SUG algebra. Use of f* rather tha*
*n i*
loses no homotopical information.
Lemma 2.3. For nonequivariant spectra F 2 S U# , there is a natural weak equi*
*v
alence between i*F and f*F .
Proof.One compares the functors i* and f* to the twisted halfsmash product fun*
*c
tor determined by a path h : I ! I (UG ; U# ) connecting i to f. One so obtai*
*ns
homotopy equivalences i*E  ! h n E  f*E for a class of spectra E 2 S UG
that includes all CW spectra. Conjugation from left to right adjoints gives the*
* con
clusion, since a simple diagram chase shows that the conjugate natural maps ind*
*uce
isomorphisms of homotopy groups. See [8, pp. 5962] and [3, I.2.5] for details.*
* ___ 
Following the philosophy of [3], it is more natural to consider the twisted f*
*unction
spectrum F [I (UG ; U# ); F ) than to make the arbitrary choice of an isomorphi*
*sm f.
Similar arguments show that this too is weakly equivalent to i*F for any F 2 S *
*U# .
However, to retain structure, one must restrict F to be an Lspectrum and repla*
*ce
the function spectrum by an operadic version obtained by equalizing a pair of m*
*aps
F [I (UG ; U# ); F_)_//_//_F [I (U# ; U# ) x I (UG ; U# ); F );
EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA 7
one arrow is induced by composition I (U# ; U# ) x I (UG ; U# ) ! I (UG ; U# )*
* and
the other is suitably induced from the action of I (U# ; U# ) on F ; compare [3*
*, I.7.5].
However, there is#no need to introduce this construction: it gives the right ad*
*joint
to the functor IUUG: S UG [L ] ! S U# [L ], and we know from [4, 1.3] that the*
* right
# UG
adjoint of IUUGis IU# . By Lemma 2.1 and the specialization to G = e of the fol*
*lowing
G
observation, we again lose no homotopical information if we replace i* by IUU#,*
* and
this functor too carries nonequivariant SU# algebras to SUG algebras.
Lemma 2.4. Let f : U  ! U0 be an isomorphism of Guniverses. Then there are
natural isomorphisms
0 * 0 U 0
f*E ~=IUUE and f E ~= IU0E
for E 2 GS U[L ] and E02 GS U0[L 0].
Proof.Regard f as a Gmap from a point into I (U; U0). Then the following com
posite is a homeomorphism of Gspaces over I (U; U0):
fxid 0 O 0
{*} x I (U; U)____//_I (U; U ) x I (U;_U)_//_I (U; U ):
By [8, VI.3.1(iii)], there results a natural isomorphism
f*(I (U; U) n E) ~=I (U; U0) n E:
Passing to coequalizers, we obtain
f*E ~=f*(I (U; U) nI (U;U)E) ~=I (U; U0) nI (U;U)E:
Since f* = f1*, the second isomorphism follows from the first. ___ 
We now have two possible interpretations of the underlying nonequivariant S
algebra R associated to an SG algebra RG : we can take f*R#Gor, more canonical*
*ly,
G #
IUU#RG . We prefer to keep our options open, and this leads us to the following
definition.
Definition 2.5.A commutative SG algebra RG is split as an algebra if there is a
commutative Salgebra R and a map j : IUUGR ! RG of SG algebras such that j is
a (nonequivariant) equivalence of spectra and the natural map ff : i*R ! IUUGR*
* is
an (equivariant) equivalence of Gspectra. We call R the (or, more accurately,*
* an)
underlying nonequivariant Salgebra of RG .
Since the composite j O ff is a nonequivariant equivalence and the natural map
R  ! i*i*R is a weak equivalence (provided that R is tame, [8, II.1.8] and [3,
I.2.5]), R is weakly equivalent to i*R#G. Thus R is a highly structured version*
* of the
underlying nonequivariant spectrum of RG . Of course, RG is split as a Gspectr*
*um
with splitting map j O ff.
We have a parallel definition for modules.
8 J. P. MAY
Definition 2.6.Let RG be a commutative SG algebra that is split as an algebra
with underlying Salgebra R and let MG be an RG module. Regard MG as an IUUGR
module by pullback along j. Then MG is split as a module if there is an Rmodule
M and a map O : IUUGM ! MG of IUUGRmodules such that O is a (nonequivariant)
equivalence of spectra and the natural map ff : i*M ! IUUGM is an (equivari
ant) equivalence of Gspectra. We call M the (or, more accurately, an) underlyi*
*ng
nonequivariant Rmodule of MG .
Again, M is a highly structured version of the underlying nonequivariant spec*
*trum
of MG , and MG is split as a Gspectrum with splitting map O O ff. The ambigui*
*ty
that we allow in the notion of an underlying object is quite useful: it allows *
*us to
arrange the condition on ff in the definitions if we have succeeded in arrangin*
*g the
other conditions. The proof of this depends on the closed model category struct*
*ures
on all categories in sight that is given in [3, VIIx4].
Lemma 2.7. Let RG be a commutative SG algebra.
(i)Suppose given a commutative Salgebra R0 and a map j0 : IUUGR0 ! RG
of SG algebras such that j is a (nonequivariant) equivalence of spectra.*
* Let
fl : R ! R0 be a weak equivalence of Salgebras, where R is a qcofibrant
Salgebra, and define j = j0O IUUGfl : IUUGR ! RG . Then RG is split wi*
*th
underlying nonequivariant Salgebra R and splitting map j.
(ii)Let MG be an RG module and suppose given an R0module M0 and a map
O0 : IUUGM0 ! MG of IUUGR0modules such that O0 is a (nonequivariant)
equivalence of spectra. Regard M0 as an Rmodule by pullback along fl, let
# : M ! M0 be a weak equivalence of Rmodules, where M is a qcofibrant
Rmodule, and define O = O0O IUUG# : IUUGM ! MG . Then MG is split with
underlying nonequivariant Rmodule M and splitting map O.
Proof.It is immediate from [4, 0.2] that ff : i*R ! IUUGR and ff : i*M ! IUUG*
*M are
equivalences#of Gspectra.# Thus we need only observe that, ignoring the Gacti*
*on,
the maps IUUGfl and IUUG# are weak equivalences, and it is immediate from the c*
*ase
# __
G = e of Lemma 2.2 that the functor IUUG preserves weak equivalences. _ 
As observed in [3, VII.1.3], the splitting map j : IUUGR ! RG of a split com*
*mu
tative RG algebra is the unit of a structure of IUUGRalgebra on RG . The comp*
*osite
id^j OE
RG ^SG IUUGR ____//_RG ^SG RG____//RG
gives RG a structure of right Rmodule in the sense prescribed in Definition 1.*
*6, and
RG is an (RG ; R)bimodule with left action of RG induced by the product OE of *
*RG .
Proof of Theorem 0.2.Let M be an Rmodule. It is evident that RG ^R M, as defin*
*ed
in Definition 1.7 with U = U0 and UG = U00= U000, is an RG module with action
EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA 9
induced by the left action of RG on itself. We abbreviate notation by setting
MG = RG ^R M:
Since the functor IUUG : MR  ! GMIUUR is monoidal by [4, 0.1] and the functor
G
RG ^IUUR(?) : GMIU R ! GMRG is monoidal by an easy elaboration of [3, III.3.10*
*],
G UG
the functor RG ^R (?) is monoidal.
As observed in the previous lemma, we can and may assume that our given un
derlying nonequivariant Salgebra R is qcofibrant as an Salgebra. Let M be a *
*cell
Rmodule. Then M is qcofibrant, and the condition on ff in the definition of *
*an
underlying Rmodule is satisfied. Define
O = j ^ id: IUUGM ~=IUUGR ^IU R IUUGM ! RG ^IU R IUUGM = MG :
UG UG
Clearly O is a map of IUUGRmodules, and we must prove that it is an equivalence
of spectra. Recall from [3, IIIx1] that we have a free functor FR from spectra*
* to
Rmodules given by
FRX = R ^S (S ^L L X) ~=R ^L L X;
here L and L refer to the universe UG , but we have a similar free functor FRG *
*from
Gspectra to RG modules based on use of the linear isometries operad for U, and
similarly for IUUGR. If we forget about Gactions and compare definitions, we f*
*ind by
use of an isomorphism f : UG  ! U# that, nonequivariantly,
#
IUUGFRX ~=FIU# RX:
UG
Recalling the definition of cell Rmodules from [3, III.2.1], we see that cell *
*Rmodules
are built up via pushouts and sequential colimits from the free Rmodules gener*
*ated
by sphere spectra and their cones. The functor IUUG is a left adjoint, whether*
* we
interpret it equivariantly or nonequivariantly. We conclude that this functor c*
*arries
cell Rmodules to IUUGRmodules that, with Gactions ignored, are nonequivariant
#
cell IUUGRmodules. Now [3, III.3.8] gives that O is an equivalence of spectra *
*since j
is an equivalence of spectra.
It remains to consider the passage to derived categories, and there is a slig*
*ht sub
tlety here: our functor on modules was constructed as the composite of two func*
*tors,
but, as we saw in [4, x2], it does not follow that the induced functor on deriv*
*ed
categories factors as a composite. The solution is simple: we ignore the interm*
*ediate
category GDIUUR, as it is of no particular interest to us. Since cell approxim*
*ation
G
of Rmodules commutes up to equivalence with smash products, passage to derived
categories preserves smash products. ___ 
10 J. P. MAY
3.Global I*functors and split SG algebras
We must prove Theorem 0.3. The notion of a global I*FSP, or G I*FSP, was
defined in [7, xx5,6]. We shall only sketch the definition here, referring the*
* reader
to [7] for more details. In fact, we only need a tiny fraction of the structure*
* that is
present on SG algebras that arise from G I*FSP's. In what follows, we could w*
*ork
with either real or complex inner product spaces, and of course the complex cas*
*e is
the one relevant to complex cobordism; see [7, 6.5].
Let G I* be the category of pairs (G; V ) consisting of a compact Lie group G*
* and
a finite dimensional Ginner product space V ; the morphisms (ff; f) : (G; V ) *
* !
(G0; V 0) consist of a homomorphism ff : G ! G0of Lie groups and an ffequivar*
*iant
linear isomorphism f : V  ! V 0. Let G T be the category of pairs (G; X), whe*
*re G
is a compact Lie group and X is a based Gspace; the morphisms (ff; f) : (G; X)*
* !
(G0; X0) consist of a homomorphism ff : G ! G0 and an ffequivariant based map
f : X ! X0. Let So : G I* ! G T be the functor that sends a pair (G; V ) to *
*the
based Gspace SV .
A G I*FSP T is a continuous functor T : G I* ! G T over the category G of
compact Lie groups together with continuous natural transformations
j : So ! T and ! : T ^ T ! T O
such that the appropriate unity, associativity, and commutativity diagrams comm*
*ute.
Since T is a functor over G , we may write T (G; V ) = (G; T V ), and we requir*
*e that
(3.1) T (ff; id) = (ff; id) : (G; T V 0) ! (G0; T V 0)
for a homomorphism ff : G  ! G0 and a G0inner product space V 0regarded by
pullback along ff as a Ginner product space. Henceforward, we abbreviate notat*
*ion
by writing T (G; V ) = T V on objects. For each object (G; V ), we are given a *
*Gmap
j : SV  ! T V:
For each pair of objects (G; V ) and (G0; V 0), we are given a G x G0map
! : T V ^ T V 0! T (V V 0);
by pullback along the diagonal, we regard ! as a Gmap when G = G0.
We insert some observations that show the power of condition (3.1) and are at*
* the
heart of our work. Let e denote the trivial group and let : e ! G and " : G *
*! e
be the unique homomorphisms.
Lemma 3.2. If V has trivial Gaction, then T V also has trivial Gaction. *
*For a
general Ginner product space V , if V # denotes V regarded as an espace, then*
* T V #
is the space T V with its action by G ignored.
EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA 11
Proof.For the first statement, the functor T carries the morphism ("; id) : (G;*
* V ) !
(e; V ) of G I* to the morphism ("; id) of G T , so that the identity map on T *
*V must
be "equivariant. For the second statement, the functor T carries the morphism
(; id) : (e; V ) ! (G; V ) to the identity map on the space T V . ___ 
For a compact Lie group G and a Guniverse U, we obtain a Gprespectrum T(G;U)
indexed on U with V th Gspace T V . The structural maps are given by the compo*
*sites
id^j !
T V ^ SWV ____//_T V ^ T (W  V_)__//_T W
for V W . Write R(G;U)for the Gspectrum LT(G;U), where L is the spectrificati*
*on
functor of [8, I.2.2].
Now suppose given Guniverses U and U0. Then there is a canonical map of G
spectra indexed on U0
(3.3) i : I (U; U0) n R(G;U)! R(G;U0):
Indeed, if f : U ! U0 is a linear isometry and V is an indexing space in U, th*
*en the
maps T f : T V  ! T f(V ) specify a map of prespectra T(G;U)! f*T(G;U0)indexed
on U. By adjunction, T f gives a map of prespectra i(f) : f*T(G;U)! T(G;U0)ind*
*exed
on U0; see [8, p.58]. We record the following observation for later reference.
Lemma 3.4. If f : U ! U0 is an isomorphism, then i(f) : f*T(G;U)! T(G;U0)is
an isomorphism; if f is a Gmap then i(f) is a Gmap.
Intuitively, the twisted halfsmash product I (U; U0) n R(G;U)is obtained by *
*gluing
together the spectrifications f*R(G;U)of the f*T(G;U), and the maps i(f) glue t*
*ogether
to give the Gmap i. This sort of argument first appeared in [10, IV.1.6, IV.2*
*.2],
before the twisted halfsmash product was invented, and it was formalized in cu*
*rrent
terminology in [8, VI.2.17].
Fixing G and U, a precisely similar argument, formalized in [8, VI.5.5, VII.2*
*.4,
and VII.2.6], shows that the maps
Tf
j(f) : T V1 ^ . .T.Vj!!T (V1 . . .Vj) ! T f(V1 . . .Vj)
for linear isometries f : Uj ! U give rise to maps
j : L (j) n (R(G;U))j ! R(G;U)
that give R(G;U)a structure of L Gspectrum. When the universe U is complete,
so that its linear isometries operad L is an E1 operad of Gspaces, this means
that R(G;U)is an E1 ring Gspectrum. The map 1 gives R(G;U)an action of L (1) =
I (U; U), and functoriality implies that 2 factors through the coequalizer that*
* defines
the operadic smash product, giving
(3.5) : R(G;U)^L R(G;U)= L (2) nL (1)2(R(G;U))2 ! R(G;U):
12 J. P. MAY
As explained in [3, II.3.3 and IIx4], the maps j for j 3 can be reconstructed *
*from
the maps for j = 1 and j = 2.
Lemma 3.6. The map i of (3.3) factors through the coequalizer to give a map
0 0
i : IUUR(G;U)= I (U; U ) nI(U;U)R(G;U)! R(G;U0);
and i is a map of L 0Gspectra, where L 0is the linear isometries operad of U0.
Proof.The factorization is clear from functoriality. To check that i is a map o*
*f L 0
Gspectra, we must show that the following diagram commutes:
0 i2dni2 0 2
L 0(2) nL 0(1)2(IUUR(G;U))___//_L (2) nL 0(1)2(R(G;U0))
 
fflffl fflffl
0
IUUR(G;U)_________i__________//R(G;U0):
Using Lemma 1.3 to identify the upper left corner of the diagram and chasing th*
*rough
the definitions, we see that both composites coincide with the following one:
I (U U; U0) nI (U;U)2(R(G;U))2
idn!
fflffl
I (U U; U0) nI (UU;UU) R(G;UU)
i
fflffl
R(G;U);
here ! is induced by passage to spectra from the evident map of prespectra. __*
*_ 
Returning to our fixed G and a complete Guniverse U, we consider R(e;UG )and
R(G;U). We deduce from Lemma 3.2 that
(3.7) R(e;UG=)R(G;UG ) and R#(G;U)= R(e;U#:)
That is, R(G;UG )is R(e;UG )regarded as a Gtrivial Gspectrum indexed on the G
trivial universe UG , and R(G;U)regarded as a nonequivariant spectrum indexed on
U# is R(e;U# )
The first of these equalities allows us to specialize the map i to obtain a m*
*ap of
E1 ring Gspectra
(3.8) i : IUUGR(e;UG=)I (UG ; U) nI(UG ;UGR)(G;UG)! R(G;U):
The second of these equalities allows us to identify the target of the underlyi*
*ng map
i# of nonequivariant spectra with R(e;U#.)
EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA 13
Lemma 3.9. The map i# is an isomorphism of spectra.
Proof.Choose an isomorphism f : UG  ! U# . It is immediate that the composite
~= # i#
f*R(e;UG )___//_IUUGR(e;UG_)_//R(e;U# )
coincides with the isomorphism i(f) of Lemma 3.4; here the unlabelled isomorphi*
*sm
is given by the case G = e of Lemma 2.4. ___ 
To pass to SG algebras, we let R be the Salgebra S ^L R(e;UG )and RG be the
SG algebra SG ^L R(G;U)(where L refers respectively to UG and to U). By [4, 1.*
*4],
we have an isomorphism of SG algebras
IUUGR ~=SG ^L IUUGR(e;UG;)
and this allows us to define an isomorphism of SG algebras
(3.10) j = id^i : IUUGR ! RG :
At this level of generality, we cannot expect to prove that ff : i*R ! IUUGR i*
*s an
equivalence of Gspectra, although it seems plausible that this holds in the ex*
*amples
of interest. However, we can appeal to qcofibrant approximation, as in Lemma 2*
*.7,
to complete the proof of Theorem 0.3, thereby losing that j is an isomorphism in
order to make sure that ff is an equivalence.
References
1. G. Carlsson. A survey of equivariant stable homotopy theory. Topology 31 (19*
*92), 127.
2. A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Modern foundations of s*
*table homotopy
theory. Handbook of Algebraic Topology, edited by I.M. James. North Holland.*
* 1995.
3. A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Rings, modules, and alg*
*ebras in stable
homotopy theory. Preprint, 1995.
4. A.D. Elmendorf and J.P. May. Algebras over equivariant sphere spectra. Prepr*
*int, 1995.
5. J.P.C. Greenlees and J.P. May. Equivariant stable homotopy theory. Handbook *
*of Algebraic
Topology, edited by I.M. James. North Holland. 1995.
6. J. P. C. Greenlees and J. P. May. Generalized Tate cohomology. Memoirs Amer.*
* Math. Soc. Vol
113, No 543. 1995.
7. J.P.C. Greenlees and J.P. May. Localization and completion theorems for MUm*
*odule spectra.
Preprint, 1995.
8. L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J.E. McClure*
*). Equivariant
stable homotopy theory. Springer Lecture Notes in Mathematics Vol.1213. 1986.
9. J.P. May. Equivariant homotopy and cohomology theory. NSFCBMS Regional Conf*
*erence Pro
ceedings. To appear.
10.J. P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). E1 rin*
*g spaces and
E1 ring spectra. Springer Lecture Notes in Mathematics Volume 577. 1977.
Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA
Email address: may@math.uchicago.edu