EQUIVARIANT UNIVERSAL COEFFICIENT AND KU"NNETH
SPECTRAL SEQUENCES
L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
Abstract.We construct hyper-homology spectral sequences of Z-graded and
RO(G)-graded Mackey functors for Extand Torover G-equivariant S-algebras
(A1 ring spectra) for finite groups G. These specialize to universal coe*
*fficient
and K"unneth spectral sequences.
Introduction
In the non-equivariant context, universal coefficient and K"unneth spectral s*
*e-
quences provide important tools for computing generalized homology and coho-
mology. EKMM [5] constructs examples of these types of spectral sequences for
theories that come from "S-algebras" (or, equivalently, A1 ring spectra). These
spectral sequences are special cases of the general "hyper-homology" spectral s*
*e-
quences for computing "Tor" (the homotopy groups of the derived smash product)
and "Ext" (the homotopy groups of the derived function module) for the category
of modules over an S-algebra. The purpose of this paper is to construct versions
of these hyper-homology spectral sequences for the category of modules over an
G-equivariant S-algebra indexed on a complete universe, where G is a finite gro*
*up.
In Section 1, we derive a number of equivariant universal coefficient and K"unn*
*eth
spectral sequences from these equivariant hyper-homology spectral sequences.
In this context, the homotopy groups of a G-spectrum X form a graded Mackey
functor. There is an abelian group (_ssqX)(G=H) = ssHqX = ssq(XH ) for each sub-
group H of G, and for each subgroup K of H there are maps ssHqX ! ssKqX (induced
by the inclusion of fixed points) and ssKqX ! ssHqX (the transfer), satisfying *
*vari-
ous natural relations. This "homotopy Mackey functor" can be regarded as graded
over either the integers or the real representation ring of G. Both choices le*
*ad
to equivariant generalizations of the spectral sequences of EKMM [5]. Although
these generalizations appear formally similar and are developed here in paralle*
*l,
the spectral sequences have a very different calculational feel.
As reviewed in Section 2, the categories of Z-graded and of RO(G)-graded
Mackey functors have symmetric monoidal products denoted " #" and " *" re-
spectively. The products have adjoint function object functors "<- , ->#" and
"<- , ->*". There is an obvious notion of a graded Mackey functor ring (in ei-
ther context), which consists of a graded Mackey functor __R* with an associati*
*ve
and unital multiplication __R* __R* ! __R*. We have the evident notion of a *
*left
(resp. right) __R*-module as a graded Mackey functor __M* with an associative a*
*nd
____________
Date: October 4, 2004.
1991 Mathematics Subject Classification. Primary 55N91; Secondary 55P43,55U2*
*0,55U25.
The second author was supported in part by NSF grant DMS-0203980.
1
2 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
unital map __R* __M* ! __M* (resp. __N* __R* ! __N*). The usual coequaliz*
*er de-
fines a functor taking a right __R*-module __N* and a left __R*-module __M* to *
*a graded
Mackey functor __N* __R*_M*. Similarly, the usual equalizer defines a functor *
*taking
a pair of left __R*-modules __L* and __M* to the graded Mackey functor <__L*, _*
*_M*>__R*. In
Section 4, standard homological algebra is used to construct the associated der*
*ived
functors. We define ___Tor_R*s(__N*, __M*) as the s-th left derived functor of *
*__N* __R*_M*
and ___Exts_R*(__L*, __M*) as the s-th right derived functor of <__L*, __M*>__R*
**. These are re-
garded as bigraded Mackey functors with the usual conventions: ___Tor_R*s,o(__N*
**, __M*) =
(___Tor_R*s(__N*, __M*))o and ___Exts,o_R*(__L*, __M*) = (___Exts_R*(__L*, __M**
*))-o. The homological
grading in s is always over the non-negative integers whereas the internal grad-
ing in o is over Z or RO(G). When __R* is commutative, __R*becomes a closed
symmetric monoidal product on the category of left __R*-modules, and ___Tor_R*s*
*and
___Exts_R*become graded __R*-modules. More generally, when M is an __R*-bimodul*
*e (has
commuting left and right module structures), ___Tor_R*s(__N*, __M*) and ___Exts*
*_R*(__L*, __M*)
become graded right __R*-modules via the "unused" __R*-module structure on __M*.
Similarly, when __N* and __L* are __R*-bimodules, ___Tor_R*s(__N*, __M*) and __*
*_Exts_R*(__L*, __M*)
become graded left __R*-modules.
The equivariant stable category has a symmetric monoidal product "^", and
the homotopy Mackey functor _ss* is a lax symmetric monoidal functor into either
Z-graded or RO(G)-graded Mackey functors. In other words, there is a suitably
associative, commutative, and unital natural transformation
_ss*X _ss*Y ___//_ss*(X ^ Y ).
For formal reasons, we obtain an adjoint natural transformation
_ss*F (X, Y ) __//_<_ss*X, _ss*Y>,
where F denotes the function spectrum. Further, when R is a homotopical ring
spectrum (i.e., a ring spectrum in the equivariant stable category), _ss*R is a*
* Mackey
functor ring, commutative when R is. Likewise, when M is a homotopical R-
module, _ss*M becomes a _ss*R-module. When R is an equivariant S-algebra, we
have a smash product over R and an R-function spectrum functor. In this context,
it is convenient to extend the conventions of EKMM [5] by using ___TorR*(N, M) *
*and
___Ext*R(L, M) for the homotopy Mackey functors of the derived functors. The na*
*tural
transformations above descend to natural transformations
__N* __R*_M* ! ___TorR*(N, M) and ___Ext*R(L, M) ! <__L*, __M*>__R*,
where __R* = _ss*R, __L* = _ss*L, __M* = _ss*M, and __N* = _ss*N. When M is a *
*weak
bimodule, i.e., has a homotopical right R-module structure in the category of l*
*eft
R-modules, ___TorR*(N, M) and ___Ext-*R(L, M) are naturally right __R*-modules *
*and the
maps above are maps of right R-modules. Similar statements hold for L and N.
Our main results are the following two theorems.
Theorem. (Hyper-Tor Spectral Sequences) Let G be a finite group, R be an equi-
variant S-algebra indexed on a complete universe, and M and N be a left and a
right R-module. There are strongly convergent spectral sequences
E2s,o= ___Tor_R*s,o(__N*, __M*) =) ___TorRs+o(N, M),
of Z-graded Mackey functors and of RO(G)-graded Mackey functors which are nat-
ural in M and N. The edge homomorphisms are the usual natural transformations
EQUIVARIANT SPECTRAL SEQUENCES 3
__N* __R*_M* ! ___TorR*(N, M). When R is commutative or M is a weak bimodule,
these are spectral sequences of right __R*-modules. When N is a weak bimodule, *
*they
are spectral sequences of left __R*-modules.
Theorem. (Hyper-Ext Spectral Sequences) Let G be a finite group, R be an equi-
variant S-algebra indexed on a complete universe, and L and M be left R-modules.
There are conditionally convergent spectral sequences
Es,o2= ___Exts,o_R*(__L*, __M*) =) ___Exts+oR(L, M),
of Z-graded Mackey functors and of RO(G)-graded Mackey functors which are nat-
ural in L and M. The edge homomorphisms are the usual natural transformations
___Ext*R(L, M) ! <__L*, __M*>__R*. When R is commutative or M is a weak bimodu*
*le,
these are spectral sequences of right __R*-modules. When L is a weak bimodule, *
*these
are spectral sequences of left __R*-modules.
To be precise about the differentials, the Hyper-Tor spectral sequence has r-*
*th
differential
drs,t:Ers,t_//_Ers-r,t+r-1.
Thus, in the Z-graded context, it is a homologically graded right half-plane sp*
*ectral
sequence of Mackey functors. In the RO(G)-graded context, it essentially consis*
*ts of
one homologically graded right half-plane spectral sequence of Mackey functors *
*for
each element of the subgroup gRO(G) of RO(G) consisting of virtual representati*
*ons
of dimension zero. The Hyper-Ext spectral sequence has r-th differential
ds,tr:Es,tr_//_Es+r,t-r+1r.
In the Z-graded context, it is a cohomologically graded right half-plane spectr*
*al
sequence of Mackey functors. In the RO(G)-graded context, it consists of one
cohomologically graded right half-plane spectral sequence of Mackey functors for
each element of gRO(G).
In the Z-graded context, "strong" convergence means that for every subgroup
H of G, the spectral sequence of abelian groups Ers,t(G=H) converges strongly. *
*In
the RO(G)-graded context, it means that for every subgroup H of G and every
"oin gRO(G), the spectral sequence consisting of the abelian groups Ers,"o+t(G=*
*H)
converges strongly. Conditional convergence is defined analogously in terms of *
*the
abelian groups Es,tr(G=H) and Es,"o+tr(G=H).
We emphasize again that, although we construct them in parallel, the Z-graded
and RO(G)-graded versions of these spectral sequences are typically very differ*
*ent.
For example, if L = R ^ SV for a finite dimensional G-representation V , then _*
*_L*
is projective as an RO(G)-graded Mackey functor __R*-module, but is usually not
projective as a Z-graded Mackey functor __R*-module. The RO(G)-graded spec-
tral sequence collapses at E2 and is concentrated in homological degree zero; t*
*his
typically does not happen in the Z-graded spectral sequence.
The peculiarities in the homological algebra of Mackey functors that occur wh*
*en
G is not finite or when the universe is not complete (cf. Lewis [7]) make it wo*
*rth-
while to observe that the homological behavior of the graded Mackey functors in
the present context is perfectly standard.
Remark. Let __R* be a Z-graded or RO(G)-graded Mackey functor ring.
4 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
(a) If __M* is a projective left __R*-module or __N* is a projective right __*
*R*-module,
then ___Tor_R*s(__N*, __M*) = 0 in homological degree s > 0. In this case, the*
* Hyper-
Tor spectral sequence above collapses at E2 and the edge homomorphism is an
isomorphism.
(b) If __L* is a projective left __R*-module or __M* is an injective left __R*
**-module,
then ___Exts_R*(__L*, __M*) = 0 in cohomological degree s > 0. In this case, th*
*e Hyper-
Ext spectral sequence above collapses at E2 and the edge homomorphism is an
isomorphism.
(c) For any left __R*-modules __L*, __M*, the usual (Yoneda) Ext groups can be
identified as Exts_R*(__L*, __M*) = (___Exts,0_R*(__L*, __M*))(G=G). Evaluatio*
*n of a Mackey
functor at G=G is exact. Thus, our Hyper-Ext spectral sequence specializes to a
conditionally convergent spectral sequence
Es,t2= Exts,t_R*(__L*, __M*) =) HoGMRs+t(L, M).
Here, HoGMRs+t(L, M) is the abelian group of G-maps from L to s+tM in the
homotopy category of R-modules.
The final topic considered in this paper is that of composition pairings. The
internal function object FR (-, -) has a composition pairing
FR (M, N) ^ FR (L, M) __//_FR (L, N).
that induces a pairing
E___xtsR(M, N) ___ExttR(L, M) __//___Exts+tR(L, N)
If R is commutative, then this composition pairing descends to ^R , becoming a
map of R-modules, and the pairing on ___Extdescends to __R*, becoming a map of
__R*-modules. Likewise, for graded Mackey functor modules, the internal function
object <- , ->__R*has a composition pairing that induces a "Yoneda pairing" on
___Ext-objects
___Exts_R*(__M*, __N*) ___Extt_R*(__L*, __M*) __//___Exts+t_R*(__*
*L*, __N*).
We prove the following theorem regarding these pairings.
Theorem. The Yoneda pairing on ___Ext*_R*induces a pairing of Hyper-Ext spectral
sequences that converges (conditionally) to the composition pairing on ___Ext*R*
*. When
R is commutative, this is a pairing of spectral sequences of __R*-modules.
In the case when G is the trivial group, our argument corrects an error in EK*
*MM
[5, IVx5].
Organization. In section 1, we construct various universal coefficient and K"un*
*neth
spectral sequences from the spectral sequences described above. Section 2 conta*
*ins
a brief review of the categories of graded Mackey functors, graded box products*
*, and
graded function Mackey functors. The definitions of graded Mackey functor rings
and modules and the definitions and basic properties of the graded box product
and function object over a graded Mackey functor ring are reviewed in Section 3.
In Section 4, we discuss projective and injective graded Mackey functor modules
and use them to define ___Extand ___Tor. Our work in equivariant stable homoto*
*py
theory begins in Section 5 with a discussion of R-modules whose homotopy Mackey
functors are projective or injective __R*-modules. The spectral sequences intro*
*duced
above are constructed in Section 6. Our results on the naturality of these spec*
*tral
EQUIVARIANT SPECTRAL SEQUENCES 5
sequences and on the Yoneda pairing are proven in Section 7. Finally, in the ap*
*pen-
dix, we prove the folk theorem that the RO(G)-graded homotopy Mackey functor
is a lax symmetric monoidal functor, i.e., that the smash product of equivariant
spectra is compatible with the graded box product.
Notation and conventions. Throughout this paper, G is a fixed finite group, and
the "equivariant stable category" means the derived category (obtained by forma*
*lly
inverting the weak equivalences) of one of the modern categories of equivariant
spectra, i.e., the category of equivariant S-modules, the category of equivaria*
*nt
orthogonal spectra, or the category of equivariant symmetric spectra. Except in
Section 1, which is written from the point of view of homology and cohomology
theory, the indexing universe for spectra is always understood to be complete. *
*The
term "G-spectrum" is used as an abbreviation for "object in the equivariant sta*
*ble
category".
Except in Section 7, we phrase all statements and arguments in the equivariant
stable category and in derived categories of modules. The undecorated symbol "^"
means the smash product of spectra in the equivariant stable category or the sm*
*ash
product with a spectrum in the derived category of R-modules. We write "^S"
when we mean the point-set smash product. Likewise F (-, -) denotes the derived
function spectrum. The point-set level functor adjoint to ^S is denoted FS(-, -*
*).
If R is an equivariant S-algebra, then TorR(N, M) is the derived balanced smash
product of a left R-module M and a right R-module N. Also, ExtR(L, M) is the
derived function spectrum (or R-module) of left R-modules L and M. Further,
___TorR*(M, N) = _ss*(TorR(N, M)) and ___Ext*R= _ss-*(ExtR (L, M)). We reserve *
*(-) ^R
(-) and FR (-, -) for the point-set level functors.
Grading conventions. The usual conventions for homology and cohomology the-
ories and for derived functors require the introduction of "homological" and "c*
*oho-
mological" grading. For a (homologically) graded Mackey functor __M*, the corre-
sponding cohomologically graded Mackey functor __M* is defined by __Mff= __M-ff*
*for
all ff. When __M* is a left __R*-module, __M* is a left __R*-module. The choice*
* of whether
to regard the category of left __R*-modules and the category of left __R*-modul*
*es as
different categories or the same category with two different grading conventions
is a philosophical one: The same formulas hold provided we are consistent about
which grading we use in the latter case. For example, there are canonical natur*
*al
isomorphisms
___Exts,o_R*(__L*, __M*) ~=___Exts,o_R*(__L*, __M*)
* * * R*
T___or_Rs,o(__N , __M ) ~=___Tor_s,o(__N*, __M*)
* R*
(where ___Tor_Rs,o(__N*, __M*) = (___Tor_s(__N*, __M*))-o). For this reason, e*
*xcept in the
statements of the spectral sequences in Section 1 and in the final construction*
* of
them in Section 6, the main body of the paper treats graded Mackey functors
exclusively in homologically graded terms.
1. Universal Coefficient and K"unneth Spectral Sequences
The hyper-homology spectral sequences of the introduction lead formally to sp*
*ec-
tral sequences for computing equivariant homology and cohomology. In particular,
we obtain universal coefficient spectral sequences for computing __E*(X) and __*
*E*(X)
when either the theory E is represented by an R-module or the spectrum X is
6 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
represented by an R-module. Likewise, there are K"unneth spectral sequences for
computing __R*(X^Y ) and __R*(X^Y ). Since the equivariant homology and cohomol-
ogy theories involved are RO(G)-graded, the arguments apply to the construction
of both RO(G)-graded and Z-graded spectral sequences. The resulting RO(G)-
graded spectral sequences are new even in the case when the theories involved a*
*re
ordinary (Bredon) homology and cohomology. In fact, the motivation for the re-
search leading to this paper was the need for such spectral sequences in [6]. T*
*he
K"unneth spectral sequences are new for equivariant K-theory.
In the statements below, R is a fixed equivariant S-algebra indexed on a comp*
*lete
universe. Given a G-spectrum E indexed on a complete universe and a G-spectrum
X indexed on any universe U0, the E-homology of X is the RO(G)-graded homotopy
Mackey functor of (i*X)^E. Here, i* is the left adjoint of the forgetful functo*
*r from
the equivariant stable category indexed on the complete universe to the equivar*
*iant
stable category indexed on U0. The spectrum (i*X) ^ E is sometimes denoted
X ^ E to compactify notation. The E-cohomology of X is the RO(G)-graded
Mackey functor
__Eff(X)(G=H) = [X ^ G=H+ , ffE]G~=[X, ffE]H
or more accurately [i*X ^ G=H+ , ffE]G. Of course, all functors used here shou*
*ld
be understood as the derived functors.
The forgetful functor from the derived category of left R-modules to the equi-
variant stable category on a complete universe has a left adjoint functor that *
*we
denote as R. We write Rop for the analogous functor into the derived category
of right R-modules. These functors have the usual properties expected of derived
free R-module functors (q.v. [8]). For example, since R-modules are necessari*
*ly
homotopical R-modules, we have natural maps
R ^ X ! RX and X ^ R ! RopX
in the equivariant stable category (on a complete universe). These maps are alw*
*ays
isomorphisms in the equivariant stable category. We can compose the unit map
X ! RX with the canonical comparison maps for the derived smash product and
derived function spectra to get natural maps
X ^ M ! TorR(RopX , M) and ExtR(RX , M) ! F (X, M)
in the equivariant stable category for any left R-module M. These maps are also
always isomorphisms. Just as R ^ X and X ^ R are naturally homotopical R-
bimodules (R-bimodules in the equivariant stable category), RX and RopX are
naturally weak bimodules in their respective categories: RX is a homotopical ri*
*ght
R-module in the derived category of left R-modules and RopX is a homotopical
left R-module in the derived category of right R-modules. Moreover, the four
maps in the equivariant stable category displayed above are maps of homotopical
R-modules.
As a consequence, by taking N = RopX in the Hyper-Tor spectral sequence
and L = RX in the Hyper-Ext spectral sequence, we obtain the following spectral
sequences of __R*-modules.
Theorem 1.1. (Universal Coefficient Spectral Sequence) Let X be a G-spectrum
and M be a left R-module. There is a natural strongly convergent homology spect*
*ral
sequence of __R*-modules
E2s,o= ___Tor_R*s,o(__R*X, __M*) =) __Ms+oX
EQUIVARIANT SPECTRAL SEQUENCES 7
and a natural conditionally convergent cohomology spectral sequence of __R*-mod*
*ules
Es,o2= ___Exts,o_R*(__R*X, __M*) =) __Ms+oX.
The forgetful functor from the derived category of left R-modules to the equi-
variant stable category also has a right adjoint functor. This adjoint is denot*
*ed R]
and has the usual properties expected from a derived cofree R-module functor [8*
*].
In particular, if E is a G-spectrum indexed on a complete universe, then the ca*
*non-
ical map R]E ! F (R, E) and the canonical comparison map of derived function
spectra
ExtR(M, R]E) __//_F (M, E)
are isomorphisms in the equivariant stable category. Once again, R]E is natural*
*ly
a weak bimodule and the above maps in the equivariant stable category are maps *
*of
homotopical R-modules. Plugging R]E into the Hyper-Ext spectral sequence then
gives the cohomological spectral sequence in the theorem below. Under the natur*
*al
isomorphism
__R*E = _ss*(E ^ R) ~=_ss*(R ^ E) = __E*R,
the homological spectral sequence in this theorem coincides with the homological
spectral sequence of the previous theorem.
Theorem 1.2. Let E be a G-spectrum indexed on a complete universe and M be
a left R-module. There is a natural strongly convergent homology spectral seque*
*nce
of __R*-modules
E2s,o= ___Tor_R*s,o(__E*R, __M*) =) __Es+oM
and a natural conditionally convergent cohomology spectral sequence of __R*-mod*
*ules
Es,o2= ___Exts,o_R*(__M*, __E*R) =) __Es+oM.
K"unneth spectral sequences for computing the homology and cohomology of
X^Y arise as special cases of our universal coefficient spectral sequences. Let*
* X and
Y be G-spectra indexed on the same universe U0. The change of universe functor
i* is symmetric monoidal; that is, there is an isomorphism i*X ^ i*Y ~=i*(X ^ Y*
* )
in the equivariant stable category. The observations preceding Theorem 1.1 yield
a canonical isomorphism
i*(X ^ Y ) ^ R ~=(i*X) ^ (i*Y ) ^ R ~=(i*X) ^ R ^ (i*Y ) ~=TorR(Rop(i*X), R(i*Y*
* ))
in the equivariant stable category. Taking M = Ri*Y in the first spectral seque*
*nce
of Theorem 1.1 gives the first spectral sequence in Theorem 1.3 below. There is*
* a
similar canonical isomorphism
Ext R(Ri*X, F (i*Y, R)) ~=F (i*X, F (i*Y, R)) ~=F (i*(X ^ Y ), R),
in the equivariant stable category. Via this isomorphism, the second spectral s*
*e-
quence of Theorem 1.1 for M = F (i*Y, R) gives the second spectral sequence bel*
*ow.
Theorem 1.3. (K"unneth Theorem) Let X and Y be G-spectra indexed on the
same universe. There is a natural strongly convergent homology spectral sequence
of __R*-modules
E2s,o= ___Tor_R*s,o(__R*X, __R*Y ) =) __Rs+o(X ^ Y ),
and a natural conditionally convergent cohomology spectral sequence of __R*-mod*
*ules
Es,o2= ___Exts,o_R*(__R-*X, __R*Y ) =) __Rs+o(X ^ Y ).
8 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
Plugging M = RX into the spectral sequences of Theorem 1.2 yields the followi*
*ng
spectral sequences.
Theorem 1.4. Let X be a G-spectrum and let E be a G-spectrum indexed on
the complete universe. There is a natural strongly convergent homology spectral
sequence of __R*-modules
E2s,o= ___Tor_R*s,o(__E*R, __R*X) =) __Es+o(R ^ X),
and a natural conditionally convergent cohomology spectral sequence of __R*-mod*
*ules
Es,t2= ___Exts,t_R*(__R-*X, __E*R) =) __Es+t(R ^ X).
The spectral sequences above can be combined with Spanier-Whitehead dual-
ity to obtain additional spectral sequences. If X is a finite G-spectrum, then *
*the
Spanier-Whitehead dual of i*X is a G-spectrum DX (indexed on the complete uni-
verse). There are canonical isomorphisms E*DX ~=E-*X and E*DX ~=E-*X,
natural in E and in X. Plugging DX into the spectral sequences of Theorem 1.1
gives us cohomology to cohomology and cohomology to homology universal coeffi-
cient spectral sequences.
Theorem 1.5. (Dual Universal Coefficient Spectral Sequence) Let X be a finite G-
spectrum, and M a left R-module. There is a natural strongly convergent homology
spectral sequence of __R*-modules
* * * -(s+o)
E2s,o= ___Tor_Rs,o(__R X, __M ) =) __M X
and a natural conditionally convergent cohomology spectral sequence of __R*-mod*
*ules
Es,o2= ___Exts,o_R*(__R-*X, __M*) =) __M-(s+o)X.
These dual universal coefficient spectral sequences imply the following K"unn*
*eth
spectral sequences.
Theorem 1.6. (Dual K"unneth Theorem) Let X and Y be G-spectra indexed on the
same universe, and assume that X is finite. There is a natural strongly converg*
*ent
homology spectral sequence of __R*-modules
* * * -(s+o)
E2s,o= ___Tor_Rs,o(__R X, __R Y ) =) __R (X ^ Y ),
and a natural conditionally convergent cohomology spectral sequence of __R*-mod*
*ules
Es,o2= ___Exts,o_R*(__R-*X, __R*Y ) =) __R-(s+o)(X ^ Y ).
2. Graded Mackey functors
This section introduces the categories of RO(G)-graded and Z-graded Mackey
functors. Box products and function objects for these categories are defined in
terms of box products and function objects for ungraded Mackey functors. In this
discussion, some familiarity with (ungraded) Mackey functors is assumed. How-
ever, certain key definitions are reviewed in order to fix notation or to expla*
*in our
perspective.
Recall that the Burnside category BG may be defined as the essentially small
additive category whose objects are the finite G-sets and whose abelian group of
morphisms between the finite G-sets X and Y is the abelian group
BG (X, Y ) = [ 1 X+ , 1 Y+]G,
EQUIVARIANT SPECTRAL SEQUENCES 9
of morphisms between the associated suspension spectra in the equivariant stable
category. Disjoint union of finite G-sets provides the direct sum operation giv*
*ing
BG its additive structure. A rather simple, purely algebraic description of the
morphisms in BG can be found in [9, xV.9]; however, the approach to the category
of Mackey functors best suited to our purposes is in terms of the stable homoto*
*py
theoretic description of BG .
Definition 2.1. The category M of Mackey functors is the category of contravari-
ant additive functors from the Burnside category BG to the category Ab of abeli*
*an
groups.
Spanier-Whitehead duality provides an isomorphism of categories between BG
and BopGthat is the identity on the objects. The category of Mackey functors co*
*uld
therefore be defined equivalently as the category of covariant functors from BG*
* to
Ab. Mackey functors can also be described in terms of orbits. Since every finite
G-set is a disjoint union of orbits, every object of BG is isomorphic to a dire*
*ct sum
of the canonical orbits G=H. Let OG be the full subcategory of BG whose objects
are the orbits G=H associated to the subgroups H of G. The inclusion of OG into
BG induces an equivalence between the category of contravariant additive functo*
*rs
from OG into Ab and the category M of Mackey functors. Thus, the category
of Mackey functors could be defined equivalently as the category of contravaria*
*nt
additive functors from OG to Ab. Other equivalent definitions may be found in [*
*9,
xV.9] and [14].
Definition 2.2. An RO(G)-graded Mackey functor __M* consists of a Mackey func-
tor __Mo for each o 2 RO(G). A map of RO(G)-graded Mackey functors __M* ! __N*
consists of a map of Mackey functors __Mo ! __No for each o 2 RO(G). Z-graded
Mackey functors and maps of Z-graded Mackey functors are defined analogously.
The categories of RO(G)-graded and Z-graded Mackey functors are denoted by M*
and M# , respectively.
The "shift" functor fftakes a graded Mackey functor __M* to the graded Mackey
functor given by
( ff_M*)o= __Mo-ff,
for o 2 RO(G) (or o 2 Z).
The categories of RO(G)-graded Mackey functors and Z-graded Mackey functors
have all limits and colimits; these are formed degree-wise and object-wise. In *
*other
words, for any diagram d 7! __M*(d) of graded Mackey functors,
(Colimd__M*(d))o(X) = Colimd(__Mo(d)(X)),
and likewise for the limit.
Proposition 2.3. The categories of RO(G)-graded and Z-graded Mackey functors
are complete and cocomplete abelian categories satisfying AB5.
The box product of graded Mackey functors is constructed from the box product
of Mackey functors, which we now review briefly. The cartesian product of finite
G-sets together with the canonical isomorphism 1 X+ ^ 1 Y+ ~= 1 (X x Y )+ in
the equivariant stable category provides BG with a symmetric monoidal structure.
The unit for this structure is the one-point G-set G=G. The category M inherits
a symmetric monoidal closed structure from BG . This product is most easily
10 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
described by the coend formula
Z Y,Z2BG
(__M __N)(X) = __M(Y ) __N(Z) BG (X, Y x Z).
This box product is characterized by the universal property that maps of Mackey
functors from __M __N to __P are in one-to-one correspondence with natural tr*
*ansfor-
mations of contravariant functors BG x BG ! Ab from __M __N to __P O x. The u*
*nit
for the box product is the Burnside ring Mackey functor __B = BG (-, G=G).
The internal Hom functor adjoint to the box product is denoted <__M, __N> an*
*d is
given by the end formula
Z
<__M, __N>(X) = Hom (__M(Y ) BG (Z, X x Y ), __N(Z)).
Y,Z2BG
It may also be computed using the formulae
<__M, __N>(X) ~=M(__M, __NX ) ~=M(__MX , __N),
where __NX denotes the Mackey functor __N(- x X). Day's general results [4] abo*
*ut
monoidal structures on functor categories specialize to give that the box produ*
*ct
and the internal function object <- , -> provide the category M with a closed
symmetric monoidal structure.
Definition 2.4. Let __M* and __N* be RO(G)-graded Mackey functors. Define the
RO(G)-graded Mackey functor __M* *__N* by
M
(__M* *__N*)o= __Mff __Nfi.
ff+fi=o
Define the RO(G)-graded Mackey functor <__M*, __N*>*by
Y
<__M*, __N*>o= <__Mff, __Nfi>.
fi-ff=o
For Z-graded Mackey functors, the functors (-) # (-) and <- , ->#are defined
analogously.
Graded box products and graded function objects provide the categories M* and
M# with symmetric monoidal closed structures. The unit is the RO(G)-graded
(or Z-graded) Mackey functor __B* which is __B in degree zero and zero in all o*
*ther
degrees. The unit and associativity isomorphisms for the graded box products are
induced by the unit and associativity isomorphisms for the ordinary box product.
Further, the adjunction relating graded box products and graded function objects
is easily obtained from the adjunction relating ordinary box products and funct*
*ion
objects. This describes the monoidal closed structure on M* and M# . For Z-
graded Mackey functors, the symmetry isomorphism is just as straightforward. On
the summands of (__M# # __N# )tand (__N# # __M# )tfor t = m+n, this isomorphi*
*sm is
the composite of the Mackey functor symmetry isomorphism __Mm __Nn ~=__Nn __*
*Mm
with multiplication by (-1)mn .
For RO(G)-graded Mackey functors, the symmetry isomorphism is more delicate
because more complicated "signs" are needed for compatibility with our topologi*
*cal
applications. These signs are units in the Burnside ring __B(G=G) of G. Any ele*
*ment
a in the Burnside ring of G defines a natural transformation in the category of
Mackey functors from the identity functor to itself. The Yoneda lemma identifies
EQUIVARIANT SPECTRAL SEQUENCES 11
__B(G=G) as the endomorphism ring of __B, and the natural transformation on a
Mackey functor __M associated to a is the composite
id_Ma
__M ~=__M __B _______//__M __B ~=__M.
When a is a unit in __B(G=G), this natural transformation is an isomorphism. Ea*
*ch
unit a in __B(G=G) satisfies a2 = 1 so these units can be thought of as general*
*ized
signs.
Given a function oe from RO(G) x RO(G) to the group of units of __B(G=G), we
can define a natural isomorphism
coe:__M* *__N* ~=__N* *__M*
by taking the map on the summands associated to ff+fi = o to be the composite of
the symmetry isomorphism __Mff __Nfi~=__Nfi __Mffand the automorphism oe(ff, *
*fi).
An easy diagram chase shows that coemakes the box product into a symmetric
monoidal product exactly when oe is (anti)symmetric, that is,
oe(ff, fi)oe(fi, ff) = 1
and bilinear, that is,
oe(ff + fi, fl) = oe(ff, fl)oe(fi, fl).
The appropriate choice of oe is dictated by the definition of the homotopy Ma*
*ckey
functor _ss* and depends on the topological choices in that definition. Because
of the required bilinearity of oe, it suffices to specify oe(ff, fi) in only th*
*ose cases
where ff and fi are irreducible real G-representations. In Appendix A, we show
that, if ff and fi are non-isomorphic, then we can assume that oe(ff, fi) = 1. *
*The
element oe(ff, ff) of __B(G=G) is the automorphism of S = 1 G=G+ obtained by
stabilizing the multiplication by -1 map on Sff. This definition of oe complet*
*es
the construction of the symmetry isomorphism in the RO(G)-graded case and so
finishes the construction of the symmetric monoidal product.
Proposition 2.5. The categories M* and M# are closed symmetric monoidal
abelian categories.
3.Graded Mackey functor rings and modules
This section is devoted to a discussion of rings and modules in the categories
of graded Mackey functors. In particular, the "box over __R*" ( __R*) and the *
*"__R*-
function" (<- , ->__R*) constructions for modules over a graded Mackey functor *
*ring
__R* are defined, and their basic properties are described. For simplicity, all*
* defini-
tions are given in the context of the category M* of RO(G)-graded Mackey functo*
*rs.
However, with the obvious notational modifications, these definitions apply equ*
*ally
well to the category M# of Z-graded Mackey functors. We begin with the (usual)
definitions of rings and modules in a symmetric monoidal abelian category.
Definition 3.1. An RO(G)-graded Mackey functor ring consists of an RO(G)-
graded Mackey functor __R* together with unit i: __B* ! __R* and multiplication
12 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
~: __R* *__R* ! __R* maps for which the unit and associativity diagrams
i *id// ido*io ~ *id//
__B* *__R*_MM_R* *__R*_ __R*r *__B*__R* *__R* *__R*__R* *__R*
MMM ~| rrrr | |~
~=MMMMM | rrr~=r id *~| |
M&&fflffl|xxrr |fflffl fflffl|
__R* __R* *__R*_~_____//_R*
commute. The ring __R* is said to be commutative if the symmetry diagram
~=
__R* *__R*__________//I_R* *__R*
III uuuu
~IIII uu~u
I$$ zzuu
__R*
also commutes. In these diagrams, the unlabeled isomorphisms are the unit and
symmetry isomorphisms of the symmetric monoidal category M*.
Definition 3.2. A left module over an RO(G)-graded Mackey functor ring __R*
consists of an RO(G)-graded Mackey functor __M* and an action map , :__R* __M**
* !
__M* for which the unit and associativity diagrams
i *id// ~ *id//
__B* *__M*__MM_R* *__M*__R* *__R* *__M*___R* *__M*
MMM | | |
~=MMMMM |, id *,| |,
M&&fflffl| fflffl| fflffl|
__M* __R* *__M*__,____//__M*
commute. A right module __M* over __R* is defined analogously in terms of an ac*
*tion
map i :__M* __R* ! __M*. The categories of left and right modules over an RO(*
*G)-
graded Mackey functor ring __R* are denoted __R*-Modand Mod-__R*, respectively.
For our study of the additional structure carried by some of our spectral se-
quences, we also need the notion of __R*-bimodule.
Definition 3.3. An __R*-bimodule is a graded Mackey functor __M* having left and
right __R*-actions (, and i, respectively) for which the diagram
, *id
__R* *__M* *__R*//__M* *__R*
id *i|| |i|
fflffl| fflffl|
__R* *__M*__,____//__M*
commutes. If __R* is commutative, then every left module __M* carries a bimodu*
*le
structure in which the right action map i is the composite
,
__M* *__R* ~=__R* *__M* ___//_M*.
A bimodule structure of this form is called symmetric.
When __R* is an RO(G)-graded Mackey functor ring and __K* is an RO(G)-graded
Mackey functor, the RO(G)-graded Mackey functors __R* *__K* and <__R*, __K*>*c*
*arry
__R*-bimodule structures coming from the left and right actions of __R* on itse*
*lf.
Regarded simply as left __R*-modules, __R* * __K* and <__R*, __K*>*are called *
*the free
and cofree left __R*-modules generated by __K*. The free and cofree right __R*-*
*modules
EQUIVARIANT SPECTRAL SEQUENCES 13
generated by __K* are constructed similarly. A purely formal argument indicates
that free and cofree __R*-modules have the expected universal properties.
Proposition 3.4. Let __R* be an RO(G)-graded Mackey functor ring. The functor
__R* *(-): M* ! __R*-Modis left adjoint to the forgetful functor __R*-Mod! M*.
The functor <__R*, ->*:M* ! __R*-Modis right adjoint to the forgetful functor
__R*-Mod! M*.
These two adjunctions may be used to identify the category of left __R*-modul*
*es as
both the category of algebras over a monad on M* and the category of coalgebras
over a comonad on M*. These identifications together with the corresponding ide*
*n-
tifications for the category of right __R*-modules and the category of __R*-bim*
*odules
are the key to proving the following result.
Proposition 3.5. Let __R* be an RO(G)-graded Mackey functor ring. The cate-
gories of left __R*-modules, right __R*-modules, and __R*-bimodules are complet*
*e and
cocomplete abelian categories that satisfy AB5.
We now move on to the main constructions of this section, the functors (-) _*
*_R*
(-) and <- , ->__R*.
Definition 3.6. Let __L* and __M* be left __R*-modules and let __N* be a right *
*__R*-
module.
(a) The graded Mackey functor __N* __R*_M* is defined by the coequalizer dia*
*gram
i__*id//_ q
__N* *__R* *__M*__//_N* *__M*__//_N* __R*_M*.
id *,
Here , is the action map for __M* and i is the action map for __N*.
(b) The RO(G)-graded Mackey functor <__L*, __M*>__R*is defined by the equaliz*
*er
diagram
j __*_//_
<__L*, __M*>__R*//_<__L*,___M*>*//_<__R* *__L*,~__M*>*=<__L*,.<__R*, __M**
*>*>*
",*
Here the map * is induced from the action map for __L*, and the map ",*is in*
*duced
from the coaction map ",:_M*! <__R*, __M*>*adjoint to the action map for __M*.
The following proposition follows directly from these definitions and the pro*
*per-
ties of * and <- , ->*.
Proposition 3.7. Let __M* and __N* be a left and a right __R*-module, respectiv*
*ely.
(a) The functors __N* __R*(-):_R*-Mod! M* and (-) __R*_M*:Mod-_R*!
M* preserve all colimits and are therefore right exact.
(b) The functor <__M*, ->__R*:_R*-Mod! M* preserves all limits and is therefo*
*re
left exact.
(c) The functor <- , __M*>__R*:_R*-Modop! M* converts colimits in __R*-Modinto
limits in M* and is therefore left exact.
For any graded Mackey functors __M*, __M0*, and __M00*, the closed symmetric *
*mon-
oidal structure on M* provides a composition pairing
<__M0*, __M00*>**<__M*,___M0*>*//<__M*, __M00*>*
that is both associative and unital. This pairing restricts to a pairing for t*
*he
category of left __R*-modules.
14 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
Proposition 3.8. Let __M*, __M0*, and __M00*be left __R*-modules. The composit*
*ion
pairing on <- , ->*restricts to a pairing
<__M0*, __M00*>__R**<__M*,/__M0*>__R*/_<__M*, __M00*>__R*
that is both associative and unital.
Although the constructions __L* __R*_M* and <__M*, __N*>__R*typically yield *
*only
RO(G)-graded Mackey functors, in some important special cases they yield __R*-
modules.
Proposition 3.9. Let __L* and __M* be left __R*-modules, and let __N* be a righ*
*t __R*-
module.
(a) If __L* is an __R*-bimodule, then <__L*, __M*>__R*is naturally a left __R*
**-module.
(b) If __M* is an __R*-bimodule, then __N* __R*_M* and <__L*, __M*>__R*are n*
*aturally
right __R*-modules.
(c) If __N* is an __R*-bimodule, then __N* __R*_M* is naturally a left __R*-*
*module.
When __R* is commutative, we can always consider the symmetric __R*-bimodule
structure on any (left or right) __R*-module to obtain __R*-module structures on
(-) __R*(-) and <- , ->__R*. In fact, regarding either __M* or __N* as the bi*
*module
yields the same __R*-module structure on __M* __R*_N*, and similarly for <__M**
*, __N*>__R*.
In this context the following stronger version of the previous results hold.
Proposition 3.10. Let __R* be a commutative graded Mackey functor ring.
(a) The module category __R*-Modis a closed symmetric monoidal abelian cate-
gory with product __R*and function object <- , ->__R*.
(b) The free functor __R* *(-):M*! __R*-Modis strong symmetric monoidal
and the forgetful functor __R*-Mod! M* is lax symmetric monoidal.
(c) The composition pairing
<__M0*, __M00*>__R*_R*<__M*,/__M0*>__R*/_<__M*, __M00*>__R*
coming from the closed symmetric monoidal structure on __R*-Modis the obvious
one derived from the composition pairing of Proposition 3.8.
4. The homological algebra of graded Mackey functor modules
This section is devoted to homological algebra for the categories of Mackey f*
*unc-
tor modules over a graded Mackey functor ring. Our first objective is to show t*
*hat
these categories have enough projectives and injectives. These objects are then*
* used
to construct the derived functors ___Tor_R**and ___Ext*_R*in terms of resolutio*
*ns and to
show that they have the expected properties. Some notation is needed to constru*
*ct
the desired injective and projective objects.
Definition 4.1. For each finite G-set X, let __BX denote the Mackey functor
BG (-, X). For any abelian group E, let _I(X, E) denote the Mackey functor
Hom (BG (X, -), E). The corresponding graded Mackey functors concentrated in
degree zero are denoted __BX*and _I(X, E)*.
The enriched Yoneda Lemma gives natural isomorphisms
M(__BX , __M)~=_M(X) M(__M, _I(X, E))~=Hom(__M(X), E)
<__BX, __M>(Y~)=_M(X x Y ) <__M, _I(X, E)>(Y~)=Hom(__M(X x Y ), E)
EQUIVARIANT SPECTRAL SEQUENCES 15
of abelian groups. A coend argument dual to the end argument used to prove the
Yoneda Lemma provides the well-known isomorphism
(__BX __M)(Y ) ~=__M(X x Y ).
In the graded context, these isomorphisms give the following proposition:
Proposition 4.2. Let __M* be a graded Mackey functor. There are natural isomor-
phisms of abelian groups
< o__BX*, __M*>ff(Y ) ~=__Mo+ff(X x Y )
( o__BX* *__M*)ff(Y ) ~=__M-o+ff(X x Y )
<__M*, o_I(X, E)*>ff(Y ) ~=Hom (__Mo-ff(X x Y ), E).
In particular, o__BX*is a projective object in M* and the functors < o__BX*, -*
*>*and
__BX* *(-) are exact. Also, if E is an injective abelian group, then o_I(X, E*
*)* is
an injective object in M*, and the functor <- , o_I(X, E)*>is exact.
It follows from this proposition that the category of graded Mackey functors *
*has
enough projectives and injectives. Let __M* be a graded Mackey functor. For each
o in RO(G) and each subgroup H of G, choose a surjection Po,H ! __Mo(G=H)
whose domain is a free abelian group and an injection __Mo(G=H) ! Io,H whose
range is an injective abelian group Io,H. Then the previous proposition provides
an epimorphism M
__P* = o__BG=H* Po,H___//_M*
o,H
and a monomorphism
Y o
__M* __//_ _I(G=H , Io,H)* = _I*.
o,H
When __M* is a left __R*-module for some graded Mackey functor ring __R*, then *
*the
induced epimorphism __R* *__P* ! __M* of left __R*-modules has domain a projec*
*tive
left __R*-module. Likewise, the induced monomorphism __M* ! <__R*, _I*>*of left*
* __R*-
modules has codomain an injective left __R*-module. Similar observations apply *
*in
the case of right modules. Thus, we have proven:
Proposition 4.3. Let __R* be a graded Mackey functor ring. The categories of le*
*ft
and right __R*-modules have enough projectives and injectives.
Since an epimorphism from one projective onto another or a monomorphism
from one injective into another is split, the preceding argument also provides *
*char-
acterizations of projective and injection modules.
Proposition 4.4. An __R*-module is projective if and only if it is a direct sum*
*mand
of a direct sum of __R*-modules of the form __R* * o__BG=H*. Also, an __R*-mod*
*ule is
injective if and only if it is a direct summand of a product of __R*-modules of*
* the
form <__R*, o_I(G=H , I)*>*in which I is an injective abelian group.
Proposition 4.2 and this characterization of projectives and injectives have *
*some
important implications for the exactness of the functors __R*and <- , ->__R*. *
*In the
terminology of Lewis [7], these exactness results assert that projectives and i*
*njec-
tives are respectively internal projective and internal injective and that proj*
*ective
implies flat.
16 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
Theorem 4.5. Let __R* be a Mackey functor ring.
(a) If __P* is a projective left __R*-module, then <__P*, ->__R*is an exact f*
*unctor from
left __R*-modules to graded Mackey functors.
(b) If _I* is an injective left __R*-module, then <- , _I*>__R*is an exact fu*
*nctor from
left __R*-modules to graded Mackey functors.
(c) If __P* is a projective left __R*-module, then (-) __R*_P* is an exact f*
*unctor
from right __R*-modules to graded Mackey functors.
(d) If __Q* is a projective right __R*-module, then __Q* __R*(-) is an exact*
* functor
from left __R*-modules to graded Mackey functors.
In the category of left __R*-modules, as in any abelian category with enough
projectives and injectives, every object has both projective and injective reso*
*lutions.
To be consistent with the usual grading conventions for spectral sequences, we *
*use
the inner degree for the resolution degree in projective resolutions. Thus, for*
* any
projective resolution __P*,*, __Ps,*is an __R*-module for each s 0 and __P*,o*
*is a chain
complex of Mackey functors for each o 2 RO(G) (or Z). When _I**is an injective
resolution, _Is*is an __R*-module for each s 0 and _I*ois a cochain complex o*
*f Mackey
functors for each o 2 RO(G) (or Z).
If __P*,*is a projective resolution of a left __R*-module __M* and __Q*,*is a*
* projective
resolution of a right __R*-module __N*, then the maps
__N* __R*_P*,*oo_ __Q*,* __R*_P*,*__//__Q*,* __R*_M*
are quasi-isomorphisms of chain complexes of graded Mackey functors by Theo-
rem 4.5. The graded Mackey functor ___Tor_R*s(__N*, __M*) is defined to be the *
*s-th ho-
mology of any of these cochain complexes. The Mackey functor (___Tor_R*s(__N*, *
*__M*))o
is usually denoted ___Tor_R*s,o(__N*, __M*).
If __O*,*is a projective resolution of a left __R*-module __L* and _I**is an *
*injective
resolution of a left __R*-module __M*, then the maps
<__O*,*, __M*>__R*//_<__O*,*,o_I**>__R*o_<__L*, _I**>__R*
are quasi-isomorphisms of cochain complexes of graded Mackey functors by Theo-
rem 4.5. The graded Mackey functor ___Exts_R*(__L*, __M*) is defined to be the *
*s-th coho-
mology of any of these cochain complexes. The Mackey functor (___Exts_R*(__L*, *
*__M*))-o
is usually denoted ___Exts,o_R*(__L*, __M*).
The standard homological arguments imply that ___Tor_R**and ___Ext*_R*behave *
*as one
would expect.
Proposition 4.6. The constructions ___Tor_R**and ___Ext*_R*are well-defined, na*
*tural
in each variable, and convert short exact sequences in each variable to long ex*
*act
sequences. Moreover, there are canonical natural isomorphisms
R* ~
T___or_0(__L*, __M*) = __L* __R*_M*
and
___Ext0_R*(__M*, __N*) ~=<__M*, __N*>__R*.
As with __R*and <- , ->__R*, if one of the arguments of ___Tor_R**or ___Ext**
*_R*is an __R*-
bimodule, then ___Tor_R**or ___Ext*_R*inherits an __R*-module structure from th*
*at bimod-
ule. In the context of ordinary rings, this assertion is a purely formal conseq*
*uence
of the functoriality of ___Tor_R**and ___Ext*_R*. However, since __R* is a gra*
*ded Mackey
functor ring, to obtain this result, we must either show that the functors ___T*
*or_R**
EQUIVARIANT SPECTRAL SEQUENCES 17
and ___Ext*_R*are enriched over M* or provide a direct construction of the __R**
*-action.
Either of these approaches requires the following lemma. Its proof, which uses *
*the
right exactness of __R*, is essentially identical to that of the corresponding*
* result
for chain complexes of abelian groups.
Lemma 4.7. Let __C*,*and __D*,*be chain complexes of graded Mackey functors,
and __C**and __D**be cochain complexes of graded Mackey functors. Then there are
natural transformations
H*(__C*,*) *H*(__D*,*) __//_H*(__C*,* * __D*,*)
and
H*(__C**) *H*(__D**) __//_H*(__C** * __D**)
which are unital and associative in the appropriate sense.
Taking one of the complexes in this lemma to be __R* (concentrated in homolog*
*ical
or cohomological degree zero) and the other to be the complex used to compute
___Tor_R**or ___Ext*_R*, we obtain the desired actions of __R*.
Theorem 4.8. Let __L* and __M* be left __R*-modules, and let __N* be a right __*
*R*-module.
(a) If __L* is an __R*-bimodule, then ___Exts_R*(__L*, __M*) is naturally a r*
*ight __R*-module.
(b) If __M* is an __R*-bimodule, then ___Tor_R*s(__N*, __M*) and ___Exts_R*(_*
*_L*, __M*) are nat-
urally right __R*-modules.
(c) If __N* is an __R*-bimodule, then ___Tor_R*s(__N*, __M*) is naturally a l*
*eft __R*-module.
The second natural transformation in Lemma 4.7 may also be used to construct
the Yoneda pairing for ___Ext*_R*. Let __M*, __M0*, and __M00*be left __R*-modu*
*les, __P*,*be
a projective resolution of __M*, and _I**be an injective resolution of __M00*. *
*Then the
Yoneda pairing
___Ext*_R*(__M0*, __M00*) *E___xt*_R*(__M*, __M0*) __//___Ext*_R*(_*
*_M*, __M00*)
is the composite
H*(<__M0*, _I**>__R*) H*(<__P*,*,/__M0*>__R*)/_H*(<__M0*, _I**>__R**<__P*,**
*,)__M0*>__R*
__//_H*(<__P*,*, _I**>__R*).
Here, the first map is the second natural transformation from the lemma and the
second map comes from the composition pairing of Lemma 3.8. The usual homo-
logical arguments imply that this pairing on ___Ext*_R*behaves as expected.
Proposition 4.9. The Yoneda pairing is well-defined and associative. Moreover,
in cohomological degree zero, it agrees with the usual composition pairing.
5.Homotopy Mackey functors and the homological algebra of
equivariant R-modules
This section is devoted to a discussion of the functor _ss* from the equivari*
*ant
stable category to the category of graded Mackey functors. This functor connects
the derived smash product and function object constructions for modules over an
equivariant S-algebra R to the corresponding _ss*Rand <- , ->_ss*Rconstructio*
*ns.
As a first step in our analysis of the link between the homotopy theory of R-mo*
*dules
and the homological algebra of _ss*R-modules, we consider here those R-modules
whose homotopy Mackey functors are projective or injective as _ss*R-modules.
18 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
We begin by reviewing the construction of _ss*. The first step in this proces*
*s is
selecting a model So of the o-sphere for each element o of RO(G). It is conveni*
*ent
to take S0 = S. More generally, when o is the trivial representation of dimensi*
*on
n > 0, So is taken to be the smash product of S with the standard n-sphere spac*
*e.
For all other o, the object So may be chosen arbitrarily from the appropriate
homotopy class. Then, _ss* is defined for any G-spectrum M and any finite G-set*
* X
by
_sso(M)(X) = [So ^ X+ , M]G.
The naturality of this construction in stable maps of X gives _sso(M) the struc*
*ture
of a Mackey functor. Further, its naturality in M makes _ss* a functor from the
equivariant stable category to M*. Restricting the index o so that it lies in Z
rather than RO(G) gives a functor, which we also denote _ss*, from the equivari*
*ant
stable category to M# .
Appendix A contains a complete proof of the following folk theorem:
Theorem 5.1. The functor _ss* is a lax symmetric monoidal functor from the equi-
variant stable category to the category of RO(G)-graded (or Z-graded) Mackey fu*
*nc-
tors.
In other words, we have a suitably associative, symmetric, and unital natural
transformation _ss*N * _ss*M ! _ss*(N ^ M). By comparing the diagrams used to
define homotopical ring and module spectra in the equivariant stable category w*
*ith
those used to define graded Mackey functor rings and modules, it is easy to see*
* that
_ss* takes homotopical ring and module spectra to graded Mackey functor rings a*
*nd
modules. We apply this to equivariant S-algebras and modules over an equivariant
S-algebra R in a modern category of spectra, which pass to homotopical ring and
module spectra in the equivariant stable category. A weak bimodule in the cate-
gory of left R-modules is defined to be left R-module together with a homotopic*
*al
right R-module structure in the derived category of left R-modules. Clearly, the
underlying spectrum of a weak bimodule is a homotopical bimodule. Weak bimod-
ules in the category of right R-modules are defined analogously; their underlyi*
*ng
spectra are also homotopical bimodules. These observations are summarized in the
following corollary of Theorem 5.1:
Corollary 5.2. Let R be an equivariant S-algebra. Then _ss*R is a graded Mackey
functor ring which is commutative if R is. The functor _ss* refines to a funct*
*or
from the category of left (or right) R-modules to the category of left (or righ*
*t)
_ss*R-modules. Both refinements take weak bimodules to _ss*R-bimodules.
Let M and N be left and right modules, respectively, over an equivariant S-
algebra R. Recall that the functor TorR is the derived functor of ^R and that
___TorR*(N, M) denotes _ss*TorR (N, M). There is a canonical comparison map
N ^ M ! TorR(N, M)
derived from the natural map N ^S M ! N ^R M. The two composites
N ^ R ^ M ! N ^ M ! TorR(N, M)
coming from the actions of R on M and N coincide. Thus, the universal property
defining _ss*Rprovides a natural transformation
_ss*N (_ss*R)_ss*M __//___TorR*(N, M).
EQUIVARIANT SPECTRAL SEQUENCES 19
There is a canonical natural isomorphism
TorR(N, M) ^ X __//_TorR(N, M ^ X)
in the equivariant stable category. This isomorphism is associative in the obvi*
*ous
sense and makes the diagram
N ^ M ^JX
uuuu JJJ
uuu JJJ
zzuu J$$J
TorR(N, M) ^ X _______//_TorR(N, M ^ X)
commute. There is an analogous isomorphism in the other variable with analogous
properties. These isomorphisms, and their asserted properties, come from the fa*
*ct
that TorR(N, -) is enriched functor over the equivariant stable category [8]. T*
*aking
X = R, we see that, when M or N is a weak bimodule, TorR(N, M) is naturally a
homotopical R-module and the comparison map N ^ M ! TorR(N, M) is a map
of homotopical R-modules. The implications of these observations are summarized
in the following result:
Theorem 5.3. Let R be an equivariant S-algebra, M be a left R-module, and N
be a right R-module. There is a natural transformation of graded Mackey functors
_ss*N (_ss*R)_ss*M __//___TorR*(N, M).
If M is a weak bimodule, this is a map of right _ss*R-modules. If N is an weak
bimodule, it is a map of left _ss*R-modules.
Likewise, the functor ExtR is the derived functor of FR , and ___Ext-*R(L, M)*
* =
_ss*Ext R(L, M) for left R-modules L and M. The natural map FR (L, M) !
FS(L, M) induces a map ExtR(L, M) ! F (L, M). The two maps from ExtR(L, M)
to F (R ^ L, M) coming from the actions of R on L and M coincide and so induce
a natural transformation
___Ext-*R(L, M) __//_<_ss*L, _ss*M>(_ss*R).
The functors ExtR(L, -) and ExtR(-, M) are also enriched over the equivariant
stable category, and so we have natural maps
ExtR(L, M)^X ! ExtR(L, M ^X) and ExtR(L^X, M) ! F (X, ExtR(L, M))
in the equivariant stable category. The second of these maps is always an iso-
morphism, but the first map generally is not. These maps satisfy the evident
associativity conditions and are compatible with the canonical maps
F (L, M) ^ X ! F (L, M ^ X) and F (L ^ X, M) ! F (X, F (L, M)).
These observations yield the following result:
Theorem 5.4. Let R be an equivariant S-algebra, and L and M be left R-modules.
There is a natural transformation of graded Mackey functors
___Ext-*R(L, M) __//_<_ss*L, _ss*M>(_ss*R).
If L is a weak bimodule, this is a map of left _ss*R-modules. If M is a weak bi*
*module,
it is a map of right _ss*R-modules.
20 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
The remainder of this section is devoted to the statements and proofs of two
results about R-modules whose homotopy Mackey functors are projective or injec-
tive as __R*-modules. The behavior of the natural maps of Theorems 5.3 and 5.4 *
*for
such R-modules is of particular interest to us. For this discussion, we denote *
*_ss*R
by __R*.
Theorem 5.5.
(a) If __P* is a projective left __R*-module, then there exists a left R-modu*
*le P such
that _ss*P ~=__P*.
(b) If __Q* is a projective right __R*-module, then there exists a right R-mo*
*dule Q
such that _ss*Q ~=__Q*.
(c) If _I* is an injective left __R*-module, then there exists a left R-modul*
*e I such
that _ss*I ~=_I*.
Theorem 5.6. Let L and M be left R-modules, and let N be a right R-module.
(a) If _ss*L is projective or _ss*M is injective as a left __R*-module, then *
*the natural
map ___Ext-*R(L, M) ! <_ss*L, _ss*M>__R*is an isomorphism.
(b) If _ss*M is a projective left __R*-module or _ss*N is a projective right *
*__R*-
module, then the natural map _ss*N __R*_ss*M ! ___TorR*(N, M) is an isomorphis*
*m.
For any two left R-modules L and M, ___Ext0R(L, M)(G=G) is canonically isomor-
phic to the abelian group of maps from L to M in the derived category of left
R-modules. Likewise. for any left __R*-modules __L* and __M*, <__L*, __M*>__R*
**0(G=G) is
canonically isomorphic to the abelian group of maps from __L* to __M* in the ca*
*tegory
of left __R*-modules. This implies the following corollary of Theorem 5.6.
Corollary 5.7. Let L and M be left R-modules. If _ss*L is projective or _ss*M is
injective as a left __R*-module, then maps from L to M in the derived category *
*of
left R-modules are in one-to-one correspondence with maps from _ss*L to _ss*M in
the category of left __R*-modules.
We begin the proof of Theorems 5.5 and 5.6 with the following special case. We
state and prove it in the case of left R-modules but the analogous result holds*
* for
right R-modules.
Lemma 5.8. Let X be a finite G-set and let o be an element of RO(G). Then
there exists a left R-module Ro[X] with _ss*Ro[X] ~=__R* * o__BX*. Moreover, *
*for
any left R-module M and any right R-module N, the natural maps
(_ss*N) __R*(_ss*Ro[X]) __//___TorR*(N, Ro[X]),
___Ext-*R(Ro[X], M) __//_<_ss*Ro[X], _ss*M>__R*
are isomorphisms.
Proof.By taking Ro[X] = R0[X] ^ So, it suffices to consider the case when o = 0.
Then we take R[X] = R0[X] = R 1 X+ . Theorem 5.1 gives us a canonical map of
__R*-bimodules
__R* *__BX*= _ss*R * _ss0( 1 X+ ) __//__ss*R[X],
and an easy Spanier-Whitehead duality argument shows that this map is an isomor-
phism. The argument generalizes to show that the map _ss*N *__BX*! _ss*(N^X+ )*
* is __
an isomorphism for all N, and the rest of the proof is an easy check of diagram*
*s. |__|
We generalize this to other projective modules in the following lemma.
EQUIVARIANT SPECTRAL SEQUENCES 21
Lemma 5.9. Let __P* be a projective left __R*-module. There exists a left R-mod*
*ule
P with _ss*P ~= __P* such that the natural maps _ss*(-) __R*_P* ! ___TorR*(-, *
*P ) and
___Ext-*R(P, -) ! <__P*, _ss*(-)>__R*are isomorphisms.
Proof.Using Proposition 4.4, we can find an epimorphism f :__F* ! __P*, where _*
*_F*
is a direct sum of __R*-modules of the form __R* o__BX*. Choose a splitting*
* map
g :__P* ! __F* for f. By the previous lemma, there is a left R-module F , which*
* is a
wedge of modules of the form Ro[X], such that _ss*F ~=__F*. Also, there is a se*
*lf-map
h: F ! F in the derived category of left R-modules that induces g O f on _ss*. *
*Form
the left R-module P = h-1F as the telescope of the self-map h. Then _ss*P ~=__P*
**.
For any right __R*-module __N* and any right R-module N, the natural maps
Colim __N* __R*_F* __//__N* __R*(Colim __F*) ~=__N* __R*_P*,
Colim ___TorR*(N, F ) __//___TorR*(N, TelF ) = ___TorR*(N, P )
associated to the sequential colimits over the self-maps gOf and h are isomorph*
*isms.
By Lemma 5.8, the natural map _ss*(-) __R*_F* ! ___TorR*(-, F ) is an isomorph*
*ism,
and so the natural map _ss*(-) __R*_P* ! ___TorR*(-, P ) is also an isomorphis*
*m.
For any left R-module M, we have the usual short exact sequences of homotopy
groups
0 ! Lim1ssH*+1ExtR(F, M) ! ssH*ExtR(P, M) ! LimssH*ExtR(F, M) ! 0.
associated to a telescope. Since (g O f) O (g O f) = (g O f), Lemma 5.8 implies*
* that
hOh = h. It follows that the towers of abelian groups in question are Mittag-Le*
*ffler,
and so Lim1 = 0. Thus, the natural map
___Ext-*R(P, M) ! LimE___xt-*R(F, M)
is an isomorphism. For any left __R*-module __M*, the functor <- , __M*>__R*co*
*nverts
colimits to limits, and so the natural map <__P*, __M*>__R*! Lim <__F*, __M*>__*
*R*is an
isomorphism. By Lemma 5.8 again, the natural map ___Ext-*R(F, -) ! <__F*, _ss*(*
*-)>__R*
is an isomorphism. The natural map ___Ext-*R(P, -) ! <__P*, _ss*(-)>__R*is_ther*
*efore also
an isomorphism. |__|
If P 0is any other left R-module with _ss*P 0isomorphic to a projective left *
*__R*-
module __P*, then P 0is isomorphic in the derived category of left R-modules to*
* the
left R-module P of the previous lemma. To see this, note that a special case of
the isomorphism ___Ext-*R(P, P 0) ~=<_ss*P, _ss*P>0_R*of the previous lemma ind*
*icates
that there is a one-to-one correspondence between maps from P to P 0in the de-
rived category of left R-modules and maps from _ss*P to _ss*P 0in the category *
*of
left __R*-modules. Choosing an isomorphism _ss*P ~= __P* ~=_ss*P 0, we obtain a*
* map
P ! P 0inducing an isomorphism on homotopy groups. This proves the following
proposition.
Proposition 5.10. If P is a left R-module such that __P* = _ss*P is a projectiv*
*e left
__R*-module, then the natural maps _ss*(-) __R*_P* ! ___TorR*(-, P ) and ___Ex*
*t-*R(P, -) !
<__P*, _ss*(-)>__R*are isomorphisms.
This gives half of part (a) of Theorem 5.6. The other half is given by the fo*
*llowing
lemma.
Lemma 5.11. If I is a left R-module such that _I* = _ss*I is an injective left *
*__R*-
module, then the natural map ___Ext-*R(-, I) ! <_ss*(-), _I*>__R*is an isomorph*
*ism.
22 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
Proof.Let CI denote the class of left R-modules L for which the map ___Ext-*R(L*
*, I) !
<_ss*L, _I*>__R*is an isomorphism. Clearly CI is closed under arbitrary wedge p*
*roducts
and under suspension by any element o of RO(G). Also, an R-module L is in CI
if any R-module isomorphic to L in the derived category is in CI. By Lemma 5.8,
the modules Rn[G=H] are in CI. Theorem 4.5 indicates that the functor <- , _I*>*
*__R*
is exact. Thus, if
. .!. n-1C ! nA ! nB ! nC ! n+1A ! . . .
is a cofibration sequence, then applying either of the functors <_ss*(-), _I*>_*
*_R*or
___Ext-*R(-, I) to this sequence produces a long exact sequence of graded Mackey
functors. It follows that the cofiber of a map between modules in CI is a modul*
*e_
in CI. From this, we conclude that every left R-module is in CI. |*
*__|
The right module parts of Theorems 5.5 and 5.6 can be proven by arguments
analogous to those already given for left R-modules. Thus, to complete the proo*
*fs
of these two results, it suffices to prove part (c) of Theorem 5.5. The constru*
*ction
of the required R-modules with injective homotopy groups is completely analogous
to the construction of Brown-Comenetz dual spectra.
Let _I* be an injective left __R*-module. We define a contravariant functor F*
*I from
the derived category of left R-modules to the category of abelian groups by let*
*ting
FI(M) be the abelian group of maps of left __R*-modules from _ss*M to _I*. The
derived category of left R-modules and the functor FI satisfy the hypotheses for
the abstract form of the Brown representability theorem in Brown [3]. It follows
that there exists a left R-module I representing FI, i.e., the abelian group of*
* maps
in the derived category of left R-modules from M to I is naturally isomorphic to
FI(M). In particular, letting M range over the left R-modules Ro[X], we see that
_ss*I ~=_I*. This completes the proof of Theorem 5.5. Note that Lemma 5.11 impl*
*ies
that the R-module I is unique up to isomorphism in the derived category of left
R-modules.
6. The Construction of the Spectral Sequences
The two spectral sequences described in the introduction are constructed in
this section. Throughout this construction, R is a fixed equivariant S-algebra *
*and
__R* = _ss*R. The results in the previous section allow us to construct "resolu*
*tions" of
an R-module M in the derived category of R-modules corresponding to projective
and injective __R*-module resolutions of _ss*M. These resolutions are the equiv*
*ariant
generalization of the resolutions constructed in Section IV.5 of [5].
The following definition formalizes the relationship between our topological *
*res-
olutions of M and algebraic resolutions of _ss*M.
Definition 6.1. Let M be a left R-module, and let __M* = _ss*M. A projective
topological resolution of M consists of collections of left R-modules Ms and Ps
together with cofiber sequences
sPs js-!Ms is+1---!Ms+1 ks-! s+1Ps,
in the derived category of left R-modules for all s 0. These objects must sat*
*isfy
the conditions that M0 = M, each _ss*Ps is a projective left __R*-module, and _*
*ss*js is
an epimorphism.
A projective topological resolution (Ms, Ps) of M is compatible with a projec*
*tive
resolution __P*,*of __M* if there are isomorphisms __Ps,*! _ss*Ps under which _*
*ss*j0
EQUIVARIANT SPECTRAL SEQUENCES 23
coincides with the augmentation __P0,*! __M* and _ss*(ks O js+1) coincides with*
* the
suspension s+1ds+1 of the differential ds+1:__Ps+1,*! __Ps,*.
An injective topological resolution of M consists of collections of left R-mo*
*dules
Ms and Is together with fiber sequences
s+1 js
s+1Is ks-!Ms+1 i---!Ms -! sIs,
in the derived category of left R-modules for all s 0. These must satisfy the
conditions that M0 = M, each _ss*Is is an injective left __R*-module, and _ss*j*
*s is a
monomorphism.
An injective topological resolution (Ms, Is) of M is compatible with a given
injective resolution _I**of __M* if there are isomorphisms _Is*! _ss*Is under w*
*hich _ss*j0
coincides with the augmentation __M* ! _I0*and _ss*(js+1 O ks) coincides with t*
*he
desuspension s+1ds of the differential ds:_Is*! _Is+1*.
Projective topological resolutions of right R-modules are defined analogously*
*. In
the projective context, the modules Ms are analogous to the quotients X=Xs-1
for a nice filtration X0 X1 X2 . . .X of a space X. As discussed in
Boardman [1, 12.5ff], the cohomological spectral sequence constructed from the
cofiber sequences
X=Xs-1 __//_X=Xs __//_Xs=Xs-1
is isomorphic to the more usual spectral sequence constructed from the cofiber
sequences
Xs-1 __//_Xs __//_Xs=Xs-1,
but has better structural properties. In the derived category context, Verdier*
*'s
octahedral axiom converts "filtrations" (like the sequence X0 ! X1 ! X2 ! . . .
or the sequence M~0! M~1! M~2! . .u.sed in the next section) into "resolutions"
(like X ! X=X0 ! X=X1 ! . .o.r M0 ! M1 ! M2 ! . .).and vice-versa, but
not canonically.
As noted below, the requirement in the definition of projective topological r*
*eso-
lution that each _ss*js is an epimorphism ensures that projective topological r*
*esolu-
tions induce projective algebraic resolutions. An analogous observation applies*
* to
injective topological resolutions.
Proposition 6.2. If (Ms, Ps) is a projective topological resolution of M, then *
*_ss*Ps
is a complex of __R*-modules with differential ds+1 = -(s+1)_ss*(ks O js+1) an*
*d is a
projective __R*-module resolution of _ss*M with augmentation _ss*j0.
It is somewhat less obvious but quite important for us that there is a topolo*
*gical
resolution corresponding to every projective or injective algebraic resolution.
Lemma 6.3. Let M be a left R-module. Then every projective __R*-module resolu-
tion of __M* = _ss*M has a compatible projective topological resolution. Also, *
*every
injective __R*-module resolution of __M* has a compatible injective topological*
* resolu-
tion.
Proof.We treat the projective case in detail; the injective case is entirely an*
*alogous.
Let __P*,*be a projective resolution of __M*. First apply Theorem 5.5 to choose*
* left
R-modules Ps with _ss*Ps ~=__Ps,*. By Corollary 5.7, there is a unique map j0: *
*P0 !
M0 = M in the derived category that corresponds on passage to homotopy Mackey
functors to the augmentation __P0,*! __M*. Take M1 to be the cofiber of j0, and*
* let
i1 and k0 be the appropriate maps from the resulting cofiber sequence. Now assu*
*me
24 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
by induction that the resolution has been constructed up to Ms and that _ss*ks-1
is injective and induces an isomorphism of _ss*Ms with the submodule sIm (ds) *
*of
s__Ps-1,*. Then, by Corollary 5.7, there exists a map js: sPs ! Ms whose induc*
*ed
map on homotopy Mackey functors corresponds to the map ds: __Ps,*! Im (ds)
under the chosen isomorphisms. Take Ms+1 to be the cofiber of js, and let is+1 *
*and
ks be the induced maps. Since the map js induces an epimorphism on homotopy
Mackey functors, the long exact sequence associated to this cofiber sequence is
short exact. It follows that the map on homotopy Mackey functors induced by ks
provides an isomorphism between _ss*Ms+1 and s+1Im (ds+1). This completes_the
induction. |__|
Now we are ready to construct the spectral sequences. Let M be a left R-module
and N be a right R-module. Let __P*,*be a projective resolution of __M* = _ss**
*M,
and let (Ms, Ps) be a projective topological resolution of M compatible with __*
*P*,*.
Extend the collection of cofiber sequences to negative s by setting Ps = * and
Ms = M for s < 0 (with is+1 = idand js and ks the trivial map). Applying the
functor ___TorR*(N, -) to these cofiber sequences yields a homologically graded*
* exact
couple with
__Ds,o= ___TorRs+o(N, Ms)
(6.4)
__Es,o= ___TorRo(N, Ps).
The maps
. ._.i//__Ds-1,*i__//__Ds,*i_//_Ds+1,*_i__//__Ds+2,*i//_. . .
]];;; YY33 ]];;;
;; 33 ;;
j ;; k j 33 k j ;; k
. . . __Es-1,* __Es,* __Es+1,* . . .
in this exact couple come from the analogously named maps in the topological
resolution. Both i and j preserve the total degree s + o, and k lowers it by on*
*e.
Alternatively, given a projective resolution __Q*,*of __N* = _ss*N and a compat*
*ible
projective topological resolution (Ns, Qs) of N, setting
__Ds,o= ___TorRs+o(Ns, M)
(6.5)
__Es,o= ___TorRo(Qs, M)
gives another homologically graded exact couple of exactly the same form.
For the Ext spectral sequence, let L and M be left R-modules. Also, let __O*,*
be a projective resolution of __L* = _ss*L and let (Ls, Os) be a compatible pro*
*jective
topological resolution. Again set Ls = L, Os = * for s < 0. Apply the functor
___Ext*R(-, M) to the cofiber sequences relating Os and Ls, and let
(6.6) __Ds,o= ___Exts+oR(Ls, M)
__Es,o= ___ExtoR(Os, M).
This yields a cohomologically graded exact couple of the form
. ._.i//__Ds+2,*i___//_Ds+1,*i_//_Ds,*__i__//_Ds-1,*i__//. . .
]];;; [[777 ff [[777
;; 77 ffff 77
k ;; j k 77 ffjffffk77 j
. . . __Es+1,* __Es,* __Es-1,* . ...
EQUIVARIANT SPECTRAL SEQUENCES 25
In this exact couple, the maps i and j preserve the total cohomological degree *
*s+o,
and k raises it by one. Alternatively, let _I**be an injective resolution of __*
*M* and
(Ms, Is) be a compatible injective topological resolution of M. Extend the fib*
*er
sequences of the topological resolution by setting Is = * and Ms = M for s < 0.
Applying ___Ext*R(L, -) to the fiber sequences relating Ms and Is and setting
(6.7) __Ds,o= ___Exts+oR(L, Ms)
__Es,o= ___ExtoR(L, Is)
gives a cohomologically graded exact couple of the same form as above.
These exact couples lead to spectral sequences in the usual way. These spectr*
*al
sequences are clearly natural in the unresolved variable. Theorem 5.6 identifie*
*s the
E1- and E2-terms. When the unresolved variable is a weak bimodule, Theorems 5.3
and 5.4 imply that the exact couple is an exact couple of __R* or __R*-modules.*
* The
resulting spectral sequence is therefore a spectral sequence of __R* or __R*-mo*
*dules.
These assertions are summarized in the following result:
Theorem 6.8. Let L and M be left R-modules and N be a right R-module.
(a) The spectral sequence derived from the exact couple (6.4) has E1 complex
canonically isomorphic to __N* __R*_P*,*and E2s,o-term canonically isomorphic *
*to
___Tor_R*s,o(__N*, __M*). The spectral sequence is natural in N. If N is a weak*
* bimodule,
this is a spectral sequence of left __R*-modules.
(b) The spectral sequence derived from the exact couple (6.5) has E1 complex
canonically isomorphic to __Q*,* __R*_M* and E2s,o-term canonically isomorphic *
*to
___Tor_R*s,o(__N*, __M*). The spectral sequence is natural in M. If M is a weak*
* bimodule,
this is a spectral sequence of right __R*-modules.
(c) The spectral sequence derived from the exact couple (6.6) has E1 complex
canonically isomorphic to <__O*,*, __M*>__R*and Es,o2-term canonically isomorph*
*ic to
___Exts,o_R*(__L*, __M*). The spectral sequence is natural in M. If M is a weak*
* bimodule,
this is a spectral sequence of right __R*-modules.
(d) The spectral sequence derived from the exact couple (6.7) has E1 com-
plex canonically isomorphic to <__L*, _I**>__R*and Es,o2-term canonically isomo*
*rphic
to ___Exts,o_R*(__L*, __M*). The spectral sequence is natural in L. If L is a w*
*eak bimodule,
this is a spectral sequence of left __R*-modules.
This theorem leaves unresolved the question of whether our spectral sequences
are natural in the resolved variable. It is also not obvious that these spectr*
*al
sequences are independent of the resolution chosen to form them. To settle these
questions, we prove the following result in the next section.
Theorem 6.9. The identity map on ___Tor_R**,*(__N*, __M*) induces an isomorphis*
*m be-
tween the spectral sequences derived from (6.4) and (6.5). Similarly, the ident*
*ity
map on ___Ext*,*_R*(__L*, __M*) induces an isomorphism between the spectral seq*
*uences de-
rived from (6.6) and (6.7).
The issue of convergence for these spectral sequences must still be discussed.
Since limits and colimits of Mackey functors are formed object-wise, convergence
of spectral sequences of Mackey functors works just like convergence of spectral
sequences of abelian groups. The spectral sequences derived from (6.4) and (6.5)
are homological right half-plane spectral sequences. Thus, by Boardman [1, 6.1],
26 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
to prove that they converge strongly to
___TorRo(N, M) = Lims__D-s,s+o,
it suffices to show that Colims__Ds,-s+o= 0. In the case of (6.4), this amounts*
* to
showing that Colims___TorRo(N, Ms) = 0. For this, consider the cofiber sequence
` `
Ms __//_ Ms __//_TelMs
defining the telescope of a sequence of maps in the derived category. The canon*
*i-
cal map Colim_ss*Ms ! _ss*(TelMs) is an isomorphism, and it follows that TelMs
is trivial. Since the derived smash product over R preserves cofiber sequences,
TelTorR(N, Ms) is isomorphic to TorR(N, TelMs) and is therefore trivial. In par-
ticular, Colim ___TorR*(N, Ms) = 0. Since the edge homomorphism in this case is
induced by j0, we obtain the following result.
Theorem 6.10. The spectral sequence derived from the exact couple (6.4) converg*
*es
strongly to ___TorR*(N, M). Its edge homomorphism is the canonical map __N* __*
*R*_M* !
___TorR*(N, M).
For cohomologically graded right half-plane spectral sequences like those der*
*ived
from (6.6) and (6.7), conditional convergence to
___ExtoR(L, M) = Colims__D-s,s+o
is defined (see, for example, Boardman [1, 5.10]) to mean that
Lims__Ds,-s+o= 0 and Lim1s_Ds,-s+o= 0.
In the context of (6.7), __Ds,-s+o= ___ExtoR(L, Ms). The limit and Lim1 term t*
*hat
must vanish are defined by an exact sequence
Y Y
0 ! Lims__Ds,-s+o! ___ExtoR(L, Ms) ! ___ExtoR(L, Ms) ! Lim1s_Ds,-s+o! 0.
Consider the fiber sequence
Y Y
Mic Ms __//_ Ms __//_ Ms
defining the microscope of a sequence of maps in the derived category. Each map
is+1:Ms+1 ! Ms induces the zero map on homotopy Mackey functors, and so
MicMs is trivial. The derived function spectrum functor ExtR(L, -) preserves
fiber sequences, and so
Y Y
Ext R(L, MicMs ) __//_ ExtR (L, Ms) __//_ ExtR (L, Ms)
is a fiber sequence. Since MicMs is trivial, so is ExtR(L, MicMs ). The induced*
* long
exact sequence on homotopy Mackey functors now gives conditional convergence.
Since the edge homomorphism in this context is induced by j0, we obtain the
following result.
Theorem 6.11. The spectral sequence derived from the exact couple of (6.7) con-
verges conditionally to ___Ext*R(L, M). Its edge homomorphism is the canonical *
*map
___Ext*R(L, M) ! <__L*, __M*>__R*.
This completes the construction of the Hyper-Tor and Hyper-Ext spectral se-
quences described in the introduction.
EQUIVARIANT SPECTRAL SEQUENCES 27
7.Uniqueness, Naturality, and The Yoneda Pairing
In the previous section, we constructed a pair of Hyper-Tor spectral sequences
and a pair of Hyper-Ext spectral sequences. This section contains the proof of
Theorem 6.9, which asserts that the pair of Hyper-Tor spectral sequences are is*
*o-
morphic and so are the pair of Hyper-Ext spectral sequences. The technical work
needed to prove this result also suffices to construct construct the Yoneda pai*
*ring
of Hyper-Ext spectral sequences mentioned in the introduction. This construction
is described in Theorem 7.12.
The formation of our spectral sequences in the last section did not require a*
*ny
constructions more complicated than cofiber and fiber sequences, telescopes, and
microscopes. However, here we need more complicated homotopy colimits and lim-
its. Because of this, the constructions described in this section must be carri*
*ed out
in a point set category rather than the associated derived category. Neverthele*
*ss,
these constructions are of such a general nature that they can be carried out i*
*n any
of the modern categories of equivariant spectra (i.e., [5, 11, 12]).
For the remainder of the section, fix left R-modules L and M and a right R-
module N. Without loss of generality, it can be assumed that each of these obje*
*cts
is cofibrant and fibrant in the appropriate module category. Also, fix project*
*ive
topological resolutions (Ls, Os), (Ms, Ps), and (Ns, Qs) of L, M, and N, respec-
tively, and an injective topological resolution (Ms, Is) of M. We prove Theorem*
* 6.9
by constructing isomorphisms between the pairs of spectral sequences derived fr*
*om
these specific resolutions.
As noted in the remarks preceding the definition of a projective topological
resolution, the sequence of R-modules
M = M0 __//_M1 __//_._._.//Ms __//_. . .
is analogous to the sequence of quotients of M by a sequence of progressively l*
*arger
submodules of M,
M = M=M~-1 __//_M=M~0___//. ._.//_M=M~s-1___//. ...
To form the homotopy limits and colimits needed for the proof of Theorem 6.9, we
must "reconstruct" these missing submodules. Specifically, we construct a seque*
*nce
of (point-set level) cofibrations of R-modules
* = M~-1___//~M0~-1-!~M1//_._././_~Ms-1~-s-!~Ms//_. . .
together with compatible point-set level R-module maps fs-1:M~s-1 ! M whose
behavior in the derived category is what one would expect from an appropriate
filtration of M with quotients Ms. In particular, for each s, we choose isomorp*
*hisms
C(fs-1) ~=Ms and C(~-s) ~= sPs in the derived category that are compatible in
the following sense: They provide an isomorphism in the derived category between
the induced maps of homotopy cofibers
(7.1) C(~-s) __//_C(fs-1) __//_C(fs) __//_C( ~-s) ' C(~-s)
and the cofiber sequence
(7.2) sPs __//_Ms __//_Ms+1___// s+1Ps
that is a part of our chosen projective topological resolution. Here and for t*
*he
rest of the section, we understand the point-set model for homotopy cofibers to*
* be
28 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
formed using the usual cone construction; that is,
C(~-s) = M~s[M~s-1(M~s-1^ I+ ) [M~s-1*
and
C(fs-1) = M [M~s-1(M~s-1^ I+ ) [M~s-1*.
Note that since the model categories in [5, 11, 12] are simplicial and the obje*
*cts
M~s-1and M are cofibrant, these are cofibrant objects and are cofibers in the s*
*ense
of Quillen. Moreover, sequence (7.1) represents a cofiber sequence in the deriv*
*ed
category. The apparent shift in indexing in the cofiber sequence
M~s-1-fs-1--!M __//_Ms __//_ M~s-1
(Ms is the cofiber of M~s-1) is for consistency with the grading conventions of
Boardman [1, x12]. This shift also leads to cleaner formulas in the work below.
The "reconstruction" of the modules M~sessentially amounts to a point-set re-
finement of Verdier's octahedral axiom. The standard proof of this axiom is act*
*ually
strong enough to provide the refinement we need. Let M~-1 = *. Next choose a
cofibrant R-module M~0weakly equivalent to P0 and a point-set map f0: ~M0! M
of R-modules such that the induced cofiber sequence
M~0___//M __//_C(f0) __//_ M~0
is isomorphic in the derived category to the given cofiber sequence
P0 __//_M __//_M1 __//_ P0
by an isomorphism that is the identity on M. For the inductive step, assume that
the sequence of cofibrations and compatible maps into M has been constructed up
to the R-module M~s. Choose a fibrant R-module X and a point-set level R-module
map C(fs) ! X such that the induced cofiber sequence is isomorphic in the deriv*
*ed
category to the given cofiber sequence
Ms+1___//Ms+2___// s+2Ps+1 ! Ms+1
by an isomorphism that restricts to the previously chosen isomorphism C(fs) !
Ms+1. Let F be the homotopy fiber of the composite M ! C(fs) ! X defined
using the usual path space construction,
F = M xX XI+ xX *.
Since M and X are fibrant, this represents the fiber in the sense of Quillen. *
*By
construction, the composite M~s! M ! X factors through the composite M~s!
CM~s ! C(fs). This factorization provides a null homotopy of the map from M~s
to X and so a lift M~s! F of the map M~s! M. Choose M~s+1by factoring the
map M~s! F as a cofibration ~-s+1:~Ms! M~s+1followed by an acyclic fibration
M~s+1-~!F . Let fs+1:M~s+1 ! M be the composite map M~s+1! F ! M. The
EQUIVARIANT SPECTRAL SEQUENCES 29
objects and maps we have chosen fit into the diagram
~Ms__fs_//_M_____//_C(fs)
~-s+1|| |||| ||
fflffl|f || fflffl|
~Ms+1____/s+1/_M_//_C(fs+1)
O z<< GG O
~ O zz GG O~
fflfflOzz G##fflfflO
F X,
which commutes on the point-set level. The map C(~-s+1) ! X is a weak equiv-
alence by [5, I.6.4], [11, 3.5.(vi)], or [12, 5.8]. Thus, this procedure constr*
*ucts an
isomorphism in the derived category between cofiber sequence (7.1) and cofiber
sequence (7.2) that restricts to the previously chosen isomorphism C(fs) ! Ms+1.
This completes our construction, which is described formally in the following p*
*ropo-
sition.
Proposition 7.3. Let (Ms, Ps) be a projective topological resolution of an R-mo*
*dule
M. Then there is a sequence
* = M~-1___//~M0~-1-!~M1//_._././_~Ms-1~-s-!~Ms//_. . .
of cofibrations of R-modules together with R-module maps fs-1:M~s-1 ! M, com-
patible on the point-set level, such that C(fs-1) ~=Ms and C(~-s) ~= sPs in the
derived category. Moreover, these isomorphisms are compatible in the sense that
they provide an isomorphism in the derived category between the cofiber sequenc*
*es
C(~-s) __//_C(fs-1) __//_C(fs) __//_C( ~-s) ' C(~-s)
and
sPs __//_Ms __//_Ms+1___// s+1Ps.
In this same manner, choose sequences of cofibrations
* = L-1 __//_~L0~-1-!~L1//_._././~Ls-1~-s-!~Ls//_. . .
and
* = N-1 __//_~N0~-1-!~N1//_._././_~Ns-1~-s-!~Ns//_. . .
together with compatible maps fs:L~s! L and fs:N~s ! N consistent with the
chosen projective resolutions of L and N. When it is desirable to indicate which
of the objects L, M, and N is associated to a particular map ~-sor fs, the symb*
*ols
~-Ms, fMs, etc., are employed.
An analogous sequence of fibrations
s-1 ~-1
. ._._//Ms ~----!Ms-1 __//_._._.//~M0-!~M0__//_M-1 = *
together with compatible maps fs: M ! Ms can be constructed using the "Eck-
mann-Hilton" dual of the argument above (that is, reverse the direction of all *
*of
the arrows, and switch cofibrations and fibrations, colimits and limits, and (-*
*)^I+
and (-)I+).
To make use of the R-modules just constructed, we must introduce a family of
very simple categories.
30 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
Definition 7.4. Let D denote the category which has as objects the ordered pairs
of natural numbers (s, t) and as maps a unique map (s, t) ! (s0, t0) whenever s*
* s0
and t t0. Let Dn denote the full subcategory of D consisting of objects (s, t*
*) with
s + t n
The (point-set) smash products ~Ns^R ~Mtmay be regarded as a functor from D to
equivariant S-modules, orthogonal spectra, or symmetric spectra, as appropriate.
Likewise, the function spectra FR (~Ls, ~Mt) form a contravariant functor from *
*D to
equivariant S-modules, orthogonal spectra, or symmetric spectra. Our main tools
in this section are homotopy limits and colimits of these functors. Our argument
requires constructions of homotopy limits and colimits that are functorial on t*
*he
point-set level. Any functorial constructions should be adequate. However, at s*
*ome
points in the argument, we assume that the "bar construction" models of homotopy
limits and colimits (described, for example, in [5, Xx3]) are used in order to *
*fill in
certain details.
Definition 7.5. Let Tn = HocolimDn(N~s^R M~t) and Un = HolimDnFR (~Ls, ~Mt).
There are canonical maps
gn+1: Tn __//_Tn+1 and gn+1 :Un+1 __//_Un .
induced by the inclusion of categories Dn ! Dn+1. Similarly, there are maps
hn :Tn ! N ^R M and hn :FR (L, M) ! Un
induced by the maps M~s! M and the analogous maps for the N~s, ~Ls, and M~s
sequences.
There are also maps
Tn __//_N ^R M~n Tn __//_~Nn^R M,
FR (~Ln, M) __//_Un FR (L, ~Mn) __//_Un
induced by the maps M~s! M~s+1and the analogous maps for the ~Ns, ~Ls, and M~s
sequences. Because we are using a functorial construction of homotopy colimits
and limits, these are point-set level maps. Moreover, the diagrams
Tn ______//_N ^R M~n Tn _______//~Nn^R M
gn+1|| id^~-Mn+1|| gn+1|| ||~-Nn+1^|id|
fflffl| fflffl| fflffl| fflffl|fflffl|
Tn+1 ____//_N ^R M~n+1 Tn+1_____//~Nn+1^R M,
and the analogous diagrams for the homotopy limits Un commute on the point-set
level.
Since the diagrams above commute on the point-set level, they induce canonical
maps
C(gs+1) _________//_C(hs)________//C(hs+1)________//_ C(gs+1)
| | | |
| | | |
fflffl| fflffl| fflffl| fflffl|
C(idN ^~-Ms+1)__//_C(idN ^fMs)___//C(idN ^fMs+1)__// C(idN ^~-Ms+1)
EQUIVARIANT SPECTRAL SEQUENCES 31
and
C(gs+1) __________//_C(hs)________//_C(hs+1)________// C(gs+1)
| | | |
| | | |
fflffl| fflffl| fflffl| fflffl|
C(~-Ns+1^ idM)___//C(fNs^ idM)____//C(fNs+1^ idM)___// C(~-Ns+1^ idM)
of cofiber sequences in the derived category. The constructions N ^R (-) and
(-) ^R M preserve cofiber sequences. Thus, the isomorphisms characterizing the
terms in cofiber sequence (7.1) can be used to identify the bottom cofiber sequ*
*ences
in the two diagrams above with those defining the exact couples (6.4) and (6.5),
respectively. Setting
__Ds,o= _sss+oC(hs)
(7.6)
__Es,o= _sss+oC(gs)
gives a third homologically graded exact couple. Moreover, the commuting dia-
grams above provide maps of exact couples from (7.6) to both (6.4) and (6.5).
The following result about these three exact couples implies the part of Theo-
rem 6.9 applicable to exact couples (6.4) and (6.5).
Theorem 7.7. The spectral sequence derived from (7.6) has its E1 complex canon-
ically isomorphic to the total complex of __Q*,* __R*_P*,*. Moreover, under the*
* canon-
ical isomorphisms, the maps of this E1 complex to the E1 complexes of (6.4) and
(6.5) coincide with the augmentations
__Q*,* __R*_P*,*__//__N* __R*_P*,* and __Q*,* __R*_P*,*__//__Q*,* __R*_M*
**,
respectively.
In a similar fashion, by setting
__Ds,o= _ss-s-oF (hs)
(7.8)
__Es,o= _ss-s-oF (gs),
we obtain a cohomologically graded exact couple and maps of exact couples from
both (6.6) and (6.7) to (7.8). The relation between these three exact couples *
*is
described by the following result, which implies the remaining claims made in T*
*he-
orem 6.9.
Theorem 7.9. The spectral sequence derived from (7.8) has its E1 complex canoni-
cally isomorphic to the total complex of <__O*,*, _I**>__R*. Moreover, under th*
*e canonical
isomorphisms, the maps into this complex from the E1 complexes of (6.6) and (6.*
*7)
coincide with the augmentations
<__O*,*, __M*>__R*//_<__O*,*,a_I**>__R*nd<__L*,/_I**>__R*/_<__O*,*,,_I***
*>__R*
respectively.
For the proofs of these results, recall that, in the bar construction model o*
*f the
homotopy colimit, Tn is the geometric realization of a simplicial object which *
*in
simplicial degree d is the wedge product, indexed on the set of d composable ar*
*rows
(s0, t0) . . .(sd, td)
in Dn, of the summands ~Nsd^R ~Mtd. The face maps are given by dropping arrows *
*on
either end, composing arrows in the middle, and by the action of Dn on ~Nsd^R ~*
*Mtd.
The degeneracy maps simply insert identity maps. Likewise, the homotopy limit Un
32 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
is the geometric realization (or Tot) of a cosimplicial object which in cosimpl*
*icial
degree d is the product, indexed on the set of d composable arrows in Dn, of the
factors FR (~Lsd, ~Mtd). Here, the face maps are given by dropping or composing
arrows and by the contravariant action of Dn on FR (~Lsd, ~Mtd). The degeneracy
maps again simply insert identity maps.
Proof of Theorems 7.7 and 7.9.We give the details of the proof of Theorem 7.7.
The proof of Theorem 7.9 is quite similar. The canonical maps from the homotopy
colimits to the point-set colimits induce a map T0 ! M~0^R N~0and, for s > 0,
maps
Ts=Ts-1___//(N~0) ^R (M~s=M~s-1) _ . ._.(N~s=N~s-1) ^R (M~0).
These maps induce maps
` `
(7.10) C(gs) __//_ TorR(C(~-Nn), C(~-Mm)) ~= TorR( nQn , m Pm).
n+m=s n+m=s
This gives a canonical map from E1 to __Q*,* __R*_P*,*. It is easy to see that *
*this is
a map of chain complexes and that the maps
_ss*C(gs) __//__ss*C(idN ^~-Ms) ~=__N* __R*_Ps,*,
_ss*C(gs) __//__ss*C(~-Ns^ idM) ~=__Qs,* __R*_M*
factor through the augmentations
__Q*,* __R*_P*,*__//__N* __R*_P*,* and __Q*,* __R*_P*,*__//__Q*,* __R*_M*
as required.
It remains to see that (7.10) induces an isomorphism on homotopy groups. Let
Dn,m denote the full subcategory of Dn whose objects are the pairs (s, t) with
s m. Also, let Tn,m = HocolimDn,mN~s^R M~t. Standard homotopy theory
arguments show that the canonical map from the homotopy colimit to the colimit
induces weak equivalence of the cofiber of Ts-1,0! Ts,0with ~N0^M~sand the cofi*
*ber
of the map Ts,m-1! Ts,mwith (N~m=N~m-1) ^R M~s-m. A filtration argument now_
finishes the proof. |__|
We close this section with an explanation of the Yoneda pairing of Hyper-Ext
spectral sequences. Assume that K is yet another left R-module, and let __K* = *
*_ss*K.
Also, let __Es,or(L, K) denote the spectral sequence for ___Ext*R(L, K) derived*
* from (6.6),
__Es,or(K, M) denote the spectral sequence for ___Ext*R(K, M) derived from (6.7*
*), and
__Es,or(L, M) denote the spectral sequence for ___Ext*R(L, K) derived from (7.8*
*). A map
of complexes
M
(7.11) __Em,or(K, M) __E`,or(L, K) __//__Es,or(L, M)
`+m=s
induces in the usual way a map
M m,o `,o
__Er+1(K, M) __Er+1(L, K)
`+m=s M
= Hm (__E*,or(K, M)) H`(__E*,or(L, K))
`+m=s
___//Hs(__E*,or(K, M) __E*,or(L, K))
___//Hs(__E*,or(L, M))
= __Es,or+1(L, M).
EQUIVARIANT SPECTRAL SEQUENCES 33
of graded Mackey functors for Er+1. A pairing of spectral sequences is map of
complexes (7.11) for E1 such that the induced map on E2 is a map of complexes,
and, inductively, the induced map on each Er+1 is a map of complexes.
Theorem 7.12. The composition pairing
<__K*, _Im*>__R*<__O`,*,___K*>__R*//_<__O`,*, _Im*>__R*
induces a pairing of the Hyper-Ext spectral sequences. On E2, this pairing agre*
*es
with the Yoneda pairing. Also, this pairing agrees on E1 with the associated gr*
*aded
of the composition pairing for ___Ext*R.
Proof.The maps
(7.13) FR (K, ~Mm) ^ FR (~L`, K) __//_F (~L`, ~Mm) __//_U`+m
of S-modules, orthogonal spectra, or symmetric spectra induce maps
ExtR(K, m Im) ^S ExtR( `O` , K) __//_F (g`+m )
in the stable category. On homotopy groups, these maps induce the pairing on E1
described in the theorem, which on E2 is the Yoneda pairing, by definition. The
composition maps (7.13)also induce maps
ExtR(K, F (M~m ! M~m-r)) ^ ExtR(C(~L`-r! ~L`), K)
___//F (U`+m ! U`+m-r )
in the stable category relating the fiber and cofiber of the canonical maps M~m*
* !
M~m-r and ~L`-r! ~L`, respectively, to the fiber of the canonical map U`+m !
U`+m-r . An inductive argument using these maps indicates that the pairing on E*
*r__
preserves the differential. The statement about E1 is clear. *
*|__|
When R is a commutative S-algebra, the pairing described in Theorem 7.12 de-
scends to a pairing defined in terms of __R*rather than . Moreover, the ident*
*ifi-
cation of the pairing on the E1 level respects the __R*-module structures. The*
*orem
7.12, together with these observations, establishes the properties of the Yoneda
pairing of spectral sequences noted in the introduction.
Appendix A. The smash product and the box product
The purpose of this appendix is to prove Theorem 5.1, which asserts that the
homotopy Mackey functor is lax symmetric monoidal. Since the argument in the
case of Z-graded Mackey functors is well-known, we concentrate on the RO(G)-
graded case.
As in Section 5, choose a model So for the o-sphere for each element o of RO(*
*G)
with the restrictions that S0 = S and that So is the smash product of S with
the standard n-sphere space whenever o is the trivial representation of dimensi*
*on
n > 0. For all other o, the object So may be chosen arbitrarily from the approp*
*riate
homotopy class. When o = ff+fi, Sff^Sfiand So are isomorphic in the equivariant
stable category. Thus, we can choose an isomorphism
fff,fi:Sff+fi__//Sff^ Sfi.
Such a choice gives a homomorphism
[Sff^ X+ , M]G [Sfi^ Y+, N]G __//_[Sff^ X+ ^ Sfi^ Y+, M ^ N ]G
__//_[Sff+fi^ (X x Y )+, M ^ N,]G
34 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
which is natural in M and N in the equivariant stable category and X and Y in t*
*he
Burnside category. The universal property of the box product then gives a natur*
*al
map
OE: _ss*M * _ss*N __//__ss*(M ^ N).
There is also a canonical natural map ': __B*! _ss*Sthat includes __B as _ss0S.*
* This
map does not require any choices. Theorem 5.1 is merely a less detailed version*
* of
the following result.
Theorem A.1. The maps fff,fimay be chosen so that OE together with ': __B*! _ss*
**S
provide the functor _ss* with a lax symmetric monoidal structure.
In other words, the maps fff,fimay be chosen so that the unit diagrams
id*'// ' *id//
__B* *_ss*N____ss*S * _ss*N _ss*M * __B*____ss*M * _ss*S
(A.2) ~ || OE|| ae|| |OE|
fflffl| fflffl| fflffl| fflffl|
_ss*Noo__ss*~_ss*(S_^ N) _ss*Moo___ss*_ss*(Ma^eS),_
the associativity diagram
OE
_ss*(L;^;M) *_ss*N_______//__ss*((L<^ M) ^ N)
OE *idvvvv <<~=<
vvv <<
vv OEOE<
(A.3) (_ss*L *_ss*M) *_ss*N _ss*(L ^ (M ^ N)),
HH @@
HHH
~=HHH##H OE
_ss*L *(_ss*M * _ss*N)id*OE//__ss*L *_ss*(M ^ N)
and the symmetry diagram
~=
_ss*M * _ss*N__//_ss*N * _ss*M
(A.4) OE|| |OE|
fflffl| fflffl|
_ss*(M ^ N)_~=__//__ss*(N ^ M)
commute. The redundant right unit diagram has been included because, with it,
diagrams (A.2)and (A.3)together imply that _ss* is lax monoidal. This is verifi*
*ed
before the question of commutativity is addressed. A direct argument in terms
of specifying how to choose the isomorphisms fff,fiappears to be possible, but
would require checking a long list of complicated details. Instead we take a mo*
*re
abstract approach that reduces to checking the vanishing of a certain character*
*istic
cohomology class defined in [2, x7]. The definition of that class in our specif*
*ic case
is reviewed below.
Since S0 = S has been chosen as the unit for the smash product in the equivar*
*iant
stable category, the canonical choices for the maps f0,oand fo,0are the inverse*
*s of
the unit isomorphisms ~: S0 ^ So ! So and ae: So ^ S0 ! So. This is the unique
choice making the unit diagrams (A.2) commute. This choice is assumed in the
remainder of our argument.
EQUIVARIANT SPECTRAL SEQUENCES 35
It is easy to verify that the required associativity diagram (A.3) commutes i*
*f and
only if the diagram
kSff+fi+flSfff+fi,flS
fff,fi+flkkkkk SSS
uukkk SSS))S
(A.5) Sff^ Sfi+fl Sff+fi^ Sfl
id^ffi,flfflffl|| ffff,fi^fidlffl||
Sff^ (Sfi^ Sfl)______~=_______//_(Sff^ Sfi) ^ Sfl
commutes for all ff,fi, and fl in RO(G). In general, the failure of this diagr*
*am
to commute can be measured by a unit in the Burnside ring. To see this, recall
that, in any symmetric monoidal additive category, the abelian group of maps
between any two objects is a module over the ring of endomorphisms of the unit
object (see for example the discussion of "signs" in Section 2). The abelian gr*
*oup
[Sff+fi+fl, Sff^ Sfi^ Sfl]Gis a one-dimensional free module over [S, S]G= __B(G*
*=G)
and is generated by any isomorphism. The right-hand composite isomorphism in
diagram (A.5) is therefore the product of left-hand composite isomorphism and
a well-defined unit aff,fi,flin __B(G=G). Equivalently, aff,fi,flcan be define*
*d as the
unique unit in __B(G=G) such that
(A.6) aff,fi,fl. idSff+fi+fl= f-1ff,fi+flO (id^ffi,fl)-1 O a O (fff,fi^ id) *
*O fff+fi,fl.
Here, a denotes the associativity isomorphism in the equivariant stable categor*
*y.
Clearly, diagram (A.5) commutes if and only if aff,fi,fl= 1. Thus, we have prov*
*en:
Proposition A.7. The associativity diagram (A.3) commutes for all L, M, N if
and only if aff,fi,fl= 1 for all ff, fi, fl 2 RO(G).
The isomorphism aff,fi,fl.idSff+fi+flhas a straight-forward structural interp*
*retation
in terms of the choice of models So and isomorphisms fff,fi. Consider the full
subcategory S of the equivariant stable category consisting of objects isomorph*
*ic
to spheres So (for o 2 RO(G)), and let C be the full subcategory consisting of
the chosen models So. The inclusion of C in S is obviously an equivalence of
categories. The smash product on the equivariant stable category restricts to a
symmetric monoidal product on S. The isomorphisms fff,fican be used to construct
an equivalent symmetric monoidal smash product0on C. There, the0smash product
of Sffand Sfiis Sff+fi.0For0maps g :Sff! Sff and h: Sfi! Sfi, the induced map
g ^ h: Sff+fi! Sff +fiin C is the composite
fff,fi ff fi g^h ff0 fi0f-1ff0,fi0ff0+fi0
Sff+fi____//_S ^ S _____//S ^ S _______//S .
Because of our restriction that f0,oand fo,0are the inverses of unit isomorphis*
*ms,
the unit isomorphisms in C are the appropriate identity maps. The associativ-
ity isomorphisms in C are precisely the maps aff,fi,fl. idSff+fi+fl. Since __B*
*(G=G) is
commutative and composition in C is bilinear over __B(G=G), the pentagon and tr*
*i-
angle laws [10, VIIx1] for the monoidal structure on C translate into the follo*
*wing
assertion about the elements aff,fi,fl:
Proposition A.8. The elements aff,fi,flsatisfy the cocycle condition
aff,fi,fl+ffi. aff+fi,fl,ffi= afi,fl,ffi. aff,fi+fl,ffi. aff,fi*
*,fl,
36 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
in the group of units of the Burnside ring and are normalized in the sense that
aff,fi,fl= 1 when any of ff, fi, or fl are 0.
This may be rephrased as the assertion that the collection {aff,fi,fl| ff, fi*
*, fl 2
RO(G)} specifies a normalized 3-cocycle for the group cohomology of RO(G) with
coefficients in group A* of units of the Burnside ring.
For the proof of Theorem A.1, we must consider how this cocycle changes when
a different collection of isomorphisms fff,fiis selected. Let f0ff,fi:Sff+fi! S*
*ff^ Sfi
be another such collection satisfying our restriction on the maps f00,oand f0o,*
*0. The
composite f0ff,fiO f-1ff,fi:Sff+fi! Sff+fiis bff,fi. idSff+fifor a well defined*
* unit bff,fiin
the Burnside ring. It is easy to verify that the cocycle (a0ff,fi,fl) associate*
*d to the
isomorphisms f0ff,fiis given by
a0ff,fi,fl= bff,fi. bff+fi,fl. b-1ff,fi+fl. b-1fi,fl. aff,fi,fl.
When the collection (bff,fi) is regarded as a 2-cochain for group cohomology, i*
*ts
boundary satisfies
(db)ff,fi,fl= bff,fib-1ff,fi+flbff+fi,flb-1fi,fl.
Changing the choice of the maps fff,fitherefore changes the class (aff,fi,fl) b*
*y a
coboundary, from which it follows that a determines a well-defined element of
H3(RO(G); A*).
Conversely, given any normalized 2-cochain (bff,fi) for RO(G) with coefficien*
*ts
in A*, the rule f0ff,fi= bff,fi. fff,figives a collection of isomorphisms f0ff,*
*fi:Sff+fi!
Sff^ Sfiwith f00,oand f0o,0the inverses of the unit isomorphisms, and with asso*
*ci-
ated cocycle a0= db . a. An easy computation indicates that changing the models
So does not alter the cohomology class represented by the collection (aff,fi,fl*
*). These
observations are summarized in the following result:
Proposition A.9. There exists a cohomology class ~ain H3(RO(G); A*) such that,
for any fixed choice of models So, the association (A.6) defines a surjection f*
*rom
the set of collections of isomorphisms fff,fi:Sff+fi! Sff^ Sfi(with f0,oand fo,0
the inverses of the unit isomorphisms) onto the set of normalized 3-cocycles in*
* the
cohomology class ~a.
This result indicates, if the cohomology class ~avanishes, then any collectio*
*n of
isomorphisms f0ff,fican be adjusted via some normalized 2-cochain into a collec*
*tion
fff,fimaking diagram (A.5)commute. This reduction of the proof of Theorem A.1
to a question in cohomology opens the way for us to restrict our attention to a*
*ctual,
rather than virtual, real representations. Let RO+ (G) be the commutative monoid
of isomorphism classes of actual real representations of G. It follows from [13*
*, 4.1]
that the inclusion of RO+ (G) into RO(G) induces a homotopy equivalence from
B(RO+ (G)) to B(RO(G)). The associated restriction map
H3(RO(G); A*) __//_H3(RO+ (G); A*)
is therefore an isomorphism. Thus, it suffices to show that the image ~a+of the
cohomology class ~aof Proposition A.9 is trivial in H3(RO+ (G); A*).
The process just described for associating a normalized 3-cocycle in the coho-
mology of RO(G) to any collection of maps fff,fiindexed on ff, fi 2 RO(G) works
equally well to associate a normalized 3-cocycle in the cohomology of RO+ (G) to
any collection of maps fff,fiindexed on ff, fi 2 RO+ (G). Moreover the resulti*
*ng
class in the cohomology of RO+ (G) is clearly the restriction of the class in t*
*he
EQUIVARIANT SPECTRAL SEQUENCES 37
cohomology of RO(G). This allows us to turn the whole argument backwards: we
prove that the class ~a+is trivial by showing that we can choose model spheres
So for o 2 RO+ (G) and isomorphisms fff,fifor ff, fi 2 RO+ (G) such that the
diagrams (A.5) commute.
Let ae1, ae2, . . . , aer be an enumeration of the irreducible real represent*
*ations of
G. For each aei, select an associated sphere Saei. Any nonzero o 2 RO+ (G) can *
*be
written as a sum
o = n1ae1 + . .+.nraer
in which at least one of the niis nonzero. Let So be S ^ (Sae1)(n1)^ . .^.(Saer*
*)(nr).
Here, (Saei)(ni)denotes the ni-fold smash product of copies of Saei. For each p*
*air
ff, fi in RO+ (G), select the map ~fff,fi:Sff+fi! Sff^ Sfito be that which uses*
* the
associativity and commutativity isomorphisms of the equivariant stable category*
* to
rearrange the spheres (Saei)(ni)appearing in Sff+fiinto the proper order for Sf*
*f^Sfi
with as few transpositions as possible. Clearly this yields a collection of map*
*s which
are appropriately unital and associative. The associated cocycle is then identi*
*cally
1, and the cohomology class ~a+is therefore trivial. We have therefore proven t*
*he
following result.
Proposition A.10. The cohomology class ~aof Proposition A.9 is trivial. Thus,
we can choose the isomorphisms fff,fifor ff, fi 2 RO(G) so that OE satisfies (A*
*.2)
and (A.3).
We still need to show that the natural transformation OE is symmetric, that i*
*s,
that it satisfies (A.4). Let u(ff, fi) be the unique unit in the Burnside ring *
*such that
the composite map
fff,fi ff fi fi ff f-1fi,ffff+fi
Sff+fi____//_S ^ S ~=S ^ S _____//_S
is u(ff, fi) . idSff+fi. By examining the definition of OE and the symmetry is*
*omor-
phism for the box product, it is easy to see that OE is symmetric if and only
if u(ff, fi) = oe(ff, fi). Clearly u is antisymmetric; that is, u(ff, fi)u(fi,*
* ff) = 1.
The commutativity of the diagram (A.5) implies that u is bilinear; that is, that
u(ff + fi, fl) = u(ff, fl)u(fi, fl). Thus, to see that our choice of signs oe i*
*n Section 2 is
consistent with our choice of the maps fff,fi, it suffices to check that u(ae, *
*ae) = oe(ae, ae)
for each irreducible representation ae and that u(ae, ae0) = 1 whenever ae and *
*ae0 are
distinct irreducible representations. This follows from the discussion above an*
*d the
observation that the twist map on Sae^ Saeis homotopic to the map
-1 ^ id: Sae^ Sae_//_Sae^ Sae.
Remark A.11. We hinted in Section 2 that there is some flexibility in the units
u(ff, fi) determined by the symmetry isomorphism Sff+fi! Sfi+ffand therefore in
the choice of oe(ff, fi). The analysis leading to Proposition A.9 shows that, *
*for a
fixed choice of model spheres So, the set of collections of isomorphisms fff,fi*
*(with
f0,oand fo,0the inverses of the unit isomorphisms) is a torsor for the normaliz*
*ed
2-cochains of RO(G) with coefficients in A*. Thus, the set of collections that
satisfy (A.5) is a torsor for the normalized 2-cocycles. If we adjust the maps *
*fff,fi
by a 2-cocycle b, then the element u(ff, fi) changes by a factor of b-1fi,ff. b*
*ff,fi. It
follows that, for any collection of isomorphisms satisfying (A.5), u(ae, ae) mu*
*st be
the unit described above. However, we can find a collection that takes on any
38 L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL
desired set of values for u(ae, ae0) for distinct irreducibles ae, ae0, subject*
* only to the
restriction u(ae, ae0)u(ae0, ae) = 1.
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Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150
E-mail address: lglewis@syr.edu
DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
E-mail address: M.A.Mandell@dpmms.cam.ac.uk