RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY
THEORY.
J.P.C.Greenlees
The author is grateful to the Nuffield Foundation for its support.
Author addresses:
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk
78
Part II
Algebraic counterparts to standard constructions.
80
CHAPTER 7
Introduction to Part II
This chapter begins in Section 7.1 with a summary of the models from Part I, an*
*d a broad
summary of the contents of Part II. The following three sections provide more s*
*ubstantial
introductions to the three main types of example treated in Part II. Finally, i*
*n Section 7.5
we formalise a straightforward argument that is needed several times.
7.1. General outline.
In Part I we showed that various categories of rational T-spectra had algebra*
*ic models.
Specifically we showed (i) that the category of T-spectra over H is equivalent *
*to the derived
category of torsion Q[cH ]-modules, (ii) that the category of F-spectra is equi*
*valent to the
derived category of F-finite torsion OF-modules and (iii) that the category of *
*semifree
spectra is equivalent to the derived category of the category of Q[c]-morphisms*
* N - !
Q[c; c-1] V which become isomorphisms when c is inverted and (iv) that the cat*
*egory of
all rational T-spectra is equivalent to the derived category of the one dimensi*
*onal abelian
category A whose objects are morphisms N -! tF* V with F-finite torsion kernel *
*and
cokernel.
The main purpose of Part II is to identify the algebraic counterparts of vari*
*ous well
known topological functors. Most interesting functors F : T-Spec - ! T-Spec ari*
*se by
passage to stable homotopy from a point set level functor f. It is therefore na*
*tural to expect
that the algebraic counterpart will be the total right derived functor Rf0 : DA*
* -! DA
of a functor f0 : A -! A, or its left derived counterpart. (Many may be content*
* to view
Rf0as defined by the formula Rf0X = f0(X^) where ^Xis a fibrant approximation t*
*o X; for
further discussion the reader is referred to Appendix B on Quillen closed model*
* category
structures). This expectation proves well founded in all cases we have analyzed*
*. There are
two caveats to this statement. Firstly, it sometimes seems necessary to replace*
* A by the
torsion model Atin order to find an algebraic counterpart at the level of abeli*
*an categories.
More important is the fact that at present there seems no prospect of showing t*
*hat the
equivalence of topological and algebraic categories arises from a string of adj*
*unctions of
Quillen closed model categories. Accordingly there is no prospect of showing th*
*at f and f0
correspond, and we must proceed in an indirect way.
Thus the work of this part divides into two. Firstly we have to to do enough*
* algebra
to understand the behaviour of the functors f0 which arise, and thus to underst*
*and their
81
82 7. INTRODUCTION TO PART II
associated right derived functors. Secondly we have to show how to describe the*
* functor
F using only terms which we can model exactly. Of course, if we can identify F *
*X by its
homotopy (as happens for the standard model), it is relatively easy to verify t*
*he model on
objects. There is then the difficulty of indeterminacy in the modelling of morp*
*hisms, and
we may have to argue rather obliquely to verify this part of the model. Howeve*
*r, some
interesting functors are hard to describe exactly, even on objects. In this ca*
*se we may
attempt to construct objects X from basic objects X0using sums and cofibre sequ*
*ences in
such a way that the value F X0 of F on a basic object X0 is modelled by Rf0(M),*
* and so
that at each stage we can identify the maps with zero indeterminacy.
Because of Part I, a great deal of the work in Part II is purely algebraic. Of*
* course it is
necessary to do a little topology from time to time if we wish to know which to*
*pological
phenomena we are modelling. On the other hand, there may be readers who feel th*
*at after
Part I they do not need to talk about actual T-spectra any more, and that they *
*can work
entirely in the algebraic category. Despite the obvious dangers of this attitud*
*e, its benefits
make it desirable to make it possible to read Part II without intimate acquaint*
*ance with
Part I. It is thus helpful to recall the model before undertaking a full scale *
*algebraic study.
We therefore begin by recalling that we work almost exclusively overQthe grade*
*d ring
OF, which we may view as obtained from the ring (OF)0 = C(F; Q) ~= H Q by adjoi*
*ning
an indeterminate c of degree -2. We let eH denote the idempotent of (OF)0 corre*
*sponding
to the projection onto the Hth factor, and we let cH = eH c. We next need the t*
*wisting
module tF*which we have approached in Part I through the short exact sequence
0 -! OF -! tF*-! 2I -! 0
L 2 -1
with I = HIH , and IH = Q[cH ; cH ]=Q[cH ]. We shall shortly justify a more *
*instructive
construction, as follows. First we need to define Euler classes in suitable gen*
*erality, general-
izing the elements ck12 H*(BT+) for k 0. For any finite subset OE F we have a*
*n associ-
ated idempotent eOE2 OF, and we have define the Euler class cOE= eOEc+(1-eOE), *
*which is not
a homogeneous element of OF. The effect of cOEon an OF-module M = eOEM (1 - eO*
*E)M
is to multiply by c on the first factor and do nothing to the second: thus the *
*result of
inverting cOEon M is again a graded module. We therefore consider the set
E= {ckOE|OE F finitek 0}
of Euler classes. We shall see that tF*= E-1OF. Next,Lan OF-module M is said *
*to be
F-finite if the natural inclusion is an isomorphism H eH M ~= M, and it is to*
*rsion if
M[c-1] = 0. It turns out that M is an F-finite torsion module if and only if E-*
*1M = 0.
We now consider the category A whose objects are morphisms fi : N -! tF* V who*
*se
kernel and cokernel are Euler torsion. Morphisms are required to be the identit*
*y on the
tensor factor tF*. We call N the nub, and V the vertex of the object; the map f*
*i is called
the basing map. The category A is abelian and one dimensional, and its derived *
*category
DA is the standard model for rational T-spectra (Theorem 5.4.1). Thus there is *
*a short
exact sequence for calculating morphisms in either DA or in T-Spec in terms of*
* Hom and
Ext in A, and this sequence is split. This is stated explicitly as Theorem 5.4.*
*5.
7.1. GENERAL OUTLINE. 83
Whilst DA is technically very convenient, not all geometrically interesting *
*aspects of
T-spectra are immediately apparent. Thus, even when a construction can be perfo*
*rmed in
DA , the result may need reinterpretation before it answers interesting questio*
*ns. There
also appear to be cases where DA is not the natural way of modelling T-spectra*
*. We
therefore also need to consider the torsion model category At whose objects are*
* OF-maps
tF* V -! T with T an F-finite torsion module. This is abelian and two dimension*
*al and
(Theorem 6.3.1) its derived category is equivalent to DA . This completes the s*
*ummary of
the algebraic models.
The rest of Part 2 is concerned with modelling various geometric construction*
*s. The
examples are divided into two groups: those of a general nature and those that *
*change
the ambient group of equivariance. It seemed logical to begin with smash produ*
*cts and
function spectra, as the most fundamental. Unfortunately they are also the har*
*dest to
understand, and we only succeed in identifying them on objects, except for the *
*important
special case of products, which we identify precisely. The special case of func*
*tional duals
is made explicit in Section 15.5. We consider the product spectrum, both becaus*
*e of its
own importance, and because it is used as an essential ingredient in our identi*
*fication of
the model of the function spectrum. The topological reader may prefer to skip C*
*hapters
8 and 9 to begin with, and return to the product, smash product and function sp*
*ectrum
after the other examples, because of the substantial amount of algebra they req*
*uire.
Amongst the functors which change equivariance the trick of identifying a fun*
*ctor by
identifying its adjoint is both effective and essential, since it happens sever*
*al times that
only one of a pair of adjoint functors is directly accessible. We begin by cons*
*idering the
forgetful induction and coinduction functors, all of which are quite straightfo*
*rward. Both
types of T-fixed point functors are built into our models. The most interestin*
*g functors
are those which relate T-spectra to T=K-spectra for finite subgroups K. As wou*
*ld be
expected, the geometric K-fixed point functor K is rather easy to describe. How*
*ever both
the inflation functor from T=K spectra to T-spectra and its adjoint, the Lewis-*
*May K-fixed
point functor are rather subtle, even at the purely algebraic level. It is perh*
*aps only to be
expected that quotient functors are only identified on reasonably free objects.
We then move on to the homotopy Mackey functor and Eilenberg-MacLane spectra.
In the context of compact Lie groups of positive dimension there are two other *
*vari-
ants of Eilenberg-MacLane spectra: co-Eilenberg-MacLane spectra representing or*
*dinary
homology, and Brown-Comenetz spectra representing the Mackey duals of homotopy.*
* Since
one is used to these coinciding with Eilenberg-MacLane spectra rationally for f*
*inite groups
of equivariance, it seems worth giving a detailed analysis to show how differen*
*t they are in
the present case.
Although the work of Part II is of interest in its own right, it has further *
*justification
in the applications of Part III. In practice the author only undertook the ana*
*lysis of a
functor when forced to by the applications. The resulting clarification and sim*
*plification
in the special case was most gratifying. However, since there is now a substant*
*ial amount
of algebra, which may perhaps be of interest in its own right, we have collecte*
*d it together
in chapters of its own. This has the unfortunate effect of separating the alge*
*bra from
84 7. INTRODUCTION TO PART II
the topology which motivated it. Therefore, to help the reader navigate Part I*
*I we will
continue this introduction with more complete sketches of the analyses and moti*
*vations
for the algebraic constructions: we feel justified in talking heuristically at *
*this stage. The
structure of the analyses is that we develop the necessary algebra and then exp*
*lain the the
properties of topological constructions which relate it to the algebra. Finally*
* the algebraic
and topological functors are shown to correspond using these two parallel struc*
*tures. The
last section of this chapter outlines the general methods we employ for studyin*
*g functors.
7.2. Modelling of the smash product and the function spectrum.
In Sections 8.3 to 9.3 we identify the algebraic model of the smash product fu*
*nctor and
its right adjoint, the function spectrum functor. In essence the smash product *
*is modelled
by the left derived tensor product, and the function spectrum by its right adjo*
*int. However
there are several important caveats. Firstly, the left derived tensor product d*
*oes not exist
in all the categories modelling spectra: one may instead interpret it as the ri*
*ght derived
torsion product. The nomenclature is justified since the construction is the l*
*eft derived
tensor product in a larger category. The next important point is that the right*
* adjoint of
the left derived tensor product is not the right derived internal Hom functor, *
*although it is
closely related. This fact is a consequence of the caveat about left derived te*
*nsor products.
Both complications are essential in order to get models of the geometric constr*
*uctions.
Finally, we are only able to describe the models on objects, and not as functor*
*s; neither
the smash product nor its adjoint preserve pure parity objects in general, so o*
*ne cannot
get a firm grip on morphisms. This means that once we have shown the smash prod*
*uct is
modelled by the left derived torsion product it does not follow formally that t*
*he function
spectrum is modelled by its adjoint. Nonetheless it is true, which seems strong*
* evidence
that our models are in fact functorial. To prove functoriality in the present f*
*ramework, one
would need a natural splitting into even and odd pieces. A more satisfactory r*
*esolution
would be to prove the equivalence of homotopy categories from a string of equiv*
*alences
at the level of model categories, following the example of Quillen [21, 22]. Ho*
*wever this
approach appears inaccessible at present.
We also spend some time considering the modelling of arbitrary products of spe*
*ctra.
Section 8.2 identifying the product in the torsion model and its derived catego*
*ry is an
instructive introduction to the difficulties faced for the function spectrum. *
*However the
identification of the product in the standadard model and its derived category *
*in Section
8.6, is an essential tool in the analysis of the function spectrum. The point *
*is that we
can identify the model of the product of spectra as a functor, becauseWthe prod*
*uctQis right
adjoint to the diagonal functor. This gives us control over F ( iYi; Z) ' iF *
*(Yi; Z) when
F (Yi; Z) is known for all i.
Returning to the smash product and the function spectrum, we shall discuss fou*
*r different
contexts in parallel. The model of free spectra given by torsion Q[c]-modules, *
*the model
of F-free spectra, given by F-finite torsion modules, the model of semifree spe*
*ctra, and
finally the model of all T-spectra given by the standard model. Considerations *
*are slightly
different in the four cases, but the strategy is common to them all.
7.2. MODELLING OF THE SMASH PRODUCT AND THE FUNCTION SPECTRUM. 85
For the smash product, the first observation is that it preserves F-spectra a*
*nd F-
contractible spectra. More precisely X ^ Y ^ EF+ ' (X ^ EF+) ^ (Y ^ EF+) and
T(X ^ Y ) ' TX ^ TY . Since T(X ^ Y ) = T(X) ^ T(Y ), it is immediate that
ss*(T(X ^ Y )) = ss*(TX) ss*(TY );
and we shall show that ssT*(EF+ ^ X ^ Y ) is the `left derived tensor product' *
*of ssT*(EF+ ^
X) and ssT*(EF+ ^ Y ). This might suggest the use of the torsion model, but no*
*te that
EF+ ^ EF+ ' 2EF+; the extra suspension makes the gluing map inaccessible. We
therefore use the standard model. On the other hand, since tF*= E-1OF it is cle*
*ar that
the definition
(M -! tF* U) (N -! tF* V ) = (M N -! (tF* U) (tF* V ) = tF* U V )
preserves objects of A . There are enough flat objects in A , and the left der*
*ived torsion
product " can be calculated using these. The caveats mentioned above suggest al*
*ternative
ways of doing this which are unavoidable in some of the contexts.
The function spectrum F (X; Y ) is easily described if Y is F-contractible, a*
*s F (TX; TY )^
E"F. Similarly if X is F-contractible, so is the function spectrum, and F (X; Y*
* ) is described
by its homotopy groups. However in the general case, even if X and Y are both F*
*-spectra,
F (X; Y ) is not: in fact the right adjoint of the smash product with an F-spec*
*trum X (re-
garded as a functor from F-spectra to F-spectra) is Y 7-! F (X; Y )^EF+. This e*
*xpression
should prepare for the algebraic surprise in modelling function spectra.
The function spectrum is recognized as the right adjoint of the smash product*
*, and hence
the natural expectation (which proves to be correct on objects) is that the mod*
*el should
be given by the right adjoint to the total left derived functor ". The basic wa*
*rning is that
this right adjoint is not the total right derived functor of the internal Hom f*
*unctor.
To illustrate the issues in the simplest context, we continue the discussion *
*by concen-
trating on free spectra, modelled by torsion Q[c]-modules. To begin with we mus*
*t identify
the internal Hom functor on the category of torsion modules. This functor is ch*
*aracterized
by the adjunction
Hom(R S; T ) = Hom(R; IntHom(S; T ));
and hence the internal Hom functor between torsion modules S and T is given by
IntHom(S; T ) = cHom(S; T ):
In fact the right adjoint of " is R0Hom := RcRHom and not RIntHom. This bewilde*
*red
the author for some time. First we give an example where RIntHom(S; T ) does no*
*t model
the function spectrum whilst RcRHom(S; T ) does. This shows in particular that *
*the two
algebraic functors are different, and concentrates our attention on the correct*
* one, which
we shall show gives a model in general.
Example 7.2.1. Let X = Y = ET+, so that we have models S = T = I. We ob-
serve that RIntHom(S; T ) = cQ[c] = 0. On the other hand Hom(S; T ) = Q[c] is *
*not
injective, so that to calculate Rc RHom(S; T ) = Rc Hom(S; T ) we must take an *
*injec-
tive approximation fibre(Q[c; c-1] -! 2I) to Q[c] before applying f. We thus ob*
*tain
RcRHom(S; T ) = I, in agreement with F (ET+; ET+) ^ ET+ ' ET+. __|_|
86 7. INTRODUCTION TO PART II
The explanation of this is that the internal Hom functor is only recognized as*
* such by
virtue of the intermediate category of all modules. In fact we consider the pai*
*rs
i S
torsQ[c] - mod AEQ[c] - mod AEQ[c] - mod
of adjoint functors (with left adjoints displayed at the top). The composite of*
* the two lower
functors is IntHom (S; .) = cHom(S; .) with its domain extended to all Q[c]-mod*
*ules. In
fact the right adjoint of "S : D(Q[c]) -! D(Q[c]) is R0Hom(S; .) := RcRHom(S; .*
*). And
if S and T model the free spectra X and Y , then R0Hom(S; T ) models F (X; Y ) *
*^ ET+.
We follow our standard operating procedure, by spending Chapter 8 developing t*
*he
algebra necessary to understand the model. Section 9.1 provides topological inp*
*ut to the
final comparison, and finally, in Section 9.2 we prove smash products are model*
*led by left
derived tensor products, and in Section 9.3 we prove that function spectra are *
*modelled by
its right adjoint.
7.3.Modelling functors changing equivariance.
Now that we can model categories of G-spectra when G is any subquotient of the*
* circle
T, we may attempt to understand the functors between these categories in algebr*
*aic terms.
Of course the categories of H-spectra are very simple when H is a finite subgr*
*oup,
so functors involving these are rather straightforward to understand: thus we *
*begin in
Section 10.1 by modelling the induction and coinduction functors from H-spectra*
* to T-
spectra. There is still a potential difficulty in understanding morphisms, but *
*we recognize
the algebraic functors as being left and right adjoint to the forgetful functor*
*, which is easy
to identify. It is clear in this case that the vertex plays no real part so it *
*is unimportant
whether we work with the standard or the torsion model.
More important are the fixed point functors. Of course there are two types of*
* fixed
point functors: the geometric fixed point functors and the Lewis-May fixed poin*
*t functors.
Whichever we use, if we take T-fixed points the result is a non-equivariant spe*
*ctrum, and
hence determined simply by its ordinary homotopy groups. These are easily deduc*
*ed from
the model. We therefore concentrate on K-fixed points regarded as a functor fro*
*m T-spectra
to T=K-spectra.
The geometric K-fixed point functor is designed to pick out the isotropy parts*
* corre-
sponding to subgroups of T containing K: itWis nearly obvious that this means t*
*he vertex is
unchanged and that the effect on EF+ ^X ' H E^X is to throw away the summan*
*ds
indexed by the finite subgroups H not containing K. There is an obvious way of*
* doing
this algebraically for OF-modules: one simply applies the idempotent e 2 C(F; Q*
*) with
support consisting of subgroups H containing K, and identifies this set of subg*
*roups of T
with the subgroups of T=K in the obvious way. We prove in Section 10.2 that thi*
*s not only
gives the right functor for F-spectra, but that simply applying e to the standa*
*rd model
category models the geometric fixed point functor.
The Lewis-May K-fixed point functor K is well known to be much more complicate*
*d,
and we spend Chapters 11 and 12 in studying it. It is well known that at one ex*
*treme, the
geometric K-fixed point functor can be obtained from the Lewis-May fixed point *
*functor
7.3. MODELLING FUNCTORS CHANGING EQUIVARIANCE. 87
by K (X) = K (X ^ E[6 K]+). At the other extreme, the Lewis-May fixed point fu*
*nctor
on a free spectrum is the quotient. One might hope that an understanding of the*
*se two
extreme pieces of behaviour would lead to an identification of the algebraic mo*
*del of the
functor. Because of its intuitive appeal we present this approach in Section 1*
*2.3, but it
appears only to give the answer on objects, and not as a functor.
Instead, we find it better to recognize the Lewis-May fixed point functor as *
*the right
adjoint of the inflation functor. Terminology is not quite consistent in the l*
*iterature, so
we must explain that if U is a complete T-universe we mean the functor which ta*
*kes a
T=K-spectrum X indexed on UK , regards it as a T-spectrum indexed on UK by pull*
*back
along the quotient map q : T -! T=K, and then builds in representations by the *
*relative
stabilization functor corresponding to the inclusion j : UK -! U. The resulting*
* spectrum
is often written j*q*X, q# X or simply j*X. However, since we want to concentra*
*te on the
important features we write simply X 7-! infTT=KX for this functor. The behavio*
*ur of the
inflation functor is very simple on suspension spectra in the sense that infTT=*
*KX = X for a
space X, but its general behaviour is rather complicated, and this is reflected*
* algebraically.
Actually, its behaviour is simple on F-contractible spectra, in the sense that *
*it is essentially
the identity. It is also simple on F-spectra, in thatWit simply increases the m*
*ultiplicity of
the various factors. We may view an F-spectrum Y ' H Y (H) as classified by a *
*function
H 7-! N(H), where N(H) = ssT*(Y (H)) is a_Q[c] module._ Now, if X is an almost *
*free
T=K spectrum classified by the function H 7-! M(H ), then infTT=KX is classifie*
*d by the
__ __ __
function H 7-! M(H ), where H is the image of H in T=K. Thus the summand X(H ),
which occurs once in X, is replaced in infTT=KX by one copy for each subgroup H*
* of T
__
whose image in T=H is H . The difficulty comes in splicing these two pieces tog*
*ether, and
this leads us on an algebraic digression to the relatively well behaved categor*
*y of Hausdorff
OF-modules. This enables us to identify an algebraic inflation functor on the t*
*orsion model
category, and then to show that it models the topological inflation functor. It*
* then follows
that the right adjoint of the geometric inflation functor models the Lewis-May *
*fixed point
functor. Following standard operating procedure, we begin in Chapter 11 by stud*
*ying the
algebraic inflation and its adjoints. In Section 12.1 we establish the essentia*
*l features of
topological inflation and the Lewis-May fixed point functor. Finally we show th*
*e algebraic
and topological inflation correspond in Section 12.2; it follows that the left *
*adjoint of the
algebraic inflation models the Lewis-May fixed point functor. Finally we spend*
* Section
12.4 considering the functor on the standard model: of the functors we have con*
*sidered
this is the only one for which we do not have a good description in the standar*
*d model.
The topological quotient functor is only approachable on K-free spectra, but *
*in that case
it is left adjoint to inflation. It follows that the left adjoint to the algebr*
*aic inflation functor
models passage to the quotient. Since we are working rationally one might hope *
*to extend
the domain of good behaviour of the quotient functor to all F-spectra. Although*
* we do not
do this, supporting evidence comes from the fact that the algebraic inflation f*
*unctor does
have a left adjoint on the category of all F-finite torsion OF-modules: we disc*
*uss these
matters in Section 12.5.
88 7. INTRODUCTION TO PART II
7.4.Modelling Eilenberg-MacLane spectra and related objects.
A cohomology theory k*G(.) satisfying the dimension axiom is characterised by *
*the Mackey
functor M : G=H 7-! k0G(G=H+) describing its values on homogeneous spaces. It i*
*s repre-
sented by the Eilenberg-MacLane spectrum HM. A homology theory lG*(.) satisfyin*
*g the
dimension axiom is characterised by the coMackey functor N : G=H 7-! lG0(G=H+) *
*describ-
ing its values on homogeneous spaces. It is represented by the co-Eilenberg-Mac*
*Lane spec-
trum JN. Finally, given an injective Mackey functor I we can form the Brown-Com*
*enetz
cohomology theory by taking hInT(X) = Hom(ss_Tn(X); I), and this is represented*
* by a spec-
trum hI. Since an arbitrary Mackey functor M admits an injective resolution of *
*length 1,
we can also define the spectrum hM. For finite groups the orbit category is sel*
*f-dual, so
that there is a natural way of identifying Mackey functors and co-Mackey functo*
*rs. Thus
a Mackey functor M gives rise to three representing spectra, HM; JM and hM. It*
* is
a deeply ingrained fact that the these three spectra coincide. However, it rel*
*ies on the
facts that G=H is a 0-dimensional manifold, and that H*(BH; Q) is concentrated *
*in degree
0. Both these facts fail for positive dimensional compact Lie groups. We shal*
*l identify
the Eilenberg-MacLane, co-Eilenberg-MacLane and Brown-Comenetz spectra for the *
*circle
group, and it is apparent that they are completely different in general. Howeve*
*r it is inter-
esting to observe that all three classes of spectra are formal in the torsion m*
*odel: in fact
their torsion parts are injective.
We begin with ordinary cohomology, and the usual Eilenberg-MacLane spectrum. An
Eilenberg-MacLane spectrum is recognized by the fact that its homotopy groups a*
*re only
nonzero in a single degree. It is characterized by the homotopy groups in that*
* degree,
regarded as a Mackey functor. We therefore begin in Section 13.1 by making exp*
*licit
how to recover the Mackey functor homotopy groups from the standard model. By *
*way
of illustration we deduce a functorial construction of the model of an Eilenber*
*g-MacLane
spectrum from a Mackey functor. In Section 13.2 we take the opposite approach a*
*nd begin
with ordinary cohomology and deduce a model, and provide decompositions of Eile*
*nberg-
MacLane spectra which are interesting from a topological point of view. We foll*
*ow this by
Section 13.3 which gives an analogous approach to the analysis of co-Mackey fun*
*ctors. It
does not seem worthwhile to make the algebraic approach to co-Eilenberg-MacLane*
* spectra
explicit.
We find Brown-Comenetz spectra to be particularly interesting, and devote Sect*
*ion 13.4
to them. One reason for interest is that by definition these spectra are well s*
*uited to the
construction of Adams spectral sequences, and the author's first approach to th*
*e algebraic
modelling of T-spectra therefore used them. However the extreme complexity of t*
*he Brown-
Comenetz spectra seems to make the resulting Adams spectral sequence impractica*
*l except
in certain special cases. On the other hand the spectra do form a natural class*
* of unbounded
spectra, and they therefore provide an interesting test case for the present me*
*thods.
7.5. Functors between split triangulated categories.
We conclude the introduction to Part II by explaining our general method for m*
*odelling
functors.
We repeatedly want to construct algebraic counterparts of topological functors*
* between
7.5. FUNCTORS BETWEEN SPLIT TRIANGULATED CATEGORIES. 89
categories of spectra. In other words we have a functor F : Spec1 -! Spec2 betw*
*een two
categories of spectra, and we want to find the algebraic counterpart F 0: DA 1-*
*! DA 2, in
the sense that we complete the square
Spec1 -! Spec2
p #' '# p
DA 1 . .>. DA 2
so that it commutes up to natural isomorphism. Typically, it is reasonably easy*
* to guess
a candidate algebraic functor F 0: DA 1 -! DA 2, and in fact F 0is usually the *
*total
right derived functor of some functor f0 : A 1-! A 2. The difficulty arises be*
*cause the
equivalences p were only defined indirectly and by making certain choices. Thus*
* it is only
easy to identify the homotopy of objects, and the d-invariant of morphisms. The*
*re are two
grades of conclusion that we may hope to achieve. We may only be able to prove *
*that F 0
is correct on objects, in the sense that if X is modelled by M then F X is mode*
*lled by
F 0M. However, in especially favourable cases, we will be able to prove that it*
* is correct on
morphisms in the sense that the above square commutes up to natural isomorphism.
Since the relevant functors F and F 0preserve cofibre sequences, the idea is *
*to show that
F and F 0agree on certain basic objects X0, and that arbitrary objects X can be*
* formed
from these basic objects using only constructions in which every step can be mo*
*delled
without indeterminacy. It is often sufficient to use mapping cones of maps f : *
*X01-! X02
in which F (f) is determined by its d-invariant. If this can be done for objec*
*ts we shall
say that F is object-accessible, and if it can also be done for morphisms we sh*
*all we shall
simply say that F is accessible.
Our only comment about object accessibility is that, if we can use a one dime*
*nsional
model category A 2, it suffices to calculate the homology of the image object, *
*since then
all objects are formal in the sense that they are determined by their homotopy.*
* In the
torsion model object-accessibility is not always clear, and, even in the standa*
*rd model,
considerable work is involved in showing that the smash product and function sp*
*ectrum
functors are both object accessible.
It is often possible to work with one dimensional models, and it is therefore*
* valuable to
record the minimal data that needs to be checked to ensure the diagram commutes*
* up to
natural isomorphism. In practice we compare the two composites F1 = pF and F2 =*
* F 0p
from the top left to the bottom right.
Theorem 7.5.1. Given a split linear triangulated category Spec and a one dime*
*nsional
split abelian category A , two functors F1; F2 : Spec -! DA are naturally isom*
*orphic
provided the following conditions are satisfied.
(1) Both functors preserve (a) triangles, (b) injectives, and (c) pure parity o*
*bjects
(2) The functors agree up to isomorphism (a) on enough injective objects and (b*
*) on maps
into injective objects.
We remark that in the motivating case it is enough to check Condition 1 for t*
*he functors
F and F 0since the functors p are equivalences of triangulated categories, and *
*used to define
the notions of parity and injectivity.
90 7. INTRODUCTION TO PART II
Proof: First note that for any object Y we have F1(Y ) ' F2(Y ). Indeed we may *
*choose
an injective resolution Y - ! I -d! J, and thus a triangle FiY - ! FiI -Fid!FiJ*
*. Since
FiJ is injective by (1)(b), it follows that the morphism Fid : FiI -! FiJ is de*
*termined by
its d-invariant. Once the objects are identified using (2)(a) the two maps F1d *
*and F2d are
homotopic by (2)(b). Hence F1Y ' F2Y as required.
By property (1)(c) it is enough to check that the two maps [X; Y ] -! [F X; F *
*Y ] agree
when X and Y are of even parity. In this case the Adams short exact sequence fo*
*r [X; Y ]
is split by parity. Now suppose we have an Adams resolution Y - ! I -! J with i*
*mage
F Y - ! F I -! F J. Suppose first that f : X -! Y is of even degree, and consid*
*er the
composite X -! Y -! I. Since F I is injective by (1)(b), the two images of the *
*composite
F X -! F Y -! F I are equal. However the indeterminacy in the map F X -! F Y wi*
*th
such a composite is the image of the group [F X; -1F J], which is zero since F *
*X and F J
are of even parity and F J is injective. Since there is no indeterminacy F1f = *
*F2f.
Next, if f : X -! Y is of odd degree, it factors as X -! -1J -! Y , and the im*
*age
of this is the composite F X -! -1F J -! F Y . The first map is a map into an i*
*njec-
tive, and so F1 and F2 agree; the second map is part of the triangle F Y -! F I*
* -! F J. __|_|
CHAPTER 8
Basic algebra for models and their derived categories.
This chapter contains the most substantial algebra of the entire work. However *
*the first
section is elementary and will interest all readers. It gives the long-awaited *
*discussion of
Euler classes in the global setting. In particular, it gives the elementary pro*
*of that F-finite
torsion modules are exactly the Euler-torsion modules. The rest of the chapter*
* is only
necessary in our analysis of the product, smash product and function spectrum, *
*and some
readers will want to skip to Chapter 10.
In Section 8.2 we warm up with a treatment of products in the torsion model a*
*nd its
derived category, before giving an abstract treatment of the tensor-Hom adjunct*
*ion in
Section 8.3, illustrated by the rather straightforward case of categories of to*
*rsion modules.
Before applying this to deal with the derived category of the standard model, w*
*e need to
understand the tensor product and internal Hom on the standard model itself: we*
* study
these constructions in Section 8.4, and it involves a little work, principally *
*in the analysis
of the appropriate `torsion functor'. After that, it is not hard to verify in S*
*ection 8.5 that
the standard model fits into the framework layed out in Section 8.3. Finally, i*
*n Section 8.6
we complete the analysis of products in the standard model and its derived cate*
*gory.
8.1. Euler classes and F-finite torsion OF-modules.
In this section we explain the occurrence of F-finite torsion OF-modules, by *
*showing
that they are precisely the modules annihilated when Euler classes are inverted*
*. We also
summarise some elementary but useful algebra, and introduce associated notation.
For brevity, let C denote the category of all OF-modules, and let Cf denote t*
*he full
subcategory of F-finite modules. Furthermore we let Ct denote the category of m*
*odules
M for which each element is c-power torsion, and we let Cftdenote the F-finite *
*torsion
modules. We thus have inclusions of full subcategories in the arrangement
Cft -! Cf
# #
Ct -! C
In fact these inclusions are all left adjoint to useful functors. For an arbi*
*trary module
MLwe let cM denote the submodule of c-power torsion elements, and we define OEM*
* =
H eH M. The following statements are readily verified.
91
92 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
Lemma 8.1.1. (i) OEcM = cOEM
(ii) If L is F-finite then
Hom(L; M) = Hom(L; OEM)
(iii) If T is a torsion module then
Hom(T; M) = Hom(T; cM): __|_|
We conclude that OE provides a right adjoint to both verticals, and that c pro*
*vides a
right adjoint to both horizontals. By the first part, there is only one composi*
*te to consider,
and it is worth considering it in more detail.
We recall that there is an Euler class associated to any representation V of T*
* with V T= 0.
This is based on the map e(V ) : S0 -! SV , and exists when we have Thom isomor*
*phisms.
We let O(V ) denote the associated Euler class. Provided the requisite Thom iso*
*morphisms
exist, the fact that e(V W ) = 1 ^ e(V ) e(W ) shows they are multiplicative i*
*n the sense
that O(V W ) = O(V )O(W ): Evidently, if V T6= 0 then O(V ) = 0, since e(V ) i*
*s null.
Let us turn to the particular case of F-spectra. So far we have restricted att*
*ention to the
effect on T-spectra over H when V = V (H), but now we need to take all isotropy*
* groups
into account. Thus we consider the map e(V )^1 : S0^EF+ -! SV ^EF+, and note th*
*at,
if we split EF+ into its summands E, we obtain cv(H)H: E -! 2v(H)E whe*
*re
v(H) = dimC(V H). For any complex representation V with V T= 0 the Euler class *
*O(V )
is multiplication by cv(H)Hon the H'th summand. If V T6= 0 then O(V ) = 0. Note*
* also that
V H = 0 for almost all H, so that any OF-module M splits as M(H1) . . .M(Hn) *
*M0
where V H = 0 unless H = H1; : :;:Hn-1 or Hn, and thus the Euler class acts on *
*any
OF-module. To avoid undue dependence on representations we will use an alterna*
*tive
notation.
Definition 8.1.2.If OE F we let eOEbe the idempotent with support OE, and we *
*let
cOE= eOEc + (1 - eOE). Thus, if V is one dimensional with kernel H, we have O(V*
* ) = cOEwhere
OE = {K | K H}. More generally, if v : F -! Z0 is any function, we may let cv*
* 2 OF
be the non-homogeneous element which is cv(H)Hover H. For any representation V *
*we may
associate the function v(H) = dimC(V H), and then we have O(V ) = cv.
It is also natural to use the corresponding notation in topology, so that, for*
* any dimension
function v, we define Sv by the cofibre sequence
_
E(2v(H))-! S0 -! Sv:
H
Evidently, we can also make analogous definitions in algebra and topology when *
*v also
takes negative integer values, provided it is only negative on finitely many su*
*bgroups.
Finally, we let E denote the multiplicative set of all Euler classes:
E = {ckOE| OE F finitek 0};
with associated torsion functor
fM = {x 2 M | ex = 0 for some nonzero Euler class}e:
8.1. EULER CLASSES AND F-FINITE TORSION OF-MODULES. 93
Lemma 8.1.3. We have equalities
fM = cOEM = OEcM
Proof: It is sufficient to note that no element of M with infinitely many coord*
*inates non-
zero can be annihilated by an Euler class. __|_|
We may now formalise the connection between the discussion in the introductio*
*n and
the work in Part I by observing that
E-1OF = tF*:
This lets us regard objects (N - ! tF* V ) of the standard model category A as*
* OF-
modules N together with a basis of E-1N (i.e. a subspace V of E-1N giving an is*
*omorphism
E-1N ~=tF* V ).
Lemma 8.1.4. The functor f is left exact, and only has one right derived func*
*tor; this
is given by R1fM = E-1M=M.
Proof: First note that Euler-torsion modules may be resolved by Euler-torsion i*
*njectives,
so that RifT = 0 for i > 0, if T is Euler-torsion. Thus the short exact sequenc*
*e 0 -!
fM -! M -! M=fM -! 0 shows that for i 1 we have RifM ~=RifM=fM.
Now, by definition
0 -! fM -! M -! E-1M
is exact, and this gives the short exact sequence
0 -! M=fM -! E-1M -! E-1M=M -! 0
Since E-1M=M is an F-finite torsion module, which proves the lemma, once we obs*
*erve
that RifE-1M = 0 for all i. This is clear when i = 0, and follows in general s*
*ince an
Euler-local module admits an injective resolution by Euler-local modules. __|_|
The reason for care is that the inclusions Cft -! Ct and Cf -! C do not prese*
*rve
products or injectives.QFrom the above lemma we see that the product of F-finit*
*eQmodules
Mffin Cf is OE ffMff, and similarly, for torsion modules Mffthe product is c *
*ffMff.
Example 8.1.5. Let I be an injective F-finite torsion module, zero below degr*
*ee -2k.
As usual, injectivity may be tested by extending maps : J -! I defined on an i*
*deal J over
the whole ring OF. Now an ideal is specified by the sequence of vector spaces J*
*0; J-2; : :,:
and if we identify (OF)2nwith (OF)0by using multiplication by cn the sequence o*
*f subspaces
is increasing. We first extend over the ideal corresponding to the constant se*
*quence of
subspaces J-2k; J-2k; J-2k; J-2k; : :u:sing c-divisibility of I. We are thus r*
*educed to the
problem of showing that I0 is an injective module over C(F; Q).
However this need not be the case. For example the functionL on the ideal (e1*
*; e2; : :):
defined by (ei) = (0; : :;:0; 1; 0 : :):takes values in HQH , but does not ex*
*tend over QF
because there is no common bound to the length of the vectors (v) for v in the *
*ideal. __|_|
94 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
The main positive result is that because injectivity of F-finite modules is de*
*tected by
divisibility, the inclusion Cft-! Cf does preserve injectives.
Lemma 8.1.6. If M is an F-finite module, injective in the category Cftthen it *
*is also
injective in the category Cf. __|_|
8.2. Products in the torsion model.
It is well known that, although it is easy to show that products exist in cate*
*gories of
spectra, it can be hard to understand anything about them that the categorical *
*properties
of products do not guarantee. However the category of T-spectra is simple enou*
*gh that
we can give a complete analysis, and the author has found the process of unders*
*tanding
extremely instructive. In any case, the analysis of products is important for *
*two other
reasons: firstly the case of the standard model allows us to give an analysis o*
*f function
spectra (on objects), and secondly it allows one to understand inverse limits o*
*ver a sequence
of spectra. The algebraic and categorical issues which are important here occur*
* again in
a more complicated form in our discussion of the tensor and Hom functors. This *
*section
may therefore be helpful preparation for Section 8.3.
QSuppose then that A is an indexing set, and that we wish to understand the pro*
*duct
ff2AYffof T-spectra Yffin terms of the algebraic models of the variousQspectra*
* Yff. The
reason products are amenable to analysis is that the the functor C: CA -! C i*
*s right
adjoint to the very well behaved diagonal functor : C -! CA. Since we know th*
*at
the model of the topological diagonal is the algebraic diagonal, we know that t*
*heir right
adjoints correspond, and hence we need only identify the product in the algebra*
*ic context.
Now suppose that C = dgA for one of our finite dimensional abelian categories *
*A (this
includes the standard and torsion model categories as well asQthe categories of*
* torsion
k[c] and OF-modules). We shallQsee that there is a product A : AA - ! A, whi*
*ch
therefore induces a functor dgA : dgAA - ! dgA, which is also the product fun*
*ctor.
SinceQthis functor preserves the homotopy relation, it follows that we can defi*
*ne a functor
DA : DA A -! DA by applying Q dgAto a fibrant approximation of an object (ie *
*Q DA=
QdgA Q dgA
R ). Furthermore, since is a right adjoint, it preserves injectives an*
*d pullbacks,
and hence it is still rightQadjoint to the diagonal functor. In other words if *
*Mffis a model
of Yffthen a model of ffYffis
Y DA Y dgA
ff(Mff) = ff ^Mff:
The remaining problem is to find the homology of the model. Although this descr*
*iption
makes the construction appear straightforward, the reader may find some surpris*
*es along
the way. In this section we treat only the F-finite torsion modules and the tor*
*sion model: we
shall apply the same method to the standard model in Section 8.6 when we have e*
*stablished
the necessary algebraic foundations.
We begin with the category F-finite torsion modules, leaving the easier catego*
*ry of
torsion k[c]-modules to the reader. This is an ingredient in the discussion of *
*the torsion
model, and also illustrates the points which arise. In fact, one simply notes t*
*hat if Yffis
8.2. PRODUCTS IN THE TORSION MODEL. 95
*
* Q
a F-spectrum for each ff then the product in the category of F-spectra is EF+ ^*
* ffYff:
indeed, if T is a F-spectrum,
Y Y
[T; Yff]T = [T; EF+ ^ Yff]T:
ff ff
Part (i) of the following proposition is immediate once we note that the inclus*
*ion i :
torsOfF-mod-! OF-mod has right adjoint f : OF-mod -! torsOfF-modgiven by
taking the sub-module of elements annihilated by some Euler class.
PropositionQ8.2.1.IfQA is the category of F-finite torsion OF-modules, then
(i) QAffMff= f ffMffandQ
(ii) DAffMff= Rf ffMff.
Proof: It remains only to show that right derived functors respect composition.*
* For this
we need onlyQcheck that the relevant functors preserve injectives. This is a li*
*ttle delicate,
because (as a functor on families of F-finite torsion modules) is not a right*
* adjoint.
However, we saw in Section 8.1 that
Y Y
f = OE :
Q
Now OE is the product in the category of F-finite modules, and thusQa right a*
*djoint.QIt is
exact becauseQit is the composite of exact functors. Thus we have R(f ) = ROR(*
*OE ) =
R O (OE ), and the result follows. __|_|
Since objects of dgA are formal in this case, it is obviously useful to recor*
*d the homology
of the product.
Corollary 8.2.2. The homology of the product in the derived category is calcu*
*lated
by a split short exact sequence
Y Y Y __
0 -! R1f H*(-1Mff) -! H*( DAffMff) -! f H*(Mff) -! 0: |_|
ff ff
QThus, for example,Qif Mffis a dg object in even degrees, then the product has*
* model
f ffMff -1R1f ffMff.
Next consider the torsion model A = At. QWe may reflect that we already know *
*the
torsion part of the model for the product ffYff, and we almost know the vertex*
*. Indeed
Y Y
(Yff^ EF+) -! Yff
ff ff
is an F-equivalence, and therefore the torsion part of the product is given by *
*the product
of the torsion parts in the category of F-finite torsion modules. The vertex ca*
*n almost be
calculated from the exact sequence
Y Y Y
. .-.! ssT*(EF+ ^ Yff) -! ssT*( Yff) -! ssT*(E"F ^ Yff) -! . .:.
ff ff ff
However, the categorical procedure makes this a redundant prop to our confidenc*
*e.
96 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
Proposition 8.2.3. The product of objects Mff= (tF*Vff-! Tff) in the tors*
*ion model
Q F s
category isQ ffMff= (t* V -! T ) where
(i)T = f ffTff,
(ii) The graded vector space V and structure map are described by the pul*
*lback square
Q
V -! ffVff
# Q # Q
Hom(tF*; f ffTff)-! Hom(tF*; ffTff)
Q Q
where the lower horizontal is induced by the inclusion f ffTff-! ffTff. *
*The behaviour
on morphisms is as implied by the above categorical description.
Proof: It is convenient to replace the objects (tF* Vff-! Tff) by the corr*
*esponding ob-
jects (Vff-! Hom(tF*; Tff)). It is then immediate that, in theQcategory of*
* objectsQV - !
Hom(tF*; M) with M an arbitrary OF-module, the product is ( ffVff-! Hom(t*
*F*; ffTff)).
The fact that V is as claimed follows immediately from the universal prop*
*erty of the pull-
back. __|_|
As remarked above, this automatically gives a model for the product of T-*
*spectra. How-
ever we would like to know the homology of the product and, more exactly, *
*its vertex and
torsion part. We shall let V and T denote the functorsQgiving the vertex*
* and torsion
part of the product. For the torsion part T ffMff= DATff, and so we alre*
*ady have an
answer, provided by Corollary 8.2.2 above.
Now we turn to the vertex. We note that, as in Section 8.1, it may be use*
*ful to factorise
f = OE. We would then be concerned with pullback along the inclusions i a*
*nd j as
displayed in the following diagram:
Q *Q Q
i*j* ffVff -! j ffVff - ! ffVff
# # #
Q i Q j Q
Hom(tF*; OE ffTff)-! Hom(OE ffTff)-! Hom( ffTff):
The advantage is that if, forQexample,Qall torsion objects were supported *
*on the same finite
set of subgroups, then OE ffMff= ffMff. It is sometimes convenient to le*
*t k : fM -! M
denote the composite ji.
Again we work with the adjoint form of torsionQmodel objects. We learn f*
*rom the
torsion case, and begin with the composite OE . In the following display *
*we use the maps
Q i Q j Q
OE ffTff-! OE ffTff-! ffTff. We display the functor as a composite:
0 0 1 1 0 Q 1 0 Q 1
Vff Q j* ffVff i*j* ffVff
B@B@ # CA CAOE7-!B@ # C f B # C * *Y
Q A 7-! @ F Q A -! i j *
* Vff:
Hom(tF*; Tff)ff Hom(tF*; OE( ffTff)) Hom(t*; OE( ffTff)) ff
To understand derived functors, we must specify the categories concerned. *
* We consider
objects V -! Hom(tF*; M); the category of these with M arbitrary will be d*
*enoted Abt, the
category with M F-finite will be denoted A"t. We view these inclusions as *
*giving functors
Abt-j*!"Ati*-!At
8.3. THE TENSOR HOM ADJUNCTION. 97
We begin by noting that the functors i* and j* are both left exact. It is obvio*
*us that they
preserve monomorphisms. Next, notice that given V - ! Hom(tF*; M) we have v 2 *
*i*V
if and only if (v)(1) 2 OEM. This is because elements of tF*of positive degree *
*have finite
support. Similarly v 2 j*V if and only if (v)(1) 2 M. The half exactness then f*
*ollows
since if M N then OEM = M \ OEN and similarly M = M \ N. Since both OE and
are right adjoints, they both preserve injectives, and so R(i*j*) = Ri*Rj*. Als*
*o, both i*
and j* have the property that they are the identity on torsion free injectives.
It is now rather straightforward to calculate the right derived functors. If *
*tF* V -! M
is the torsion object and 0 -! M -! I -! J -! 0 is an injective resolution we o*
*btain
the following two exact sequences for the vertex of the right derived functors *
*of j*.
0 -! Hom(tF*; M) -! A -! Hom(tF*; M) -! Ext(tF*; M) -! V 2 -! 0
and
0 -! V 0 -! V Hom(tF*; I) -! A -! V 1 -! 0
There are precisely analogous exact sequences for either the right derived func*
*tors of i* and
of i*j*.
There is also a product in the standard model, and for use in our approach to*
* functionQ
spectra this will be more important. As in the cases we have treated here we fi*
*nd A=
^ Q^A, where ^ : ^A- ! A is right adjoint to the inclusion and Q DA = R^ R Q^A.*
* The
work is involved in making these descriptions more explicit. However the right *
*adjoint to
the inclusion functor i : A -! ^Ais much harder to describe in this case, so we*
* defer the
treatment until Section 8.6.
8.3. The tensor hom adjunction.
Once again we are led to consider an abstract categorical situation because i*
*t arises in
several algebraic ways. Thus we let A denote an abelian sub-category of an abel*
*ian category
^A. We shall treat four examples, all of which arise as algebraic models.
Example 8.3.1. A is the category of torsion k[c]-modules, for a field k and ^*
*Ais the
category of all k[c]-modules.
Example 8.3.2. A is the category A of k[c]-morphisms (N -s! t V ) which beco*
*me
an isomorphism when c is inverted, where t = k[c; c-1] and ^Ais the category of*
* all maps
(N -! t V ).
Example 8.3.3. A is the category of F-finite torsion OF-modules, ^Ais the cat*
*egory of
all F-finite OF-modules.
Example 8.3.4. A is the category A of objects (N -s! tF* V ) which become an
isomorphism when E is inverted, ^Ais the category of all such maps.
We shall illustrate the general discussion with the two single object example*
*s, and return
to the examples with objects of the form (N -! t V ) in the next section, sinc*
*e they are
much more substantial.
We begin by summarising the framework in which we operate. We shall show in *
*due
course that the above examples all fit in this framework. Firstly, suppose that*
* the inclusion
98 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
i : A -! ^Ais exact, and has a right adjoint ^ : ^A-! A. This assumption holds *
*in all
the examples above for general reasons, but we shall identify the adjoints expl*
*icitly below.
Next, we assume that for any object N of ^Athere is a tensor product functor N *
*: ^A-! ^A,
and that, if N is in A, this functor restricts to a functor on A. Finally, we s*
*uppose that
the tensor product functor is associative, and has a right adjoint Hom (N; .) *
*: ^A-! ^A; we
use bold face to emphasise that Hom (N; P ) is is an object of ^A, and not jus*
*t the abelian
group of homomorphisms. Of course, it then follows that if N is an object of A *
*the functor
N : A -! A has right adjoint IntHom (N; .) = ^ Hom (N; .). To summarise, we h*
*ave
the diagram of adjoint pairs of functors
i N
A AEA^AE ^A;
where the left adjoints are displayed at the top.
Example 8.3.1 continued: We see that the assumptions are satisfied. The right a*
*djoint to
inclusion is the c-power torsion functor: for an arbitrary k[c]-module ^Nwe hav*
*e ^N = cN.
Here Hom is the usual Hom functor, and the internal Hom functor on the catego*
*ry of
torsion k[c]-modules is
IntHom (N; P ) = cHom(N; P ): __|_|
The next example is very little different.
Example 8.3.3 continued: We see that the assumptions are satisfied. The right a*
*djoint to
inclusion is the E-torsion functor: for an arbitrary F-finite module ^Nwe have *
*^N = fN.
Here Hom is the F-finite part of the usual Hom functor:
Hom (N; P ) = OEHom(N; P );
and the internal Hom functor on the category of torsion F-finite OF-modules is
IntHom (N; P ) = f Hom(N; P ): __|_|
The point of factorizing the internal Hom functor is twofold. Firstly, as the *
*examples
illustrate, the functor Hom is much closer to the usual group of homomorphisms*
*, and thus
easier to identify. Much more important is the second reason: we wish to pass t*
*o derived
categories and retain the adjointness of functors.
To proceed further, we shall need the following condition to be satisfied: we *
*see below
that this covers our examples.
Condition 8.3.5. (i) The category A has enough injectives and has injective di*
*mension
1.
(ii) The category ^Ahas enough injectives and enough flat objects, and has both*
* injective
and flat dimension 1.
(iii) Either A has enough flat objects, or I M = 0 for all injectives I in A.
(iv) For any object C of A R1^(iC) = 0, and hence R^(iX) = X for any object X o*
*f DA.
We begin by stating the result and then discuss what is involved in making sen*
*se of it.
8.3. THE TENSOR HOM ADJUNCTION. 99
Proposition 8.3.6.Suppose given adjoint pairs of functors in the above framew*
*ork,
and suppose in addition that Condition 8.3.5 is satisfied. For objects M; N and*
* P of A we
have the natural isomorphism
[M L N; P ] = [M; R^ RHom (N; P )]:
The two points to notice here concern the way the composite functors have bee*
*n treated.
The first is that it is not immediately clear how to interpret M L N as an obje*
*ct of A,
since A may not have enough projective or flat objects. The second point is tha*
*t on the
right hand side
R^ RHom (N; P ) 6= RIntHom (N; P )
in general. This can happen because the functor Hom (N; .) does not preserve i*
*njectives,
as we saw in Example 7.2.1.
Proof: We now outline the proof of the proposition. It turns out that both pair*
*s l : D AE
E : r of adjoint functors do indeed pass to adjoint functors Ll : DD AE DE : Rr*
*. For
the pair i, e this is because i is exact, and therefore the functor on dg categ*
*ories preserves
both cofibrations and acyclic cofibrations. For the tensor-Hom pair the reason *
*is different.
First note that RHom exists since ^Ahas enough injectives. If ^Ahas enough pr*
*ojectives
we know that L exists, but it also suffices that ^Ahas enough flat objects. Now*
* consider
the functor of three variables
Hom(M N; P ) = Hom(M; Hom (N; P ))
to graded abelian groups. It follows from associativity of the tensor product *
*that, if M
is flat M preserves flat objects, and so the total right derived functor on the*
* left can
be calculated as a composite. Similarly, if P is injective Hom (.; P ) takes f*
*lat objects to
injectives so that the functor on the right can also be calculated as a composi*
*te. This gives
the equality
RHom(M L N; P ) = RHom(M; RHom (N; P ))
and the required equality comes by taking 0th homology. This gives
DA^(M L N; P ) = DA(M; R^ RHom (N; P )):
Finally if M; N and P are in A then
DA(M L N; P ) = DA^(M L N; P ):
Note first that the meaning of M L N on the left is unclear if A does not have *
*enough flat
objects. However, the tensor product functor is right exact and has only one le*
*ft derived
functor, the torsion product functor. In the cases that A does not have enough *
*flat objects
we have the equality
M L N = RTor(M; N)
in D^Aprovided M or N is in A, by Condition 8.3.5 (iii). Now the category A doe*
*s have
enough injectives, so the right hand side can be interpreted in DA. Furthermor*
*e in the
relevant cases, A-injective objects are also ^A-injective, so that RTor(M; N) h*
*as the same
meaning in ^Aand in A.
100 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
It therefore suffices to show that DA^(L; P ) = DA(L; P ) when L and P are in *
*A. For
this it suffices to note that it 0 -! P -! ^I-! ^J-! 0 is an injective resoluti*
*on of P
in ^Athen 0 -! P -! ^^I-! ^J^-! 0 is also an injective resolution. Since ^ is a*
* right
adjoint the terms are injective, and by Condition 8.3.5 (iv) the final map is s*
*urjective. __|_|
We end this section by completing the justification for the two single space e*
*xamples.
Example 8.3.1 continued. (The modifications for Example 8.3.3 are minimal.)
The short exact sequence
0 -! k[c] -! k[c; c-1] -! 2I -! 0
immediately gives the following.
Lemma 8.3.7. For any k[c]-module M there is an exact sequence
0 -! Tor(M; 2I) -! M -! M[c-1] -! M 2I -! 0;
and hence, for any torsion module T , we have T 2I = 0 and Tor(T; 2I) = T . _*
*_|_|
Note that this means in particular that we may calculate Tor*(S; T ) for torsi*
*on modules
S and T using an injective resolution of S. Indeed if 0 -! S -! I -! J -! 0 is*
* an
injective resoloution using sums of suspensions of I we obtain the exact sequen*
*ce
0 -! Tor(S; T ) -! Tor(I; T ) -! Tor(J; T ) -! S T -! 0:
Our problem is that there are not enough projectives amongst torsion modules, s*
*o that it is
not clear that the left derived torsion product exists. However the category of*
* k[c]-modules
is one dimensional, and thus the torsion product functor is left exact, so we m*
*ay use the
right derived torsion product instead.
Lemma 8.3.8. On the category of k[c]-modules, the left derived torsion product*
* is equiv-
alent to the suspension of the right derived torsion product on torsion modules*
*. More
precisely, if M or N is a torsion module
M L N = RTor(M; N):
Proof: It is enough to establish that k[c] L N = RTor(k[c]; N). Consider the e*
*xact
sequence
0 -! k[c] -! k[c; c-1] -! 2I -! 0:
Since the first two terms are flat, it gives the calculation
Tor(2I; N) = ker(N -! N[c-1]):
On the other hand, k[c; c-1] is both flat and injective, so that RTor(k[c; c-1]*
*; N) = 0.
Thus RTor(k[c]; N) = -1RTor(2I; N). Moreover, since I is injective RTor(2I; N)*
* =
Tor(2I; N). __|_|
8.4. HOM, TENSOR AND TORSION FUNCTORS IN STANDARD MODELS. 101
We write M " N for this functor, and refer to it as the left derived tensor p*
*roduct. If M
and N are torsion modules it should be interpreted as the suspension of the rig*
*ht derived
torsion product: by the lemma this is not ambiguous.
Lemma 8.3.9. The right adjoint of "B : D(torsk[c]) -! D(torsk[c]) is R0Hom(B;*
* .) :=
RcRHom(B; .).
Proof: The adjoint pair i : k[c] - mod o torsk[c] - mod : c passes to derived c*
*ategories
since i is exact. The adjoint pair B : k[c] - mod o k[c] - mod : Hom(B; .) pas*
*ses
to derived categories by use of projectives. This shows the required adjunction*
*, or more
properly, that
D(k[c])(A " B; C) = D(torsk[c])(A; RcRHom(B; C)):
The remaining point is to show that
D(torsk[c])(A " B; C) = D(k[c])(A " B; C)
when A; B and C are torsion. The point is that we may calculate A " B = RTor(A;*
* B) on
the right using torsion injectives. __|_|
8.4.Hom, tensor and torsion functors in standard models.
In this section we treat Examples 8.3.2 and 8.3.4, showing that they satisfy *
*the basic
assumptions of the previous section. In the following section we shall deal wit*
*h passage to
derived categories.
We have already considered the objects e(V ) = (t V -! t V ) and f(T ) = (T*
* -! 0)
which are left adjoints to the vertex and nub functors. We now need to consider*
* objects good
as domains. Inspired by topology, we consider spheres analogous to the the repr*
*esentation
spheres SnV with V the natural representation:
-n
Sn := (2nO c-! t):
Of course we also permit the notation Sm+n = m Sn . Because t is Q in each ev*
*en
dimension we must emphasize that degree 0 is special, and t is an abbreviation *
*of t Q.
-n
Thus m Sn = (m+2nO c-! t m Q).
fl
We begin by investigating the functor Sn corepresents, so let C = (P -! t *
*W ) be
some object of ^A.
Lemma 8.4.1.
Hom(Sn ; C) = fl-1(c-n -2nV ):
Proof: A map
2nO -! P
c-n # # fl
1OE
t -! t W
is obviously determined by (2n1). However it is subject to the constraint that *
*fl((2n1)) =
(1 OE)(c-n 1) = c-n OE(1). __|_|
102 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
Thus we can begin to reconstruct C from its subspaces P (c-n) = fl-1(c-n V );*
* this will
motivate the construction of of the functor ^ : ^A-! A below. In fact it is con*
*venient to
introduce a dummy variable O of degree 0, so we may write pOn for an element p *
*2 P if we
consider it as an element of P (cn). The action of c is then given by c.(pO-n) *
*= (cp)O-(n-1),
and this map is co-represented by the map c : S(n-1) -! Sn
2n+2O -c! 2nO
c-(n+1)# # c-n
t -1! t
The only elements of the nub P which lie in several distinct P (cn) are those l*
*ying in the
kernel of fl. Of course, if C is in A, the kernel of fl is c-power torsion, and*
* can be detected
by mapping out of objects (T -! 0), and the vertex of C can also be calculated *
*as follows.
L *
*n c
Lemma 8.4.2. For an arbitrary object C of A, we may recover tW as lim!( nP (c*
* ) -!
L n c 4L n c
2 nP (c ) -! nP (c ) -! . .).. Furthermore, the vertex W and the map W =
1 W -! t W may be recovered as W = im(P (c0) -! lim!(2nP (cn))). __|_|
The advantage of the category ^Ais that it has certain extra objects. It is ob*
*vious that
the object ^E= (0 -! t) co-represents the vertex functor. To co-represent the *
*nub one
would want to consider the dual t0= Hom(t; k) and an object of the form (O -! t*
*t0) with
1 7-! cn cn; however the sum is infinite and so this putative object must be r*
*eplaced
by the spheres Sn of which it would be a limit.
We may now describe the functors that concern us. The right adjoint ^ to the i*
*nclusion
of categories i : A -! ^Acan be described using direct limits as follows. For a*
*ny object C
in ^Awe may consider the category of objects x : T -! C over C with T in A. Res*
*tricting
to the sub-category of objects with cardinality less than card(t)card(C) so as *
*to obtain a
category with small skeleton, we may take the direct limit ^(C) = lim!T . The f*
*ollowing
description is more helpful.
fl 0 fl0 0
Lemma 8.4.3. If C = (P -! t W ) then ^(C) = (P -! t W ) may be described
L n
as follows. The nub of ^C is P 0= n2ZP (c )=K where K is the submodule genera*
*ted by
differences p(Om -On) with p being c-power torsion. The vertex of ^C is W 0= im*
*(P (c0) -!
lim!(2nP (cn)). The natural map t W 0-! lim(2nP (cn)) is an isomorphism and the
n ! n
structure map is the natural map.
Proof: We may see that this definition is forced since for A in A we have Hom(A*
*; C) =
Hom(A; ^C). Taking A = Sn we see that (P 0)(cn) = P (cn). We then find t W *
*~=
P 0[c-1] = lim!2nP 0(cn).
We write pOn for an element of P 0represented by p 2 P with fl(p) 2 cnW . The *
*elements
of W 0are thus represented by elements of form, pO0 and the map tW 0-! lim!(2nP*
* (cn))
n
8.4. HOM, TENSOR AND TORSION FUNCTORS IN STANDARD MODELS. 103
is given by cn pO0 7-! cnpOn. This is obviously surjective; to see it is injec*
*tive it suffices
to restrict to c0 W 0 t W 0, and this is clearly injective.
If C is in A then ^C = C, and we may take the unit of the adjunction to be th*
*e identity.
The counit i^C -! C of the adjunction is given on nubs by the inclusions P (cn)*
* -! P ,
and this extends uniquely over t W 0since t W is c-local. The extension takes*
* 1 W 0
into 1 W by construction. Note that ^i^ = ^ and i^i = i; the triangular identi*
*ties are
easily checked. __|_|
Example 8.4.4. Although we have chosen the notation ^ to suggest the analogy *
*with
the torsion functor, the functor can behave in quite different ways. Thus it an*
*nihilates the
object ^Eco-representing the vertex ^(0 -! t) = 0. On the other hand,
i ___j
^(M -! 0) = M[O; O-1]=(cM . (Om - On)|m; n 2 Z) -! t M;
___ __
where M = M=cM. |_|
Note also that the functor ^ does admit a description analagous to the descri*
*ption
lim!Hom(A=In; .) of the I-power torsion functor of an ideal I in a commutative *
*ring A.
n
Unfortunately it is not given by an internal Hom, but rather by taking the dire*
*ct limit in nub
and torsion parts. More precisely, we consider the inverse system and . .-.! f(*
*O=c3) -!
f(O=c2) -! f(O=c), and adjoin to the diagram the maps Sk -! 2kf(O=cl) for all k
and l. Note that the spheres Sk have no incoming arrows. Thus if we apply Hom(*
*.; C)
and take direct limits we obtain the nub of ^C. For the vertex we have the inve*
*rse system
. .-.! S-2 -! S- -! S0, which admits a comparison map to the nub system; appl*
*ying
Hom(.; C) and taking direct limits we obtain the vertex part of ^C.
For the tensor Hom adjunction we need objects A = (M -! t U), B = (N -! t V*
* )
and C = (P -! t W ). The tensor product is the obvious one: A B = (M N -!
(t U) (t V ) = t (U V )). This clearly preserves A.
Next, we describe the internal Hom functor in ^A. In fact we define an object*
* Hom (B; C) =
(Q -! t H), and then show Hom (B; .) gives the right adjoint to B. Of course*
* we
identify H and Q by assuming the defining adjunction
Hom(A B; C) = Hom(A; Hom (B; C)):
Taking A = E^ = (0 -! t) we see H = Hom(V; W ). Taking A = Sn we see that
Q(c-n) = 2nHom(Sn B; C). Furthermore, the action of c on Q is corepresented *
*by
c : S(n-1) -! Sn . The structure map is given by the map Q(c-n) = 2nHom(Sn
B; C) -! cn Hom(V; W ) taking a homomorphism Sn B -! C to its vertex part. We
could summarise all this by saying that Hom (B; C) is obtained by applying Hom(*
*.B; C)
104 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
to the diagram
. . .-! S- -! S0 -! S - ! . . .
- " %
E^
We may therefore describe the unit j : A -! Hom (B; AB) and counit ffl : Hom (*
*B; C)
B -! C. On vertices we simply use the familiar maps U -! Hom(V; U V ) and eval*
*ua-
tion Hom(V; W ) V - ! W . On nubs we may consider the part over c-n where we h*
*ave
Hom(Sn ; A) -! Hom(Sn B; A B) obtained by tensoring with B. For the counit
we consider the part arising from i + j = n, giving Hom(Si B; C) Hom(Sj ; B) *
*-!
Hom(Sn ; C), and since Si Sj ~= Sn we may use composition. It is readily verif*
*ied that
the nub and vertex components are compatible.
Lemma 8.4.5. The unit j : A -! Hom (B; A B) and counit
ffl : Hom (B; C) B -! C
just described, show that Hom (B; .) is right adjoint to B.
Proof: The triangular identities on vertices are the familiar ones, and on nubs*
* they are
easily checked. __|_|
This completes the account of the tensor product, Hom and ^ functors in Exampl*
*e 8.3.2.
In the next section we will discuss passage to derived categories.
We now turn to the full standard model, and a certain amount of extra work is *
*required.
As in the simpler model, we begin with objects which are good as domains. Inspi*
*red by
topology we should perhaps expect these objects to be analogous to spheres, and*
* we have
chosen the notation to emphasize this analogy.
Recall from Section 8.1 that if V T= 0, there is an associated Euler class O(V*
* ). Alge-
braically the analogue of the representation V is the dimension function v : F *
*-! Z0 ,
given by v(H) = dimC(V H), which is only positive on the finitely many subgroup*
*s fixing a
non-zero vector in V .
Now let v : F -! Z be an arbitrary function. Consider elements of tF*in each d*
*egree as
functions on F, and take the sub-OF-module OF(v) consisting in degree 2n of fun*
*ctions
on F which are only non-zero at H if n v(H). For any such a dimension function*
* we
consider the object
Sv = (OF(v) -! tF*)
of ^A, where the map is inclusion. Note that, if v has finite support, the map *
*becomes an
isomorphism when E is inverted, and hence Sv is an object of A. We extend the n*
*otation
to include suspensions in the obvious way: Sn+v = nSv. Note that if is the dim*
*ension
function of the natural representation (restricted to the trivial subgroup) thi*
*s notation is
consistent with the notation used for the semifree case. Note also that if cv i*
*s cv(H)Hover H
and we view OF as a submodule of tF*, then OF(v) = c-vOF.
8.5. HOM, TENSOR AND TORSION FUNCTORS ON DERIVED CATEGORIES. 105
Lemma 8.4.6. For any representation V with V T= 0, and dimension function v(H*
*) =
dimC(V H) the algebraic object Sv models the topological stable sphere SV . __*
*|_|
In general, it is quite hard to describe the functor Sv corepresents, but the*
* idea is that
Hom(Sv; C) is specified by elements of the nub of C which are cv-divisions of e*
*lements of
fl-1(c0 W ). For example S0 co-represents fl-1(c0 W ). The other essential case*
* is when
v is constant at n > 0 on a finite set OE of subgroups and zero on its compleme*
*nt, so that
cv = cnOE= eOEcn + (1 - eOE). In this case
Hom(Sv; C) = fl-1(c-nOE -2nW ) x fl-1(c0F\OE W )):
This suffices to find P (o) = fl-1(o W ) for any homogeneous o 2 tF*. Indeed *
*if o is a
monomial of positive degree it is cnOEfor some n, and hence P (cnOE) = eOE2nHom*
*(Sv; C). On
the other hand if o is of negative degree P (o) = oP (c0) = oHom(S0; C).
8.5.Hom, tensor and torsion functors on derived categories.
We now take the functors on A and ^Aand show that they pass to derived catego*
*ries
where they continue to give adjoint pairs. Condition 8.3.5 explains what is nec*
*essary.
For the i-^ adjunction this is immediate, since i is an exact functor. Thus i*
*t preserves
cofibrations, and acyclic cofibrations so that i = Li. However for the tensor-H*
*om adjunction
a little more comment is necessary. In one respect the situation is simpler tha*
*n in the single
object examples.
Lemma 8.5.1. The category A has enough flat objects, and it is of flat dimens*
*ion 1.
Proof: It is obvious that Sn is flat for all n. By the description of Hom(Sn *
*; B), it is
easy to construct a map from a sum of spheres to B which is surjective on nubs.*
* It is then
necessarily surjective on vertices since inverting c is exact. This shows there*
* are enough
flat objects.
Next, if is a sum of spheres, any subobject R is flat. Indeed, the nub of R *
*is a subobject
of a free k[c]-module and hence free, and the vertex is necessarily flat. __|_|
Obviously spheres remain flat in ^A. Furthermore the object ^Eco-representing*
* the vertex
is flat, so there are also enough flat objects in ^A, which is also of flat dim*
*ension 1.
Lemma 8.5.2. For B 2 A the left derived tensor product functor B : DA -! DA
exists, and may be calculated by flat resolutions. The analogous statement is t*
*rue for the
category ^A.
Proof: In the absence of projective objects, we must say something to establish*
* the exis-
tence of the left derived tensor product. We have seen that any epimorphism F *
*- ! A
with F flat gives a flat resolution 0 - ! R - ! F -! A - ! 0, and if F 0-! A
is a second, we may compare the two using the sum F F 0-! A. This shows that
cofibre(R B - ! F B) gives a well defined functor on the derived category. T*
*he
rest of the proof that this is the total left derived functor proceeds as with *
*cofibrant ap-
proximations: the essential point is that tensor product with a flat object pre*
*serves weak
106 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
equivalences. __|_|
We continue the tradition of writing " for this functor. It follows directly *
*as in the
general discussion that the tensor-Hom adjunction passes to derived categories.*
* There is
one last point.
Lemma 8.5.3. For objects A; B and C of A we have
DA(A " B; C) = DA^(A " B; C)
Proof: In the semifree case this is immediate since, on the one hand, the notat*
*ion A " B
is unambiguous, and, on the other hand, i does preserve injectives. __|_|
Now in the general case, Example 8.3.4, we begin by remarking that Sv is flat *
*when v
takes finitely many values. Indeed the module OF(v) is projective as a OF-modul*
*e, since
it is a finite sum of suspensions of eOEOF where OE F is a set on which v is c*
*onstant.
Corollary 8.5.4. There are enough flat objects in the standard model A. __|_|
Lemma 8.5.5. The standard model A has flat dimension 1.
Proof: Since the category has injective dimension 1 it suffices to show that th*
*e standard
injectives have flat dimension 1. The injectives e(V ), are themselves flat. *
*Observe next
that f(T ) is flat if T is flat as a OF-module, and so a flat resolution of T g*
*ives one of
f(T ). __|_|
However the last point needs a little more care than in previous cases.
Lemma 8.5.6. For objects A; B and C of the standard model A,
DA(A " B; C) = DA^(A " B; C)
Proof: Firstly the object A " B is again unambiguous and lies in DA, since ther*
*e are
enough flat objects in A , and they remain flat in A^t. However injectives in *
*A are not
necessarily injective in A^t: it suffices to prove the following lemma instead.
Lemma 8.5.7. If C is an object of A then R1^(C) = 0, and hence for any object *
*X of
DA , we have R^ iX = X.
Proof: For an object C of A , take an injective resolution 0 -! C -! ^I-! ^J-! *
*0 in
^At. We obtain an exact sequence 0 -! C -! ^^I-! ^J^, and we claim it is an inj*
*ective
resolution in A . The injectivity of ^^Iand ^J^is immediate since ^ is a right *
*adjoint. It
remains to see that the last map is surjective. To see this, take an injective *
*resolution of C
in A . This will involve objects e(V ) and f(I). Those of the first type remain*
* injective in
^At, but those of the second have L HI(H) replaced by QH I(H). However this is *
*sufficient
to give an injective resolution in A^t, and applying ^ we recover the original *
*resolution in
A. __|_|
8.6. PRODUCTS IN THE STANDARD MODEL. 107
__
|_|
This completes the proof that the left derived tensor product has a right adj*
*oint.
We conclude by continuing Example 8.4.4.
Example 8.5.8. We evaluate R^f(M) for an arbitrary OF-module M. Indeed, we co*
*n-
sider the fibre sequence M -! E-1M -! E-1M=M, and apply R^f(.). Since E-1M is
torsion free and E-local we have ^f(E-1M) = e(E-1M). Furthermore, an E-local mo*
*dule
admits an injective resolution by E-local injectives, so R^ = ^ on the category*
* of ob-
jects f(N) with N E -local. Finally, since E-1M=M is E-torsion it is in A and*
* we find
R^f(E-1M=M) = ^f(E-1M=M) = f(E-1M=M) by 8.5.7. Thus we have a fibre sequence
R^f(M) -! e(E-1M) -! f(E-1M=M). __|_|
8.6. Products in the standard model.
The main purpose of this short section is to record the product in the standa*
*rd model and
its derived category. It completes the work of Section 8.2, which calculated th*
*e products
for other standard examples and the torsion model. As in that case, the product*
* in DA
automatically models the product of T-spectra, since both are adjoint to their *
*respective
diagonal functors which obviously correspond.
The present section is almost a special example ofQthe Hom functor,Wsince of *
*course
products of a single object can be constructed as iC = Hom ( iS0; C). Nonethe*
*less the
product functor will have special significance because it is the only case of t*
*he function
spectrum for which we have a model as a functor. This will be essential in iden*
*tifying the
model of the function spectrum on objects.
The method of the present section is the same as that of Section 8.2, and exp*
*loited also for
Hom in Section 8.3. By definition the product is the right adjoint to the diago*
*nal functor.
*
* Q ^A
As in the case of the Hom functor it is convenient to identify the product func*
*tor in
^Aand then note that ^ Q^Ais the product in A. However, this time there is no p*
*roblem
whatever in identifying the maps in the derived category, since the diagonal fu*
*nctor is
Q^A
exact. Thus the productQon DA is R^ R . To avoid confusion we shall only us*
*e the
undecorated symbol for a functor which is the product on underlying sets.
Proposition 8.6.1.Example 8.3.1 revisited. The product of objects Miin the de*
*rived
category of torsion k[c]-modules is
Y __
Rc Mi: |_|
i
The modifications for the second single-object model are straightforward, and*
* were made
in 8.2.1.
The two-object models are a little more complicated. Let us begin with the c*
*ase of
k[c]-modules, with ^Abeing the category of k[c]-morphisms N -! t V . As befor*
*e we
swiftly calculate the product in the category ^A. Thus if Ci= (Pi- ! t Wi), an*
*d if we
108 8. BASIC ALGEBRA FOR MODELS AND THEIR DERIVED CATEGORIES.
QA^ Q
assume i Ci= (Q -! t H) then weQapply Hom(E^; .) to deduce H = iWi. Applying
Hom(Sn ; .)Qwe deduceQQ(c-n) = iPi(c-n). We give a more direct construction. W*
*e let
ffi : t iWi- ! i(t Wi) denote the diagonal, and we let ffi* be the pullback*
* functor:
Q
. -! iCi
# #
Q ffi Q
t iWi -! i(t Wi)
QA^ *Q
Lemma 8.6.2. The product on the category ^Ais = ffi .
Proof: Immediate from the universal property of pullbacks. __|_|
Q *
* *Q
Proposition 8.6.3. The product on the semifree category DA is R(^ffi*) = R^R*
*ffi .
Proof: Since the product of sets is exact, the only point at issue is that ffi**
* preserves in-
jectives. This is immediate since the injectives are either of the form (N -! N*
*), which ffi*
takes to a map of the same form, or of form (I -! 0), on which ffi* is the iden*
*tity. __|_|
All is just the same for the full standard model.
QA^ * Q __
Lemma 8.6.4. The product on the category ^Ais = ffi OE . |_|
Proposition 8.6.5. TheQproduct on theQderived category of the standard model c*
*ate-
gory DA is R(^ffi*) = R^Rffi*OE . __|_|
CHAPTER 9
Products, smash products and function T-spectra.
This chapter begins with a section fundamental to the structure of the derived *
*category of
the standard model category: it gives factorizations for all maps between basic*
* injectives.
This is then applied in Section 9.2 to show that the smash product is modelled *
*by the left
derived tensor product, and in Section 9.3 to show the function spectrum is mod*
*elled by
the right adjoint of the left derived tensor product.
9.1. Maps between injective spectra.
We have shown that every T-spectrum Y admits an Adams resolution
ffi(Y )
Y -! I(Y ) -! J(Y );
W W
and furthermore I(Y ) = ffIffand J(Y ) = fiJfiwhere the spectra Iffand Jfiare*
* sus-
pensions of the standard spectra E for various H 2 F and E"F. Furthermore, *
*no
F-contractible summands are requiredWin J(Y ). Thus the map ffi(Y ) is describ*
*ed by its
components ffiff: Iff-! J(Y ) = fiJfi. Each such component is thus a map E *
*-! J
or "EF -! J for an F-free injective J. It is the purpose of the present section*
* to show
that such maps can be described as composites of inclusions of direct summands *
*and a few
maps of very special form.
Note first that [E; E]T*= Q[cH ], so that any graded self-map of E i*
*s a rational
mulitple of cn : E -! 2nE for some n 0. On the other hand [E"F; EF+]T*= *
*tF*;
in degree 0 and negative degrees, the maps factor through the quotient map E"F *
*- !
EF+, and are thus described by a map EF+ -! 2n+1EF+ for some n 0. However
in positive degrees each element has a finite support. We thus choose a subgro*
*up H
containing all the supporting subgroups, and the element factors through the co*
*mposite
E"F ' "EF ^S-nV (H)-! EF+ ^S-nV (H). Here EF+ ^S-nV (H)is a wedge of suspensions
-2vKE , where v(K) = n if K H and v(K) = 0 otherwise.
Lemma 9.1.1. If J is a wedge of suspensions of spectra E for various H then
(i) Any map E -! J factors as
E -! 2nE -! J
109
110 9. PRODUCTS, SMASH PRODUCTS AND FUNCTION T-SPECTRA.
for some n, where the first map is a scalar multiple of cn and the second is th*
*e inclusion of
a direct summand.
(ii) Any map "EF -! J factors as
_ _ 0
"EF -! EF+ ^ S-nV (H)' -2v(K)+1E- ! -2v (K)+1E-! J
K K
0(K)
where the0first map is as described above, the second has components cv(K)-v *
*: E -!
2v(K)-2v (K)E and the third is the inclusion of a direct summand.
Proof: (i) First note that the map is described by its effect in homotopy. Next*
* note that
the image in homotopy is a quotient of I(H) = ssT*(E), and hence is of the f*
*orm 2nI(H).
This is injective, and so the inclusion of the image is split. Since all object*
*s concerned are
injective we may realise the splitting geometrically.
(ii) The map is described by a homomorphism : tF*-! ssT*(J). Of course the im*
*age of
1 is torsion, and hence cn(1) = 0 for some n. Hence factors through one of the*
* maps
"EF -! -2n+1EF+ described above; this is clear algebraically, and thanks to inj*
*ectivity
may be realized geometrically. The resulting map -2n+1EF+ -! J is described by *
*its
components, each of which can be described as in Part (i). __|_|
9.2. Models of smash products.
In this section we show that the left derived tensor product as discussed in S*
*ections 8.4
and 8.5 is indeed the model of the smash product, in the category of free, semi*
*free, almost
free or arbitrary T-spectra, as appropriate. Unfortunately, we are only able to*
* describe the
model on objects; neither the smash product nor its adjoint preserve pure parit*
*y objects
in general, so one cannot get a firm grip on morphisms. To resolve this difficu*
*lty in the
present framework, one would need a natural splitting into even and odd pieces.*
* A more
satisfactory resolution would be to prove the equivalence of homotopy categorie*
*s from a
string of equivalences at the level of model categories, following the example *
*of Quillen [22].
The first observation is that the smash product preserves F-spectra and F-cont*
*ractible
spectra. More precisely X^Y ^EF+ ' (X^EF+)^(Y ^EF+) and T(X^Y ) ' TX^TY .
Since T(X ^ Y ) = T(X) ^ T(Y ), it is immediate that
ss*(T(X ^ Y )) = ss*(TX) ss*(TY );
and we shall show that ssT*(EF+ ^X ^Y ) is the left derived tensor product of s*
*sT*(EF+ ^X)
and ssT*(EF+ ^Y ). This might suggest the use of the torsion model, but we warn*
* that there
is a suspension which makes the gluing map inaccessible.
The connection with topology arises from the fact that I " M = M for any F-fin*
*ite
torsion module M, where " denotes the left derived torsion product functor. Ind*
*eed, we
have E ^ Y = Y , and hence
ssT*(E ^ Y ) = Tor(ssT*(E); ssT*(Y )) = ssT*(E) " ssT*(Y ):
We may now prove the special case of the theorem in which both spectra are F-s*
*pectra.
9.2. MODELS OF SMASH PRODUCTS. 111
Proposition 9.2.1.For F-spectra X and Y , their smash product is modelled by *
*the
dg OF-module ssT*(X) " ssT*(Y ). In particular there is a short exact sequence
0 -! ssT*(X) ssT*(Y ) -! ssT*(X ^ Y ) -! Tor(ssT*(X); ssT*(Y )) -! 0:
Proof: We have observed that the case when Y is injective is immediate. For th*
*e gen-
eral case use an Adams resolution Y - ! I(Y ) -! J(Y ), realising ssT*(Y ) -! I*
* -d! J
in homotopy, and obtain a cofibre sequence X ^ Y -! X ^ I(Y ) -! X ^ J(Y ). T*
*he
model of X ^ Y is thus the fibre of (S " I - ! S " J) = (Tor(S; I) -! Tor(S; J*
*)).
Providing we identify the map as 1 " d = Tor(1; d) the proof is complete. Howev*
*er both
I(Y ) and J(Y ) are wedges of suspensions of E, so it suffices to show for a*
*n arbitrary
map f : nE -! E that the map 1 ^ f : nX ^ E -! X ^ E induces
1 " p(f). But we have classified all such maps f: they are multiplications by c*
*mHfor some
m. Identifying X ^ E with X, the map 1 ^ f is identified with cmHby definiti*
*on of the
Q[cH ]-module structure. __|_|
Theorem 9.2.2. For arbitrary rational T-spectra X and Y , their smash product*
* is mod-
elled by ssA*(X) " ssA*(Y ). In particular there is a short exact sequence
0 -! ssA*(X) ssA*(Y ) -! ssA*(X ^ Y ) -! Tor(ssA*(X); ssA*(Y )) -! 0:
Proof: We have observed that the case when Y is injective is immediate. For the*
* general
case use an Adams resolution Y -! I(Y ) -! J(Y ), realising ssA*(Y ) -! I - d!*
*J in
homotopy, and obtain a cofibre sequence X ^ Y -! X ^ I(Y ) -! X ^ J(Y ). The mo*
*del
of X ^ Y is thus the fibre of (ssA*(X) " I -! ssA*(X) " J). It remains to ident*
*ify the map
as 1 " d.
However I(Y ) is a wedge of suspensions of spectra E, so it suffices to de*
*al with one
factor ffi : E -! J(Y ) where J(Y ) is a F-free injective. In Section 9.1 we*
* identified
such maps as composites of certain standard maps and retractions. The smash pr*
*oduct
and its model respect retracts, so it suffices to check that the standard maps *
*are accurately
modelled. These are of two types. The first type are the maps cn : E -! 2nE<*
*H>,
which is correctly modelled since the model is built with OF-modules. The secon*
*d type are
the maps g : "EF -! EF+ ^S-nV (H), which is the cofibre of the map f : S-nV (H)*
*-! "EF
which is the identity on geometric fixed points. Of course the codomain of f is*
* injective,
so f is determined by its induced map. To describe the model let v be the funct*
*ion with
v(K) = n when K H, and v(K) = 0 otherwise, and let OF(-v) be the submodule of
tF*, which in degree m consists of the product of the factors with m -v(K). Th*
*us g is
modelled by the map depicted thus
tF* -! tF*=OF(-v)
# #
tF* -! 0:
In other words, this is the cokernel of the inclusion OE : S-v -! e(Q), in the *
*notation of
Section 8.3. The modelling process preserves cofibre sequences, so it suffices *
*to show that
1 ^ OE : ssA*(X) " S-v -! ssA*(X) " e(Q) models 1 ^ f : X ^ S-nV (H)-! X ^ "EF.*
* It suffices
112 9. PRODUCTS, SMASH PRODUCTS AND FUNCTION T-SPECTRA.
to deal with the case when X is even. However in this case X ^ S-nV (H)and X ^ *
*"EF are
also both even, so the map 1 ^ f is determined by its induced map. Furthermore,*
* since
X ^ "EF is F-contractible, this is determined by its effect on vertices, which *
*is the identity.
This is faithfully modelled by 1 " OE. __|_|
9.3.Models of function spectra.
The preceding sections we have identified the candidate for an algebraic model*
* of function
spectra, and established various formal properties. In this section we establi*
*sh that the
function spectrum is modelled by the right adjoint R0Hom to the left derived t*
*ensor product
in all four of our examples.
Theorem 9.3.1. If Y and Z are rational T-spectra modelled by B and C in the st*
*andard
model, then F (Y; Z) is modelled by R^ RHom (B; C).
The analogous results hold for free, almost free and semi-free spectra. We sha*
*ll give the
proof in the free case by way of motivating the proof of 9.3.1. The other cases*
* are similar.
Proposition 9.3.2. If Y and Z are free spectra modelled by torsion Q[c]-modul*
*es S
and T then the internal function spectrum F (Y; Z) ^ ET+ is modelled by the dg *
*torsion
module RcRHom(S; T ).
Proof: We note that this result could be proved from the general case using a l*
*ittle algebra,
since the theorems on function spectra and smash products assert that F (Y; Z) *
*^ ET+ is
modelled by R^ RHom (f(S); f(T )) " I. Nonetheless it seems worth giving a m*
*ore
direct proof, by way of preparation for the case of arbitrary spectra. Special *
*cases of the
proposition admit simpler proofs, but the rather indirect approach does appear *
*necessary.
The above discussion showed that we ought to concentrate on the case when S pre*
*served
cofibrations. Whilst this only happens in degenerate cases, the torsion produc*
*t functor
preserves cofibrations when S is injective.
This suggests that it is natural to identify F (Y; Z) ^ ET+ as the cofibre of *
*F (I(Y ); Z) ^
ET+ - F (J(Y ); Z) ^ ET+ as might otherwise seem rather unnatural: we refer to*
* this as
the Worms method. In our case we know that both I(Y ) and I(Z) are wedges of su*
*spensions
of ET+, and so F (I(Y ); Z) ^ ET+ and F (J(Y ); Z) ^ ET+ are internal products *
*of suspen-
sions of spectra F (ET+; Z)^ET+ ' Z ^ET+. Since we have modelled products, it s*
*uffices
to identify the map between the products. However the map I(Y ) -! J(Y ) is spe*
*cified
by its components, and any map ET+ -! J(Z) factors as ET+ -! 2nET+ -! J(Z)
for some n where the first map is multiplication by cn and the second is the in*
*clusion of a
direct summand. The remaining verifications will be treated in more detail in t*
*he proof of
the general case. __|_|
However this gives very little indication of the general answer. The vertices *
*are especially
hard to describe since they involve both torsion and torsion free parts. For ex*
*ample if X
and Y are both F-spectra we may calculate [X; Y ]T*using the Adams short exact *
*sequence,
and F (X; Y ) ^ "EF has homotopy groups E-1[X; Y ]T*. Roughly speaking this su*
*ggests
9.3. MODELS OF FUNCTION SPECTRA. 113
a contribution E-1RHom(S; T ) from the torsion parts of a general spectrum. It*
* is not
immediately clear how this should be mixed with the vertex contribution Hom(V; *
*W ).
Remark 9.3.3. We have proved that if Y and Z are F-spectra modelled by F-fin*
*ite
torsion modules, the vertex of F (Y; Z) is modelled by E-1RHom(S; T ). We clai*
*m that
it is also modelled by RHom(tF*; RfRHom(S; T )). This is a manifestation of War*
*wick
duality.
To be precise, if E is a multiplicatively closed set in a commutative ring A *
*we define the
chain complex K(E) by the fibre sequence
K(E) -! A -! E-1A:
Denoting projective approximation by the letter P, Warwick duality is the state*
*ment that
there is a quasi-isomorphism
Hom(P E-1A; K(E) X) ' Hom(P K(E); X) E-1A:
The derived analogue of this is
RHom(tF*; RfX) ' E-1RHom(I; X):
The required equivalence thus follows provided X = Hom(S; T ) is complete in th*
*e sense
that the natural map
X = RHom(OF; X) -! RHom(I; X)
is an equivalence. However for X = Hom(S; T ) we may use the tensor Hom adjunct*
*ion to
establish completeness:
RHom(OF; RHom(S; T )) -! RHom(I; RHom(S; T )) -'! RHom(I"S; T ) = RHom(S; T ): *
* __|_|
The obvious conjecture is that the function spectrum corresponds to the inter*
*nal Hom
functor before passage to homotopy, and hence that in the homotopy category the*
* function
spectrum is modelled by something closely related to RHom. The case of F-spectr*
*a shows
that the model will not be RHom itself, but should be recognized as the right a*
*djoint
of the total left derived tensor product. Of course, this would follow immedia*
*tely if the
identification of the smash product with the total left derived functor of tens*
*or product was
functorial. Nonetheless, we have identified the right adjoint R0Hom of the tota*
*l left derived
functor of tensor product. It is a homotopy functor that preserves cofibre sequ*
*ences, and
it suffices to show that the function spectrum is object-accessible and is mode*
*lled by this
functor on generators.
Proof of Theorem 9.3.1: One might expect the proof to use the cofibration F (Y;*
* Z) -!
F (Y; I(Z)) -! F (Y; J(Z)), where Z -! I(Z) -! J(Z) is an Adams resolution. How*
*ever
it is not clear how to eliminate indeterminacy in the description of the map F *
*(Y; I(Z)) -!
F (Y; J(Z)). Instead, we therefore use our analysis of products to apply the Wo*
*rms method
as follows.
In the usual way, we are going to prove that the functor F (Y; Z) is object-a*
*ccessible, by
constructing it from particular cases F (Y 0; Z0), by taking mapping cones of m*
*aps which can
be identified with no indeterminacy at any stage. Of course the algebraic model*
*, R0Hom is
114 9. PRODUCTS, SMASH PRODUCTS AND FUNCTION T-SPECTRA.
a functor on the homotopy category and preserves cofibres, so it suffices to sh*
*ow that it has
the correct value on the building blocks F (X0; Y 0). We begin by outlining the*
* argument.
Step 1: For an arbitrary Z we give a description of F (E"F; Z) and F (E; Z) *
*for all
H. This is reasonably straightforward. The description is not claimed to be nat*
*ural in Z;
however, it is automatically natural in the first variable, since all maps betw*
*een spectra
of the form "EF or E are essentially given by multiplication by powers of c *
*or Euler
classes as we saw in Section 9.1.
Step 2: W Consider an AdamsWresolution Y -! I(Y ) - ! J(Y ) of Y . Thus both
I(Y ) = ffIffand J(Y ) = fiJfiare wedges of suspensions of E"F and E for*
* vari-
ous H. Furthermore J(Y ) does not involve any copies of "EF.
StepQ3: Using the description of productsQwe can describe the models of F (I(Y *
*); Z) =
ffF (Iff; Z) and F (J(Y ); Z) = fiF (Jfi; Z). We therefore have a cofibre seq*
*uence
Y ffi*Y
F (Y; Z) - F (Iff; Z) - F (Jfi; Z):
ff fi
Step 4: Describe the map ffi* between the products. For this, note that the m*
*ap ffi :
I(Y ) -! J(Y ) is categorically described by its components ffiff: Iff-! J(Y ),*
* so the map
of function spectra is also described by its components F (ffiff; Z). It there*
*fore suffices to
describe these.
Step 5: Since Iffis either a suspension of "EF or of E for some H, we have s*
*hown
in Section 9.1 that any map into a spectrum of the form J(Y ) can be described *
*as the
composite of a very explicit map followed by the inclusion of a direct summand.*
* The de-
scription in Step 1 is natural for these maps, and the product description is c*
*ategorical.
The point of this analysis is that it can be carried out in parallel in the al*
*gebraic and
topological categories, and the primitive pieces and constructions necessarily *
*correspond.
We proceed with Step 1.
Lemma 9.3.4. The algebraic model of F (E"F; Z) is correctly described by the t*
*heorem.
Consequently, if the model of Z is C = (P -! tF*W ) with torsion part T then th*
*e model
of F (E"F; Z) is torsion free and the vertex is the homology of the fibre of a *
*certain map
W -! RHom(tF*; T );
whose composite with the projection to Hom(tF*; T ) is adjoint to the homomorph*
*ism tF*
W -! T .
Proof: We take the model e(Q) of E"F, and establish the lemma in turn for Z bei*
*ng
F-contractible, F-free and arbitrary.
If Z is F-contractible F (E"F; Z) ' Z. On the other hand C = e(W ), which is a*
*lready
injective, so that RHom (e(Q); C) = Hom (e(Q); C) = e(W ). Since this too is*
* injective
R^e(W ) = ^e(W ), and e(W ) is already in the standard model, so ^ does not alt*
*er it.
9.3. MODELS OF FUNCTION SPECTRA. 115
If Z is an F-spectrum in even degrees then F (E"F; Z) is F-contractible, and *
*has ho-
motopy Hom(tF*; T ) in even degrees and Ext(tF*; T ) in odd degrees. On the oth*
*er hand
C = f(T ), and we may choose an injective resolution f(T ) -! f(I) -! f(J). Thus
RHom (e(Q); f(T )) = fibre (Hom (e(Q); f(I)) -! Hom (e(Q); f(J))):
Since tF*is flat, the two terms are both injective, so that
R^RHom (e(Q); f(T )) = fibre(^Hom (e(Q); f(I)) -! ^Hom (e(Q); f(J))):
Of course Hom (e(Q); f(I)) has zero vertex, and Hom(tF*; I) is torsion free, s*
*o that
^Hom (e(Q); f(I)) = e(E-1Hom(tF*; I)):
However, Euler classes are already invertible, so the functor E-1 may be omitte*
*d. The
same comments apply to J. Hence R^RHom (e(Q); f(T )) is Hom(tF*; I) in even de*
*grees,
and Ext(tF*; I) in odd degrees, as required.
For the general case we must consider the structure map qZ : Z ^ "EF -! Z ^ E*
*F+.
Now note that the induced map F (E"F; q) : F (E"F; Z ^ "EF) -! F (E"F; Z ^ EF+)*
* is
determined by the diagam
F(E"F;q)
F (E"F; Z ^ "EF)- ! F (E"F; Z ^ EF+)
'# #
qX 0
F (S0; Z ^ "EF) - ! F (S ; Z ^ EF+)
since all maps from an F-contractible spectrum to F (EF+; T ) are null. This a*
*rgument
applies equally well in the algebraic category, where it describes the map indu*
*ced in the
model. __|_|
Lemma 9.3.5. The algebraic model of F (E; Z) is correctly described by the*
* theorem.
Remark 9.3.6. For calculational purposes we care more about knowing the the t*
*or-
sion part and vertex of F (E; Z), both of which are easy. Indeed, the torsio*
*n part is
RcRHom(I(H); T ) where T is the torsion part of Z, and the vertex is E-1RHom(I(*
*H); T ).
On the other hand, this does not serve the present purpose.
Proof: Note first that both the algebraic and geometric model are trivial when *
*Z is F-
contractible. Indeed F (E; Z ^ "EF) ' *, and on the other hand e(W ) is inj*
*ective,
so that RHom (I(H); e(W )) = Hom (I(H); e(W )) = 0. We may thus assume Z is an
F-spectrum, and C = f(T ). Similarly, we may assume that Z is a T-spectrum ove*
*r H.
For notational simplicity we assume H = 1 and drop the subscripts. Now if Z is *
*semifree,
the function spectrum F (E; Z) is semifree, and similarly in algebra. We may*
* therefore
conduct the rest of the proof with semifree objects: it is clear that the alge*
*braic and
topological functors including semifree categories in the full categories corre*
*spond.
Now take an injective resolution 0 -! T -! I -! J -! 0. We thus find
i j
R^RHom (f(I); f(T )) = fibre R^ Hom (f(I); f(I)) -! R^ Hom (f(I); f(J)):
116 9. PRODUCTS, SMASH PRODUCTS AND FUNCTION T-SPECTRA.
Now Hom (f(I); f(I)) has zero vertex and is thus f(H(I)) where H(I) = Hom(f(I)*
*; I).
Since H(I) is torsion free, we can easily construct an injective resolution 0 -*
*! H(I) -!
H(I)[c-1] -! K(I) -! 0, and hence
i j
R^f(H(I)) = fibre ^f(H(I)[c-1]) -! ^f(K(I)):
However ^f(H(I)[c-1]) = e(H(I)[c-1]), and ^f(K(I)) = K(I); in particular we hav*
*e cal-
culated R^RHom (f(I); I), and it is even; similarly R^RHom (f(I); J) is even,*
* and the
map between them is classified by its effect in homotopy. Since the relevant m*
*aps are
composites of retractions and collapse maps cn : E -! 2nE, it is immediat*
*e that
they are modelled by the corresponding algebraic maps. __|_|
This completes Step 1. Steps 2, 3 and 4 require no further comment. We proce*
*ed to
Step 5.
Lemma 9.3.7. The identifications in 9.3.4 and 9.3.5 are natural for the maps i*
*n 9.1.1.
Proof: The identifications are certainly natural for retractions. This leaves t*
*he question of
naturality for the maps cn : E -! 2nE and "EF -! EF+ ^S-nV (H). Naturality
for those of the first type is clear since the modelling is all done in the cat*
*egory of OF-
modules.
For those of the second type we may compose with projection onto the Kth facto*
*r, and
it is sufficient to prove naturality for these. There are two types, depending *
*on whether K
is contained in H or not. If K is not contained in H, the map is equal to the c*
*omposite
"EF -! EF+ -! E , and thus a special case of the second type.
We may therefore suppose that f : E"F -! -2n+1E is given, and that we want
to understand the map f* : F (-2n+1E; Z) -! F (E"F; Z). Applying F (f; .) t*
*o the
cofibre sequence Z ^ EF+ -! Z -! Z ^ "EF, we obtain the diagram
F (-2n+1E; Z ^ EF+) -'! F (-2n+1E; Z) -! F (-2n+1E; Z ^ "EF)
# # #
F (E"F; Z ^ EF+) -! F (E"F; Z) -! F (E"F; Z ^ "EF);
in which the three verticals are induced by f. Since F (E; Z ^ "EF) ' *, the*
* induced
map is determined by the composite ff : F (-2n+1E; Z ^ EF+) -! F (E"F; Z ^ "*
*EF).
Next, the codomain of ff is F-contractible, and so ff is determined by its effe*
*ct in homotopy.
Splitting Z ^ EF+ into even and odd parts, we see that the Adams short exact se*
*quence
shows that the effect of f in homotopy determines the effect of ff in homotopy.*
* Since the
model is based on the effect of f in homotopy, this completes the proof. __|_|
CHAPTER 10
Induction, coinduction and geometric fixed points.
This chapter treats the simplest change of groups functors. In Section 10.1 we *
*give algebraic
models of the forgetful functor from T-spectra to H-spectra, and its adjoints, *
*the induction
and coinduction functors. In Section 10.2 we give the algebraic model of the g*
*eometric
K-fixed point functor.
10.1. Forgetful, induction and coinduction functors.
It is the purpose of this section to describe the algebraic counterparts of t*
*he forgetful
functor U : T-Spec -! H-Spec , the induction functor ind : H-Spec - ! T-Spec ,
X 7-! TnH X, and the coinduction functor coind : H-Spec - ! T-Spec, X 7-! FH [T*
*; X).
The induction and coinduction functors are left and right adjoint to the forget*
*ful functor
respectively, so that if X is a T-spectrum and Y an H-spectrum
[T nH Y; X]T = [Y; UX]H and [UX; Y ]H = [X; FH [T; Y )]T:
We must begin by describing the category H-Spec . First, we recall that an ar*
*bitrary
rational H-spectrum splits as a product of Eilenberg-MacLane spectra; thus H-Sp*
*ec is
equivalent to the derived category of the 0-dimensional abelian category of rat*
*ional H-
Mackey functors.
For any finite group H, the rational Mackey functors are easy to describe, an*
*d are sums
of the functors arising from representations of the Weyl groups NH (K)=K for K *
* H. In
our case H is abelian, so we need only explain that a module V for the quotient*
* H=K gives
rise to a Mackey functor RHK(V ) defined by
(
V L if L K
RHK(V )(L) = 0 if L 6 K
The restriction maps are given by inclusions of fixed point sets, and the trans*
*fer maps are
given by coset sums. We let MH denote the category of all rational Mackey func*
*tors, and
MtrivHdenote the full subcategory of Mackey functors with trivial Weyl group ac*
*tion (i.e.
sums of functors of form RHK(Q). Thus
H-Spec ' D(MH ):
117
118 10. INDUCTION, COINDUCTION AND GEOMETRIC FIXED POINTS.
Because the category of rational Mackey functors is of global dimension 0, and *
*the free
functors are realizable, all spectra are generalized Eilenberg-MacLane spectra *
*and it is easy
to see that the equivalence is given by X 7-! ss_H*(X). Therefore the algebraic*
* counterpart
of the forgetful functor T-Spec -! H-Spec is the functor DA -! D(MH ) given *
*by the
the same condition. However, it is perhaps clearer to compose with the equivale*
*nce
Y
DMH -'! QH=K - mod
K
whose Kth component is M 7-! eK M(K).
Summary 10.1.1.QThe forgetful map T-Spec -! H-Spec corresponds to the func-
tor DA -! KD(QH=K - mod) with Kth component M 7-! [LK ; M], where LK =
(Q(H) -! 0) is the algebraic counterpart to the basic cell oe0K. __|_|
We have two methods open to us to identify the algebraic induction and coinduc*
*tion
functors. We could calculate the behaviour of the geometric functors, but inste*
*ad we shall
simply guess the answers and prove they have the requisite adjointness properti*
*es. Actually,
we `guess' the functors by a calculation on geometric objects. Because H-spectr*
*a split as
a wedge of Eilenberg-MacLane spectra, and the induction functor commutes with w*
*edges,
the induction will be determined by its values on Eilenberg-MacLane spectra. Fi*
*nally, it
is useful to use the fact that FH [T; X) ' -1T nH X: Now we simply observe that*
* any
induced spectrum T nH X is an F-spectrum, and is thus determined by its homotopy
groups. ssT*(T nH Y ) = ssT*(FH [T; Y )) = ssH*(Y ).
Lemma 10.1.2. IfLthe H-Mackey functor M corresponds to V = (VK ) with VK a QH=*
*K-
module (ie M = KRHKVK ), then
M __
M(H) = (VK )H : |_|
K
Definition 10.1.3.If VK is a graded QH=K-module for each K H, then we define
induction and coinduction on V = (VK ) by
M
ind(V ) = ( (VK )H -! 0)
K
and M
coind(V ) = ( (VK )H -! 0);
K
where the Chern class c acts as zero in both cases.
Q
Evidently these are both exact functors KQH=K - mod -! A , and hence induce
functors on the derived category.
Proposition 10.1.4.The inductionQand coinduction functors are left and right a*
*djoint
to the forgetful functor A -! K QH=K -mod, and the same holds at the level of *
*derived
categories.
Before sketching the proof we record the desired corollary.
10.2. GEOMETRIC FIXED POINTS. 119
Corollary 10.1.5. The algebraic and topological induction and coinduction fun*
*ctors
correspond in the sense that the diagrams
FH[T;.)
H-Spec TnH-!T-Spec H-Spec -! T-Spec
'# #' and '# #'
Q ind Q coind
KQH=K - mod -! DA DA K QH=K - mod -! DA
are commutative up to natural isomorphism. __|_|
Proof: As usual we must record the units and counits, and we leave the verifica*
*tion of the
triangular identities to the reader. We must first recall the natural exact seq*
*uence
0 -! eK M=ceK M -! [LK ; M] -! -1ann(c; eK M) -! 0:
To describe the counit indUM -! M suppose M = (N -! tF* W ). It is enough to
describe the map of nubs since the domain is c torsion and hence maps to zero i*
*n tF* W ,
so this is necessarily consistent on vertices. The counit has Kth component th*
*e evident
map
([LK ; M]) -! (-1ann(c; eK N)) -! eK N:
The unit V -! UindV has as its Kth component the composite
VK -! (VK )H -! [LK ; indV ] ~=(VK )H (VK )H
where the first map is given by coset averages, and the second uses the fact th*
*at the short
exact sequence for [LK ; indV ] splits naturally in V .
Next, to describe the unit M -! coindUM it is obviously enough to discuss nub*
*s, since
the codomain has zero vertex. Its Kth component is
eK N -! eK N=ceK N -! [LK ; M]:
The counit UcoindV -! V has Kth component the composite
[LK ; coind(V )] -! (VK )H -! VK : __|_|
10.2. Geometric fixed points.
We give an analysis of the geometric H-fixed point functor, regarded as a fun*
*ctor from
T-spectra to T=K-spectra for a finite subgroup K. The functor T is corresponds *
*to taking
the vertex, and is an integral part of our analysis. It therefore_requires no f*
*urther comment.
To minimize confusion in the coming_discussion we let T = T=K and use bars to*
* indicate
reference_to the_ambient_group T; when necessary, we denote_the_quotient map by*
* q_: T -!
T . For example F is the family_of_finite_subgroups of T, and H is the image in*
* T of the
subgroup H of T. Note that the T-space EF + may be regarded as a T-space, and a*
*s such
it is EF+. Nonetheless, we shall use use the notation which best indicates whic*
*h group is
acting.
120 10. INDUCTION, COINDUCTION AND GEOMETRIC FIXED POINTS.
In one sense, the functor K is obvious from our construction. Indeed, we analy*
*ze X
using the cofibre sequence
X -! X ^ "EF -! X ^ EF+;
and if we apply K we obtain
K X -! K (X ^ "EF) -! K (X ^ EF+):
__
Now we use the fact that X ^ "EF = (TX) ^ "EF so that K (X ^ "EF) = (TX) ^ "EF.
On the other hand, we have conducted our analysis using the stable splitting
_
EF+ ' E
H
where E is the space which is the cofibre of the universal map E[ H]+ -! E[ *
*H]+.
Since K is the extension of the K-fixed point space functor the basic fact is *
*the following.
Lemma 10.2.1. For any finite subgroups H and K of T, we have the equivalence
( __
E if H K
(E)K =
* if H 6 K
__
of based T-spaces.
Proof:_The characterization of universal spaces_by their_fixed point spaces giv*
*es (EH)K =
EH for any family H of subgroups of T, where H = {H | H 2 H}. __|_|
Restated in the stable language, this states that, whenever K H,
__
K E ' E:
The natural guess is now that the invariants for K (X ^ EF+) are obtained from*
* the
sequence of spectra X(H) = X ^ E and their characteristic Q[cH ]-modules ssT*
**(X(H))
by ignoring those terms with H 6 K. This turns out to be correct. To make sense*
* of it,
we need to observe the there is a natural identification of the rings of operat*
*ions Q[cH ]and
Q[c__H].
The following observation is repeatedly useful.
Lemma 10.2.2. If K H and Y is any T-spectrum over H then K gives an isomorph*
*ism
~= K K _T
K : [X; Y ]T*-! [ X; Y ]*:
Proof: Since _
[K X; K Y ]T*= [X; "E[6 K]^ Y ]T*;
and we have the cofibre sequence
E[6 K]+ -! S0 -! "E[6 K];
it is enough to show E[6 K]+ ^ Y ' *. However Y ' Y ^ E, and, by considering*
* fixed
points, E[6 K]+ ^ E is a contractible space. __|_|
10.2. GEOMETRIC FIXED POINTS. 121
__
Using the fact that K E ' E, we obtain an identification of rings of *
*operations.
Corollary 10.2.3. Whenever K H, K induces an isomorphism
~= __ __ _T __
Q[cH ]= [E; E]T*-! [E; E]* = Q[c__H]: |_|
Furthermore, we can now confirm intuition.
Corollary 10.2.4. If K H then K induces an isomorphism
_ __
ssT*((K X)(H )) = ssT*(X(H))
of Q[cH ]-modules.
Proof: We calculate
__ __ __
(K X)(H ) = (K X) ^ E= K (X ^ E) ' K (X(H)): |_|
This shows how to construct the algebraic analogue. We consider the cofibre s*
*equence
q^M
M -! e(VM ) -! f(TM )
and apply K to obtain
Kq^MK
K M -! K e(VM ) -! f(TM ):
Thus,_it is enough to identify K on e(VM ) and f(TM ). Since X(T) = TX ^ "EF*
* and
TX = TK X, we take K e(VM ) = _e(VM ); in other_words,_vertices are_identical*
* VKM =
VM . On the other hand, we take K f(TM ) = f(TM ), where TKM = TM is obtaine*
*d by
deleting the factors not containing K. More succinctly, we let e_denote the ide*
*mpotent
of_OF with support the set of subgroups containing_K. Now view O __Fas eOF_and *
*take
TM = eTM . Also, once we have observed that tF*~=etF*we can say ^qKM = ^qMis e^*
*qM
It remains to identify this as a functor on arbitrary objects.
Definition 10.2.5.Let e 2 OF be the idempotent whose_support is the_set of fi*
*nite
subgroups containing K. Using the identifications O __F~=eOF and tF*~=etF*, we *
*define the
functor
__
K : A -! A
by
K (NM -rM!tF* VM ) := (eNM -erM!etF* VM );
or more briefly K M := eM . __|_|
We may now set about showing the algebraic functor has the desired properties.
122 10. INDUCTION, COINDUCTION AND GEOMETRIC FIXED POINTS.
Theorem 10.2.6. The algebraic functor K induces a functor
__
K : DA -! DA
so that the diagram
K __
T-Spec -! T-Spec
'# #'
K __
DA -! DA
commutes up to natural isomorphism.
Before we prove this, we should verify that the identifications
__ __
O __F~=eOF and tF*~=etF*
correspond to suitable geometric statements. However, some care is necessary at*
* this point:
several naive expectations are false._
First, note that we may view a T-space Y as a T-space, and_hence we may form t*
*he
dual DTY of Y as a T-spectrum; we may also form the dual T-spectrum D_TY and vi*
*ew it
as a T-spectrum by building in representations. (The inflation functor functor *
*building in
representations is often written j* where j : UK - ! U is the inclusion of univ*
*erses, but in
this section we shall follow the convention for suspension spectra and omit the*
* notation j*,
which will always be made clear from the context. When_notation is required for*
* emphasis
we write infD_TY .) Furthermore, by regarding the T-map D_TY ^ Y -! S0 as a T-m*
*ap, we
obtain a comparison map
: D_TY -! DTY
__
of T-spectra. We warn that this is definitely not an equivalence for Y = EF +; *
*the easiest_
way to see this is to observe that the T-equivariant homotopy_groups of "E[6 K]*
*^ D_TEF +
are zero in positive degrees, whilst those of "E[6 K]^ DTEF + are not. This mea*
*ns that
__
K DEF+ 6' DEF + :
However, the following positive result is what we need.
Lemma 10.2.7. The map
__
: D_TEF +-! DTEF+
is an F-equivalence.
Proof: Apply the forgetful functor to the construction of ; since EF+ is H-equi*
*variantly
S0 for all finite subgroups H, the lemma_follows from the space level construct*
*ion of the
Spanier-Whitehead dual of a finite T-complex. __|_|
We may now find the geometric basis for the algebraic definition of K on nubs.
10.2. GEOMETRIC FIXED POINTS. 123
__
Lemma 10.2.8. There is a natural equivalence eDEF+ ' D_T(EF + ) ^ "E[6 K], of*
* T-
spectra and hence
__
X ^ eDTEF+ ' K X ^ D_TEF +^ "E[6 K];
and _
__
essT*(X ^ DTEF+) ~=ssT*(K X ^ D_TEF +):
Proof: First observe that EF+ = eEF+ _ (1 - e)EF+, and that (1 - e)EF+ ' E[6 K]+
whilst eEF+ ' EF+ ^ "E[6 K]. Thus E[6 K]+ ^ eDTEF+ ' (1 - e)EF+ ^ eDTEF+ ' *,
so that eDTEF+ ' "E[6 K]^ eDTEF+.
We may therefore consider the natural map which is the composite
__ ^1
D_TEF +^ "E[6 K]-! DTEF+ ^ "E[6 K]-! eDTEF+ ^ "E[6 K]' eDTEF+:
Since is an F-equivalence, it follows from the definition_of_e that is an F-e*
*quivalence.
For the rest, we want to show that the map "E[6 K]^ D_TEF +-! eDTEF+ induces a
bijection of [S0; .]T*. Now diagrams
S0
__ . &
"E[6 K]^ D_TEF + -! eDTEF+
correspond under the adjunction to diagrams
eEF+
__ . &
"E[6 K]^ D_TEF +^ eEF+ -! S0:
Finally,
"E[6 K]^ D_TE__F+^ eEF+ ' "E[6 K]^ eEF+ ' eEF+;
and the horizontal is induced by the collapse map EF+ -! S0. The cofibre of the*
* horizon-
tal thus becomes (1 - e)EF+ when smashed with EF+, and hence the horizontal ind*
*uces
a bijection in [eEF+; .]T*as required. __|_|
Proof of 10.2.6: For the first statement, we need only observe that the above d*
*efinition
gives an exact functor on A ; it then induces a functor on the category of dg A*
* -objects,
and preserves homology isomorphisms. The existence of the functor then follows *
*from the
universal property of a category of fractions.
To see the algebraic K is compatible with the topological one we apply the *
*functor
comparison theorem 7.5.1. Note first that Condition 1 holds. Indeed, it is ob*
*vious that
both functors preserve cofibre sequences. It is also obvious that the algebrai*
*c functor
preserves both parity and injectives; the analogous fact for the topological fu*
*nctor follows
from the agreement of the functors on objects used to motivate the definition. *
*This gives
(2)(a), and finally we must check (2)(b), that if X and Y are each either F-con*
*tractible_or
injective F-spectra, then for any map f : X -! Y we have K (ssA*(f)) = ssA*(K *
* f).
124 10. INDUCTION, COINDUCTION AND GEOMETRIC FIXED POINTS.
If Y is F-contractible, we use the diagram
~= T T
[X; Y ]T* -! Hom(ss*( X); ss*( Y ))
K #~= _ #=
~= _TK _TK
[K X; K Y ]T*-! Hom(ss*( X); ss*( Y ));
_
which commutes since TK = T. If Y is F-free and injective we use the diagram
~= T T
[X; Y ]T* -! Hom(ss*(X ^ DEF+); ss*(Y ))
K # _ # K
~= T K __ _TK
[K X; K Y ]T*-! Hom(ss*( X ^ DEF + ); ss*( Y )):
We must explain why the diagram commutes. First note that, since Y is an_F-spec*
*trum e,
is defined as a self-map of Y and eY ' Y ^ "E[6 K]so that [K X; K Y ]T*= [X; e*
*Y ]T*. The
diagram would therefore commute if the right hand vertical was replaced by Hom(*
*ssT*(X ^
DEF+); ssT*(Y )) -! Hom(ssT*(X ^ DEF+); essT*(Y )). It therefore suffices to p*
*rove that
multiplication by e on nubs corresponds to the geometric construction, which wa*
*s 10.2.8
above. __|_|
It seems worth recording one consequence of the above discussion.
Lemma 10.2.9. If Y -! I(Y ) -! J(Y ) is an Adams resolution, then it remains s*
*o after
taking fixed points. __|_|
CHAPTER 11
Algebraic inflation and deflation.
This chapter develops the algebra necessary for modelling the Lewis-May fixed p*
*oint func-
tor. Section 11.1 introduces the class of Hausdorff OF-modules, which is large*
* enough
to cover the modules that concern us, and small enough to allow a suitable defi*
*nition of
inflation. In Section 11.2 we show the inflation and deflation functors have th*
*e requisite
adjointness properties on the category of Hausdorff OF-modules, and in Section *
*11.2 this
is extended to the torsion model category.
11.1.Algebraic inflation and Hausdorff OF-modules.
Given a quotient homomorphism G -! G=N one obtains various change of group re-
sults, and we generically refer to functors from G=N-equivariant structure to G*
*-equivariant
structure as inflation, and to functors in the reverse direction as deflation. *
* This section
is devoted to a particular algebraic case of this which will be important in th*
*e analysis of
Lewis-May fixed points. The reader principally interested in the topological ap*
*plications
may prefer to look at the motivation in Section 12.3 before reading further.
We suppose given a finite subgroup K of T and follow the conventions_of Secti*
*on 10.2 on
notation by using bars to indicate the_ambient group is the quotient T = T=K. F*
*irst, note
that the quotient map q : T_-!_T=K = T induces a map of subgroups, and in parti*
*cular__
a surjective map_q*_: F -! F . Since any subgroup mapping to a fixed subgroup H*
* of T
lies inside q-1(H ), it is clear that the fibres of q* are finite.
Accordingly, since OF = C(F; Q[c]) the map q* induces a map
__
q* : O__F-! OF;
and since the fibres of q* are finite this extends to
__
q* : tF*-! tF*:
Explicitly, q* is the diagonal map on each factor in the sense that q*(e__H) = *
*_L=__HeL.
Pullback along q* induces a deflation functor
__
def : OF - mod -! O__F- mod:
If we think of an F-finite module M as corresponding to a sequence of modules_e*
*LM,_then
the effect of deflation is to collect together all the summands eLM with L = H *
*and make
125
126 11. ALGEBRAIC INFLATION AND DEFLATION.
this the summand e__HdefM. After this section we shall often omit notation for *
*deflation,
since the underlying set is unchanged.
We also want to obtain an inflation functor
__
inf : O__F- mod -! OF - mod;
__
in the reverse direction. On_F-finite_modules we would want to define inf(N ) t*
*o be the F-
finite_module with eH inf(N ) = e__HN. Thus_the_inflation functor simply gives *
*each summand
e__HNthe multiplicity of the fibre of q-1*(H ):
__ M __
inf(N ) = e__HN;
H
so the word inflation is also suggestive of the construction. However, the cons*
*truction is
not given by pullback along a ring homomorphism, so it is not obvious how to ex*
*tend it to
arbitrary OF-modules. For our purposes it is sufficient to define it on all mod*
*ules occurring
as nubs. Q
For any OF-module M, we may form the module M^ = H eH M. We have chosen the
notation since this can be viewed as a completion, and it is therefore natural *
*to call a
module M for which the natural map
M -! M^
is injective, a Hausdorff module. The convenient property of a Hausdorff modul*
*e M is
that maps into M are determined by theirQcomponentLmaps into eH M. An example *
*of
a non-Hausdorff module is the quotient HQH = H QH , where OF acts on QH via *
*the
idempotent eH . We shall define the inflation functor on all Hausdorff modules.*
* Note that,
by construction, the completion functor M 7-! M^ is exact.
Lemma 11.1.1. (i) All F-finite OF-modules are Hausdorff.
(ii) The ring OF is Hausdorff as a module.
(iii) Any submodule of a Hausdorff module is Hausdorff.
(iv) The class of Hausdorff modules is closed under extensions, in the sense th*
*at if 0 -!
M0- ! M -! M00-! 0 is an exact sequence of OF-modules with M0 and M00Hausdorff,
then M is also Hausdorff.
(v) The class of Hausdorff modules is closed under arbitrary direct sums or dir*
*ect products.
Proof: Parts (i) and (ii) are clear. Parts (iii) and (iv) follow from the Snake*
* Lemma by
exactness of completion. In Part (v), closure under direct products is obvious.*
* For direct
sums the inclusion M Y Y
: Mi- ! eH Mi
i i H
is injective since it is the composite of the map
M M Y
Mi- ! eH Mi;
i i H
injective by hypothesis, and the inclusion of the sum in the product. However, *
* also fac-
tors through completion, which is therefore also injective. __|_|
11.1. ALGEBRAIC INFLATION AND HAUSDORFF OF-MODULES. 127
Corollary 11.1.2. The modules N and tF* V occurring in any object N -! tF* V
of the standard abelian category A are both Hausdorff.
Proof: Since N is an extension of an F-finite module by a submodule of tF* V i*
*t is
sufficient, by Parts (iii) and (iv) of the lemma, to observe that tF* V is Haus*
*dorff. The
fact that tF*is Hausdorff follows from Part (iv), since tF*is the extension of *
*OF over the
F-finite module 2I, and the fact that tF* V is Hausdorff now follows from Part *
*(v). __|_|
We are now equipped to give a sufficiently general definition of inflation.
__ __ ^_
Definition 11.1.3.If N is a Hausdorff O __F-module with completion N we proce*
*ed as
__ _^_ Q __
follows. On the complete module ^Nwe define inflation by infN = H e__HNin the*
* obvious
__ _^_
way.__Now define infN to be the set of sequences (nH ) 2_infN_ so that for all*
* sections
s : F -! F of q* the sequence (ns(__H)) is an element of N .
__
Note that each section s defines a ring homomorphism s* : OF -! O __F, and he*
*nce we
__ ^_ _^_ *
* ^_
may view N and N as modules over OF. Furthermore s induces a projection infN -*
*! N ,
__
and this is a map of OF-modules when ^Nis a module by pullback along s*. Taking*
* the
product over all sections we see that the definition states that there is an ex*
*act sequence
__ _^_ Y _^___
0 -! infN -! infN -! N =N ;
s
__
or equivalently that infN is a pullback
__ _^_ Y __
infN = infN xQ ^_ N :
sN s
__
This establishes that infN is indeed a OF-module.
Lemma 11.1.4. The above definition extends to a left exact functor
__ Haus
inf : O__F - mod -! OHausF- mod:
Proof: The observation just made ensures that inf is a functor on Hausdorff mod*
*ules.
__ _^_ __
Evidently infN -! infN is completion, so that infN is Hausdorff.
The_functor_is certainly exact on complete modules. Given a short exact sequ*
*ence
0 __ __00
0 -! N -! N -! N -! 0 of Hausdorff modules, the sequence of quotients 0 -!
_^_N0=__N0-! ^_N=__N-! ^_N00=__N00-! 0 is also exact by the Snake Lemma. Left e*
*xactness also
follows from the Snake Lemma, using the exact sequence formulation of the defin*
*ition. __|_|
128 11. ALGEBRAIC INFLATION AND DEFLATION.
11.2. Algebraic inflation and deflation of OF-modules.
We now establish that inflation, as defined in the previous section, has the p*
*roperties we
require: it behaves well on model categories, and has good adjointness properti*
*es.
__
Lemma 11.2.1._(i) For any graded_vector space_V weLhave_inf(tF* V ) = tF* V .
(ii) For any F-finite module N we have inf(N ) = He__HN. __|_|
Note in particular that this means inflation gives an exact functor on the tor*
*sion model
category At. More surprising is that it is also exact on modules occurring in t*
*he standard
model (i.e. on modules isomorphic modulo Euler-torsion to those of the form tF**
* V ).
Lemma 11.2.2._Inflation_is exact on modules occurring as nubs, and hence induc*
*es an
exact functor A -! A.
Proof: Before beginning, we warn that the category of Hausdorff modules is not *
*abelian.
However the category of modules occurring as nubs is abelian, and furthermore, *
*F-finite
injective modules and modules of the form tF* V are injective in this category.*
* We may
therefore use right derived_functors_in_the_course of the proof. *
*__
_If_we_have an object_fi:_N_-!_tF*V of_A, there are short exact_sequences_0_-! *
*K -!
N -! I -! 0 and 0 -! I -! tF*_V_- ! C -! 0 as usual,_where the K and C are
F-finite torsion modules and Iis_the image of fi. Since inflation is exact on t*
*he category of
F-finite modules we_find R1infK = 0, and so it suffices, by the first short ex*
*act_sequence,_
to show that R1infI = 0. From the second exact sequence_and_exactness_on module*
*s_tF*V
we see that it is enough to check surjectivity of inf tF*_V -! infC_. Since inf*
*C is F-finite,
this is true provided_each idempotent_summand eH inf tF* V -! infC is surjecti*
*ve. But
this map is e__HtF* V -! e__HC, which is surjective by hypothesis. __|_|
The following adjunction is the algebraic core of the section.
Proposition 11.2.3.The functors inf and def are both left and right adjoint to*
* each
other when restricted_to the subcategories_of Hausdorff modules. Thus if M is a*
* Hausdorff
OF-module and N is a Hausdorff O __F-module we have isomorphisms
__ __ __ __
Hom(defM; N) = Hom(M; infN ) and Hom(infN ; M) = Hom(N ; defM):
Proof: We shall give the unit and counit of each adjunction and leave the reade*
*r to verify
the triangular identities. We begin by considering_the_F-finite case, for whic*
*h we may
restrict_attention_to a single subgroup H of T, and to the finite set of subgro*
*ups L with
L= H . __ __
Letting R = Q[c], and supposing_there are k subgroups L with L = H , we are th*
*us
concerned with R-modules N and Rk-modules M and the diagonal homomorphism R -!
Rk. It is probably clearest to think of M as a k-tuple (M1; M2; : :;:Mk) of R-m*
*odules. The
inflation map is thus pullback along the diagonal ring homomorphism q* : R -! R*
*k with
11.2. ALGEBRAIC INFLATION AND DEFLATION OF OF-MODULES. 129
__ __ *
* __
associated map def(M1; : :;:Mk) = M1. .M.k. On the other hand inf(N ) = (N ; : *
*:;:N).
The unit j : M -! inf O defM of the first adjunction is the map
(M1; : :;:Mk) -! (M1 . . .Mk; : :;:M1 . . .Mk)
given in the ith_term_Mi_-! M1 . . .Mk by the canonical injection. The counit
ffl : def O infN -! N of the first adjunction is the folding map
__ __ __
N : : :N-! N :
__ __
The unit j : N -! def O infN of the second adjunction is the diagonal map
__ __ __
N -! N . . .N:
The counit ffl : inf O defM -! M of the second adjunction is the map
(M1 . . .Mk; : :;:M1 . . .Mk) -! (M1; : :;:Mk)
given in the ith coordinate by the projection M1 . . .Mk -! Mi.
The above descriptions of units and counits all extend_in an obvious way to c*
*omplete
modules, by taking products over the various subgroups H . This makes it obvio*
*us that
the triangular identities continue to hold in this context. It remains to cons*
*ider general
Hausdorff modules, and we simply need to verify that if an element lies in the *
*relevant
uncompleted module then so does its image. We use here the fact that completion*
* commutes
with inflation by definition, and with deflation since the fibres of q*_are_fin*
*ite.
To use the above work_it is convenient to enumerate the subgroups of T and ar*
*range
1 __2 __i
them in_a_row,_say H ; H ; : :.:Below each subgroup H we enumerate the subgro*
*ups L
with L = H , say as Li1; Li2; : :;:Lik(i). Thus we have an infinite number of c*
*olumns with
__ __
a finite and bounded number of rows in each column. Now, for a O __F-module N ,*
* we may
__1 __2
consider its idempotent summands N ; N ; : :a:rranged in a row, and for an OF-m*
*odule
M we consider its idempotent summands Mijas arranged in an array patterned afte*
*r the
subgroups. The above_discussion was_essentially the case_of_a single_column. No*
*w for a
i
Hausdorff module N the module infN has components (infN )ij= N independent o*
*f j.
A section s is simply a function selecting a row s(i) for each column i, and th*
*e condition
__ __ *
* ^_
that an element (xij) 2 infN^ lie in infN is that for each section s the eleme*
*nt (xis(i)) 2 N
__
actually lies in N .
We may now verify that the units and counits restrict to give maps between un*
*completed
modules. For the unit of the first we suppose (xij) 2 M; then j(x)ij= xij2 Mi1.*
* .M.ik(i).
But for any s we find (j(x)is(i)) = (x1s(1); x2s(2); : :):does lie in defM sinc*
*e it is esx, where
es 2 OF is the idempotent with support the image of s.
For the counit_of the first adjunction we must be a little_more careful. We *
*choose
i __i
y 2 def O infN , which has ith component yi = (yij) 2 N . . .N. Then we have
ffl(y)i= yi1+ . .+.yik(i). Suppose that the maximum number of rows in a column *
*is k; we
__
shall express ffl(y) as a sum of k elements known to be in N . Indeed, we divid*
*e the columns
130 11. ALGEBRAIC INFLATION AND DEFLATION.
into sets C1; C2; : :;:Ck where Cj is the set of columns with exactly j rows. N*
*ow choose
sections sj1; : :s:jjso that they exhaust the j rows in each column from Cj, an*
*d let
y(j) = eCj((yisj) + (yij ) + . .+.(yij ))
1(i) s2(i) sj(i)
__ __ *
* __
The idempotent eCj lies in O __F, and the j terms lie in N since y lies in def *
*O infN . Now
ffl(y) = y(1) + . .+.y(k).
It is trivial to verify_that_the unit of the second adjunction_preserves_the u*
*ncompleted
submodule, since if y 2 N then j(y)i = (yi; : :;:yi) 2 N . . .N. However the c*
*ounit
is again quite tricky. Suppose that x 2 inf O defM. Then ffl(x)ij= ssj(xij) w*
*here xij2
(inf O defM)ij= Mi1 . . .Mik(i)and ssj is projection onto the jth factor. In ef*
*fect, for
__i
each subgroup H we have a k(i) x k(i) square of entries. The hypothesis is tha*
*t however
we select a row from each such square we obtain an element (xis(i)) 2 defM, and*
* we need
to know that if we select the diagonal elements of each square we obtain an ele*
*ment of M.
However if we let e1; e2; : :;:ek be the idempotents defined by requiring eiis *
*supported on
the ith element along each diagonal and if we let j denote the section which se*
*lects the jth
row whenever it exists and the first row otherwise, then
ffl(x) = e1(xi1) + . .+.ek(xik);
and for each fixed j, (xij) 2 defM by hypothesis. __|_|
11.3. Inflation and its right adjoint on the torsion model category.
The use we make of the results of the_previous section is in constructing func*
*tors to
correspond to the inflation functor T- Spec -! T - Spec building in representat*
*ions, and
its right adjoint. This right adjoint is the Lewis-May fixed point functor [18,*
* II.4.4]. It is
difficult to identify the Lewis-May fixed point functor directly, since it does*
* not satisfy all
the hypotheses of Theorem 7.5.1; instead we shall identify the algebraic inflat*
*ion functor,
which does satisfy the relevant conditions, and then identify the Lewis-May fix*
*ed point
functor as its adjoint. The second curious feature is that it seems extremely *
*difficult_to
define the Lewis-May fixed point functor using the standard_abelian models A a*
*nd A .
Instead we define it using the torsion models At and At; this means that our co*
*ntrol over
its behaviour on morphisms in DA is somewhat indirect. Because we are forced to*
* approach
the Lewis-May fixed point functor through the torsion model, we do the same for*
* inflation.
The definition of the inflation functor
__
inf : At- ! At
is the obvious termwise extension of the functor on Hausdorff modules:
__
inf(tF* V -! T ) = (tF* V = inf(tF* V ) -! infT ):
We have seen in 11.2.2 this is an exact functor.
We now define its right adjoint,
__
K : At- ! At;
11.3. INFLATION AND ITS RIGHT ADJOINT ON THE TORSION MODEL CATEGORY. 131
which we call the algebraic Lewis-May fixed point functor.
Suppose that M = (tF* V -! T ) we take
__ q*1
K M = (tF* V -! tF* V -! T )
__
where tF*and T are regarded as O__F-modules by pullback along q*. This definiti*
*on extends
in an obvious way to morphisms, and is visibly exact. The reason for naming thi*
*s after the
Lewis-May fixed point functor will appear in Section 12.1 below, although the f*
*ollowing
proposition may seem sufficient explanation in view of [18, II.4.4].
Proposition 11.3.1.The algebraic Lewis-May fixed point functor
__
K : At- ! At
is right adjoint to the inflation functor
__
inf : At- ! A:
Thus we have natural isomorphisms
__ __ K
Hom(infN ; M) = Hom(N ; M)
__ __
for any object N of At and any object M of At.
Proof: As usual it suffices to construct the unit
__ K __
j0: N -! infN
and the counit
ffl0: infK M -! M
*
* __
and verify_the_triangular identities. Before we begin, notice that the map q* :*
* tF*-! tF*=
def O inftF*is the unit j of the second adjunction of 11.2.3.
__ __ _s __
For the unit, suppose N = (tF* V -! T) and define j0by the diagram
__ 1 __
tF* V -! tF* V
| # q* 1
_s | tF
* V _
_#_ # inf(s)
j __
T -! def O inf(T )
where the lower horizontal is the unit of the adjunction of 11.2.3. The commuta*
*tivity of
the square states j O_s= inf(_s)O(q*1), which follows from the triangular ident*
*ity of 11.2.3
together with naturality of j. For the counit suppose M = (tF* V - s!T ) and de*
*fine ffl0
by the diagram
tF* V -!1 tF* V
inf(q*) 1 # |
inf O def(tF*) V | s
# #
inf O def(T )-!ffl T
132 11. ALGEBRAIC INFLATION AND DEFLATION.
where the lower horizontal is the counit of the adjunction of 11.2.3. The commu*
*tativity
of the square states s = inf(s) O (inf(q*) 1), which follows from the triangul*
*ar identity
of 11.2.3 together with the naturality of ffl. The triangular identities are im*
*mediate from
those of 11.2.3. __|_|
We remark that we cannot use a similar construction for a left adjoint of inf *
*because
of the role of q*. We would need to know that the unit_tF*-! inf O def tF*of t*
*he first
adjunction of 11.2.3 was the inflation of some map tF*-! def tF*, which is not *
*the case.
In fact, inf does not have a left adjoint, since it does not preserve arbitrary*
* products. We
can substantiate this when we have studied products a little more, but the clai*
*m is the
algebraic counterpart of the fact that the natural map
Y __ Y __
inf( -nET +) -! (inf -nET +)
n n
is not an equivalence, where n runs through the natural_numbers. It will be eas*
*ier to verify
this later. The convenient fact is that ssT*(Y ) = ssT*(Y ^ K S0)_and, since th*
*e Lewis-May
fixed point spectrum K S0 is a wedge of S0 and terms including ET +, we need to*
* consider
the comparison map
__ Y -n __ Y __ -n __
ET +^ ET + -! ET + ^ ET +:
n n
This is not an equivalence since its fibre is
Y __
"E_T^ -nET +:
n
Q 1-n_
The homotopy groups of this fibre are obtained from n I by inverting c1, an*
*d are
thus nonzero.
However we shall see later that inflation does have a left adjoint on suitable*
* subcategories.
We end this section with_a_discussion of the induced_functors on derived categ*
*ories. The
point is that both inf : A t-! At and K : At -! A tare exact functors, and hence
the functors on dg categories that_they induce preserve homology_isomorphisms. *
* They
therefore induce functors inf : DA t-! DAt and def : DAt -! DA ton derived cate*
*gories
in the usual way. It is reasonable to use the notation inf and def, as we have *
*done, because
of the exactness of the functors on abelian categories. However, for certain pu*
*rposes we
also want to view these as functors on derived categories of the standard model.
Now the Lewis-May fixed point functor is not defined in an obvious way on A , *
*and so
it is not_misleading or ambiguous to continue to use the same notation for the *
*functor
DA - ! DA . But notice that if M is an object of A , then K M is calculated by *
*taking
an injective approximation, moving it into dgHAt , moving into dgAt, applying i*
*nf and
moving back to dgAt. We shall make the end result of this process more explicit*
* on objects
in Section 12.3.
11.3. INFLATION AND ITS RIGHT ADJOINT ON THE TORSION MODEL CATEGORY. 133
__
However inflation is also defined as a functor A - ! A by applying_inf termwi*
*se, and
it is still exact by 11.2.2. It therefore induces a functor DA -! DA on deri*
*ved cate-
gories. It will be valuable to know that this corresponds with the original fun*
*ctor under
the equivalence between standard and torsion model categories; this_is obvious_*
*by the fibre
construction of the equivalences fib : dgAt -'! dgA and fib : dgA t-'! dgA .
Lemma 11.3.2. The diagram
__ inf
DA -! DA
'# #'
__ inf
DA t -! DAt
commutes up to natural isomorphism. __|_|
134 11. ALGEBRAIC INFLATION AND DEFLATION.
CHAPTER 12
Inflation, Lewis-May fixed points and quotients.
We begin in Section 12.1 by recording the formal properties of the inflation an*
*d Lewis-
May fixed point functors in topology, and deducing the facts we need later in t*
*he chapter.
There are then two natural routes through the chapter: to simply obtain a model*
* for the
Lewis-May fixed point functor, continue with Section 12.2. There we apply the *
*Functor
Comparison Theorem 7.5.1 to deduce that the algebraic inflation functor models *
*the topo-
logical one, and conclude that their right adjoints, the Lewis-May fixed point *
*functors, also
correspond. However this route is rather indirect, and some readers will prefe*
*r a more
concrete approach. We therefore give a more direct analysis of the Lewis-May fi*
*xed point
functor in Section 12.3, by considering the F-free and F-contractible pieces se*
*parately.
In the same spirit, Section 12.4 makes the functor explicit in the standard mod*
*el: this is
very much based at the derived category level, and does not approach morphisms *
*at all.
However, it does illustrate the behaviour of the functor, and it gives a splitt*
*ing theorem.
The chapter ends with the very short Section 12.5, drawing attention to the f*
*act that
we have also modelled the quotient on sufficiently free spectra, since it is le*
*ft adjoint to
inflation.
12.1.The topological inflation and Lewis-May fixed point functors.
We continue to use the notation_introduced in Section 10.2, with K being a fi*
*nite sub-
group of T, and q : T -! T the quotient_map. We have explained_in Section 7.3 t*
*hat the
inflation functor inf = infT_T: T-Spec -! T-Spec regarding a T-spectrum_as T-sp*
*ectrum
is the composite of the pull-back along the quotient q : T -! T and the functor*
* building
in representations which are not K-fixed.
The purpose of this section is to provide the various facts we shall need abo*
*ut the topo-
logical inflation and Lewis-May fixed point functors. This will give us enough *
*information
to apply the Functor Comparison Theorem 7.5.1 to the algebraic and geometric in*
*flation
functors. Although we are first concerned with inflation, there are various po*
*ints where
it seems convenient to use properties of Lewis-May fixed points, so we discuss *
*the two
functors together. Because of its important place in the discussion, we include*
* notation for
the inflation functor throughout this section.
135
136 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
__
The fundamental fact [18, II.4.4] is that for any T-spectrum Y , there is an a*
*djunction
_
[inf Y; X]T*= [Y; K X]T*:
The_crudest implication, which we shall often use, is that, since S0 is an infl*
*ation, ssT*(X) =
ssT*(K X). We shall also need to understand the fixed point functor on an infla*
*ted spectrum.
Thus if we take X = inf Y , the T-equivariant identity map on the left correspo*
*nds to the
unit j : Y - ! K inf Y of the adjunction. Accordingly for any T-spectrum Y we *
*may
consider the composite
K (X) ^ Y -! K (X) ^ K (inf Y ) -! K (X ^ inf Y ):
The following curious lemma is the key to the analysis.
__
Lemma 12.1.1. If Y is a T-spectrum then the natural map
K (X) ^ Y -! K (X ^ inf Y )
__
is an equivalence of T-spectra.
Proof: We have a natural map which commutes with direct limits and cofibre sequ*
*ences ____
in the variable Y ,_so_it_is enough to verify it is an equivalence_when Y is a *
*cell, T=H +
for some subgroup H T. However, the_lemma for any finite T-complex Y is clea*
*r by
playing with adjunctions, since the T-dual of Y regarded as a T-spectrum is the*
* T-dual. __|_|
The particular case X = S0 is often useful:
__
Corollary 12.1.2. For any T-spectrum Y we have a natural equivalence
K (S0) ^ Y -'! K inf Y;
where K S0 denotes the Lewis-May fixed point spectrum. __|_|
This becomes most useful in conjunction with tom Dieck's calculation of K S0, *
*which
we recall in Example 12.4.6 below.
Now we turn to the behaviour of inflation, and begin by considering_how_basic *
*cells
behave. Recall that there are only finitely many subgroups L with L = H .
__
Lemma 12.1.3. If we inflate the basic T-cell oe0_Hto a T-spectrum we find an e*
*quivalence
_
infoe0_H' oe0L
_L=__H
of T-spectra.
__ __
Proof: Let_H = q-1(H ) and consider the_quotient map q : H -! H . This induces *
*a map
q* : A(H ) -! A(H), and for any x 2 A(H ), whenever L H, we have OEL(q*(x)) = *
*OE__L(x).
Thus in particular
q*(e__H) = _L=__HeL:
The lemma follows. __|_|
12.1. THE TOPOLOGICAL INFLATION AND LEWIS-MAY FIXED POINT FUNCTORS. 137
__ __ __
Since the the T-space EF + is EF+ when regarded as a T-space (i.e. inf(EF + )*
* ' EF+),
we immediately deduce the required consequence.
__
Corollary 12.1.4. If Eis regarded as a T-spectrum we have an equivalence
__ _
E ' E
_L=__H
of T-spectra. __|_|
It follows from this and the calculation of the self maps of E, that the i*
*nflation map
__ __ _T __ __ T Y
Q[c__H]= [E; E]* -! [E; E]* = Q[cL]
_L=__H
is the diagonal inclusion c__H7-! (cL)L.
Corollary 12.1.5. The inflation functor
__ __ __ _T T
O __F= [EF + ; EF +]* -! [EF+; EF+]* = OF
induces the ring homomorphism q*. __|_|
Since inf is exact on F-finite modules we deduce the following.
__
Corollary 12.1.6. For an F-spectrum X
_
ssT*(infX) = inf ssT*(X):
Proof: We apply inflation to an Adams resolution of X. Now use the fact that 12*
*.1.4 also
identifies inflations of maps between F-injectives with their algebraic counter*
*parts. We
will give a detailed proof of a generalization of this immediately below. __|_|
We would like to make the analogous statement for arbitrary spectra. The rem*
*aining
obstacle is the identification of maps between injectives. The key to understan*
*ding this is
a calculation. __ __
First note that we have a projection map infE_-! E, with adjoint E*
* -!
K E. Similarly the injection map E -! infE can be used to form
__ __ __ __ __ 0
E ^ infD_TE-! infE ^ infD_TE' inf(E^ D_TE) -! S
__ __
with adjoint infD_TE-! DTE; the adjoint of this is a map D_TE-! K (D*
*TE).
__
Lemma 12.1.7. The natural maps described above give equivalences of T-spectra
__ ' K __ ' K
E-! E and DE -! (DE)
138 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
Proof: Both facts follows from 10.2.2. The second begins with the calculation
_
[A; K (DTE)]T = [infA; DTE]T = [infT ^ E; S0]T = [infT ^ E; E]T;
and continues by saying this is the same as
__ _T __ 0 _T __ __ _T
[T; D_TE] = [T ^ E; S ] = [T ^ E; E] :
Under passage to geometric fixed points this is an isomorphism by 10.2.2. Since*
* E is
a space over K, geometric fixed points coincides with Lewis-May fixed points. O*
*ne must
now check that this map coincides with that induced by the map described above.*
* The
first equivalence is slightly easier. __|_|
Corollary_12.1.8. Under the_identification of Q[cH ]and Q[c__H], for any map f*
* : X -!
Y of T-spectra with Y an F-spectrum the map
infX ^ DTE -! infY ^ DTE ' infY ^ E
induces the same map in ssT*as
__ __ __
X ^ D_TE-! Y ^ D_TE' Y ^ E
_
in ssT*.
_
Proof: We use the fact that ssT*(Z) = ssT*(K Z) and K (inf Y ^ Z) ' Y ^ K Z. We*
* now
obtain a diagram
X ^ K (DTE) -! Y ^ K (E) ^ K (DTE) -! Y ^ K (E)
" __ __" __ " __
X ^ D_TE -! Y ^ E^ D_TE -! Y ^ E
in which the first and last verticals are equivalences by 12.1.7 above. The com*
*mutativity
of the left square is naturality of the equivalence K (inf Y ^ Z) ' Y ^ K Z. Fo*
*r the right
hand square, we first explain that the upper horizontal is the composite of
K E ^ K (DTE) -! K (E ^ DTE)
and the K fixed points of evaluation to S0, which lifts uniquely to E. Commu*
*tativity
now follows, since the square with Y omitted is the adjoint of the diagram
inf K (E) ^ K (DTE) -! E ^ DTE - ! E
__" " __
infE ^ DTE - - - ! infE;
whose commutativity is clear from the definitions. __|_|
12.2. CORRESPONDENCE OF ALGEBRAIC AND GEOMETRIC INFLATION MAPS. 139
__
Proposition 12.1.9.For an arbitrary T-spectrum X,
_ __
ssT*(infX ^ DTEF+) = infssT*(X ^ D_TEF +):
Hence also we have __
ssA*(infX) = infssA*(X):
Proof: We have already seen this is true if X is injective. The result follows *
*once we know
that topological inflation corresponds to algebraic inflation for maps between *
*two injective
spectra X. This is clear if both copies of X are F-spectra, so the remaining ca*
*se is covered
by considering a map __ __
f : "EF- ! nEF +:
Since we are considering maps into a Hausdorff module it suffices to consider i*
*ts idempotent
parts. In other words we need to show that the map
E"F ^ DTE ' "E_F^ eH DTEF+ -! nE__F+^ eH DTEF+ ' nE
induces the same as
"E__F^ D_TE<__H>' "E_F^ e__HD_TE__F+-! nE__F+^ e__HD_TE__F+' nE<__H>:
This is a special case of 12.1.8. __|_|
We could at this point give a direct approach to the Lewis-May fixed point fu*
*nctor on
objects. To emphasize the logical structure of the argument, we defer this to S*
*ection 12.3.
However, the reader is recommended to look at this by way of motivation for the*
* definition
of the algebraic Lewis-May fixed point functor.
12.2.Correspondence of Algebraic and geometric inflation maps.
Theorem 12.2.1. The algebraic inflation functor induces a functor
__
inf : DAt -! DA t
so that the diagram
__ inf
T-Spec -! T-Spec
'# #'
__ inf
DA t -! DAt
commutes.
Proof: The existence_of the functor on derived categories is immediate from the*
* fact the
algebraic functor At -! At of Section 11.2 is exact. __
We want to apply the Functor Comparison Theorem 7.5.1 to the two functors T-S*
*pec -!
DA . We are therefore implicitly using the equivalence DAt -'! DA to identify *
*DAt and
DA . __
We work through Condition 1, starting with the topological functor T-Spec -! *
*T-Spec -!
DA . It certainly preserves triangles. Enough topological injectives are obta*
*ined from
140 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
__ __ *
*__
wedges of suspensions of "EF and the_variousWspectra E. The inflation of "E*
*F is "EF
and, by_12.1.4,_the inflation of Eis L E where the wedge is over the su*
*bgroups L
with L = H . For the algebraic functor the analogous facts were built into the *
*definition.
To see that objects of pure parity are preserved is immediate from exactness of*
* inflation as
manifested in 11.3.2.
The fact that the topological functor preserves objects of pure parity is imme*
*diate from
12.1.9.
We now turn to Condition 2 of 7.5.1. The definition of the algebraic inflation*
* functor
was designed to agree with the topological one on enough injectives, as one see*
*s from
12.1.9. Finally it remains to prove that they agree for maps into injectives, a*
*nd we recall
that 11.2.2 showed the algebraic inflation functor on_the standard model was al*
*so termwise
application_of inflation. Suppose that X and Y are T-spectra. First consider th*
*e case when
Y is F-contractible, we use the diagram
~= A A T T
[X; Y ]T*-! Hom(ss* (X); ss* (Y~))=Hom(ss*( X); ss*( Y ))
inf "~=_ " inf "=
~= __A __A _T T
[X; Y ]T*-! Hom(ss* (X); ss* (Y~))=Hom(ss*( X); ss*( Y )):
The diagram commutes since the horizontals factor through passage to total geom*
*etric
fixed points. Both algebraic and topological_inflation functors are the identit*
*y on vertices.
Next we suppose that Y is an injective F-spectrum, and use the diagram
~= A A T T
[X; Y ]T*-! Hom(ss* (X); ss* (Y~))=Hom(ss*(X ^ DEF+); ss*(Y ))
inf " _ " inf "
~= __A __A _T __ _T
[X; Y ]T*-! Hom(ss* (X); ss* (Y~))=Hom(ss*(X ^ DEF + ); ss*(Y )):
We_must_explain_why_the diagram commutes. In other words we must_show_that if
ssA*(X)_= (N__-! tF*V ) and_the map f : X -! Y corresponds to : N -! ssT*(Y ) *
*so that
= ssT*(X^DEF + -! Y ^DEF + ' Y ) then inf() is ssT*(X^DEF+ -! Y ^DEF+ ' Y ).
A map into a Hausdorff OF-module (such as ssT*(Y )) is determined by its idempo*
*tent
parts._ It is thus_sufficient to show that ssT*(eH (X ^ DEF+) -! eH Y ) is the *
*same as
ssT*(e__H(X ^ DEF + ) -!_e__HY_), but_since_eH X ^ DEF+ ' X ^ eH DEF+ ' X ^ DE<*
*H>
and similarly e__HX ^ DEF + ' X ^ DE this follows from 12.1.8. __|_|
It is now formal to deduce the correspondence between the algebraic and topolo*
*gical
Lewis-May fixed point functors.
Theorem 12.2.2. The algebraic Lewis-May K-fixed point functor K induces a func*
*tor
__
K : DAt -! DA t
so that the diagram
K __
T-Spec -! T-Spec
'# #'
K __
DAt -! DA t
commutes.
12.3. A DIRECT APPROACH TO THE LEWIS-MAY FIXED POINT FUNCTOR. 141
Proof: The existence of the functor_on derived categories is immediate from the*
* fact the
K is an exact functor At -! At.
We have seen that the algebraic Lewis-May fixed point functor is right adjoin*
*t to infla-
tion at the level of abelian categories. The unit and counit of the adjunction *
*induce maps
of derived categories, and the triangular identities continue to hold. Since th*
*e topological
Lewis-May fixed point functor is right adjoint to inflation by [18, II.4.4], an*
*d since we have
seen in 12.2.1 that the left adjoints correspond, the right adjoints also corre*
*spond. __|_|
12.3. A direct approach to the Lewis-May fixed point functor.
In the previous section we completed the proof that the algebraic Lewis-May K*
*-fixed
point functor corresponds to the geometric one, and for logical purposes all fu*
*rther discus-
sion could take place in the algebraic models. However, we think it is importan*
*t to make
the link between algebraic and geometric forms of the functor more direct, and *
*that is the
purpose of this section. This can be viewed as motivation for the definition of*
* the algebraic
Lewis-May fixed point functor, and can be used to give an alternative proof of *
*the theorem
for objects. However it seems hard to get a sufficiently tight grasp on morphis*
*ms to give a
complete proof of the theorem by this route.
The construction of the model is reasonablyWobvious if the question is viewed*
* correctly.
The point is that our decomposition X(F) ' HX(H) follows the decomposition of*
* a
Mackey functor as an hSB -module. Thus
M
ssH*(X) = ssH*(X(L)):
LH
__
The point is that_when H K we may calculate ssH*(K X) = ssH*(X), so that whate*
*ver
the subgroup H , this group contains the summands for X(L) whenever L K. The_e*
*nd
result is that_the when we decompose (K X)(F) as aWwedge of spectra (K X)(H ), *
*the
term (K_X)(1) contains information equivalent to LK X(L), and in_general_the *
*term
(K X)(H ) contains information equivalent to X(L) for all L with L = H .
Once again we need to give a context in which to make this meaningful. The b*
*asic
idea is that there is a natural way of identifying Q[cH ]and Q[c__H]. Indeed, *
*we gave one
identification in 10.2.3 in terms of geometric fixed points; for our present pu*
*rpose we must
show the K fixed point map Q[cH ]-! Q[c__H]coincides with passage to Lewis-May*
* fixed
points.
Lemma 12.3.1. If H is a finite subgroup containing K, then passage to Lewis-M*
*ay K-
fixed points defines the isomorphism
~= __ __ _T
Q[cH ]= [E; E]T*-! [E; E]* = Q[c__H]
also described as passage to geometric fixed points above.
142 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
Proof: Since E is concentrated over K in the sense that E ' E ^ "E[6 K*
*],
Lewis-May fixed points coincide with K fixed points. __|_|
Note that by 12.1.5 the inflation map
__ __ _T __ __ T Y
Q[c__H]= [E; E]* -! [E; E]* = Q[cL]
_L=__H
is the diagonal inclusion c__H7-! (cL)L. This gives sense to the following cons*
*equence.
Corollary 12.3.2. If H is a finite subgroup of T containing K then
_ __ M __
ssT*((K X)(H )) ~= ssT*(X(L)) |_|
_L=__H
The curious Lemma 12.1.1 also lets us deduce the F-contractible part.
Corollary 12.3.3. For any T-spectrum X we have an equivalence
_
T(K X) ' TX:
In particular _
__ T
ssT*(K (X) ^ "EF) ~=ss*(X ^ "EF):
__ __ __ *
* __
Proof: Let Y = E"F in 12.1.1, to find that K (X) ^ "EF ' K (X ^ "EF). Since E*
*"F
regarded as a T-space is "EF, the result follows. __|_|
The result may seem peculiar_to begin_with, so we outline an alternative_heuri*
*stic argu-
ment. We note that ssT*(K (X) ^ "EF) is the localization of ssT*(K X) = ssT*(X)*
* so as to
__ __ ___T
invert Euler classes of all representations V of T with V . On the other hand s*
*sT*(X ^ "EF)
is the localization so as to invert all representations V of T with V T= 0. How*
*ever the Euler
class_of V divides the Euler class of V n and if K is of order n then V n is a *
*representation
of T. The two localizations are therefore equal.
We are now in a position to give the algebraic analogue of the Lewis-May fixed*
* point
functor. The basic ingredient is the ring homomorphism
__ __ __ _T T
q* : O__F= [EF + ; EF +]* -! [EF+; EF+]* = OF
defined by the inflation map. We have already seen that it is given explicitly *
*by the diagonal
maps Y
Q[c__H]-! Q[cL]:
_L=__H
Because the vertex and torsion are treated so differently, it proves difficult*
* to define the
Lewis-May fixed points in the standard model. More precisely, it is not hard to*
* identify the
homotopy type of the fixed point object, even at the level of the abelian categ*
*ory; however
it seems impossible to do this sufficiently naturally to obtain a functor. The *
*remedy is to
12.4. LEWIS-MAY FIXED POINTS ON OBJECTS IN THE STANDARD MODEL. 143
retreat to the differential graded category, and once we have done that, it is *
*much simpler to
work with the torsion model. We have given a functor fib : dgAt -! dgA which in*
*duces an
equivalence of derived categories so it is easy to translate the construction i*
*nto the standard
category once we have the torsion model form of an object or morphism. However *
*we recall
that the inverse functor takes values in the category dgHAt , so the easiest wa*
*y to get back
into At is to take homology: this loses control of morphisms, and exactly corre*
*sponds to
the difficulties described above.
We now complete the alternative approach to Lewis-May fixed points by checkin*
*g directly
that the algebraic construction is correct on objects.
Lemma 12.3.4. The algebraic Lewis-May fixed point functor is compatible with *
*the ge-
ometric one on objects in the sense that
K (p(X)) ' _p(K X)
for any T-spectrum X.
Proof: Of course we motivated the definition by arranging this was true if X is*
* either
F-free or F-contractible.
However, because the definitions on torsion and vertices was so different, th*
*e core of the
matter is that the structure maps are correct. For this we consider the diagram
__ __ __ __ ' __
K (X) ^ DEF + ^ "EF -! K (X) ^ DEF + ^ EF + - K (X) ^ EF +
'#_ __ _'#_ __
K (X ^ DEF + ) ^ "EF -! K (X ^ DEF + ) ^ EF +
# __ # __ #
K (X ^ DEF+) ^ "EF -! K (X ^ DEF+) ^ EF +
'# '#
K (X ^ DEF+ ^ "EF) -! K (X ^ DEF+ ^ EF+) -' K (X ^ EF+)
in which the equivalences come from 12.1.1. Since the top horizontal is q and t*
*he bottom
horizontal is _q, and the right hand vertical is an equivalence, it remains to *
*remark that the
central vertical induces infin homotopy. __ __
This in turn follows from the fact that DEF + -! K (DEF+) induces inf: O__F-!*
* OF,_
as is clear from the fact that it is two adjunctions from the T-map EF+ ' EF+^D*
*EF + -!
S0. __|_|
12.4. Lewis-May fixed points on objects in the standard model.
It is useful to identify the behaviour of the fixed point functor on objects,*
* using the fact
that objects of DA correspond to objects of A in the standard model. It does n*
*ot seem
possible to extend this construction to a functor on the abelian category A its*
*elf, because
of the choices involved in the definition: it is entirely based on a homotopy l*
*evel analysis.
fi F
We suppose given an object M = (N -! t* V ) of the standard model A . Lettin*
*g I
denote the image of fi we consider the modules associated to M by the exact seq*
*uences
0 -! K -! N -! I -! 0 and 0 -! I -! tF* V -! C -! 0;
144 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
the torsion part is then given by T = K -1C. The definition is based on the fa*
*ct that
the_torsion part of the Lewis-May fixed point object K M is simply T regarded a*
*s an
O__F-module by pullback along q*, and its vertex is the same as that for M.
__K __
We shall make a definition of an object M of A, which will be of the homoto*
*py type
of K M;_the notation for the associated modules will be systematically indicate*
*d by bars.
K __ * __
Thus M has torsion_module T = q T , and vertex V = V . From now on we regard
all_OF-modules as O __F-modules,_and omit the notation q*. We therefore need to*
* define an
O__F-homomorphism _s: tF* V -! T ; it is natural to use the composite
__ inf1 fi
tF* V - ! tF* V -! T:
This gives a map of short exact sequences
__ __
0 -! tF* V -! tF* V -! tF*=tF* V - ! 0
# c # # :
0 -! C -! C -! 0 - ! 0
__
In particular, we note that although T =_T , the contributions_from_the kernel *
*and cokernel
of the basing map are quite different: C is bigger than C and K is correspondin*
*gly smaller
than K. __
Now consider the map inf 1 : tF*_V_-!_tF* V ; it is injective and we view it *
*as an
inclusion, so that we may take I= (tF* V ) \ I. Thus we have an exact sequence
__ __ __
0 -! I- ! I -! (tF*=tF*) V -! D -! 0:
__
Now let N0= fi-1(I), so that we have an exact sequence
__
0 -! K -! N0- ! I- ! 0:
Definition 12.4.1.We define the crude Lewis-May_fixed point object of an objec*
*t M =
fi F __K __ fi __F __ -1__ 0 __
(N -! t* V ) of_A by_taking M =_(N_ -!_t* V ), where N = D N and fiis
the composite N = -1D N0- ! N0- ! I- ! tF* V .
Remark 12.4.2. With the above definition we have a splitting
__K -1__ K
M ~=f( D ) M;
__ __K
with D injective. Note in particular that, even if M is even, M will not usu*
*ally be even.
The fact that the splitting is not canonical is the reason this construction is*
* not sufficient
to construct the algebraic Lewis-May fixed points as a functor. __|_|
__K __
We should begin by observing that M is actually an object of A.
__ __
Lemma 12.4.3. The kernel_and cokernel of the map fijust defined are both F-fin*
*ite and
K __
torsion, and hence M is an object of A.
12.4. LEWIS-MAY FIXED POINTS ON OBJECTS IN THE STANDARD MODEL. 145
__ __ __ __
Proof:_By construction we see that,_since_C__is injective, C = C D and also K *
* =
K -1D . This establishes that K and C are F-finite and torsion. __|_|
__K
It now makes sense to claim that the construction describes Lewis-May fixe*
*d points.
__K
Proposition 12.4.4.The construction gives the Lewis-May fixed point functo*
*r on
objects in the sense that if we identify objects of DA up to isomorphism with *
*those of A,
then __
K
K M ' M:
__ _fi __ __
Proof: The_object_N_-!_ tF* V is_determined_by the image_of_fi, and its extensi*
*on class
in_Ext(im(fi); ker(fi)) = Ext(I; K -1D ). The image of fiis_the kernel of the *
*composite
tF* V -! tF*_V -! T by construction, as required. Since D is injective, the ext*
*ension
class of N is the pullback of that of N in Ext(I; K) by construction. __|_|
Combining 12.2.2, 12.4.2 and 12.4.4 we obtain a topological splitting result.
Corollary 12.4.5. There is an unnatural splitting
K X ' -1K X _ K X
__ __
with K X an injective F-spectrum. |_|
Example 12.4.6. Consider_the_algebraic_0-sphere,_LT_=_(OF -! tF*). We find_C *
*= 2I,
K = 0 and I = OF._Thus I= O__F, C = 2I, and K = I=I. More explicitly, I=I is a *
*sum __ __
of injectives I(H ), with multiplicity one less than the number_of subgroups_L *
*with L = H .
For example, if K is of order 2 there is a single summand I(H ) whenever H is o*
*f odd order.
This should be compared with the equivalence of [18, V.11.1], which gives
_
K S0 ' E[\K= = 1]+=(K=);
K
where_the_quotient of the universal_K=-free_T=-space, E[\K= = 1]+ by K= is view*
*ed
as a T-space using the isomorphism T = T=K ~=(T=)=(K=)._Note first that the sum-
mand with = K is S0, and that all other terms are F-spaces.
It seems worth making the consistency with the algebraic description explicit*
*. Let K* =
K=, and, more generally, use asterisks to denote reference to T=. We have obse*
*rved
that if = K the summand is S0, so now suppose is a proper subgroup of K. Since
K* is finite, the quotient of a K*-free universal space is again rationally a u*
*niversal space.
Indeed _
E[\K* = 1] = E;
L*\K*=1
__
and when L*\ K* = 1, E=K* = E: Thus
_ __
E[\K* = 1]+=K* ' E:
L*\K*=1
146 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
__
Thus, a subgroup L gives rise to a summand E for each subgroup K with
(|L*|; |K*|) = 1; this happens_exactly_once for each_subgroup not containing K.*
* The
multiplicity of the summand E is thus |q-1*(L )| - 1. __|_|
It seems worth recording the following.
Lemma 12.4.7. If Y -! I(Y ) -! J(Y ) is an Adams resolution, then it remains s*
*o after
taking Lewis-May fixed points.
Proof: We have seen that passage to Lewis-May fixed points preserves F-contract*
*ible
objects and F-free objects. By the classification of torsion F-finite injective*
*s we see that
the fixed points of a F-free injective is also injective.
Finally we must observe that
__ __ __
0 -! ssA*(K Y ) -! ssA*(K I(Y )) -! ssA*(K J(Y )) -! 0
is exact. This is a diagram chase using the fact that the first map is an isomo*
*rphism of
vertices, and that, because the torsion sequence is exact, the sequences of ker*
*nels and of
cokernels are also both exact. __|_|
Finally we remark that it is not hard to prove directly that d invariants of t*
*he algebraic
and topological Lewis-May fixed points agree. This falls short of a complete di*
*rect analysis
because Lewis-May fixed points do not preserve pure parity objects.
12.5.Quotient functors.
It is notorious that the quotient functor is only well behaved on sufficiently*
* free spectra.
This phenomenon has already shown itself in the algebra. Topologically, it seem*
*s impossible
even to calculate even the homotopy groups of quotient of arbitrary spectra.
*
* __ f
In fact the properties of the deflation functor show that def : D(torsOfF)-! D*
*(torsO __F)
__ f f
is left adjoint to inf : D(torsO __F)-! D(torsOF) . Hence, whenever the topolo*
*gical
quotient is left adjoint to the topological inflation functor, the algebraic de*
*flation models
the effect on homotopy groups of the topological quotient. Applying [18, I.3.8 *
*and II.2.8]
we immediately deduce the required result.
Proposition 12.5.1.The algebraic K-deflation functor corresponds to the topolo*
*gical
quotient by K on the category of K-free spectra. __|_|
It seems reasonable to refer to the topological functor =K defined by the diag*
*ram
=K __
T-Spec=F - ! T-Spec=F
'# #'
__ f
D(torsOfF) -def!D(torsO __F)
12.5. QUOTIENT FUNCTORS. 147
as the quotient by K. We have seen that it agrees with the quotient by K on K-*
*free
spectra, and it extends this functor on rational spectra to arbitrary F-spectra*
*. It would
be interesting to know if there is a topological definition of this functor.
148 12. INFLATION, LEWIS-MAY FIXED POINTS AND QUOTIENTS.
CHAPTER 13
Homotopy Mackey functors and related constructions.
This chapter investigates a number of constructions associated with a Mackey fu*
*nctor, and
the reader is advised to glance at Appendix A before reading further.
Section 13.1 makes explicit the homotopy Mackey functor applied to objects of*
* the stan-
dard model, and deduces the models of Eilenberg-MacLane spectra as a consequenc*
*e. The
following section studies Eilenberg-MacLane spectra from a topological point of*
* view, con-
centrating on cell structures and useful cofibre sequences involving them. Sect*
*ion 13.3 does
the same for coMackey functors, the representing objects of ordinary homology. *
*Finally, in
Section 13.4, we study the Brown-Comenetz spectra associated to a Mackey functo*
*r; these
are usually unbounded, and illuminate the rational Segal conjecture.
13.1. The homotopy Mackey functor on A .
We have the information to calculate the homotopy Mackey functor of any objec*
*t of
DA . We shall make use of the notation described in Section 5.5, and in partic*
*ular we
recall from Example 5.5.2 that the algebraic basic cells are LH = (Q(H) -! 0) a*
*nd
LT = (OF - ! tF*). We use the obvious injective form of the algebraic fixed ce*
*ll ^LT=
(tF*n I -! tF*) where the the semidirect product tF*n I is additively the direc*
*t sum,
but with differential given by the augmentation tF*-! 2I. We proceed in steps, *
*using
the notation ssT*(M) = [LT; M] and essH*(M) = [LH ; M] for an object M of DA . *
* The
symbol abbreviates the more descriptive eH ssH*, the point being that LH is the*
* algebraic
counterpart of the basic cell oe0Hrather than G=H+. Since M ' H*(M) in DA we *
*may
work with objects with zero differential.
Lemma 13.1.1. If M is an object with zero differential and sM is the composite
d(qM )
VM - ! tF* VM -! TM ;
then there is a natural isomorphism
ssT*(M) ~=-1cok(sM ) ker(sM ):
Proof: This is straightforward from the triangle M -! e(VM ) -! f(TM ). Indeed
[LT; e(VM )] = [VT; VM ] = Hom(Q; VM ) = VM , and because NT = OF is projective*
* we have
[LT; f(TM )] = [NT; TM ] = Hom(OF; TM ) = TM . The map is easily identified.
149
150 13. HOMOTOPY MACKEY FUNCTORS AND RELATED CONSTRUCTIONS.
The splitting is constructed as follows. If v 2 ker(sM ) then a map LT -! M is*
* explicitly
represented by the homomorphism LT -! M^ displayed in the diagram
{1v;0} F
OF - ! (t* VM ) n ^TM
# #
tF* -1v! tF* VM : __|_|
The homotopy groups ssH*(M) are a little easier, since LH is torsion.
Lemma 13.1.2. If M is an object with zero differential and there is a short ex*
*act sequence
0 -! TM (H)=cH -! essH*(M) -! -1ann(cH ; TM (H)) -! 0;
where TM (H) = eH TM is regarded as a module over Q[cH ]~=eH OF in the obvious*
* way.
The sequence splits unnaturally.
Proof: This follows from the Adams short exact sequence by calculating Hom(Q(H)*
*; TM ) =
ann(cH ; TM (H)) and Ext(Q(H); TM ) = 2TM (H)=cH . __|_|
Before continuing, let us see what this implies about objects with homotopy gr*
*oups only
nonzero in a single degree: the Eilenberg-MacLane objects.
Example 13.1.3. If M is an Eilenberg-MacLane object with zero differential and*
* homo-
topy groups ssT*(M) = W (T) and essH*(M) = W (H), concentrated in degree zero t*
*hen
M
VM = W (T) 2I(H) W (H)
H
and M
TM = 2I(H) W (H):
H
The map sM : VM - ! TM is surjective.
Proof: Suppose M is an Eilenberg-MacLane object with non-zero homotopy in degre*
*e zero.
Since any non-zero object of with zero differential has non-zero homotopy we ma*
*y suppose
M is even or odd, and the following discussion leads to the conclusion M = 0 if*
* M is odd.
We therefore suppose M is even.
Now e(W ) is an Eilenberg-MacLane object as in the statement if W is concentra*
*ted in
degree 0. If M is not torsion free, then TM 6= 0. Next, we see that TM is i*
*njective and
in odd degrees. Indeed, TM decomposes into even and odd parts; since TM is to*
*rsion, if
the even part is non-zero it gives a non-zero homotopy group in odd degree aris*
*ing from
-1ann(cH ; TM (H)) for some H, so we may suppose TM is odd. For the same reaso*
*n, if
the odd part is non-zero it definitely has homotopy groups in even degrees, and*
* hence the
odd contribution must be zero, and TM (H)=cH = 0 for all H; it is therefore div*
*isible and
hence injective.
Hence TM (H) is entirely entirely odd and TM (H) = I(H) W (H) where W (H) is
concentrated in degree zero. Now consider sM : VM -! TM = 2I(H) W (H). The
cokernel gives homotopy in odd degrees, so that sM is onto. Since ssT*(M) is co*
*ncentrated
in degree 0, sM is an isomorphism, except perhaps in degree 0. __|_|
13.1. THE HOMOTOPY MACKEY FUNCTOR ON A. 151
To determine the entire Mackey functor we need to identify the restriction ma*
*p ssT*(M) -!
essH*(M) induced by the generator pH : LH - ! LT of [LH ; LT] ~=Q. We must begi*
*n by
being explicit that pH is represented by the homomorphism
0 1
{0;H} F
i j B Q(H) - ! t*n I C
^pH: LH -! ^LT = B@ # # CA
0 - ! tF*
where H : Q(H) -! I is the inclusion of the degree 0 part of eH I. This is read*
*ily verified
from the definitions.
Proposition 13.1.4.Restriction map ssT*(M) - ! essH*(M) is the direct sum of *
*the
natural quotient map
res0: -1cok(sM ) -! TM (H)=cH
_ __
given by res0(t) = tand the map
res00: ker(sM ) -! -1ann(cH ; TM (H))
given by res00(v) = fiM (c-1H v).
Proof: The proof is by unravelling definitions. In fact we suppose given a map *
* : LT -!
M, which we have seen can actually be represented by a homomorphism of the same*
* name.
Using hats to denote fibrant approximation as usual, we then find a homomorphis*
*m ^ so
that the diagram
LT -! M
# #
^pH ^
LH - ! ^LT -! M^
commutes. Then res() is determined by ^pH (1) = ^(0; H (1)) = ^(d(c-1H; 0)) = d*
*^(c-1H; 0).
Now ^takes the form
^0 F
tF*n I -! (t* VM ) n ^TM
# #
00
tF* -! tF* VM :
Next observe that if the map of nubs is 0= {0V; 0T} : OF -! (tF* VM ) n ^TMthen*
* the
extension of 0over tF*determines ^0since the differential tF*-! I is surjective*
*, and the
map into the tF* VM factor must be 00. It remains to specify an extension ": tF*
**-! ^TM
of 0Tand verify that the result commutes with differentials. Fortunately we nee*
*d only do
this when is of two particularly simple_forms.
For res0we suppose corresponds to t2 cok(sM : VM - ! TM ) for some element t*
* 2 TM
(which may be of even or odd degree). In this case factors through f(TM ) -! M*
*. Now
suppose ^TM= fibre(I -! J), and view t as an element of I. We may then extend t*
*he map
: OF -! I to ": tF* I, and any such "gives rise to a map ^LT-! f(T^M) and henc*
*e to
^ as the composite ^LT-! f(T^M) -! M^. Therefore res0(_t) is represented by d"(*
*c-1H) 2 J,
152 13. HOMOTOPY MACKEY FUNCTORS AND RELATED CONSTRUCTIONS.
which is an element dy where y 2 I satisfies cH y = t; the result follows by co*
*nsidering the
resolution resulting from the resolution beginning 0 -! TM (H) -! TM (H)=cH I(*
*H).
For res00we use the explicit splitting given in 13.1.1 above. Suppose that co*
*rresponds to
v 2 VM , so that 0V= 1v, 0T= 0 and 00= 1v. We naturally choose "= 0, and it is *
*im-
mediate that this defines a map ^0commuting with differentials. Thus ^0(c-1H) =*
* (c-1Hv; 0);
the result follows. __|_|
This allows us to complete the discussion of Eilenberg-MacLane objects.
Example 13.1.5. We continue with the notation of Example 13.1.3. First observe*
* that
since sM is surjective res0= 0. Thus for v 2 ssT0(M) = W (T) we have res(v) = f*
*iM (c-1Hv).
The map q0M: tF* VM - ! TM may thus be chosen to have components
M
q0T: tF* W (T) -! 2I(H) W (H)
H
and
M
q0H: tF* 2I(H) W (H) -! 2I(H) W (H)
H
describedLas follows. First q0Tis the composite of projection tF* W (T) -! 2I *
*W (T) ~=
H 2I(H)W (T) and the map which is restriction on each factor. Next q0His the c*
*ompos-
ite of projection tF*2I(H)W (H) -! eH tF*2I(H)W (H) ~=Q[cH ; c-1H]2I(H)
W (H) and the map ckHclHw 7-! ck+lHw, where ck+lHis interpreted as zero if k+l *
* 0. __|_|
13.2. Eilenberg-MacLane spectra.
In the previous section we worked entirely in the algebraic model to identify *
*objects
with homotopy only in one degree. In the present section we study Eilenberg-Ma*
*cLane
spectra as is natural from a topological point of view. The identification of t*
*he model of
a topological Eilenberg-MacLane spectrum gives an alternative to the algebraic *
*deduction,
but the point of view is sufficiently different for this second method to be wo*
*rth including.
We shall use notation and terminology for Mackey functors established in Appen*
*dix A,
which the reader should refer to as necessary. In particular we recall that a M*
*ackey functor
M corresponds to a collection of vector spaces V (H), one for each finite subgr*
*oup H and
a vector space V (T) with restriction maps V (T) -! V (H).
The easy example is the Eilenberg-MacLane spectra associated to the Mackey fun*
*ctor
L[U] concentrated over T, for which we obviously have H(L[U]) ' E"F ^ S0[U]. T*
*his
deals with all F"-injectives. On the other hand, the basic F-injective is the *
*functor RK
concentrated over T and K. An arbitrary F-injective is a product of injectives *
*which are
sums of the basic injective RK , so the same is true of the corresponding Eilen*
*berg-MacLane
spectra. Therefore the following examples describe the structure of Eilenberg-*
*MacLane
spectra for all projective and injective spectra.
13.2. EILENBERG-MACLANE SPECTRA. 153
Example 13.2.1. (i) For the indecomposable projective Mackey functor PT (the *
*`con-
stant' functor) we have a cofibre sequence
E"F ^ S0[QF] -! S0 -! HPT:
(ii) For any finite subgroup K with corresponding indecomposable projective Mac*
*key func-
tor PK concentrated over K we have a cofibre sequence
E"F -! E -! HPK :
(iii) For any finite subgroup K with corresponding injective Mackey functor RK *
*we have
cofibre sequences
HRK -! oe0K-! HPK
and
HPK -! HRK -! "EF:
Proof: The case of HPT is straightforward: we kill homotopy groups in positive *
*degrees
to obtain a map S0 -! HPT, and observe its fibre is F-contractible.
In fact the same idea works for HPK : kill homotopy groups in positive degree*
*s to obtain
a map E -! HPK . The first cofibre sequence for HRK follows since oe0Kis a t*
*wo stage
Postnikov tower. The second follows from the defining extension 0 -! PK - ! RK *
*- !
L[Q] -! 0 for RK , as in A.17. __|_|
One may obtain a cell picture for an arbitrary Eilenberg-MacLane spectrum HM *
*by
taking a minimal projective resolution 0 -! P1 -! P0 -! M - ! 0 and using the
associated cofibre sequence HP1 -! HP0 -! HM.
To complete the analysis of HM in our framework we need to analyze HM ^ EF+ a*
*nd
HM ^ "EF. We let M(F) = lim!M(H) = H V (H).
H
Lemma 13.2.2. For any Mackey functor M there is an equivalence
_
HM ^ EF+ ' E ^ S0[V (H)];
H
L
and hence HM ^EF+ corresponds to the OF-module M(F)(OF)0I = HV (H)I(H).
Proof: By construction, (HM ^EF+)(K) = HM ^E . Recall that E = oe0K[ oe0K^
e2 [ oe0K^e4 [: :;:now consider the spectral sequence obtained by applying [S0;*
* HM ^o]T*
to the skeletal filtration. Because of the suspension in the Wirthm"uller isom*
*orphism we
find [S0; HM ^ oe0K]T*= V (K), and thus the spectral sequence collapses to show*
* that the
homotopy groups are V (K) in each positive odd dimension.
It remains to show that cK gives an isomorphism ssT2n+1(HM ^ E ) -! ssT2n-*
*1(HM ^
E ) for n 1. For this we use the long exact sequence of the cofibering
HM ^ E ^ oe0K-! HM ^ E ^ S0 -e!HM ^ E ^ SV (K)
in which e induces multiplication by cK . The result follows from the fact that*
* HM ^ E
is K-equivariantly an Eilenberg-MacLane spectrum, so that ssT*(HM ^ E ^ oe0K*
*) is con-
centrated in degree 1. __|_|
154 13. HOMOTOPY MACKEY FUNCTORS AND RELATED CONSTRUCTIONS.
The homotopy type of the rational spectrum TX is determined by its homotopy gr*
*oups
ss*(TX) ~=ssT*(E"F ^ X). With due caution about duality we may calculate THM for
any Mackey functor M with associated hSB -module V .
Lemma 13.2.3.
8
> M(F) ifk is even and positive
: 0 otherwise
Proof: This follows immediately from the cofibre sequence EF+ - ! S0 -! E"F and
13.2.2. __|_|
Note in particular that THM is only an Eilenberg-MacLane spectrum if M is an F*
*"-
injective. By contrast, the idempotent description of the geometric fixed point*
* functor given
in Theorem 10.2.6 shows that K HM is an Eilenberg-MacLane spectrum for all fin*
*ite K.
Corollary 13.2.4. The map HM ^ "EF - ! HM ^ EF+ is classified by the OF-
morphism
q : ss*(THM) tF*-! M(F) (OF)02I
described as follows, in which ffl : tF*-! 2I denotes the quotient. If n 1 an*
*d x 2
ss2n(THM) = M(F) then q(x y) = x ffl(y); if x 2 ss0(THM) = M(T), then c(x y)*
* =
resTF(x) ffl(y).
Q
Note that resTFin the statement is a map M(T) -! H V (H), whose codomain is t*
*he
product, but that ffl(y) is only nonzero in finitely many coordinates.
Proof: Suppose first that x is of positive degree. From the proof of 13.2.3, we*
* know that
q(x 1) = x and the result follows since non-zero elements of I(H) are uniquely*
* divisible.
For x of degree zero it is enough to show that q(x c-1H) = resTH(x). For this*
* we examine
the map
HM ^ "EF ^ oe-V (H)-! HM ^ E ^ oe-V (H)
in [S0; .]T. Transposing oe-V (H)across to its dual in the domain, and replacin*
*g the spaces
in the codomain by suitable skeleta we must examine
[oeV (H); HM ^ "E(2)]T -! [oeV (H); HM ^ oe1H]T:
Now recall that E"(2)= oeV (H), and note that if x 2 M(T), it is represented*
* in the
first group by x ^ 1 : S0 ^ oeV (H)-! HM ^ oeV (H). Its image in the second is*
* ob-
tained by composing with the map HM ^ oeV (H)-! HM ^ oe1H. This corresponds to
resTH(x) 2 M(oe0H) = eH M(H) as required. __|_|
13.3. CO-MACKEY FUNCTORS AND ORDINARY HOMOLOGY. 155
13.3. coMackey functors and spectra representing ordinary homology.
A coMackey functor is the algebraic object providing coefficients for ordinar*
*y homology.
It is therefore a covariant additive functor hSO -! Ab . By exactly the same a*
*rgument
as for Mackey functors, we see that the category of coMackey functors is equiva*
*lent to the
category of hSBop-modules. An hSBop-module W is thus described by vector spaces*
* W (T)
and W (H) together with induction maps W (H) -! W (T) for each finite subgroup *
*H.
The treatment of Mackey functors in Appendix A is the algebraic model for the*
* theory of
co-Mackey functors, so the reader should refer to it as necessary. In view of t*
*he similarity
to the structure of Mackey functors, we omit the proofs of the following facts.
One may make explicit the condition that an hSBop-module W is injective or pr*
*ojective,
and give canonical resolutions.
L
Lemma 13.3.1. (i) A hSBop-module W is projective if and only if the map HW *
*(H) -!
W (T) is a monomorphism.
(ii)A hSB -module W is injective if and only if the maps W (H) -! W (T) are all*
* epimor-
phisms. __|_|
Note in particular that if W (H) = 0 for all H then W is projective, and if W*
* (T) = 0
then W is injective. The principal projectives are (a) QT defined by QT(H) = 0 *
*for all H
and QT(T) = Q and (b) QH defined by QH (H) = Q, QH (T) = Q (with induction being
the identity), and QH (K) = 0 if H 6= K. It is not hard to see how to construct*
* canonical
projective and injective resolutions of length 1.
The existence and uniqueness of T-spectra JN representing homology with coeff*
*icients
in N is guaranteed by Brown representability. For any coMackey functor N we le*
*t N|F
denote the quotient zero at T; this has zero structure maps involving T and can*
* therefore
be regarded as a Mackey functor. In fact JN is a two stage Postnikov tower with
8
> L[N(T)] ifk = 0 ;
: 0 otherwise
where L[N(T)] is the Mackey functor concentrated at T where it takes the value *
*N(T).
There is thus a cofibre sequence
"EF ^ S0[N(T)] -! JN -! -1H(N|F):
Dually, the cofibre sequence
"EF ^ S0[M(T)] - HM - H(M|F)
can be regarded as a decomposition of HM in terms of spectra JN.
Finally we place JN in the torsion model.
156 13. HOMOTOPY MACKEY FUNCTORS AND RELATED CONSTRUCTIONS.
Lemma 13.3.2. For any coMackey functor N there is an equivalence
_
JN ^ EF+ ' -1E ^ S0[W (H)];
H
L
and hence JN ^EF+ corresponds to the injective OF-module N(F)(OF)0I = HW (H)
I(H).
Proof: The proof follows that of 13.2.2, but is slightly simpler. __|_|
The behaviour of the homological Eilenberg-MacLane spectra JN is rather differ*
*ent
from their more familiar cohomological counterparts, and this is manifested in *
*the fixed
points. Suppose N isLa coMackey functor with corresponding hSBop-module W , and*
* let
N(F) = lim!N(H) = HW (H). Let us define D(N) and E(N) by the exact sequence
H
0 -! D(N) -! N(F) -! N(T) -! E(N) -! 0:
Thus E(N) is measures the failure of an F-induction theorem for N, and D(N) mea*
*sures
the precision of such a theorem.
Lemma 13.3.3.
8
>>>E(N) ifk = 0
< D(N) ifk = 1
ssk(TJN) = >
>>:N(F) ifk is at least 3 and odd
0 otherwise
Proof: This follows immediately from the cofibre sequence EF+ - ! S0 -! E"F and
13.3.2. __|_|
Note in particular that TJN is only an Eilenberg-MacLane spectrum if N is a F"-
projective.
Because N(F) (OF)0I is uniquely divisible on all non-zero elements the assembl*
*y map
is easily identified in this case.
Corollary 13.3.4. The map JN ^ "EF -! JN ^ EF+ is classified by the OF-
morphism
q : ss*(TJN) tF*-! N(F) (OF)0I
defined by q(x y) = q(x) ffl(y), where ffl : tF*-! 2I is the quotient map. _*
*_|_|
13.4. BROWN-COMENETZ SPECTRA. 157
13.4.Brown-Comenetz spectra.
Another interesting class of spectra are the Brown-Comenetz spectra. If I is *
*any injective
Mackey functor we may define the functor hI*T(o) by
hInT(X) = Hom(ss_Tn(X); I):
Since I is injective this is an exact functor of X and hence it is represented *
*by a spectrum
hI. The reader may find the homotopy groups of hI more complicated than expecte*
*d, and
in particular hIis very rarely bounded below. For further information on Mackey*
* functors,
and associated notation the reader is referred to Appendix A. In particular, A.*
*15 shows
that any injective is a sum of those of two types, and it is enough to deal wit*
*h F-injectives
and "F-injectives.
Lemma 13.4.1. (i) If I is an F-injective Mackey functor
8
>* I|F ifk = -1
: 0 otherwise;
where I|F is the largest subfunctor of I which is zero at T.
(ii) If I = L[U] is an "F-injective Mackey functor
8
>>>L[U] ifk = 0
< CF[U] ifk = -1
ss_T*(hI) = > F
>>:L[U ] ifk is odd and -3
0 otherwise:
Here UF is the vector space of U-valued functions on F, and CF[U] is the F-inje*
*ctive
defined by CF[U](T) = UF and CF[U](H) = U with restriction maps being the relev*
*ant
projections.
Proof: First recall the homotopy functors of cells from A.16. Since ss_T0(S0) *
*= PT, and
ss_T0(oe0H) = PH , it follows from the Yoneda lemma that hI_0T= I whichever sor*
*t of injective
I is.
On the other hand, in positive degrees ss_T*(S0) is L[QF] in each odd degree *
*and 0 in each
even degree. It is easy to verifyQthat Hom(L[U]; M) = Hom(U; C(M)) where the co*
*re is
defined by C(M) = ker(M(T) -! H M(H)). Observe that C(L[U]) = U and C(I) = 0
if I is a F-injective.
For each finite subgroup we have ss_T1(oe0H) = RH , and an extension
0 -! PH -! RH -! L[Q] -! 0:
Since Hom(PH ; L[U]) = 0 and Hom(L[Q]; I) = 0 if I is F-injective, this is enou*
*gh to
calculate the homotopy groups. In the case that I is F-injective this also dete*
*rmines the
functors. Finally we need to understand I = L[U] and in particular I* := ss_T-1*
*(hI), or more
precisely the restriction maps I*(T) = Hom(QF; U) -! U = I*(H).
Unravelling definitions we see that the restriction map is induced by applyin*
*g Hom( ; L[U])
to the map ss* : RH = ss_T1(oe0H) -! ss_T1(S0) = L[QF] induced by the projectio*
*n oe0H-! S0.
Since Hom(V; L[U]) = Hom(V (T); U), it is only the part of ss* at T that is rel*
*evant, which
158 13. HOMOTOPY MACKEY FUNCTORS AND RELATED CONSTRUCTIONS.
is dealt with in the following lemma. __|_|
Lemma 13.4.2. The map ss* : Q = ssT1(oe0H) -! ssT1(S0) = QF is the inclusion o*
*f the Hth
factor.
Proof: The tom Dieck splitting isomorphism states that a certain natural map fr*
*om a direct
sum to ssT1(X) is an isomorphism, for any space X. The Hth factor is ss1(E(T=H+*
*) ^T=H
XH ) ~=ss0(XH ), and the map H (oe0H) -! (G=H+)H - ! (S0)H induces a bijection *
*of
ss0. __|_|
Note in particular that ss_T0(hJ ) = J, so that if M is an arbitrary Mackey fu*
*nctor one
may define a spectrum hM by taking a resolution 0 -! M -! J0 -! J1 -! 0 of Mac*
*key
functors and realizing it as a cofibre sequence hM -! hJ0- ! hJ1. It is easy t*
*o see there
is a short exact sequence
0 -! Ext(ss_Tn(X); M) -! hMnT(X) -! Hom(ss_Tn(X); M) -! 0:
We remark that it is extremely easy to construct an Adams resolution by spectr*
*a hJ,
and its convergence is immediate from the ordinary Whitehead theorem. Its E2 t*
*erm is
also reasonably computable. However the spectral sequence does not seem useful *
*except
perhaps for maps from bounded below spectra to finite Postnikov towers.
Lemma 13.4.3. (i) For any F-injective Mackey functor I with associated hSB -mo*
*dule
V there is an equivalence
_
hI ^ EF+ ' oe-1H^ S0[V (H)]:
H
We might reasonably write oe-1F[I] for this spectrum.
(ii) For any "F-injective Mackey functor I = L[U] there is an equivalence
hI ^ EF+ ' -1EF+[U]:
Proof: Note that
[S0; hI ^ oe0H]Tk= [oe-1H; hI]Tk= ss_Tk-1(hI)(oe0H);
and refer to 13.4.1 for the values this takes.
For Part (ii) we argue as in the proof of 13.2.2. If I is an "F-injective the *
*displayed group
is only nonzero for k = 0. The homotopy groups follow as before, and the actio*
*n of cH
follows from the fact that hI ^ E is H-equivariantly an Eilenberg-MacLane sp*
*ectrum.
For Part (i) the shape is slightly different. As above we see that ssT*(hI ^ o*
*e0H) = V (H)
V (H). Now the attaching maps oe2n+1H-! oe2nHin E are non-trivial in ssT*, a*
*nd hence
so are their duals. It follows from the definition of hI*T(o) that the map
[S2n+1; hI ^ oe2n+1H]T-![S2n+1; hI ^ oe2nH]T
k k
Hom(ss_T1(oe0H); I)-! Hom(ss_T0(oe0H); I)
k k
V (H) -! V (H)
13.4. BROWN-COMENETZ SPECTRA. 159
is an isomorphism. Hence, by induction on n, the limit ssT*(hI ^ E) is conce*
*ntrated in
degree 0, where it is V (H). __|_|
Lemma 13.4.4. (i) For any F-injective Mackey functor I with associated hSB -m*
*odule
V we have (
Q
V (H)= H V (H) ifk = 0
ssk(ThI) = 0H otherwise
(ii) For any "F-injective Mackey functor I = L[U] we have
8
>>>UF ifk is odd and 3
< ____UFifk = 1
ssk(ThI) = > F
>>:U ifk is odd and -1
0 otherwise
____
where UF is the kernel of the natural map UF -! U.
Proof: In both cases we consider the cofibre sequence EF+ -! S0 -! "EF, and use*
* 13.4.3
and 13.4.1 to give the groups in the long exact sequence in ssT*. The connectin*
*g maps are
identified using the definition of hI as it applies to identify the map hI0T(oe*
*-1H) -! hI0T(S0)
as in 13.4.3 (i). __|_|
Corollary 13.4.5. If I is an F-injective, then in the notation of 13.4.3
Y M
hI ' oe-1F[I] _ "EF ^ S0[ V (H)= V (H)]:
H H
In particular, if V (H) is only nonzero for finitely many H, then hI ' oe-1F[I]*
* is an F-
spectrum.
Proof: Since tF*and ssT*(oe-1F[I])QareLinLeven degrees, the assembly map is det*
*ected as an
element f 2 Hom(tF* H V (H)= H V (H); H V (H)). But if f 6= 0 with f(x) nonz*
*ero
in the Hth component then f(eH x) 6= 0; but eH x = cH "xfor some "xwith degree *
*2, and by
dimension f("x) = 0. __|_|
Corollary 13.4.6. If I = L[U] the map hI ^ "EF -! hI ^ EF+ is classified by t*
*he
OF-morphism
q : ss*(ThI) tF*-! U I
defined by the_fact_that q(x ckH) is x(H) if it lies in a nonzero group; this *
*applies whether
x is in UF; UF or UF .
160 13. HOMOTOPY MACKEY FUNCTORS AND RELATED CONSTRUCTIONS.
Proof: This follows from the above discussion unless k is negative. Furthermor*
*e, since
U I is uniquely divisible in positive degrees, the result follows unless x is *
*of negative
degree.
Let us therefore suppose x 2 UF comes from ss-2k+1(ThI), with k 1. It suffice*
*s to
show that q(x c-kH) is x(H). For this we must examine the map
hI ^ "EF ^ oe-kV (H)-! hI ^ 1E ^ oe-kV (H)
in ssT-2k+1= [S-2k+1; .]T. Transposing oe-kV (H)across into its dual in the dom*
*ain, and using
the fact that "E-! E is the direct limit of maps oelV (H)-! E(2l-2)wit*
*h dual
oe-lV (H)- -2lE(2l-2)we must examine the direct system
[oekV (H)^ oe-lV (H); 2k-1hI]T -! [oekV (H)^ -2lE(2l-2); 2k-1hI]T:
Of course [X; 2k-1hI] = Hom(ssT2k-1(X); U) because I = L[U], so we only need to*
* under-
stand
oe(k-l)V'(H)oekV^(H)oe-lV-(H)oekV (H)^ -2lE(2l-2)' 2k-2lE(2l-2)
in ss2k-1. It is more convenient to consider the previous term in the cofibre *
*sequence
-1 ^ oekV (H); and observe this has zero homotopy in positive odd degrees. The *
*required
map is surjective for each l and hence also in the limit. __|_|
We may describe hI for an "F-injective I in more familiar terms.
Lemma 13.4.7. If I = L[U] there is a cofibre sequence
"EF ^ S0[U] -! hI -! -1F (EF+; S0[U]):
Proof: By 13.4.3 we have a map
hI ^ EF+ = -1EF+ ^ S0[U] -! -1S0[U];
and hence, by adjunction a map hI -! -1F (EF+; S0[U]). By construction this is*
* an
F-equivalence, so that the fibre is an F-contractible spectrum. The result fol*
*lows pro-
vided we check that the map hI -! -1F (EF+; S0[U]) is an isomoprhism in negative
dimensional homotopy. This is easily verified by adjunction, since diagrams Sk *
*-! hI -!
-1F (EF+; S0[U]) correspond to diagrams Sk ^ EF+ -! hI ^ EF+ -! -1S0[U]. __|_|
*