THE STABLE HOMOTOPY CATEGORY
HAS A UNIQUE MODEL AT THE PRIME 2
STEFAN SCHWEDE
1. Introduction
The stable homotopy category has been extensively studied by algebraic topol*
*ogists for a long time.
For many applications it is convenient or even necessary to work with point set*
* level models of spectra as
opposed to working up-to-homotopy, and the outcome of a calculation can depend *
*on the choice of model.
In recent years many new models for the stable homotopy category have been cons*
*tructed. It is especially
useful to have the structure of a closed model category in the sense of Quillen*
* [Qui] and many examples
of spectra categories fit into this context [BF , Rob87, EKMM , HSS, Lyd, MMSS*
* ]. Moreover all known
examples capture the `same homotopy theory' - in technical terms one speaks of *
*Quillen equivalent model
categories [Hov, Def. 1.3.12]. Hence not only the homotopy categories, but also*
* higher order information
such as Toda brackets, homotopy colimits and homotopy types of function spaces *
*coincide. In two Quillen
equivalent model categories the answer to every homotopy theoretic question com*
*es out the same.
In a model category one can pass to the associated homotopy category by form*
*ally inverting the class
of weak equivalences. However, passage to the homotopy category loses informat*
*ion and in general the
`homotopy theory' can not be recovered from the homotopy category, see 2.1 and *
*2.2 for two examples. In
this paper we show that in contrast to the general case, the stable homotopy ca*
*tegory completely determines
the stable homotopy theory 2-locally. We prove a uniqueness theorem which says *
*that there is essentially
only one model category structure underlying the stable homotopy category of 2-*
*local spectra _ the stable
homotopy category has no `exotic' models at the prime 2.
We call a pointed model category stable if it is cocomplete and the loop and*
* suspension functors defined
on its homotopy category are inverse equivalences. The homotopy category of a *
*stable model category
is naturally triangulated with suspension and cofibration sequences defining th*
*e shift operator and the
distinguished triangles [Hov, Prop. 7.1.6].
Main Theorem: Let C be a stable model category. If the homotopy category o*
*f C and the 2-local
homotopy category of spectra are equivalent as triangulated categories, then th*
*ere exists a Quillen equivalence
between C and the 2-local model category of spectra.
We prove a stronger form of the main theorem as Theorem 3.5 below. The stron*
*ger version says that
already the subcategory of finite 2-local spectra determines the model category*
* structure up to Quillen
equivalence of model categories. In particular there is only one way to `comple*
*te' the homotopy category
of finite 2-local spectra to a triangulated category with infinite coproducts _*
* as long as some underlying
model structure exists. This gives a partial answer to Margolis' Uniqueness Con*
*jecture [Mar, Ch. 2 x1] for
the stable homotopy category, see Corollary 3.7.
The proof of the main theorem relies on the following characterization of se*
*lf-equivalences of the 2-local
stable homotopy category:
Theorem: Let F be an exact endofunctor of the homotopy category of finite 2-*
*local spectra. If F takes
the 2-local sphere spectrum to itself (up to isomorphism), then F is a self-equ*
*ivalence.
This result is a combination of Propositions 3.1 and 3.2 below. To obtain th*
*e same conclusion at an odd
prime, one has to assume in addition that the functor F does not annihilate the*
* first p-torsion element in the
stable homotopy groups of spheres, see Proposition 3.1 for the precise statemen*
*t. The reason that the prime
_____________
Date: May 2, 2000; 1991 AMS Math. Subj. Class.: 55P42, 55U35.
1
2 STEFAN SCHWEDE
2 behaves differently from the odd primes goes back to the `misbehavior' of the*
* mod-2 Moore spectrum that
its identity map has order 4. In other parts of stable homotopy theory this is *
*often a nuisance; for us it is
the key to why we can prove the uniqueness theorem for 2-local spectra. Current*
*ly we have no replacement
for this part of the argument at odd primes, see also Remark 5.1.
Our reference model is the category of spectra in the sense of Bousfield and*
* Friedlander [BF , x2]. This is
probably the simplest model category of spectra and its objects are sequences {*
*Xn}n0 of pointed simplicial
sets together with maps Xn -! Xn+1. Morphisms are given on every level and comm*
*ute strictly with
the structure maps. The weak equivalences are the stable equivalences, i.e., th*
*e morphisms which induce
isomorphisms of stable homotopy groups. For the remaining details of the model *
*structure see [BF , Thm.
2.3]. A variant of the stable model structure is the p-local model structure fo*
*r a prime p; here the weak
equivalences are those morphisms which induce an isomorphism on stable homotopy*
* groups tensored with
Z(p); see [SSa, 4.1] for details on the p-local model structure.
Acknowledgments: This paper owes a lot to several discussions with Mark Maho*
*wald. He first ex-
plained to me why every element in the homotopy groups of spheres is a `higher *
*order Toda bracket' of
Adams filtration one elements; this is now a main ingredient of the inductive a*
*rgument in Lemma 4.1.
2.Background and related results
2.1. A triangulated category with several models. In general, the triangulated *
*homotopy category
does not determine the Quillen equivalence type of a model category. As an exam*
*ple we consider the n-th
Morava K-theory spectrum K(n) for a fixed prime and some number n > 0. By a the*
*orem of Robinson
[Rob89] this spectrum admits the structure of an A1 -ring spectrum and so its m*
*odule spectra form a stable
model category. The ring of homotopy groups of K(n) is the graded field Fp[vn; *
*v-1n] with vn of dimension
2pn - 2. Hence the homotopy group functor establishes an equivalence between th*
*e homotopy category of
K(n)-module spectra and the category of graded Fp[vn; v-1n]-modules.
Similarly the homology functor establishes an equivalence between the derive*
*d category of the graded field
Fp[vn; v-1n] and the category of graded Fp[vn; v-1n]-modules. This derived cate*
*gory arises from a model cat-
egory structure on differential graded Fp[vn; v-1n]-modules with weak equivalen*
*ces the quasi-isomorphisms.
So the two stable model categories of K(n)-module spectra and DG-modules over F*
*p[vn; v-1n] have equiv-
alent triangulated homotopy categories. On the other hand they are not Quillen *
*equivalent: if they were,
then the homotopy types of the function spaces would agree [DK , Prop. 5.4]. B*
*ut for DG-modules all
function spaces are products of Eilenberg-MacLane spaces, which is not the case*
* for K(n)-modules.
2.2. Franke's algebraic model for the E(n)-local stable homotopy category. In [*
*Fra], Franke
constructs an exotic model for the homotopy category of E(n)-local spectra at a*
* `large' prime. Earlier
Bousfield [Bou] had given an algebraic description of the isomorphism classes o*
*f K-local spectra at an
odd prime. However Bousfield could not determine whether his algebraic model de*
*scribes the morphisms
between the spectra correctly. As one application of a general uniqueness theo*
*rem [Fra, Sec. 2.2 Thm.
5], Franke provides an algebraic derived category which is equivalent, as a tri*
*angulated category, to the
homotopy category of K-local spectra, see [Fra, Sec. 3.1 Thm. 6]. Franke's uniq*
*ueness theorem applies more
generally to the homotopy categories of E(n)-local spectra whenever n2+ n < 2p *
*- 2. Hence he obtains
an equivalence of triangulated categories between the derived category of an ab*
*elian model category and
the homotopy category of E(n)-local spectra, see [Fra, Sec. 3.5 Thm. 10]. By th*
*e same reasoning as in 2.1,
these two kinds of model categories are not Quillen equivalent, see also [Fra, *
*Sec. 3.1 Rem. 1].
We currently do not know whether there exist exotic models for the stable ho*
*motopy category at an
odd prime. Via Proposition 3.1 this problem reduces to a question about the ff*
*1-map. Franke's exotic
equivalences are relevant for these considerations since the ff1-map survives K*
*-localization.
2.3. Reduction to~`exotic sphere spectra'. Suppose that C is a stable model cat*
*egory and let
: Ho(Spectra) -=!Ho(C)be an equivalence of triangulated categories. Then the *
*image of the sphere
spectrum P = (S0) is a small weak generator (see [SSa, Def. 3.1]) of the homoto*
*py category of C. In
[SSb], B. Shipley and the author associate to an object P of a stable model cat*
*egory a ring spectrum
End C(P) called the endomorphism ring spectrum. The ring of homotopy groups of *
*EndC(P) is isomorphic
THE STABLE HOMOTOPY CATEGORY AT 2 *
* 3
to the graded ring of self-maps of P in the homotopy category of C. If P is a s*
*mall weak generator then we
also show that the model category C is Quillen equivalent to the category of mo*
*dules over the endomorphism
ring spectrum EndC(P).
The original equivalence establishes an isomorphism between the ring sss*of*
* stable homotopy groups of
spheres and the graded ring of self-maps of P in Ho(C); this in turn is isomorp*
*hic to the homotopy groups
of the ring spectrum EndC(P). Since is an exact functor, the isomorphism betwe*
*en sss*and ss*EndC(P)
also preserves Toda brackets. Hence the endomorphism ring spectrum EndC(P) look*
*s very much like the
sphere spectrum, and the original model category C is Quillen equivalent to the*
* category of spectra if and
only if the unit map S0 -! EndC(P) of the endomorphism ring spectrum is a stabl*
*e equivalence. In other
words: the question whether there are exotic models for the stable homotopy cat*
*egory can be reduced to
the question about the existence of `exotic sphere spectra', i.e., ring spectra*
* which are not equivalent to
the sphere spectrum, but whose derived category is equivalent to the stable hom*
*otopy category. While this
reduction gives a better idea of what possible exotic models look like, we will*
* not use the results of [SSb]
here and rather prove our uniqueness theorem directly.
2.4. An integral uniqueness result assuming additional structure. The homotopy *
*category of a
stable model category admits additional structure which one can take into accou*
*nt when proving a unique-
ness result. The homotopy category of every model category admits an action of *
*the homotopy category of
simplicial sets [Hov, Thm. 4.3.4]. If the model category is stable, then this a*
*ction induces an action of the
graded ring sss*of stable homotopy groups of spheres, see [SSa, 2.3].
In [SSa] B. Shipley and the author show that with this extra structure the s*
*table homotopy category
determines the model category structure up to Quillen equivalence. More precise*
*ly we show that if C is a
stable model category and if the homotopy category of C admits a sss*-linear eq*
*uivalence to the homotopy
category of spectra, then C is Quillen equivalent to the Bousfield-Friedlander *
*stable model category of
spectra. Hence the result of the present paper is a strengthening of the Unique*
*ness Theorem of [SSa], at
least 2-locally. While [SSa] mainly depends on model category arguments, we hav*
*e to use more information
about the structure of the stable homotopy category here.
3.Proof of the 2-local uniqueness theorem
In this section we deduce our main theorem from other results which should b*
*e of independent interest.
The first two results concern properties of the stable homotopy category. Propo*
*sition 3.1 is an elaboration
on the idea that the stable homotopy groups of spheres are generated under `hig*
*her order Toda brackets'
by the elements of Adams filtration one (see [Coh] for a precise formulation of*
* this fact). For a prime p
the mod-p Adams filtration of a map of spectra is the largest number n such tha*
*t the map can be factored
as a composite of n maps all of which induce the trivial map on mod-p cohomolog*
*y. When the prime
is understood we simply speak of the filtration of a map. Adams showed [Ada] t*
*hat the only positive
dimensional elements in ss*S0(2)which have filtration one are multiples of the *
*Hopf maps j; and oe in
dimensions 1, 3 and 7 respectively. For odd primes the only such elements are *
*in the first non-trivial
p-torsion homotopy group ss2p-3S0(p), see [Liu, Thm. 1.2.1].
An exact functor between triangulated categories is an additive functor F wh*
*ich commutes with shift and
preserves distinguished triangles. More precisely: F is endowed with a natural *
*isomorphism X : F(X[1]) ~=
F(X)[1] such that for every distinguished triangle (homotopy cofibre sequence)
X - -f---!Y ---g--!Z - -h---!X[1]
the sequence
F(X) --F(f)---!F(Y ) --F(g)---!F(Z) -XOF(h)-----!F(X)[1]
is again a distinguished triangle. An equivalence of triangulated categories is*
* an equivalence of categories
which is exact and whose inverse functor is also exact. In what follows square*
* brackets [-; -] denote
morphisms in the homotopy category of spectra, possibly graded when decorated w*
*ith a subscript.
4 STEFAN SCHWEDE
Proposition 3.1.Let p be a prime number and let F be an exact endofunctor of th*
*e homotopy category of
finite p-local spectra which takes the p-local sphere spectrum to itself (up to*
* isomorphism). If every element
of Adams filtration one in the graded endomorphism ring [F(S0(p)); F(S0(p))]* i*
*s in the image of F, then F
is a self-equivalence.
The next result says that the prime 2 is special because the Hopf maps are a*
*lways taken care of. We do
not know whether the analogue of the following proposition is true for odd prim*
*es, see also Remark 5.1.
Proposition 3.2.Let F be an exact endofunctor of the homotopy category of finit*
*e 2-local spectra which
takes the 2-local sphere spectrum to itself (up to isomorphism). Then all maps *
*of Adams filtration one in
the graded endomorphism ring [F(S0(2)); F(S0(2))]* are in the image of F.
We prove Propositions 3.1 and 3.2 in Sections 4 and 5 respectively.
In order to state the next auxiliary result we recall the notion of a compac*
*tly generated triangulated
category. An object A of a triangulated category T is called compact (also call*
*ed small or finite) if the
representable functor T (A; -) preserves infinite coproducts. A full subcategor*
*y S of a triangulated category
T is closed under extensions if whenever two of the objects X; Y and Z in a dis*
*tinguished triangle
X - ----! Y -----! Z - ----! X[1]
belong to S, the third object also belongs to S (this implies that S contains t*
*he zero objects and is closed
under isomorphisms, finite sums and shift in either direction). A triangulated*
* category T is compactly
generated if T is the only subcategory which contains the compact objects and i*
*s closed under extensions
and infinite coproducts. The compact objects of the stable homotopy category a*
*re precisely the finite
spectra, and similarly for the p-local stable homotopy category. The stable ho*
*motopy category and its
full subcategory of p-local spectra are compactly generated. However, there are*
* triangulated categories in
which the zero objects are the only small objects _ see [HS , Cor. B.13] for ex*
*amples which are Bousfield
localizations of the stable homotopy category.
The following lemma is well-known and we give the easy proof at the end of S*
*ection 4.
Lemma 3.3. Let F be an exact functor between compactly generated triangulated *
*categories with infinite
coproducts. If F preserves coproducts and restricts to an equivalence between t*
*he full subcategories of compact
objects, then F is an equivalence.
Finally, we quote a result from [SSa] which is entirely model category theor*
*etic. It roughly says that the
model category of spectra is the `free stable model category on one object'. He*
*re spectra are understood in
the sense of Bousfield and Friedlander, endowed with the stable model structure*
* of [BF , 2.3]. In particular
the weak equivalences are those maps which induce isomorphisms of stable homoto*
*py groups. The p-local
model structure is the localization of the stable model structure of spectra in*
* which the weak equivalences
are the maps inducing an isomorphism of stable homotopy groups tensored with Z(*
*p), see [SSa, 4.1] for
details. A left Quillen functor is a functor between model categories which has*
* a right adjoint and which
preserves cofibrations and acyclic cofibrations.
Proposition 3.4.[SSa, 5.1] Let C be a stable model category and X a cofibrant a*
*nd fibrant object of C.
Then there exists a left Quillen functor from the category of spectra to C whic*
*h takes the sphere spectrum
to X. If the endomorphism ring of X in the homotopy category of C is a Z(p)-alg*
*ebra, then the functor is
also a left Quillen functor with respect to the p-local stable model structure *
*for spectra.
Now we can state and prove a uniqueness result which has the Main Theorem of*
* the introduction as a
special case. This version is stronger since the hypothesis only refer to the f*
*ull subcategory of compact, or
finite, objects in the triangulated homotopy category.
Theorem 3.5. Let C be a stable model category whose homotopy category is compac*
*tly generated. Suppose
that the full subcategory of compact objects in the homotopy category of C and *
*the homotopy category of finite
2-local spectra are equivalent as triangulated categories. Then there exists a *
*Quillen equivalence between C
and the 2-local model category of spectra, whose left adjoint ends in C.
THE STABLE HOMOTOPY CATEGORY AT 2 *
* 5
Proof.Let be an equivalence of triangulated categories from the homotopy categ*
*ory of finite 2-local
spectra to the compact objects in the homotopy category of C. We choose a cofib*
*rant and fibrant object
X of C which is isomorphic to (S0(2)) in the homotopy category of C. Propositio*
*n 3.4 yields a left Quillen
functor, with respect to the 2-local stable model structure, from the category *
*of spectra to C which takes
the sphere spectrum to X. We denote the functor by X ^ -. This left Quillen fun*
*ctor has an exact total
left derived functor X ^L- on the level of homotopy categories (see [Qui, I.4 P*
*rop. 2] or [Hov, Prop. 6.4.1]).
The derived functor X ^L - takes the localized sphere spectrum to the compac*
*t object X, hence it
takes compact objects to compact objects; we denote by (X ^L -)|smallthe restri*
*ction to finite 2-local
spectra. The composite functor F = -1 O (X ^L -)|smalltakes the 2-local sphere *
*spectrum to itself, up
to isomorphism, so by Propositions 3.1 and 3.2 it is a self-equivalence of the *
*finite 2-local stable homotopy
category. Since F and -1 are equivalences of categories, so is (X ^L -)|small. *
*By Lemma 3.3, the functor
X ^L - is then an equivalence of categories, so the left Quillen functor X ^ - *
*and its right adjoint_are in
fact a Quillen equivalence [Hov, Prop. 1.3.13]. *
* |__|
Warning: The equivalence takes the 2-local sphere spectrum to the object X,*
* and the same is true
for the left derived functor X ^L -. If there was a natural transformation betw*
*een and (X ^L -)|small
which induces an isomorphism at S0(2), then the natural transformation would be*
* a natural isomorphism, so
X ^L - would also be an equivalence on compact objects. However, there is no re*
*ason why such a natural
transformation should exist, and so there is no a priori reason why X ^L - shou*
*ld be an equivalence.
In particular we do not claim that the left Quillen equivalence X ^- lifts t*
*he triangulated equivalence .
Hence we leave open the question whether there are exotic self-equivalences of *
*the 2-local stable homotopy
category, i.e., self-equivalences not induced from a Quillen equivalence (or wh*
*at is the same: self-equivalences
other than iterated (de-)suspensions).
Remark 3.6. (Margolis' Uniqueness Conjecture) In `Spectra and the Steenrod alg*
*ebra', H. R. Mar-
golis introduces a set of axioms for a stable homotopy category [Mar, Ch. 2 x1]*
*. The stable homotopy
category of spectra satisfies the axioms, and Margolis conjectures [Mar, Ch. 2,*
* x1] that this is the only
model, i.e., that any category which satisfies the axioms is equivalent to the *
*stable homotopy category.
As part of the structure Margolis requires a triangulation, infinite coprodu*
*cts and that the whole category
be generated by a single compact object. Furthermore, Margolis' Axiom 5 asks fo*
*r an equivalence between
the full subcategory of compact objects and the Spanier-Whitehead category of f*
*inite CW-complexes. So
the Uniqueness Conjecture really concerns possible `completions' of the categor*
*y of finite spectra to a trian-
gulated category with infinite coproducts. Margolis also assumes the existence *
*of a compatible symmetric
monoidal smash product, but the smash product does not enter into our present c*
*onsiderations.
Margolis shows [Mar, Ch. 5 Thm. 19] that modulo phantom maps each model of h*
*is axioms is equivalent
to the standard model. Moreover, Christensen and Strickland show in [CS] that i*
*n any model the ideal of
phantom maps is equivalent to the phantoms in the standard model.
One can consider a 2-primary analog of Margolis' Uniqueness Conjecture by mo*
*difying his Axiom 5 and
instead requiring the full subcategory of compact objects in the stable homotop*
*y category to be equivalent,
as a triangulated category, to the homotopy category of finite 2-local spectra.*
* Our main theorem proves
the following 2-primary analog of Margolis' Uniqueness Conjecture for stable ho*
*motopy categories with a
model:
Corollary 3.7.Suppose that S is a 2-primary stable homotopy category (in the se*
*nse of [Mar, Ch. 2 x1])
which is equivalent, as a triangulated category, to the homotopy category of a *
*stable model category. Then
S is triangulated equivalent to the homotopy category of 2-local spectra.
Note that we do not assume any internal smash product on the model category,*
* and the corollary does
not give that the equivalence between S and the stable homotopy category of 2-l*
*ocal spectra preserves the
smash products.
6 STEFAN SCHWEDE
4.A characterization of self-equivalences of the stable homotopy category
In this section we prove Proposition 3.1. Throughout, p denotes any prime an*
*d F is an exact endofunctor
of the homotopy category of finite p-local spectra. We assume further that F t*
*akes the p-local sphere
spectrum to itself (up to isomorphism) and that all filtration one maps of posi*
*tive dimension from F(S0(p))
to itself are in the image of F. We want to show that F is then a self-equivale*
*nce.
Cohomology will always be spectrum cohomology with mod-p coefficients. If K *
*is a finite spectrum we
denote by fi(K) (resp. o(K)) the smallest (resp. largest) dimension in which th*
*e mod-p cohomology of K
is non-trivial. As before square brackets [-; -] denote morphisms in the homoto*
*py category of spectra,
possibly graded when decorated with a subscript.
Lemma 4.1. Suppose that the map of graded rings [S0(p); S0(p)]* -! [F(S0(p)); *
*F(S0(p))]* induced by the
functor F is an isomorphism below and including dimension n for some n 0.
(1)Let K and L be two finite p-local spectra. Then the map F : [K; L] ----! *
*[F(K); F(L)] is an
isomorphism if o(K) - fi(L) < n and an epimorphism if o(K) - fi(L) =*
* n.
(2)Let K be a finite p-local spectrum satisfying o(K) - fi(K) n + 1. Then *
*there exists a finite p-local
spectrum K0with fi(K0) fi(K) and o(K0) o(K) and an isomorphism K ~=F(K0)*
* in the homotopy
category of spectra.
(3)Every map from F(Sn+1(p)) to F(S0(p)) of Adams filtration at least two is *
*in the image of F.
Proof.(1) When K and L are localized sphere spectra, the claim hold by assumpti*
*on. The general case is
obtained by cell inductions for K and L.
We first prove the claim when L is a wedge of localized sphere spectra of a *
*fixed dimension using induction
on the total dimension of the mod-p cohomology of K. If H*K is trivial, then K *
*is contractible and the
statement is true. Otherwise we can pinch off the top cells of K, i.e., we can *
*choose a distinguished triangle
_ ff _
So(K)-1----! M ----! K ----! So(K) (*)
I I
with M a finite p-local spectrum with o(M) < o(K) and with strictly smaller coh*
*omology. Hence by
induction the claim is true for M and L. Taking homomorphism groups [-; L] from*
* the triangle (*) gives
a long exact sequence of abelian groups. The functor F preserves distinguished*
* triangle, and so taking
homomorphism from the image sequence into F(L) yields a similar exact sequence *
*and F gives a map
between the sequences. Using that L is a wedge of spheres and that the claim ho*
*lds for M with estimate
increased by one, the five lemma proves the statement for K and this special L.
Now we do a similar induction on the dimension of H*L. This time we collapse*
* the bottom cells of L,
i.e., we embed L in a triangle
_ _
Sfi(L)----!L ----!L0----! Sfi(L)+1
J J
where the dimension of H*L0is strictly smaller than that of L and fi(L0) > fi(L*
*). By induction the claim
holds for the spectra K and L0, and using the five lemma and the previous parag*
*raph we deduce it for K
and L.
(2) We argue by induction on the difference o(K) - fi(K). If the cohomology *
*of K is concentrated in
at most one dimension, then K is equivalent to a (possibly empty) wedge of loca*
*lized spheres and the
statement is true. Otherwise there exists a distinguished triangle (*) as in pa*
*rt (1) with M a finite p-local
spectrum which satisfies o(M) < o(K) and fi(M) = fi(K). By induction there exis*
*ts a finite spectrum
M0 with fi(M0) fi(M); o(M0) o(M) and an isomorphism between F(M0) and M. By p*
*art (1) the
composite
_ ~= _ ff ~=
F( So(K)-1) ----! So(K)-1----! M ----! F(M0)
I I
W
is of the form F(ff0) for some ff02 [ ISo(K)-1; M0]. We let K0 be some mapping*
* cone of the map ff0,
obtained by embedding ff0in a distinguished triangle. Then we have the inequali*
*ties fi(K0) fi(M0)
THE STABLE HOMOTOPY CATEGORY AT 2 *
* 7
fi(M) = fi(K) and o(K0) max{o(M0); o(K)} = o(K). We end up with a diagram
_ _
So(K)-1 ________Mwff__________Kw _________w So(K)
I | | I
| | ||| |
| | ||| |
| | | |
~|= |~= ||| |~=
| | || |
| | ||| |
_ |u ||u ||u _ |u
F( So(K)-1)____F(M0)w________wF(K0)______wF( So(K))
I F(ff0) I
in which both rows are distinguished triangles and the left square commutes. He*
*nce we can choose a map
K -! F(K0) which makes the entire diagram commute, and this map is the isomorph*
*ism we are looking
for.
(3) We first claim that every map F(Sn+1(p)) -! F(S0(p)) of filtration at le*
*ast two can be factored through
a spectrum of the form F(K0) where K0 is a finite p-local spectrum with cohomol*
*ogy concentrated in
dimensions 1 through n. Both maps in such a factorization are in the image of F*
* by part (1), hence the
original map is also in the image.
Except for the use of the functor F the claim is a well-known argument - it *
*is e.g. used by Cohen [Coh,
Thm._4.2]_who_attributes_it to Adams. We choose an isomorphism between F(S0(p))*
* and S0(p)and denote
by HZ (p)and HFp the fibers of the composite Hurewicz maps
~= 0 Hurewicz 0 ~= 0 Hurewicz
F(S0(p)) ----! S(p)------! HZ (p)and F(S(p)) ----! S(p)------! HFp
to the p-local and mod-p Eilenberg-MacLane_spectra_respectively. A map to F(S0(*
*p)) has Adams filtration
two if and only if it lifts to the fiber HFp and if some (hence any) such lift *
*is trivial in mod-p cohomology.
Now let ff : F(Sn+1(p)) -! F(S0(p)) be a map of filtration at least two. Sin*
*ce n+ 1 > 0, ff lifts to a map
___
F(Sn+1(p)) -! HZ (p). Furthermore the lift induces the trivial map in mod-p co*
*homology since the map
_____ ___
HZ (p)-! HFp is surjective in mod-p cohomology.
Since F(Sn+1(p)) is an (n+1)-dimensional p-local sphere, the lift of ff fact*
*ors through the (n+1)-dimensional
___
skeleton_of the spectrum HZ (p). Since in addition ff is trivial in mod-p cohom*
*ology and the cohomology of
HZ _(p)is_acyclic with respect to the Bockstein operator, the lift can even be *
*factored through the n-skeleton
of HZ (p). This n-skeleton is a finite p-local spectrum with mod-p cohomology c*
*oncentrated in dimensions
1 through n. By part (2) of this lemma there exists a finite p-local_spectrum_*
*K0 with cohomology in
dimensions 1 through n and an isomorphism between the n-skeleton of HZ (p)and F*
*(K0),_which proves the
claim. *
* |__|
Now we can give the
Proof of Proposition 3.1.Suppose F is an exact endofunctor of the homotopy cate*
*gory of finite p-local
spectra which takes the p-local sphere spectrum to itself, up to isomorphism. F*
*urthermore all filtration
one maps of positive dimension from F(S0(p)) to itself are in the image of F. W*
*e claim that the map of
graded rings F : [S0(p); S0(p)]* -! [F(S0(p)); F(S0(p))]* is an isomorphism. Si*
*nce F(S0(p)) is isomorphic to S0(p)
the map is necessarily an isomorphism in non-negative dimensions, and in positi*
*ve dimensions both sides
of the map are finite groups of the same order. Suppose the claim was false and*
* let m > 0 be the smallest
dimension in which F is not bijective, hence not surjective. By Lemma 4.1 (3), *
*any element not in the
image has filtration one, which contradicts the assumptions. Hence the above ma*
*p is indeed bijective in all
dimensions.
8 STEFAN SCHWEDE
So the hypothesis of Lemma 4.1 is satisfied for arbitrarily large n; conclus*
*ion (1) of that Lemma shows
that F is full and faithful and conclusion (2) shows that F is surjective on is*
*omorphism classes._Thus F is
an equivalence of categories. *
* |__|
It remains to prove Lemma 3.3 which allows us to detect an equivalence of tr*
*iangulated categories on
compact objects.
Proof of Lemma 3.3.Let F : S -! T be an exact functor between compactly genera*
*ted triangulated
categories with infinite coproducts. Suppose that F preserves coproducts and re*
*stricts to an equivalence
between the full subcategories of compact objects. We want to show that F itsel*
*f is an equivalence.
We fix a compact object A of S and consider the full subcategory of S consis*
*ting of those Y for which
the map
F : S(A; Y ) ----!T (F(A); F(Y ))
is bijective. By assumption this subcategory contains all compact objects. Sinc*
*e F is exact, the subcategory
is closed under extensions. Since A and F(A) are compact and F preserves coprod*
*ucts, this subcategory is
also closed under coproducts. Since S is compactly generated, the map F : S(A; *
*Y ) -! T (F(A); F(Y )) is
thus bijective for all compact A and arbitrary Y .
Similarly for arbitrary but fixed Y the full subcategory of S consisting of *
*those X for which the map
F : S(X; Y ) -! T (F(X); F(Y )) is bijective is closed under extensions, coprod*
*ucts and contains the
compact objects. Hence this subcategory coincides with S which means that F is *
*full and faithful.
Now we consider the full subcategory of T of objects which are isomorphic to*
* an object in the image of
F. By assumption this category contains all compact objects, and it is closed u*
*nder coproducts since these
are preserved by F. We claim that this subcategory is also closed under extensi*
*ons. Since T is compactly
generated this shows that F is essentially surjective and hence an equivalence.
To prove the last claim we consider a distinguished triangle
X ---f-!Y ----! Z ----! X[1] :
Since the subcategory under consideration is closed under isomorphism and shift*
* in either direction we can
assume that X = F(X0) and Y = F(Y 0) are objects in the image of F. Since F is *
*full there exists a map
f0: X0-! Y 0satisfying F(f0) = f. We can then choose a mapping cone for the map*
* f0 and a compatible_
map from Z to F(Cone(f0)) which is necessarily an isomorphism. *
* |__|
5.Taking care of the Hopf maps
In this final section we prove Proposition 3.2. Here F denotes an exact endo*
*functor of the homotopy
category of 2-local spectra which takes the 2-local sphere spectrum to itself (*
*up to isomorphism). We want
to show that all maps of Adams filtration one in the graded endomorphism ring [*
*F(S0(2)); F(S0(2))]* are in the
image of F. We introduce a slight abuse of notation: after choosing an isomorph*
*ism between F(S0(2)) and
the 2-local sphere spectrum we identify the ring [F(S0(2)); F(S0(2))]* with the*
* ring of 2-local stable homotopy
groups of spheres (this identification does not depend on the choice of isomorp*
*hism). With this convention
we have to show that the Hopf maps j; and oe are in the image of F in dimensio*
*ns 1, 3 and 7 respectively.
We start by showing that the map F(j) is non-trivial. Since F is exact, both*
* rows in the diagram
F(S0(2))____wF(S0(2))x2_F(M(2))w_____F(S1(2))w
~ | |~ ||| |~
= | |= || |=
|u |u |u |u
S0(2)______wS0(2)x2_____M(2)w________S1(2)w
are distinguished triangles (here M(2) denotes the mod-2 Moore spectrum). Since*
* the left square commutes
we can choose a map between F(M(2)) and M(2) making the entire diagram commute,*
* and this map is
necessarily an isomorphism.
THE STABLE HOMOTOPY CATEGORY AT 2 *
* 9
The identity map of the Moore spectrum M(2) has additive order 4. Since F(M(*
*2)) is isomorphic to
M(2), its identity map also has order 4. Since F is additive, it does not annih*
*ilate the degree 2 map of the
Moore spectrum. On the other hand this degree 2 map factors as the composite
M(2) -pinch---!S1 --j--!S0 -incl.---!M(2) :
Hence F(j) has to be non-zero, which forces F(j) = j.
Because of the relation 4 = j3 (see e.g. [Tod, Theorem 14.1 (i)]) we know th*
*at 4 . F() = F(4) =
F(j3) = j3 = 4. Since generates the cyclic group ss3S0(2)~=Z=8, we conclude th*
*at F() = u . with u a
2-local unit. Hence is in the image of F.
For the last Hopf map oe we exploit the Toda bracket relation 8oe = <; 8; > *
*(see e.g. [Tod, Lemmas 5.13
and 5.14]) which holds without indeterminacy since the fourth stable homotopy g*
*roup of spheres is trivial.
Since F preserves triangles it takes threefold Toda brackets to threefold Toda *
*brackets and we obtain
8 . F(oe) = F(8oe) = F(<; 8; >) = u2. <; 8; > = 8u2. *
*oe :
Since the latter Toda-bracket has no indeterminacy we conclude that 8 . F(oe) =*
* 8u2 . oe. Again since oe
generates the cyclic group ss7S0(2)~=Z=16, we conclude that F(oe) coincides wit*
*h oe up to a 2-local unit, so
oe is in the image of F. This finishes the proof of Proposition 3.2.
Remark 5.1. The fact that multiplication by 2 is non-trivial on the mod-2 Moor*
*e spectrum is equivalent
to the fact that M(2) does not admit a product, i.e, there is no map M(2) ^ M(2*
*) -! M(2) in the
stable homotopy category which splits the two inclusions j ^ id; id^ j : M(2) -*
*! M(2) ^ M(2), where
j : S0 -! M(2) is the inclusion of the bottom cell.
For an odd prime p the Moore spectrum M(p) admits a unique and commutative p*
*roduct, which is
also associative for p 5. Hence M(p) is a ring spectrum in the stable homotopy*
* category. However, the
product on M(p) can not be made associative up to `coherent higher homotopy', i*
*.e., M(p) does not admit
the structure of an A1 -ring spectrum. In fact the element ff1 2 ss2p-3S0(p)is *
*the obstruction to p-th order
homotopy associativity. One could try to use this to detect ff1 and to prove th*
*e odd primary version of
Proposition 3.2.
For p = 3 the associativity obstruction can be described more concretely, an*
*d we explain why we expect
that the odd primary uniqueness problem cannot be decided by only looking at pr*
*oducts on the Moore
spectrum. Let : M(3) ^ M(3) -! M(3) denote the unique and commutative multipli*
*cation on the mod-3
Moore spectrum. This product is not associative and in fact the associator O (*
* ^ id) - O (id^ ) :
M(3) ^ M(3) ^ M(3) -! M(3) factors as the composite
M(3) ^ M(3) ^ M(3) --pinch----!S3 --ff1--!S0 ----j--! M(3) :
Suppose F is an exact endofunctor of the homotopy category of 3-local spectra w*
*hich takes the 3-local
sphere spectrum to itself. The same argument as for p = 2 yields an isomorphism*
* f : F(M(3)) -! M(3)
under the isomorphism F(S0(3)) ~=S0(3), which is unique since the first stable *
*homotopy groups of spheres
has no 3-torsion.
The multiplication splits the inclusion id^ j : M(3) -! M(3) ^ M(3), so the*
* smash product of two
copies of the mod-3 Moore spectrum is isomorphic to the wedge M(3) _M(3). Simil*
*arly for three smash
factors there is an isomorphism
~= 2
M(3) ^ M(3) ^ M(3) ----! M(3) _M(3) _M(3) _ M(3)
which we can choose in such a way that the projection onto the first summand is*
* the map O ( ^ id) :
M(3) ^ M(3) ^ M(3) -! M(3). The exact functor F commutes with wedge and suspens*
*ion, so together
with (suspensions of) the isomorphism f : F(M(3)) ~=M(3) we obtain an isomorphi*
*sm
__
f : F(M(3) ^ M(3) ^ M(3)) ----!M(3) ^ M(3) ^ M(3)
10 STEFAN SCHWEDE
such that the diagram _
F(M(3) ^ M(3) ^ M(3))____M(3)w^fM(3) ^ M(3)
F(O(^id)|) |O(^id)
| |
|u |u
F(M(3)) _________________M(3)wf
*
* __
commutes. However, it is not clear at this point if the choices can be made so *
*that the isomorphisms f
and f also compatible with the map O (id^ ) : M(3) ^ M(3) ^ M(3) -! M(3) which*
* gives the other
summand in the associator. Hence it is conceivable that the functor F annihilat*
*es the associator, and if
that happens, then F does not detect ff1.
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