.
Theorem 2.1.3. Let C be a stable homotopy category.
(a) Suppose that X is small (respectively F -small, or strongly dualizable) an*
*d Y
is strongly dualizable. Then X ^ Y is also small (or F -small, or strong*
*ly
dualizable). Thus, the categories of small, F -small, and strongly dualiz*
*able
objects are all G-ideals.
We also have the following implications.
(b) G-finite ) strongly dualizable , F -small.
(c) If C is algebraic, then small , G-finite ) strongly dualizable , F -small.
(d) If C is unital algebraic, then small , G-finite , strongly dualizable , F -
small.
(e) If C is algebraic, any G-ideal of small objects is closed under the Spanie*
*r-
Whitehead duality functor D.
Proof.(a): Suppose that X is small and Y is strongly dualizable. We then have
a a
[X ^ Y; Zi]= [X; DY ^ Zi]
a
= [X; DY ^ Zi]
M
= [X; DY ^ Zi]
M
= [X ^ Y; Zi]:
AXIOMATIC STABLE HOMOTOPY THEORY 15
Thus X ^Y is small. The proof when X is F -small is similar. Now suppose instead
that X is strongly dualizable. We find that
F (X ^ Y; Z)= F (X; F (Y; Z))
= DX ^ DY ^ Z
= F (X; DY ) ^ Z
= D(X ^ Y ) ^ Z:
Thus, X ^ Y is strongly dualizable. It is easy to see that the categories of sm*
*all,
F -small and strongly dualizable objects are all thick; as they are closed under
smashing with a strongly dualizable object, they are in fact G-ideals.
(b): The category of strongly dualizable objects is thick and contains G, so *
*it
contains all G-finite objects. If X is strongly dualizable then
a a a a
F (X; Yi) = DX ^ Yi= DX ^ Yi= F (X; Yi):
Thus X is F -small.
Conversely, suppose that X is F -small, and let D be the category of those Y *
*for
which the natural map DX ^ Y -! F (X; Y ) is an isomorphism. Using part (e) of
Theorem A.2.5, we see that every strongly dualizable object lies in D, in parti*
*cular
G D. Moreover, it is easy to see that D is localizing, so D = C. Thus X is
strongly dualizable.
(c): First note that G consists of small objects because C is algebraic, so e*
*very
G-finite object is small. The converse will be proved as Corollary 2.3.12 belo*
*w.
Thus small , G-finite. The other claims in (c) are covered by (b).
(d): Suppose that X 2 C is F -small; it is enough to show that X is small.
This follows`immediately`by applying the homology functor [S; -] to the equality
F (X; Yi) = F (X; Yi).
(e): Now suppose that C is algebraic, D is a G-ideal of small objects, and X *
*2 D.
Then X is strongly dualizable, so by Lemma A.2.6, DX is a retract of DX^X^DX.
In particular, using part (a), we find that DX is small. This means DX is G-fin*
*ite,
by part (c). Since D is a G-ideal, we conclude that DX ^ X ^ DX 2 D,_so
DX 2 D. |__|
Note that this theorem gives evidence that the choice of generators in an alg*
*ebraic
stable homotopy category is not very relevant, although we shall not attempt to
make this precise here.
One might ask whether every G-finite object lies in the triangulated category
generated by G, or whether one really needs to use retractions as well as cofib*
*ers.
Retractions are necessary when C is the derived category of the ring Q x Q (a
connective, semisimple, monogenic Brown category). However, suppose that C is
a connective stable homotopy category, and write R = ss0S. If R is Noetherian
and has finite global dimension, and finitely generated projective R-modules are
free, then every G-finite object lies in the triangulated category generated by*
* G (see
Section 7).
2.2. Weak colimits and limits. In this section, we discuss colimits and limits *
*in
a stable homotopy category. Almost never will actual colimits and limits exist,*
* but
weak versions always exist.
Definition 2.2.1.Let C be a triangulated category. Fix a small category I; we
will write i for a typical object. Given an object X of C, let cX denote the fu*
*nctor
16 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
I -! C which is constant at X. Given a map g :X -! Y in C, let cg: cX -! cY
denote the obvious natural transformation.
Given any functor i 7! Xifrom I to C, we say that a pair (X; o) is a weak col*
*imit
of (Xi) if
1. X is an object of C.
2. o is a natural transformation from (Xi) to cX .
3. Given any natural transformation ae: (Xi) -! cY , there is a map g :X -! Y
so that ae = cgo. Equivalently, the natural map
[X; Y ] -!lim-i[Xi; Y ]
is surjective for all Y .
Note that the map g need not be unique. The pair (X; o) will be called a minimal
weak colimit if the map lim-!iHXi -! HX induced by o is an isomorphism for
all homology functors H :C -! Ab . (This definition is mainly useful when C is
algebraic.)
Our definition of minimal weak colimit is not the obvious generalization of t*
*he
one given in [Mar83 , Chapter 3]. However, our definition is often equivalent t*
*o that
of Margolis. Indeed, we have the following proposition, whose proof we will def*
*er
to Proposition 4.2.1.
Proposition 2.2.2.Let C be a Brown category. Suppose that I is a small category,
i 7! Xi is a functor from I to C, and (oi:Xi -!X) is a weak colimit. Then X is
the minimal weak colimit if and only if the induced map
lim-![Z; Xi]* -![Z; X]*
is an isomorphism for all Z 2 G.
The definition could be modified in various ways when C is not a Brown catego*
*ry.
At present we know few examples in that context; our present definition handles
them better than any of the variants, but that could easily change if more exam*
*ples
come to light.
The question of existence of a minimal weak colimit for a given diagram is su*
*btle
(except when the diagram is a sequence). Even if the diagram can be rigidified *
*in
some underlying closed model category, there is no reason in general that the h*
*omo-
topy colimit should be a weak colimit in the homotopy category. See Theorem 4.2*
*.3
for a result in this direction.
We also make the following more constructive definition.
Definition 2.2.3.Given a sequence
X0 f0-!X1 f1-!X2 f2-!: : :
in a stable homotopy category C, define the sequential colimit to be the cofibe*
*r of
the map
a a
F : Xi-! Xi
that takes the summand Xi to Xiq Xi+1 by 1Xi - fi. This is often called the
telescope of the Xi, but we prefer a more consistent terminology for the various
different kinds of colimits.
AXIOMATIC STABLE HOMOTOPY THEORY 17
In particular, given a self-map f :dX -! X of an object X, we write f-1 X for
the sequential colimit of the sequence
X -f!-dX -f!-2dX : :::
We have the following proposition.
Proposition 2.2.4.Let I be a small category and C a stable homotopy category.
(a) Every functor i 7! Xi from I to C has a weak colimit.
(b) Suppose that (oi:Xi -! X) and (oei:Yi -! Y ) are weak colimits, and that
(ui:Xi -!Yi) is a natural transformation. Then there is a compatible map
u: X -! Y (typically not unique) such that the following diagram commutes:
Xi --oi--!X
? ?
ui?y ?yu
Yi --oei--!Y
(c) Suppose that C is algebraic and that (Xi -! X) is a minimal weak colimit.
Then any compatible map (as in (b)) from X to any other weak colimit X0 is
a split monomorphism, so X is a retract of X0. If X0 is also minimal, then
X is non-canonically isomorphic to X0.
(d) The sequential colimit of a sequence is a minimal weak colimit. In fact, f*
*or
any Y there is a Milnor exact sequence
0 -!lim-1i[Xi; Y ] -![X; Y ] -!lim-i[Xi; Y ] -!0:
(e) If Y 2 C and (oi:Xi-! X) is a (minimal) weak colimit then (oi^1: Xi^Y -!
X ^ Y ) is a (minimal) weak colimit.
(f) Suppose that C is algebraic and D C is a localizing subcategory. If (Xi-!*
* X)
is a minimal weak colimit in C with each Xi2 D, then X 2 D.
Proof.The proof of most of this is the same as for the analogous propositions
in [Mar83 , Chapter 3]. That is, to construct a weak colimit of (Xi), we consid*
*er
the cofiber of the map
a F a
Xdom(ff)-! Xi
ff2mor I i2ob I
where Xdom(ff)maps to Xdom(ff)by the identity and to Xcodom(ff)by -ff. It is ea*
*sy
to verify that this is a weak colimit (proving (a)), but it is almost never min*
*imal.
Part (b) follows easily.
Suppose we have a compatible map u: X -! X0 as in part (c). By (b), we also
have a compatible map v :X0 -! X. Thus vu: X -! X is compatible with the
identity map of (Xi). It follows that for any Z 2 *G, the map vu induces the
identity on [Z; X] = lim-![Z; Xi]. By Lemma 1.4.5 we see that vu is an isomorph*
*ism,
so u is a split monomorphism. If X0 is also minimal then uv is an isomorphism by
the same argument, so u and v are isomorphisms. This proves (c).
Next, consider a sequence (Xi:i 2 N) as in (d), and write X for the sequential
colimit. By applying [-; Y ] to the cofibration which defines X, we obtain a lo*
*ng
exact sequence
a F* a a F* a
[ Xi; Y ] -- [ Xi; Y ]- [X; Y ]- [ Xi; Y ] -- [ Xi; Y ]:
i i i i
18 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
From this we extract a short exact sequence A*+1Q-! [X; Y ]* -!QB*, where A*
and B* are the cokernel and kernel of the map i[Xi; Y ]* -! i[Xi; Y ]*. The*
*se
are by definition just lim1i[Xi; Y ]* and limi[Xi; Y ]*, so we get a Milnor exa*
*ct se-
quence as stated in (d). For the right hand map to be surjective means precisely
that X is a weak`colimitLof the Xi. Now suppose that H is a homology func-
tor, so`that H( Xi)`= iHXi. One can check directly that the induced map
F*: H( Xi) -! H( Xi) is injective, with cokernel lim-!iH(Xi). It follows easi*
*ly
that X is the minimal weak colimit. This proves (d).
Suppose that (oi:Xi-! X) is a weak colimit. We claim that (oi^ 1: Xi^ Y -!
X ^ Y ) is also a weak colimit. To see this, suppose we have compatible maps
Xi^ Y -! Z. By adjunction we get maps Xi -!F (Y; Z); as X is a weak colimit
of the Xi, we get a map X -! F (Y; Z); by adjunction we get a map X ^ Y -! Z.
It is easy to check that this has the required property. Suppose moreover that X
is the minimal weak colimit, and that H is a homology functor. Then H(- ^ Y ) is
also a homology functor, so H(X ^ Y ) = lim-!iH(Xi^ Y ). It follows that X ^ Y *
*is
the minimal weak colimit of the objects Xi^ Y . This proves (e).
By the proof of (a), if each object in a diagram is in a localizing subcatego*
*ry,
then that diagram has a weak colimit that is also in the localizing subcategory*
*. A
minimal weak colimit is a retract of any other weak colimit, so it will also_be*
*_in the
localizing subcategory. This proves (f). |__|
Remark 2.2.5. As observed by Boardman (see [Bou83 ]), the result of part (f) is
true for homotopy colimits. Suppose that C is a stable homotopy category that
arises from a suitable closed simplicial model category. Let {Xi} be a diagram
in this underlying category, such that each object Xi lies in a given localizing
subcategory D C. The claim is that the homotopy colimit (X, say) also lies in
D. Indeed, one can show that X is homotopy equivalent to the sequential colimit
of a sequence X(0) -! X(1) -! : :o:f cofibrations, such that X(k)=X(k - 1) is a
coproduct of suspensions of Xi's.
We pause to prove two useful facts about sequential colimits.
Lemma 2.2.6. If f :dX -! X is an isomorphism, then there is a natural iso-
morphism X -! f-1 X.
` 1
Proof.We will omit the suspensions from this proof. Write Y = i=0X, and let
Xi be the ith copy of X inside Y . Recall that f-1 X is the cofiber of the map
F :Y -! Y that takes the summand Xi to Xiq Xi+1 by (1; -f). Let J :X -! Y
be the inclusion of X0, and let Q: Y -! X be the map that is f-k :X -! X on
Xk. Finally, let G: Y -! Y be the map that sends Xk to X0 q : :q:Xk-1 by
(-f-k ; : :;:-f-1 ). One can then check that QJ = 1, QF = 0, GJ = 0, GF = 1
and F G + JQ = 1. Thus F is a split monomorphism, and J identifies X with the_
cokernel (or cofiber) of F . |__|
Let N denote the natural numbers, N = {0; 1; 2; : :}:.
Lemma 2.2.7. Let {Xk}k2N be a directed system of objects of a stable homotopy
category C. Suppose that u: N -!N is a weakly increasing map, such that u(k) -!
1 as k -! 1. Then we have an isomorphism of sequential colimits
~=
lim-!kXu(k)-! lim-!kXk
AXIOMATIC STABLE HOMOTOPY THEORY 19
Proof.We define two maps
a a
G; H : Xu(k)-! Xk
k k
as follows. The map G just sends the kth summand Xu(k)in the source to the
u(k)th summand in the target by the identity. The map H sends Xu(k)to Xu(k)q
: :q:Xu(k+1)-1; the component Xu(k)-! Xm is just the map provided by the
direct system. We also write F for the usual map, whose cofiber is by definitio*
*n the
sequential colimit. It is straightforward to check that we get a commutative sq*
*uare
as on the left of the following diagram, and thus a map
f :lim-!kXu(k)-! lim-!kXk
as indicated.
` F ` `
kXu(k)?----! k Xu(k)----!? lim-!kXu(k)----!? k Xu(k)?
H?y ?yG ?yf ?yH
` F ` `
kXk ----! kXk ----! lim-!kXk ----! k Xk
Consider the resulting Milnor exact sequences for [lim-!kXk; Y ] and [lim-!kXu(*
*k); Y ].
It is well-known that the relevant lim-and lim-1terms are isomorphic, so that
[lim-!kXk; Y ] = [lim-!kXu(k); Y ]:
As this holds for all Y , Yoneda's lemma tells us that f is an isomorphism. *
* |___|
Remark 2.2.8. One might ask whether the sequential colimit of cofiber sequences
is a cofiber sequence. Provided that there is a suitable underlying closed mod*
*el
category, it turns out that this is true, in the following weak sense. Suppose *
*that
we have cofiber sequences Xk uk-!Yk vk-!Zk wk--!Xk and commutative diagrams
Xk --uk--! Yk
? ?
fk?y ?ygk
Xk+1 -uk+1---!Yk+1
Then it is possible to choose maps hk: Zk -!Zk+1, and compatible maps
X1 -u1-!Y1 -v1-!Z1 -w1-!X1
of the sequential colimits, such that (fk; gk; hk) is a morphism of triangles, *
*and
(u1 ; v1 ; w1 ) is a cofiber sequence.
Indeed, we can make a telescope construction to replace the sequence {Xk} by
a weakly equivalent sequence of cofibrations (with Xk cofibrant). We can then
inductively modify the Y 's, u's and g's to get a weakly equivalent diagram in *
*which
the g's and u's are cofibrations and the X -Y squares commute on the nose. Havi*
*ng
done this, the claim is fairly clear.
We do not know whether this is true in an arbitrary stable homotopy category.
We also have the notion of a weak limit, dual to that of a weak colimit.
Definition 2.2.9.Let I be a small category. Given any functor i 7! Xi from I to
C, we say that a pair (X; o) is a weak limit of (Xi) if
1. X is an object of C.
2. o is a natural transformation from cX to (Xi).
20 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
3. Given any natural transformation ae: cY -! (Xi), there is a map g :Y -! X
so that ae = ocg. Equivalently, the natural map
[Y; X] -!lim-i[Y; Xi]
is surjective for all Y .
Note that the map g need not be unique.
We also have the simpler definition of a sequential limit. Recall that stable
homotopy categories always have arbitrary products, by Lemma 1.4.7.
Definition 2.2.10.Let C be a stable homotopy category Given a sequence of ob-
jects of C,
X0 -f0-X1 -f1-X2 -f2-: : :
define the sequential limit to be the fiber of the map
Y Y
F : Xi- Xi
such that ssiO F = ssi- fiO ssi+1.
We do not know a good notion of a minimal weak limit in any of our settings.
However, we do have the following two propositions, whose proofs are analogous *
*to
that of Proposition 2.2.4. We leave the proofs to the interested reader.
Proposition 2.2.11.Let I be a small category, and C a stable homotopy category.
(a) Every functor i 7! Xi from I to C has a weak limit.
(b) Suppose that (oi:X -! Xi) and (oei:Y -! Yi) are weak limits, and that
(ui:Xi -!Yi) is a natural transformation. Then there is a compatible map
u: X -! Y (typically not unique).
(c) The sequential limit of a sequence is a weak limit. In fact, for any Y the*
*re is
a Milnor exact sequence
0 -!lim-1[Y; -1Xi] -![Y; X] -!lim-[Y; Xi] -!0:
(d) If Y 2 C and (oi:X -! Xi) is a weak (resp., sequential) limit then the
diagram
(F (Y; oi): F (Y; X) -!F (Y; Xi))
is a weak (resp., sequential) limit.
Proposition 2.2.12.Suppose that (oi:Xi -! X) is a weak (resp., sequential)
colimit in a stable homotopy category C, and Y is an object of C; then the diag*
*ram
(F (oi; Y ): F (X; Y ) -!F (Xi; Y ))
is a weak (resp., sequential) limit. |___|
Sequential limits are exact in a limited sense, analogous to Remark 2.2.8.
2.3. Cellular towers and constructibility. In this section, we consider the pro*
*b-
lem of constructing an object from a given family of objects A. We first consid*
*er
the case A = *G.
Proposition 2.3.1.Suppose that C is an algebraic stable homotopy category. Then
every object X can be written as the sequential colimit of a sequence 0 = X0 -!
X1 -! : :,:in which the cofiber of each map Xk -! Xk+1 is a coproduct of objects
of *G.
AXIOMATIC STABLE HOMOTOPY THEORY 21
Proof.Suppose that X 2 C. Let X0 = X, and let S0 be the coproduct
a a
Z:
Z2*G f2[Z;X0]
There is an obvious map S0 -!X0 which induces a surjection [Z; S0] -![Z; X0] for
all Z 2 *G. Let X1 be the cofiber of this map. By iterating this construction, *
*we
get cofibrations Sk -!Xk -!Xk+1 in which Sk is a coproduct of copies of objects
in *G, and the map [Z; Xk] -![Z; Xk+1] is zero for every Z 2 *G.
Now let Xk denote the fiber of the map X -! Xk. Using the octahedral axiom,
we get a diagram as follows:
____-Xk+1_____
J
J] J
Jc J
J J
AE J J
X __________-Xk+1
J
OE J OE J J
J Jc JJ
J J J
J J^ J^ AE
Xk ___________oec___________oeXkSk
In particular, we get a sequence of maps 0 = X0 -! X1 -! : :-:!X, in which
the cofiber of Xk-1 -! Xk is Sk. Let CX be the sequential colimit. By the weak
colimit property, we get a map CX -! X compatible with the given maps Xk -! X.
Suppose that Z 2 *G. By construction, the map [Z; X] -! [Z; Xk] is zero for
k > 0, so the map [Z; Xk] -! [Z; X] is surjective. Moreover, the map [Z; Sk] -!
[Z; Xk] is surjective; after a diagram chase, we conclude that the kernel of the
map [Z; Xk] -! [Z; X] goes to zero in [Z; Xk+1]. It follows easily that [Z; CX]*
* =
lim-!k[Z; Xk] = [Z; X], and thus (by lemma 1.4.5) that the map CX -! X_is_an
isomorphism. |__|
This construction is called the cellular tower for X.
We now restate and prove Theorem 1.2.1.
Theorem 2.3.2. Let C be an enriched triangulated category. Suppose that G is a
set of small strongly dualizable objects of C. Suppose also that whenever [Z; X*
*] = 0
for all Z 2 *G, we have X = 0. Then C is an algebraic stable homotopy category.
Proof.We need to show both that the only localizing subcategory of C that conta*
*ins
G is C itself, and that every cohomology functor is representable. Suppose that
X 2 C. As in the proof of Proposition 2.3.1, we construct a sequence of objects
0 = X0 -! X1 -! X2 -! : :.: We define CX to be the sequential colimit, and
obtain a map CX -! X, with cofiber LX, say. Just as above, we find that this
induces an isomorphism [Z; CX] -![Z; X] for all Z 2 *G, so that [Z; LX]* = 0 for
all such Z. Thus LX = 0 and X ' CX. By construction, CX lies in the localizing
subcategory loc generated by G.
Now let H be a cohomology functor on C. We need to show that H is rep-
resentable. This is much the same as [Mar83 , Theorem 4.11]. We shall define
22 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
recursively a sequence of objects
X(0) i0-!X(1) i1-!X(2) i2-!: : :
and elements u(k) 2 H(X(k)) such that i*ku(k + 1) = u(k). We start with
a a
X(0) = Z:
Z2*G v2H(Z)
We take u(0) to be the element of
Y Y
H(X(0)) = H(Z)
Z2*G v2H(Z)
whose (Z; v)th component is v. We then set
T (k) = {(Z; f) | Z 2 *G; f :Z -! X(k); f*u(k) = 0}:
We define X(k + 1) by the cofiber sequence
a ik
Z -! X(k) -! X(k + 1):
(Z;f)2T(k)
By applying H to this, we obtain a three-term exact sequence (with arrows re-
versed). It is clear by construction that u(k) maps to zero in the left hand te*
*rm,
so that there exists u(k + 1) 2 H(X(k + 1)) with i*ku(k + 1) = u(k) as required.
We now let X be the sequential colimit of the objects X(k). The cofibration
defining this sequential colimit gives rise to a short exact sequence
0 -!lim-1kH(X(k)) -!H(X) -!lim-kH(X(k)) -!0:
Using this, we find an element u 2 H(X) that maps to u(k) in each H(X(k)). As
in Yoneda's lemma, this induces a natural map oU :[U; X] -! H(U). It is easy to
see that oZ is an isomorphism for each Z 2 *G (using the fact that these objects
are small). It is also easy to see that
{Z | okZ is an isomorphism for all}k
is a localizing category. It contains G, so it must be all of C; thus o_is an i*
*somor-
phism. |__|
We now prove some technical results about the cardinality of various categori*
*es.
Recall that for infinite cardinals and we have
+ = = max(; ):
Definition 2.3.3.Given X 2 C, we define a cardinal number c(X) by
X
c(X) = |[Z; X]|:
Z2*G
Note that c(X) max(|G|; @0). We also define
C = {X 2 C | c(X) }
and
X X
c(C) = c(Z) = |[W; Z]n|:
Z2G Z;W2G;n2Z
AXIOMATIC STABLE HOMOTOPY THEORY 23
It is not hard to see that C is a`thick subcategory. Moreover, if {Xi}i2Iis a *
*family
of objects in C and |I| , then iXi2 C . Finally, if max(|G|; @0) then
C = {X | |[Z; X]| for allZ 2 *G}:
Proposition 2.3.4.Suppose that C is an algebraic stable homotopy category, and
that c(C). Then
(a) X 2 C if and only if X is the sequential colimit of a cellular tower
0 = X0 -!X1 -!: : :
such that the cofiber of Xi -! Xi+1 is a coproduct of suspended generators
indexed by a set of cardinality at most .
(b) C is essentially small. That is, the isomorphism classes of objects of C *
* form
a set.
Proof.Certainly if X is such a sequential colimit, then |[Z; X]| for all Z 2 *
**G,
since the generators are small. Conversely, if X 2 C , then in the construction*
* of
the cellular tower as in Proposition 2.3.1, we only need to take coproducts over
sets of cardinality at most . These cellular towers give us explicit models for
isomorphism classes of objects of C so we can use them to construct a small
skeleton. More precisely, let A0 denote the set of all coproducts of suspension*
*s of
generators indexed by sets of size at most . Having defined An, for each map fr*
*om
an object of A0 to an object of An, choose a cofiber. Denote the set of such ch*
*oices
by An+1. Let A1 be the union of the An, and for each sequence X1 -!X2 -!: : :
in A1 , choose a sequential colimit. Let A denote the set of such choices. Then*
* the
cellular tower constructed above shows that any object of C is isomorphic_to an
element of A. |__|
Proposition 2.3.5.Let C be a stable homotopy category, and S a set of objects of
C. Then the thick subcategory generated by S is essentially small, as is the G-*
*ideal
generated by S.
Proof.Without loss of generality, we may assume that S is closed under suspensi*
*ons
and desuspensions. We shall recursively define sets Sk C for each integer k 0,
closed under suspensions and desuspensions, starting with S0 = S. Suppose that
we have constructed Sk. Every retract of an object X 2 Sk corresponds to an
idempotent e 2 [X; X], so there are only a set of these, up to isomorphism; cho*
*ose
one in each isomorphism class. Similarly, there are only a set of maps f :X -! Y
with X; Y 2 Sk, and thus only a set of cofibers, up to isomorphism; choose one
in each isomorphism class. Let Sk+1 be the union of Sk with the set of all these
choices, so that Sk+1 is againSa set, closed under suspensions and desuspension*
*s. It
is easy to see that the set kSk is equivalent to the thick subcategory genera*
*ted
by S. __
The proof for G-ideals is similar. |__|
Corollary 2.3.6.The thick subcategory of G-finite objects is essentially small.*
* |___|
We now return to the problem of constructing objects in a stable homotopy
category from a given set of small objects.
Definition 2.3.7.Let A be an essentially small thick subcategory of small objec*
*ts
in C. Let X 2 C be an arbitrary object. Write A (X) for the category whose
objects are maps (Z -u!X) with Z 2 A, and whose morphisms are maps Z -v!Z0
24 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
such that u0v = u. This again has only a set of isomorphism classes. We often
write Xfffor a typical object of A (X). If H :A -!Ab is an exact functor, write
bHA(X) = lim H(X ):
-! A(X) ff
Note that bHAis a functor on all of C. If C is algebraic, we can let A be the c*
*ategory
of G-finite objects (which is essentially small by Corollary 2.3.6). In this ca*
*se, we
write (X) for A (X) and bH(X) for bHA(X).
Before stating the properties of A (X) and bHA, we need to recall a definitio*
*n.
Definition 2.3.8.A category I is filtered if
1. For any i; j 2 I there exists an object k 2 I and maps i -!k- j.
2. Given any two maps u; v :i -! j in I, there is an object k 2 I and a map
w :j -! k with wu = wv.
A functor F :J -!I of filtered categories is cofinal if
1. For any i 2 I there exists an object j 2 J and a map u: i -!F j.
2. Given any two maps u; v :i -! F j in I, there is an object k 2 J and a map
w :j -! k with (F w)u = (F w)v.
If I is filtered, then it is well-known and easy to see that the colimit func*
*tor from
I-indexed diagrams of Abelian groups to Abelian groups is exact. If F :J -! I is
cofinal, and A: I -!Ab , then it is also well-known that
lim-!IAi= lim-!JAFj:
Proposition 2.3.9.A (X) is a filtered category, functorial in X. It has a termi-
nal object if X lies in A. Moreover, bHA is a homology functor, which agrees wi*
*th
H on A.
Proof.A is triangulated, so it has finite weak colimits. These can also be used*
* as
finite weak colimits in A (X), so A (X) is a filtered category.
Suppose we have a map f :X -! Y . This gives an evident functor A (X) -!
A (Y ), sending (U -u!X) to (U -fu!Y ). If X 2 A then it is immediate that
(X -1!X) is the terminal object in A (X).
Next we show that bHA is additive. Suppose that U 2 A and a 2 H(U). Then,
for any map u: U -! X we get an object (U; u) = (U -u!X) of A (X), and thus an
element of bHA(X), which we shall call [u; a]. Suppose that we have two differe*
*nt
maps u; v :U -! X; we claim that [u + v; a] = [u; a] + [v; a]. To see this, con*
*sider
the following diagram in A (X).
(U; u)--i0--!(U q U; u q v)-i1---(U; v)
x
??
(U; u + v)
It is clear that
[u + v; a] = [u q v; *a] = [u q v; (a; a)] = [u q v; (a; 0)] + [u q v; (0;*
* a)]:
Similarly, we have [u; a] = [u q v; i0*a] = [u q v; (a; 0)] and [U; v] = [u q v*
*; (0; a)],
which proves the claim.
AXIOMATIC STABLE HOMOTOPY THEORY 25
It follows that bHAis an additive functor. Thus, when I is finite we have
!
a M
(2.3.1) bHA Xi = bHA(Xi)
i2I i2I
On the other hand, if I is infinite then one sees (using the smallness of objec*
*ts of
A) that
! !
a a
bHA Xi = lim bH X
-! J A i
i2I i2J
where J runs over finite subsets of I. It follows that (2.3.1)holds even when I*
* is
infinite; i.e., bHAtakes arbitrary coproducts to direct sums.
We now show that HbA is an exact functor. Let X -f!Y g-!Z be a cofiber
sequence. Suppose that y0 2 bHA(Y ). Then there is an object V -v!Y of A (Y )
and an element y 2 H(V ) which represents y. Now suppose that y0 maps to
zero in bHA(Z). By examining the definitions, we see that there is a factorizat*
*ion
(V v-!Y f-!Z) = (V m-!W -w! Z) such that W 2 A and H(m)(y) = 0. Let
U -k!V be the fiber of m, so we can choose a map u: U -! X making the following
diagram commute:
U --k--! V --m--! W
? ? ?
u?y v?y w?y
X --f--! Y ---g-! Z
Because H(m)(y) = 0, we see that y = H(k)(x) for some x 2 H(U). This defines
an element x02 bHA(X), whose image in bHA(Y ) is y0. This means that bHA is an
exact functor, and in fact a homology functor.
If we restrict to A, there is an evident natural transformation bHA(X) -!H(X).
For X 2 A, the colimit of H over A (X) is just the value at the terminal_object,
in other words H(X). |__|
Remark 2.3.10. If H and A are as in Definition 2.3.7, and if H0 is any homology
functor extending H, there is a natural transformation HbA -! H0 of homology
functors that is an isomorphism on the localizing subcategory loc generated *
*by
A.
For convenience, we record the most important special case separately.
Corollary 2.3.11.Let C be an algebraic stable homotopy category, and F the sub-
category of G-finite objects. If H is an exact functor F -! Ab, then
Hb(X) = lim H(X )
-! (X) ff
defines a homology functor C -! Ab extending H. Moreover, any other extension_
of H is canonically isomorphic to bH. |__|
We can finally finish the proof of Theorem 2.1.3
Corollary 2.3.12.An object X in an algebraic stable homotopy category is small
if and only if it is G-finite.
26 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Proof.Suppose that X is small, so that [X; -] is a homology theory. It follows
by Proposition 2.3.9 that [X; X] = lim-!(X)[X; Xff]. In particular, 1X 2 [X; *
*X]
must factor through some Xff2 (X), in other words X is a retract of Xff. By
the definition of (X), Xffis G-finite, so X is G-finite. We have already seen_t*
*he
converse in Theorem 2.1.3. |__|
We would like to prove an analogous result for cohomology functors, but the
failure of the inverse limit functor to be exact prevents us from doing so in g*
*eneral.
However, there is one situation when the inverse limit is exact.
Definition 2.3.13.Given a ring R, a linear topology on an R-module M is a
topology such that the cosets U + m of open submodules U form a basis of open
sets. A module M with a linear topology is linearly compactTif it is Hausdorff,*
* and
for every family of closed cosets {Aff}, the intersection Affis empty if and *
*only if
some finite subintersection is empty (see for example [Jen72, p. 56]). In parti*
*cular,
if M is compact Hausdorff then it is linearly compact.
Note that products and closed subspaces of linearly compact modules are lin-
early compact [Jen72], so the inverse limit over a filtered category of linearl*
*y com-
pact modules under continuous maps is again linearly compact. Moreover, [Jen72,
Theoreme 7.1] implies that the inverse limit functor (taken over any filtered c*
*at-
egory) is exact on the category of linearly compact R-modules and continuous
homomorphisms. Jensen only states this theorem for inverse limits over directed
sets, but the proof works for filtered categories as well. However, we give a d*
*ifferent
proof here, because we need its stronger statement in what follows.
Proposition 2.3.14.Suppose that R is a ring, I is a filtered category, and M :f*
*f 7!
Mffis a functor from I to the category of linearly compact R-modules and contin-
uous homomorphisms. Suppose that for each object ff of I, Cffis a (necessarily
nonempty) closed coset of Mffsuch that C forms a subfunctor of M. (That is, for
any map ff -! fi, the induced map Mfi-! Mfftakes Cfito Cff.) Then lim-Cffis
nonempty.
Q
Proof.As mentioned above, Mffis linearly compact. Given any finite collection
J 6= ; of morphisms of I, let D denote the set of domains of r 2 J andQR the se*
*t of
codomains of r 2 J. For each fi 2 D[R, let Tfidenote the subsetQof Mffconsist*
*ing
of all (mff) such that mfi2 Cfi. Then Tfiis a closed cosetQin Mff. For each
morphism r :fi -! fl 2 J, let Ur denote the subset of Mffconsisting of all (m*
*ff)
such that r(mfl) = mfi. Then Ur is the kernel of the continuous homomorphism
Y (1;-r)
Mff-! Mfix Mfl----! Mfi:
In particular, UrTis a closedTsubgroup.
Now let SJ = ff2D[RTff\ r2J Ur. We claim that SJ 6= ;. Because I is
filtered, there is an object fl and maps ff -sff!fl for every ff 2 D [ R such t*
*hat
sff= sfiO r for every r :ff -! fi in J. One can now choose a class c 2 Cfland
define mff= Msff(c) for ff 2 D [ R and 0 otherwise. Clearly (mff) 2 SJ, so SJ
is nonempty as claimed. As it is a nonempty intersection of closed cosets, it i*
*s a
closed coset. Note that any finite intersection of SJ's is again an SJ. Thus,*
* by
linear compactness, the intersection of the SJ's is nonempty. This intersection*
*_is
precisely lim-Cff. |__|
AXIOMATIC STABLE HOMOTOPY THEORY 27
The exactness of the inverse limit on linearly compact modules is then immedi*
*ate:
Corollary 2.3.15.Let R be a ring, and let I be a filtered category. Consider
the (Abelian) category [Iop; M] of contravariant functors from I to the categor*
*y M
of linearly compact R-modules and continuous homomorphisms. Then the inverse
limit functor [Iop; M] -!M is exact.
Proof.Suppose that Mff-fff!Nff-gff!Pffis an exact sequence of inverse systems
of linearly compact R-modules. Suppose (nff) 2 lim-Nff, and gffnff= 0 for all f*
*f.
Let Cff= f-1ff{nff}. Then Cffis nonempty, and thus a closed coset in Mff. By the
preceding proposition, lim-Cffis nonempty. Any element in it is a class in lim-*
*Mff
mapping to (nff). |___|
We can now prove the following proposition.
Proposition 2.3.16.Suppose that A is an essentially small thick subcategory of
small objects in a stable homotopy category C. Let R be a ring, and let M be the
category of linearly compact R-modules and continuous homomorphisms. Suppose
that H :Aop -!M is an exact functor. Then
HbA(X) = lim H(X )
- A(X) ff
defines a cohomology functor Cop -!M, which agrees with H on A.
Proof.We first claim that bHtakes coproducts to products. The proof does not use
linear compactness, and is similar to the proof of the analogous part of Propos*
*i-
tion 2.3.9 (see also [Mar83 , Proposition 4.8]).
It therefore suffices to check that bH is exact. Let X -f!Y -g!Z be a cofiber
sequence. Define A (g) to be the category of commutative squares
U ----! V
?? ?
y ?y
Y --g--!Z
where U and V are in a small skeleton of A. The morphisms are commutative
diagrams. There are then obvious functors from A (g) to A (Y ) and A (Z), and
it is straightforward to verify that these are cofinal. We write Yff-! Zfffor a*
* typical
object of A (g). Thus bH(Y ) = lim-A(g)H(Yff), and similarly for bH(Z).
Now suppose that we are given a class y 2 bH(Y ) such that bH(f)y = 0. The cl*
*ass
y is given by a compatible family yff2 H(Yff) for each ff 2 A (g). Then for each
ff 2 A (g), we have a map Yff-gff!Zff, so we can let Cff= H(gff)-1(yff). We cla*
*im
that Cffis nonempty. Indeed, let Xff-fff!Yffdenote the fiber of gff. Then Xff2 *
*A,
and the induced map Xff-! Y factors through X. Therefore, since bH(f)y = 0, we
must have H(fff)(yff) = 0, so, by the exactness of H, Cffis nonempty. It is then
clear that Cffis a closed coset, so, by Proposition 2.3.14, lim-Cffis nonempty.*
* A
class in this inverse limit is a z 2 bH(Z) such that bH(g)(z) = y. *
* |___|
We now discuss when an object X can be constructed from a given set of finite
objects.
Proposition 2.3.17.Suppose that B = {Fi} is a set of small objects in a stable
homotopy category C. Let A = thick** be the thick subcategory generated by B,
28 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
which is essentially small by Proposition 2.3.5. Consider the following conditi*
*ons
on an object X of C:
(a) X is the sequential colimit of a sequence
0 = X0 -!X1 -!X2 -!: : :
such that the cofiber of Xk -! Xk+1 is a coproduct of suspensions of eleme*
*nts
of B.
(b) X is in the localizing subcategory loc**** generated by B.
(c) For every homology functor H defined on C, the natural map bHA(X) -!H(X)
is an isomorphism.
Then we have (a),(b))(c), and if C is algebraic then (c))(b). Moreover, a small
object X is in loc**** if and only if it lies in A.
Proof.Write D = loc**** = loc****.
We first make a construction for arbitrary X 2 C. This is very similar to the
construction in Proposition 2.3.1, to which we refer for more details. Let X0 =*
* X
and let A0 be the coproduct
a a
Z:
Z2*B f2[Z;X0]
There is an obvious map A0 -! X0 which induces a surjection [Z; A0] -! [Z; X0]
for all Z 2 A. Let X1 be the cofiber of this map. Iterating, we get cofibrati*
*ons
Ak -! Xk -! Xk+1 in which Ak is a coproduct of suspensions of copies of objects
in B, and the map [Fi; Xj]* -![Fi; Xj+1]* is zero for every Fi.
Now let Xk be the fiber of the map X -! Xk, and CX the sequential colimit
of the Xk, so we get a map CX -! X. As in Proposition 2.3.1, we see that
[Z; CX] ' [Z; X] for all Z 2 A. Let LX be the cofiber of CX -! X, so that
[Z; LX] = 0 for all Z 2 A. As the category of those Z for which [Z; LX]* = 0 is
localizing, we conclude that [Z; LX] = 0 for all Z 2 D.
We now turn to the main part of the proof. It is clear that (a))(b).
Suppose that (b) holds. We claim that LX = 0. Indeed, LX 2 D and [Z; LX] =
0 when Z 2 D, so [LX; LX] = 0, so LX = 0. Thus X = CX, and so (a) holds.
Suppose again that (a) holds, and that H :C -! Ab is a homology functor. To
prove that (a))(c), we need to show that HbA(X) = H(X). Both HbA and H
are homology functors, so the subcategory D0 on which the map bHA -! H is an
isomorphism, is localizing. As A D0, we see that D D0, in particular X 2 D0.
Thus (a))(c).
Now suppose that C is algebraic, and that (c) holds. Let H be a homology
theory. Because [Z; LX] = 0 when Z 2 A, we see that HbA(LX) = 0, and thus
bHA(CX) = HbA(X). On the other hand, HbA(X) = H(X) by assumption, and
bHA(CX) = H(CX) because (b))(c). Thus H(CX) = H(X) for every homology
theory H. Because C is algebraic, the functor [Z; -] is a homology theory whene*
*ver
Z 2 G, so [Z; CX] = [Z; X]. It follows that CX = X, and so (c))(a).
Finally, suppose that X is small and in D. Then [X; -] is a homology theory,
so, since (b))(c), [X; X] = lim-!(X)[X; Xff]. It follows that 1 2 [X; X] facto*
*rs_
through some Xff2 A; in other words X is a retract of Xff, so X 2 A. |_*
*_|
Remark 2.3.18.
AXIOMATIC STABLE HOMOTOPY THEORY 29
(a) The map X -! LX constructed in this proof is finite localization away from
A. See Theorem 3.3.3 for details.
(b) Suppose that C is an algebraic stable homotopy category, A is an essential*
*ly
small thick subcategory, and X 2 loc****. Then the natural map bHA(X) -!
H(X) is an isomorphism for all homology functors H defined on A, making
it look as though X should be the minimal weak colimit of the diagram
A (X). Indeed, if A (X) has a minimal weak colimit X0, then we have a
map X0 -! X which gives an isomorphism under any homology functor, so
that [Z; X0] ' [Z; X] for every Z 2 *G. It follows that X0' X. In particul*
*ar,
this holds if C is a Brown category (see Definition 4.1.4 and Theorem 4.2.*
*3).
3.Bousfield localization
We now present some basic results on Bousfield localization in a stable homo-
topy category; in particular, we define localization functors and investigate t*
*heir
properties, and we show that, in an algebraic stable homotopy category, one can
localize with respect to any homology functor. We also discuss the properties *
*of
the full subcategory of L-local objects, for various kinds of localization func*
*tors L.
The first definition of this kind of localization was given by Adams [Ada74 ]*
*. Cer-
tain set-theoretic problems with Adams' definition were cured by Bousfield [Bou*
*79a ,
Bou79b, Bou83]. See [Rav84 , Rav92] for an analysis of some particularly import*
*ant
and interesting examples in the homotopy category of spectra.
3.1. Localization and colocalization functors. See Section A.1 for the defini-
tion of an exact functor between triangulated categories.
Definition 3.1.1.
(a) Suppose that i: 1 -! L is a natural transformation of exact functors from C
to itself. We say that the pair (L; i), or just L, is a localization funct*
*or if
(i)The natural transformation Li from L to L2 is an equivalence.
(ii)For all objects X; Y the map
*
[LX; LY ] iX-![X; LY ]
is an isomorphism.
(iii)If LX = 0 then L(X ^ Y ) = 0 for all Y .
(b) Dually, suppose that q :C -! 1 is a natural transformation of exact functo*
*rs
from C to itself. We say that (C; q), or just C, is a colocalization funct*
*or if
(i)The natural transformation Cq from C2 to C is an equivalence.
(ii)For all objects X; Y the map
[CX; CY ] qY-*-![CX; Y ]
is an isomorphism.
(iii)If CX = 0 then CF (Y; X) = 0 for all Y .
(c) A morphism of localization functors is a natural transformation u: L -! L0
of exact functors such that ui = i0(and similarly for colocalization funct*
*ors).
(d) If M :C -! C is an exact functor and MX is trivial, we say that X is M-
acyclic. A map X -f!Y is called an M-equivalence if Mf is an isomorphism.
(e) If L is a localization functor and iX :X -! LX is an isomorphism, we say
that X is L-local; we let CL denote the full subcategory of L-local object*
*s.
30 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
(f) Dually, if C is a colocalization functor and jX :CX -! X is an isomorphism,
we say that X is C-colocal; we let CC denote the full subcategory of C-col*
*ocal
objects.
We start by proving a basic fact.
Lemma 3.1.2. Let L be a localization functor on a stable homotopy category C.
Then an object Y 2 C is L-local if and only if Y ' LX for some X. A similar
statement holds for colocalization functors.
Proof.If Y is local, then Y ' LY . For the converse, we need only prove that LX
is local for all X. By naturality of i, we have
(LiX ) O iX = iLX O iX :X -! L2X:
It follows from condition (ii) that LiX = iLX , so iL is an equivalence as requ*
*ired. |___|
Remark 3.1.3. Much of the theory can be developed without conditions (a)(iii)
and (b)(iii), but we do not know of interesting applications for this. We have
therefore included these conditions so as to simplify the hypotheses for various
results below.
Remark 3.1.4. We have followed the usual practice in category theory by defining
colocalization functors to be the dual thing to localization functors. Unfortun*
*ately,
this conflicts with the language used by Bousfield in [Bou79b , Bou79a]. His "c*
*olo-
calization with respect to E" is actually a localization functor in our languag*
*e. The
associated category of acyclics is the localizing subcategory loc generated *
*by E.
It turns out that there are rather few morphisms of localization functors.
Lemma 3.1.5. Let L and L0be localization functors on a stable homotopy category
C. If iL0:L0-! LL0is an isomorphism (equivalently, if L0-local objects are L-lo*
*cal),
then there is a unique morphism of localization functors u = (iL0)-1 O Li0 from*
* L
to L0; otherwise, there are no such morphisms.
Proof.Firstly, suppose that iL0is an isomorphism. (By Lemma 3.1.2, it is equiva-
lent to say that CL0 CL.) It is easy to check that u = (iL0)-1 O Li0is a morphi*
*sm
from L to L0. Suppose that v is another such morphism. Then ui = i0= vi but
L0X is L-local, so i*: [LX; L0X] -![X; L0X] is an isomorphism, so u = v.
Conversely, suppose that we have a morphism u: L -!L0. First, we claim that
(iL0) O u = Li0:L -!LL0:
As the target is L-local, it is enough to check this after composing with i. Us*
*ing
the naturality of i, we find that
(iL0) O u O i = (iL0) O i0= (Li0) O i;
as required.
Next, consider the composite
0 (i0L0)-1
w = (LL0-uL-!L0L0-----! L0):
0 w
It is easy to see that the composite L0X -iL-!LL0X -! L0X is the identity, so L*
*0X
is a retract of the L-local object LL0X and thus is L-local. Thus L0-local_obje*
*cts
are L-local. |__|
Lemma 3.1.6. Suppose that C is a stable homotopy category.
AXIOMATIC STABLE HOMOTOPY THEORY 31
(a) There is a natural equivalence between localization and colocalization fun*
*ctors,
in which L and C correspond if and only if CX -! X -! LX is a cofiber se-
quence. More precisely, consider the following category B. An object consi*
*sts
of exact functors (C; L) and morphisms (q; i; d) of exact functors such th*
*at
(C; q) is a colocalization, (L; i) is a localization, and
CX -q!X -i!LX -d!CX
is exact. The morphisms are the evident thing. Then the forgetful functors
from B to the categories of localization and colocalization functors are e*
*quiv-
alences.
In the following statements, we assume that C and L correspond as above.
(b) The following are equivalent:
(i)X is L-local.
(ii)iX :X -! LX is an equivalence.
(iii)X ' LY for some Y .
(iv)[Z; X] = 0 (or [Z; X]* = 0, or F (Z; X) = 0) for all L-acyclic (equiva-
lently, C-colocal) objects Z.
(v)X is C-acyclic.
(Of course, (i) , (ii) by definition.) Dually, the following are equivalen*
*t:
(i)X is C-colocal.
(ii)qX :CX -! X is an equivalence.
(iii)X ' CY for some Y .
(iv)[X; Z] = 0 (or [X; Z]* = 0, or F (X; Z) = 0) for all C-acyclic (equiva-
lently, L-local) objects Z.
(v)X is L-acyclic.
(c) iX :X -! LX is initial among L-local objects under X, and terminal among
L-equivalences out of X. Dually, qX :CX -! X is terminal among L-acyclic
objects over X, and initial among C-equivalences into X.
(d) The class of L-acyclics (= C-colocals) forms a localizing ideal, and the c*
*lass
of L-locals (=C-acyclics) forms a colocalizing coideal.
(e) As a functor from C to CL, L is left adjoint to the inclusion of the L-loc*
*al
objects into C; similarly, C is right adjoint to the inclusion of the C-co*
*local
objects into C. In particular, L and C are uniquely determined by the subc*
*at-
egory of L-acyclics, or by the subcategory of L-locals.
Proof.First we observe that if X is L-acyclic and Y is L-local, then [X; Y ]* =*
* 0.
Indeed, LY = Y and LX = 0 and i*: [LX; LY ]* ' [X; LY ]* so
[X; Y ]* = [X; LY ]* = [LX; LY ]* = 0:
Also, we recall from Lemma 3.1.2 that LX is always L-local.
(a): Suppose that (L; i) is a localization functor. For each X we can choose a
cofiber sequence
CX -q!X -i!LX -d!CX:
By applying L, we see that LCX = 0. On the other hand, suppose that LU = 0.
Then [U; LX]* = 0, so by applying [U; -]* to the above sequence we find that
[U; CX] ' [U; X]. It follows that CX is terminal among L-acyclic objects over X.
Next, consider a morphism f :X -! Y . The axioms for a triangulated category gi*
*ve
32 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
a commutative diagram as follows, in which g is a priori not uniquely determine*
*d.
-1LX ---d-! CX ---q-! X --i--!LX
? ? ? ?
Lf?y g?y ?yf ?yLf
-1LY ---d-! CY ---q-! Y --i--!LY
However, because of the universal property of CY -q!Y , we see that g is unique
after all. As it is unique, it is clearly functorial, and we can call it Cf. (I*
*f we simply
used the universal property to produce g in the first place, we would not know *
*that
the left square commutes.)
As L is an exact functor, it comes equipped with an equivalence L ' L. As
i is a morphism of exact functors, we have i = i under this identification. It *
*is
standard that LX --d-!CX -q! X -i!LX is a cofiber sequence. An argument
similar to the above shows that there is a unique natural equivalence C ' C
making the following diagram commute.
-1d q i
-1LX ----! CX ----! X ----! LX
?? ? ? ?
y ?y ?y1 ?y
LX ---d--!CX --q--! X --i--! LX
We now show that C is exact. Suppose that
X -f!Y -g!Z -h!X
is a cofiber sequence, and let Z0 denote the cofiber of Cf :CX -! CY . Since L *
*is
exact, Z0 is L-acyclic. We have a morphism of exact sequences
CX --Cf--!CY - -ff--!Z0---fi-!CX
? ? ? ?
q?y q?y r?y q ?y
X --f--! Y - -g--! Z ---h-! X
where we have identified CX with C(X) as above.
Applying the 5-lemma, we see that r*: [W; Z0] -! [W; Z] is an isomorphism for
all L-acyclic W . Thus Z0 is terminal among L-acyclic objects over Z, and we can
therefore identify Z0 with CZ and r with q. The proof of the functoriality of C
then shows that ff and fi are uniquely determined, and so must be Cg and Ch
respectively.
This makes C into an exact functor, and q and d into morphisms of exact func-
tors.
We need to show that C is a colocalization functor. We saw above that LC = 0.
It then follows from the cofibration CCX -qC-!CX -i!LCX = 0 that qC is an
isomorphism, verifying condition (i). We also saw above that if LCX = 0, then
[CX; CY ] = [CX; Y ], verifying condition (ii). Finally, suppose that CX = 0 and
Y is arbitrary; we need to show that CF (Y; X) = 0. As CF (Y; X) is terminal
among L-acyclics over F (Y; X), it is enough to show that CF (Y; X) q-!F (Y; X)*
* is
the zero map, or equivalently that the adjoint map Y ^ CF (X; Y ) -! X is zero.
However, CF (X; Y ) and hence Y ^ CF (X; Y ) is L-acyclic, and X is L-local, so
[Y ^ CF (X; Y ); X] = 0 as required.
AXIOMATIC STABLE HOMOTOPY THEORY 33
Next, consider a morphism u: L -! L0. An evident comparison of cofibrations
gives a map v :CX -! C0X compatible with u. The indeterminacy is measured by
[CX; -1L0X], but Lemma 3.1.5 tells us that -1L0X is L-local, so this group is
zero. Thus v is unique, and therefore functorial. Similarly, it is compatible w*
*ith
C ' C and C0' C0.
We have now seen that the functor from B to the category of localization func*
*tors
is full, faithful and essentially surjective, hence an equivalence. The argumen*
*t for
colocalization functors is dual.
(b): We prove only the first statement; the proof for the second is dual. By
definition, (i) is equivalent to (ii), which is equivalent to (iii) by Lemma 3.*
*1.2. If
(iii) holds and LZ = 0 then [Z; X] = [Z; LY ] = [LZ; LY ] = 0. We also have
L(U ^ Z) = 0 for any U, so [U; F (Z; X)] = [U ^ Z; X] = 0. By taking U = F (Z; *
*X),
we see that F (Z; X) = 0, and therefore [Z; X]* = ss*F (Z; X) = 0. Thus (iii)
implies (iv), except that we have not yet verified that L-acyclic means the same
as C-colocal. If (iv) holds then 0 = q :CX -! X but Cq :CCX -! CX is an
isomorphism, so CX = 0; thus (iv) implies (v). Suppose that (v) holds, so CX = *
*0.
Then the cofiber sequence CX -q!X -i!LX shows that iX is an isomorphism, so
X is L-local. Thus (v) implies (i). By part of the dual, we see that Z is C-col*
*ocal
if and only if it is L-acyclic, giving the equivalence of the two versions of (*
*iv).
(c): If f :X -! Y and Y is L-local, then Y = LY so i*X:[LX; Y ] ' [X; Y ], so*
* f
factors uniquely through iX . Thus iX :X -! LX is initial among L-local objects
under X. Next, suppose that g :X -! Z is an L-equivalence. By naturality we
have Lg O iX = iZ O g, so h O g = iX where h = (Lg)-1 O iZ :Z -! LX. If also
h0O g = iX then h - h0:Z -! LX factors through the cofiber of g. As this is
L-acyclic it has no nonzero maps to LX, so h = h0. It follows that iX is termin*
*al
among L-equivalences out of X. The claims about C are dual.
(d): All the functors in question are exact, so all the categories in questio*
*n are
thick. Using description (iv) above of the L-local objects, it is immediate tha*
*t CL
is closed under arbitrary products, and hence a colocalizing subcategory. Simil*
*arly,
description (iv) of CC shows that it is closed under coproducts and hence local*
*izing._
(e): This follows immediately from (c). |__|
One quite often needs to consider the fiber of a morphism between localiza-
tion or colocalization functors. This kind of situation is analyzed in the foll*
*owing
proposition.
Proposition 3.1.7.Consider a morphism u: L -!L0of localization functors. By
part (a) of Lemma 3.1.6, there is a unique morphism v :C -! C0 of the correspon*
*d-
ing colocalization functors such that the diagram
CX --q--!X ---i-! LX ---d-! CX
? ? ? ?
v?y 1?y ?yu ?yv
0 i0 d0
C0X --q--!X ----! L0X ----! C0X
commutes. Let MX be the fiber of u: LX -! L0X; then M can be made into a
functor in a canonical way, and the various composites of the above functors are
34 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
given by the following table.
________________________
| O |L| |L0C|| C0 |M |
|_____|_|_|___||____|__|__
| L |L| |L0 |0 |M |M |
|_____|_|_|___|__|__|__|_
| L0 |L|0|L0 |0 |0 |0 |
|____|_|_|___|__|___|___|
| C |0| |0 |C |C |0 |
|____|_|_|___|__|___|___|
| C0 |M|| 0 |C |C0 |M |
|____|_||____|__|___|__|_
| M | |M |0 |0 |M | M |
|___|_|__|___|__|__|___|_
(In particular, F G = GF for any two of these functors.) Moreover, there is
an octahedral diagram (as in Definition A.1.1) of natural maps as follows, so in
particular MX is also the cofiber of CX -v!C0X.
________-C0X_______
J
J] J
J J
q0 J e d0 J
v J J iC0= C0i
AE JJ J
i0 J
X ________________-L0X J
J
J
OE J J OE JJ J
q i J u Je J
J 0 J J
JJ J d L0J
J J^ = Ld J^ AE
CX ________________oeeL________________oeXMX
d qL0= Lq0
JJ]
J________________________________e
dC0= C0d
Proof.We saw in Lemma 3.1.5 that LL0 = L0 and that L0-local objects are L-
local. It follows that CL0 = 0, and thus (using the fibration C0 -! 1 -! L0)
that CC0 = C. By a dual argument, C0C = C, L0C = 0 and L0L = L0. After
identifying L0= L0L = LL0 as above, one finds that u = Li0= i0L: L -! L0 and
thus that the fiber MX of uX is LC0X = C0LX. This makes M a functor. More
precisely, given a map f :X0 -!X1 and cofibrations M0 -!LX0 -!L0X0 -!M0
and M1 -! LX1 -! L0X1 -! M1, there is a unique map M0 -! M1 compatible
with the cofibrations and the maps Lf; L0f. Thus, if we choose a fiber MX for
each map uX , then we can make M into a functor in a unique way such that the
maps -1L0 -! M -! L are natural. The resulting functor is, up to canonical
AXIOMATIC STABLE HOMOTOPY THEORY 35
isomorphism, independent of the choices made. One choice is to take MX = LC0X
and another is to take MX = C0LX, so these are canonically isomorphic. Dually,
we have v = C0q = qC0:C -! C0. It follows that the cofiber of v is C0L =
LC0, which is the same as M again. This justifies all of our table of compositi*
*ons
except for the last row and column. These follow from the rest of the table aft*
*er
substituting M = C0L = LC0. The octahedral axiom guarantees the existence
of an octahedral diagram of the stated kind, except that the bottommost region
might not commute and the arrows marked v and C0i = iC0 might be something
else. However, one can check by naturality that the bottom region commutes,
and that v and C0i = iC0 are the unique maps with the required commutativity_
properties. |__|
We conclude this section by observing that localization functors interact well
with the smash product.
Proposition 3.1.8.For any localization functor L, there is a natural map LX ^
LY -! L(X ^ Y ), which interacts with the isomorphisms S ^ X = X = X ^ S,
X ^ Y = Y ^ X, and (X ^ Y ) ^ Z = X ^ (Y ^ Z) in the obvious way. In particular,
LS is a commutative ring object in C, and every L-local object Y is a module ov*
*er
LS in a natural way.
Proof.First, observe that the map
L(iX ^ iY ): L(X ^ Y ) -!L(LX ^ LY )
is an equivalence, as one sees easily using the cofibrations X ^ CY -! X ^ Y -!
X ^LY and CX ^LY -! X ^LY -! LX ^LY . It follows that the map LX ^LY -!
L(LX ^LY ) factors uniquely through a map LX ^LY -! L(X ^Y ). We leave it to
the reader to check that this has the right coherence properties. In particular*
*, we
get a multiplication map LS ^ LS -! L(S ^ S) = LS and a unit map iS :S -! LS
which make LS into a ring object. Moreover, if Y is L-local, we get a multiplic*
*ation
map
LS ^ Y = LS ^ LY -! L(S ^ Y ) = LY = Y;
which makes Y into a module over LS. |___|
3.2. Existence of localization functors. We have so far not touched on one
basic question: given a localizing ideal D, when is there a localization funct*
*or
L such that the category of L-acyclics is precisely D? We know by part (e) of
Lemma 3.1.6 that such a functor is essentially unique if it exists.
Definition 3.2.1.If D is a localizing ideal, and there is a localization functo*
*r L
such that the category of L-acyclics is precisely D, then we shall write LD = L*
*. If
H is a homology or cohomology functor, and D = {X | H(X ^ Y ) = 0 for allY },
and LD exists, then we shall write LH = LD . We will refer to LH -acyclic and
LH -local objects as H-acyclic and H-local, respectively. Note that H(X) = 0 do*
*es
not imply that X is H-acyclic if C is not monogenic.
The first result here is due to Bousfield [Bou79b ] (see also [Bou83 ]), who *
*works
in a closed model category. Margolis [Mar83 ] has given a proof that will work *
*in
an arbitrary algebraic stable homotopy category.
See Definitions 1.1.3 and 3.1.1 for the relevant definitions in the following.
36 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Theorem 3.2.2 (Bousfield localization).For any homology functor H :C -! Ab
on an algebraic stable homotopy category C, the localization functor LH exists.
Proof.For each X 2 C, we need to construct a map X -! LX where LX is H-local
and the fiber is H-acyclic. The methods of Lemma 3.1.6 will then show that L is
automatically a functor and in fact a localization functor. The construction of*
* LX
in [Mar83 , Chapter 7], applied to the homology functor
M
X 7! H(X ^ Z);
Z2G
relies only on basic properties of triangulated categories and homology functor*
*s,
together with Corollary 2.3.11 and Brown representability of cohomology_functor*
*s.
|__|
We do not know of any example of a localizing subcategory for which a local-
ization functor can be proved not to exist. Nick Kuhn has suggested to us that
the question of whether localizations always exist may not be decidable using o*
*nly
the usual axioms of set theory. Certainly the proof in [Mar83 ] involves findin*
*g big
cardinal numbers, so maybe some of the large cardinal axioms are relevant.
In the homotopy category of spectra, Bousfield has shown [Bou79a ] that a lo-
calization functor exists for any localizing subcategory generated by a set`(no*
*t a
proper class) of objects. (If the subcategory is generated by {Ei} and E = iEi
then the localization functor is what Bousfield calls E-colocalization.) The pr*
*oof
probably generalizes to any stable homotopy category derived from a closed model
category satisfying suitable axioms.
3.3. Smashing and finite localizations. We now define a particularly important
special class of localization functors (compare [Rav84 , Definition 1.28]). Fir*
*st, we
need a lemma.
Lemma 3.3.1. Suppose that L is a localization functor on a stable homotopy cat*
*e-
gory C. Then there is a natural map ffX :LS ^ X -! LX, which is an isomorphism
when X is strongly dualizable.
Proof.The map is much as in Proposition 3.1.8: by applying L to the cofibration
CS ^ X -! S ^ X -! LS ^ X, we get an equivalence LX ' L(LS ^ X). We define
ffX to be the composite LS ^ X -i!L(LS ^ X) ' LX. Suppose that X is strongly
dualizable. Using part (b)(iv) of Lemma 3.1.6, we see that LS ^ X = F (DX; LS)_
is L-local, and thus that ffX is an isomorphism. |__|
Definition 3.3.2.A localization functor L: C -! C is smashing if it satisfies t*
*he
following equivalent conditions:
(a) The natural map ffX :LS ^ X -! LX defined in Lemma 3.3.1 is an isomor-
phism for all X.
(b) L preserves coproducts.
(c) The colocalizing subcategory CL of L-local objects is also a localizing su*
*bcat-
egory.
Proof of equivalence.It is easy to see that (a))(b))(c). Suppose that (c) holds.
Given a family of objects {Xi}, we have a cofiber sequence
a a a
CXi-! Xi-! LXi
AXIOMATIC STABLE HOMOTOPY THEORY 37
in which`the first`term is L-acyclic and the last is L-local. By applying L, we*
* see
that L( Xi) = LXi, so that (b) holds. Finally, suppose that (b) holds. It is
then easy to see that the category of those X for which ffX is an isomorphism, *
*is__
localizing. By Lemma 3.3.1, it contains G, so it is all of C. *
* |__|
As a special case of smashing localizations, we have the finite localizations*
*, first
considered by Miller [Mil92].
Theorem 3.3.3 (Finite localization).Suppose that {Xi} is a set of small objects
in a stable homotopy category C. Let A denote the G-ideal generated by {Xi} and
let D denote the localizing ideal generated by {Xi}. Then there is a smashing
localization functor L = LfA, which depends only on A, whose acyclics are preci*
*sely
D. Moreover, the small objects in D are precisely the objects of A.
Definition 3.3.4.We refer to localization functors of this type as finite local*
*iza-
tions. Note that there is a finite localization functor for every essentially *
*small
G-ideal of small objects. We refer to LfAas finite localization away from {Xi}.*
* If B
is a set of small objects of C and A is the G-ideal generated by B, we often ab*
*use
notation and write LfBfor LfA.
Proof.First, note that, by Theorem 2.1.3(a), A consists of small objects. In fa*
*ct,
it is easy to see that A is the thick subcategory generated by the set B = {Z1 ^
. .^.Zr ^ Xi} where 0 r < 1, and Zj 2 G. (In the algebraic case, we can just
take r 1.) Moreover, A is essentially small by Proposition 2.3.5. Note as we*
*ll
that D is the localizing subcategory generated by B, by Lemma 1.4.6.
Applying Proposition 2.3.17 to the set of small objects of B, we construct (f*
*or
each X 2 C) a cofiber sequence CX -q!X -i!LX, in which CX is in D and
[Z; LX] = 0 for Z 2 D. It follows that CX -q!X is terminal among objects of D
over X. It follows in turn that C can be made into a functor, and q into a natu*
*ral
transformation. Moreover qX is an isomorphism for all X 2 D, in particular qCX *
*is
an isomorphism. We also see that CX = 0 if and only if [Z; X] = 0 for all Z 2 D.
The set of such X is a coideal, because D is an ideal. By assembling these fact*
*s,
we conclude that C is a colocalization functor, and that L is the complementary
localization functor.
The collection of acyclics for L forms a localizing subcategory containing B,
and thus D. On the other hand, if LX = 0 then X = CX, which lies in D by
construction. Thus D is precisely the category of L-acyclics.
An object X is L-local if and only if CX = 0, if and only if [Z; X] = 0 for a*
*ll
Z 2 A. Using the fact that these objects Z are small, we see that the category *
*of
L-local objects is closed under coproducts, so that L is smashing.
Finally, suppose that X is small and lies in D. Then X 2 A by the last_part_of
Proposition 2.3.17. |__|
Because the functor LfAis smashing, its image is a localizing subcategory. We
can thus hope to find another localization functor, whose acyclic category is L*
*fAC.
We have quite a good understanding of this situation, as expressed by the follo*
*wing
theorem. Readers who are familiar with chromatic topology may like to bear the
38 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
following example in mind.
F (n)= a finite type-n spectrum
C = the E(n)-local category
A = the thick subcategory generated by LnF (n)
D = the nth monochromatic category
E = the K(n)-local category.
LfA= Ln-1
LA = LK(n):
Theorem 3.3.5. Let A be a G-ideal of small objects in a stable homotopy category
C. Suppose that A is essentially small, consists of strongly dualizable objects*
*, and is
closed under the Spanier-Whitehead duality functor D (all of which are automatic
if C is algebraic). Write
Z = A? = {Y | 8W 2 A [W; Y ] = 0}
D = ?Z = {X | 8Y 2 Z [X; Y ] = 0}
E = Z? = {X | 8Y 2 Z [Y; X] = 0}
Then there are (co)localization functors
CfAX -! X -! LfAX
CA X -! X -! LA X
with the following properties.
(a) LfAX = LfAS ^ X and CfAX = CfAS ^ X.
(b) LA X = F (CfAS; X) and CA X = F (LfAS; X).
(c) ker(CfA) = ker(LA ) = image(LfA) = image(CA ) = Z.
(d) ker(LfA) = image(CfA) = D, and this is the localizing subcategory generated
by A.
(e) ker(CA ) = image(LA ) = E, and this is the colocalizing subcategory genera*
*ted
by
A0= {W ^ U | W 2 A ; U 2 C}:
(f) There are isomorphisms LA CfA= LA and CfALA = CfA.
(g) The functors LA :D -!E and CfA:E -!D are mutually inverse equivalences.
Remark 3.3.6. As with the functor LfA, if B is a set of small objects in C and A
is the G-ideal generated by B, we often abuse notation and write LB for LA .
Proof.We first justify the comments at the beginning of the statement of the th*
*e-
orem. If C is algebraic, then Corollary 2.3.6 assures us that A is essentially *
*small,
and Theorem 2.1.3 implies that the objects of A are strongly dualizable, and th*
*at
A is closed under D. We shall first prove (a)-(f), and only verify afterwards t*
*hat
LA is a localization functor and CA is a colocalization functor.
Because A is a G-ideal of dualizable objects closed under D, we see (with an
obvious notation) that
Z = {Y | F (A; Y ) = {0}} = {Y | A ^ Y = {0}}
AXIOMATIC STABLE HOMOTOPY THEORY 39
and thus that Z is an ideal. It follows that
D = {X | F (X; Z) = {0}}
(which is an ideal), and that
E = {X | F (Z; X) = {0}}
(which is a coideal).
The functors LfAand CfAwere defined in Theorem 3.3.3, where it was also proved
that LfAX = LfAS ^ X; the other half of (a) follows easily, as does the fact th*
*at
LfAS = LfAS ^ LfAS
and
CfAS ^ CfAS = CfAS:
We define LA X = F (CfAS; X) and CA X = F (LfAS; X), so that (b) holds by
definition. Clearly these are idempotent exact functors, and the cofibration Cf*
*AS -!
S -! LfAS gives rise to natural cofibrations
CA X -! X -! LA X -! CA X:
It follows that ker(CA ) = image(LA ) and that ker(LA ) = image(CA ), and simil*
*arly
that ker(CfA) = image(LfA) and that ker(LfA) = image(CfA). Thus, we need only
prove half of (c),(d) and (e).
It follows from Theorem 3.3.3 that
ker(CfA) = image(LfA) = Z;
image(CfA) = ker(LfA) = D;
and that D is the localizing subcategory generated by A.
(c): All that is left to prove is that ker(LA ) = Z. Suppose that Y 2 Z. Then
{W | F (W; Y ) = 0} is a localizing subcategory containing A and therefore al*
*so
containing D = image(CfA). In particular, it contains CfAS, so LA Y = 0. Con-
versely, suppose that LA Y = 0. For W 2 A we have CfAS ^ W = CfAW = W , so
that
[W; Y ] = [CfAS ^ W; Y ] = [W; F (CfAS; Y )] = [W; LA Y ] = 0:
Thus Y 2 Z.
(d): This was all proved in Theorem 3.3.3, as remarked above.
(e): Suppose that X 2 E, so that F (Y; X) = 0 for all Y 2 Z = image(LfA). In
particular, CA X = F (LfAS; X) = 0, so that X = LA X 2 image(LA ). Conversely,
suppose that X = LA X = F (CfAS; X). Then for Z 2 Z = ker(CfA) we have
CfAS ^ Z = 0 and thus
F (Z; X) = F (Z; F (CfAS; X)) = F (Z ^ CfAS; X) = 0:
It follows that X 2 E.
We still need to show that E is the same as the colocalizing subcategory E0
generated by A0. Consider an object X = W ^ U 2 A0, so that W 2 A. We have
CfAS ^ DW = DW , so
LA (X) = F (CfAS; W ^ U) = F (CfAS ^ DW; U) = F (DW; U) = X:
40 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Thus X 2 E. It follows that E is a colocalizing subcategory containing A0, so
E0 E.
On the other hand, suppose that X 2 E, so that X = F (CfAS; X). Using an
A-based cellular tower for CfAS, we see that X lies in the colocalizing subcate*
*gory
generated by F (A; X) = D(A) ^ X A0. Thus X 2 E0.
(f): We saw in (c) that ker(LA ) = image(LfA). Thus, by applying LA to the
cofibration CfAX -! X -! LfAX, we see that LA CfA= LA . The proof that CfALA =
CfAis similar.
(g): On E = image(LA ) we have LA CfA= LA = 1 (using (f)). Similarly, on D
we have CfALA = 1. The claim follows.
We still need to show that LA is a localization functor. We have already seen*
* that
it is idempotent and exact, so we need only check that [X; LA Y ] = [LA X; LA Y*
* ],
or equivalently that [CA X; LA Y ] = 0. This holds because CA X 2 Z by (c), and
LA Y 2 E by (e), and [Z; E] = {0} by the definition of E. *
* __
It follows as in Lemma 3.1.6 that CA is a colocalization functor. *
*|__|
Theorem 3.3.7 (Algebraic localization).Let C be a unital algebraic stable homo-
topy category, and let T be a set of homogeneous elements in the graded ring
ss*S. Then there is a finite localization functor LT and a natural equivalence
ss*(LTX) = T -1ss*(X).
Proof.Each element t 2 T is a map Sd -! S, say; write S=t for the cofiber, and
A = {S=t | t 2 T }. Write LT = LfA, and CT for the corresponding colocalization
functor. Because inverting T is a coproduct-preserving exact functor on ss*(S)-
modules, we see that T -1ss*(X) is a homology functor of X. It vanishes on A, a*
*nd
thus on CTX for all X. It follows that T -1ss*(X) = T -1ss*(LTX). On the other
hand, we know that [S=t; LTX] = 0. By considering the cofibration Sd -t!S -!
S=t, we conclude that multiplication by t is an isomorphism on ss*(LTX)._ Thus
T -1ss*(LTX) = ss*(LTX). |__|
In this case, the functor LA should be thought of as a kind of completion at *
*the
ideal generated by T . See part (c) of Lemma 6.3.5 for a more precise statement
along these lines.
We now discuss the telescope conjecture. This was first stated by Ravenel
in [Rav84 ], as a conjecture about spectra. It was reformulated in many differ-
ent ways, and finally shown by Ravenel to be false. Nonetheless, it can be shown
that analogues are true in many interesting stable homotopy categories. Inciden-
tally, this conjecture is mislabeled as the smashing conjecture in [Nee92a]. S*
*ee
Definitions 1.4.3, 3.1.1, 3.3.2, and Definition 3.3.4 for the relevant terms.
Definition 3.3.8.Suppose that C is a stable homotopy category. We shall say
that the telescope conjecture holds in C if every smashing localization of C is*
* a
finite localization. If so, there is a one-to-one correspondence between essent*
*ially
small G-ideals of small objects and smashing localizations. The essentially sm*
*all
hypothesis is automatic in case C is algebraic.
3.4. Geometric morphisms. Consider a localization functor L on a stable ho-
motopy category C. Recall that CL is the category of L-local objects in C, or
equivalently the image of L. In the next section, we shall study the properties*
* of
CL, and of L considered as a functor from C to CL. The answer will be that L is
AXIOMATIC STABLE HOMOTOPY THEORY 41
a "geometric morphism" from C to CL. The purpose of the present section is to
explain this concept.
Definition 3.4.1.Let C and D be enriched triangulated categories. A geometric
morphism from C to D is an exact functor L: C -!D which admits a right adjoint
J, together with natural isomorphisms
ff: SD' LSC
: LX ^D LY ' L(X ^C Y ):
The maps and ff are required to commute in the evident sense with the symmetric
monoidal structures on C and D. If L has a right adjoint J and maps ff; as abo*
*ve
which are not necessarily isomorphisms, we say that L is a lax geometric morphi*
*sm.
If C and D are stable homotopy categories and L : C -! D is a (lax) geometric
morphism, we say that L is a (lax) stable morphism if L takes G-finite objects *
*of C
to G-finite objects of D.
The terminology is stolen from topos theory. It is justified by the fact that
a map X -! Y of schemes gives rise to a geometric morphism D(Y ) -! D(X)
of derived categories (and also a geometric morphism of the corresponding topoi
of sheaves). In some special cases, a geometric morphism (of topoi or of stable
homotopy categories) will also admit a left adjoint. The functors which arise *
*in
equivariant stable homotopy theory from a change of group or universe [LMS86 ,
Chapter II] are all either geometric morphisms or adjoints of geometric morphis*
*ms.
We point out that it is straightforward to compose (lax) geometric morphisms
and (lax) stable morphisms. We can therefore form a (very large) category of
stable homotopy categories, where the morphisms are stable morphisms, or lax
stable morphisms if we prefer.
We can now make more precise our claim that the choice of generators is not
very important in an algebraic stable homotopy category. Indeed, suppose C is
an algebraic stable homotopy category with two sets of small generators G and G*
*0.
Then the identity functor is a stable isomorphism between (C; G) and (C; G0).
Let L: C -! D be a geometric morphism, with right adjoint J. Let X and Y
denote objects of C, and U and V objects of D. By juggling adjoints, one can
construct natural maps as follows.
fi:SC -! JSD
:JU ^C JV -! J(U ^D V )
# :LFC(X; Y ) -!FD (LX; LY )
# :JFD (U; V ) -!FC(JU; JV )
ss:X ^C JU -! J(LX ^D U)
ss#:L(X ^C JU) -!LX ^D U
ae:FC(X; JU) ' JFD (LX; U)
ae#:LFC(X; JU) -!FD (LX; U)
The isomorphism ae is a sort of internal version of the adjunction [X; JU] ' [L*
*X; U].
None of the other maps need be isomorphisms. We refrain from listing any of the*
*ir
commutativity and coherence properties.
If L is merely a lax geometric morphism then we can still construct # , ae and
ae# , but ae need not be an isomorphism.
42 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Proposition 3.4.2.Let L: C -!D be a geometric morphism, with right adjoint J.
Then L preserves coproducts, and J is an exact functor which preserves products.
Proof.It is well-known that left adjoints preserve coproducts and right adjoints
preserve products. It is proved in [Mar83 , Proposition A2.11] that adjoints_of*
* exact
functors are exact. |__|
3.5. Properties of localized subcategories. In this section we show that any
localization of a stable homotopy category is a stable homotopy category.
Theorem 3.5.1. Suppose that C is a stable homotopy category, and that L: C -!C
is a localization functor. Then CL has a natural structure as a stable homotopy*
* cat-
egory, such that L: C -!CL is a stable morphism (the right adjoint is the inclu*
*sion
J :CL -! C). Considered as a functor from C to CL, L preserves the following
structure:
(a) cofibrations
(b) the smash product and its unit
(c) coproducts
(d) (minimal) weak colimits, and in particular sequential colimits
(e) strong dualizability.
(Of course, (a) and (b) are part of the claim that L is a geometric morphism.) *
*The
inclusion functor J :CL -! C preserves the following structure:
(a) cofibrations
(b) function objects
(c) products
(d) sequential limits.
The following maps are isomorphisms:
# : JFL(U; V ) -!F (JU; JV )
ss#:L(X ^ JU) -!LX ^L U
ae:F (X; JU) -!JFL(LX; U)
(where the subscript L indicates structure in CL). Moreover, LJ ' 1.
Theorem 3.5.2. Suppose in addition that C is algebraic. Then L preserves small-
ness if and only if L is smashing. Suppose that this holds. Then CL is also alg*
*ebraic,
and J is a lax geometric morphism. The following maps are isomorphisms:
:JU ^C JV -! J(U ^L V )
ss:X ^C JU -! J(LX ^L U)
(Again, the subscript L indicates structure in CL). If C is a Brown category, t*
*hen
CL is also a Brown category.
Proof of Theorem 3.5.1.First, we can triangulate CL by declaring that X -! Y -!
Z -! X is a triangle in CL if and only if it is a triangle in C (and X, Y and Z*
* lie
in CL). Using the fact that CL is a thick subcategory of C, we see that this ma*
*kes
CL into a triangulated category.
Next, we define a smash product X ^L Y on CL by X ^L Y = L(X ^ Y ). It is
easy to check (using Proposition 3.1.8) that SL = LS is a unit for this product,
and that it makes CL into a symmetric monoidal category. If X is arbitrary and
AXIOMATIC STABLE HOMOTOPY THEORY 43
Y is L-local, we can see that F (X; Y ) = F (LX; Y ), and that this object is L*
*-local.
Using this, we see that CL can be made into a closed symmetric monoidal category
by defining FL(X; Y ) = F (X; Y ). It is again easy to check that this structur*
*e is
compatible with the triangulation. `
Suppose that {Xi} is a family of objects of CL. Write Xi for their coproduct
in C. For Y 2 CL, we have
ia j a Y
[L Xi ; Y ] = [ Xi; Y ] = [Xi; Y ]:
` `
It follows that LXi= L( Xi) is the categorical coproduct in CL of the objects
Xi.
We see directly from these constructions that L (considered as a functor C -!*
*CL)
preserves cofibrations, the smash product and its unit, and coproducts. We know
from part (e) of Lemma 3.1.6 that L is left adjoint to J. It follows that L is*
* a
geometric morphism.
If H is a cohomology functor on CL, then H O L is a cohomology functor on
C (because L preserves coproducts). There is therefore an object X of C and a
natural equivalence [Y; X] -!H(LY ). By choosing Y to be L-acyclic, we find that
X is L-local, and therefore represents H as a functor on CL.
Suppose that Z 2 C is strongly dualizable. We claim that LZ is strongly dual-
izable in CL; in other words, for every Y 2 CL we claim that
FL(LZ; Y ) = FL(LZ; SL) ^L Y:
Indeed, the left hand side is just F (LZ; Y ) = F (Z; Y ). The right hand side*
* is
L(F (LZ; LS) ^ Y ). We know that F (LZ; LS) = F (Z; LS) = DZ ^ LS, which is L-
equivalent to DZ. Thus, the right hand side is L(DZ ^ Y ) = LF (Z; Y ) = F (Z; *
*Y ),
as required.
We now define GL = {LZ | Z 2 G}, thereby making L into a stable morphism.
If D is a localizing subcategory of CL which contains GL, then {X 2 C | LX 2 D}
is a localizing subcategory of C which contains G, hence all of C. It follows *
*that
D = CL. Thus, with all this structure, CL becomes a stable homotopy category.
We still need to show that L preserves (minimal) weak colimits. Suppose that
(oi:Xi -! X) is a weak colimit. This means that [X; Y ] -! lim-[Xi; Y ] is an
epimorphism for all Y . As L is a functor, we certainly have compatible maps
(Loi:LXi-! LX). If Y 2 CL then [LX; Y ] = [X; Y ] and [LXi; Y ] = [Xi; Y ], so *
*we
have an epimorphism [LX; Y ] -!lim-[LXi; Y ]. Thus LX is a weak colimit in CL of
{LXi}.
Now suppose that (oi:Xi -! X) is a minimal weak colimit, and that H is a
homology functor on CL. Then H O L is a homology functor on C, so we must have
H(LX) ' lim-!H(LXi). Thus (Loi:LXi-! LX) is a minimal weak colimit in CL.
The claims about the inclusion functor J are now easy. We have already seen
that # and ae are isomorphisms, and it is easy to see that ss# is also. As_L ' *
*1 on
CL, we see that LJ ' 1. |__|
Proof of Theorem 3.5.2.In this proof we assume that C is algebraic.
Suppose that L is smashing. Then, for any family {Xi} of objects of CL, the
coproduct in C is already local, and so is the same as the coproduct in CL. Thu*
*s,
if Z 2 C is small we have
a a a M M
[LZ; L Xi] = [Z; L Xi] = [Z; Xi] = [Z; Xi] = [LZ; Xi]
44 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
so that LZ is small in CL.
Conversely, suppose that L preserves smallness. Let {Xi} be a family of L-loc*
*al
objects. Then, for all Z 2 *G, we have isomorphisms
a M M ia j ia j
[Z; Xi] = [Z; Xi] = [LZ; Xi] = [LZ; L Xi ] = [Z; L Xi ]:
` `
Therefore the natural map Xi-! L( Xi) is an isomorphism, so L is smashing.
Suppose again that L is smashing. Define K :C -! C by KX = F (LS; X). We
claim that KX is actually L-local. Indeed, suppose that Z is L-acyclic; then
[Z; KX] = [LS ^C Z; X] = [LZ; X] = [0; X] = 0;
which implies the claim. We may thus regard K as a functor C -!CL, and as such
it is easily seen to be right adjoint to J. It follows that J, equipped with th*
*e maps
fi :SC -! JSL
:JU ^C JV -! J(U ^L V )
is a lax geometric morphism. It is immediate from the definitions that and ss *
*are
isomorphisms.
We defer to Theorem 4.3.4 the proof that CL is a Brown category when C_is a
Brown category. |__|
We next consider the local category obtained by localizing at a set of small
objects.
Theorem 3.5.3. Let A C be a thick subcategory of small objects as in Theo-
rem 3.3.5. Let L = LA :C -! CL be the localization functor constructed there,
whose category of acyclics is
Z = {Y | 8W 2 A [W; Y ] = 0}:
Then we can make CL into an algebraic stable homotopy category with G = A, and
L into a geometric morphism (but not a stable morphism in general). Moreover,
L admits a left adjoint M = CfAas well as a right adjoint J. Thus L preserves
products, function objects, and sequential limits (as well as the other structu*
*re listed
in Theorem 3.5.1). The following maps are isomorphisms, where the subscript L
indicates structure in CL.
# :LFC(X; Y ) ' FL(LX; LY )
# :JFL(U; V ) ' FC(JU; JV )
ss#:L(X ^C JU) ' LX ^L U
ae:FC(X; JU) ' JFL(LX; U)
ae#:LFC(X; JU) ' FL(LX; U)
Moreover, LM ' 1 ' LJ :CL -! CL.
Proof.We know from Theorem 3.5.1 that CL is an enriched triangulated category
with all cohomology functors representable, and that L: C -! CL is a geometric
morphism. It follows easily from Theorem 3.3.5 that L = 1 on A, so that A CL.
It also follows that if Z 2 C is such that [W; Z] = 0 for all W 2 A, then LZ = *
*0. If
in addition we have Z 2 CL, then clearly Z = 0. It now follows from Theorem 2.3*
*.2
that CL is an algebraic stable homotopy category with generators A.
AXIOMATIC STABLE HOMOTOPY THEORY 45
Recall from Theorem 3.3.5 that LX = F (MS; X) and that MU = MS ^ U. It
follows that
[U; LX] = [U; F (MS; X)] = [MS ^ U; X] = [MU; X]:
Thus M is left adjoint to L, which implies that L preserves products and sequen*
*tial
limits.
We know from Theorem 3.3.5 that ML = M, and it follows easily that MU ^X =
M(U ^ LX). We therefore have
[U; LF (X; Y=)][MU; F (X; Y )] = [M(U ^ LX); Y ]
= [U ^ LX; LY ] = [U; F (LX; LY )]:
It follows that LF (X; Y ) = F (LX; LY ), in other words that # is an isomorphi*
*sm.
It follows in turn that ae# is an isomorphism. We saw in Theorem 3.5.1 that # ,*
* ss#
and ae are isomorphisms, and that LJ = 1. We saw in Theorem 3.3.5 that LM_' L
on C, so that LM ' 1 on E = CL. |__|
There are many properties one would like CL to have that it does not enjoy in
general.
Example 3.5.4.
(a) In the homotopy category of spectra S, let L denote localization with resp*
*ect
to MU or with respect to the wedge of all the Morava K-theories K(n) (where
0 n < 1). Then there are no nonzero small objects in SL [Str].
(b) While every object in thick is strongly dualizable, there will be other
strongly dualizable objects in general. Indeed, any interesting element of*
* the
Picard group in the K(n)-local category will be strongly dualizable yet not
in thick. There are many examples already when n = 1 [HMS94 ].
Note that the subcategory of colocal objects will not, in general, form a sta*
*ble
homotopy category even though coproducts of colocal objects are always colocal.
The problem is that colocalization functors would preserve cogenerators, but th*
*ere
is no reason to expect them to preserve generators. This is the main reason that
localization functors arise more often than colocalizing functors in stable hom*
*otopy
theory. See, however, Theorem 9.1.1 for a situation in which the subcategory of
colocal objects does form a stable homotopy category.
Next we point out that any localization L, even if it is not smashing, induces
correspondences between the full subcategories of CL and certain full subcatego*
*ries
of C.
Definition 3.5.5.Suppose that C is a triangulated category and L is a localizat*
*ion
functor. We say that a subcategory D of C is L-replete if it is full, and whene*
*ver
X -! Y is an L-equivalence, then X 2 D , Y 2 D.
Lemma 3.5.6. Suppose that C is a triangulated category and L is a localization
functor. There is a bijection between replete subcategories of CL and L-replete*
* sub-
categories of C. This correspondence sends thick subcategories to thick subcate*
*gories
and localizing subcategories to localizing subcategories.
Proof.If D is an L-replete full subcategory of C, then define F (D) = D \ CL. If
E is a full subcategory of CL, then define G(E) to be the full subcategory of C
with objects {X | LX 2 E}. It is trivial to check that F O G(E) = E, and using
L-invariance that G O F (D) = D. We also leave it to the reader to check that if
46 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
D C and E CL correspond, then D has the appropriate structure (is thick
or localizing) if and only if E does. The crucial point is that L always prese*
*rves_
coproducts as a functor from C to CL. |__|
We point out that localization functors rarely preserve products (except in t*
*he
situation of Theorem 3.5.3), so L-replete colocalizing subcategories of C will *
*not
correspond to colocalizing subcategories of CL.
3.6. The Bousfield lattice. In this section we define Bousfield classes and dis*
*cuss
a few of their basic properties; see [Bou79a ] and [Rav84 ] for the proofs and *
*for other
useful results.
Definition 3.6.1.
(a) Fix an object X in a stable homotopy category C. We say that an object Y
is X-acyclic if X ^ Y = 0, and that an object Z is X-local if F (Y; Z) = 0*
* for
all X-acyclic objects Y .
(b) We define the Bousfield class of X (written ) to be the collection of X-
local objects. This forms a colocalizing coideal (Definition 1.4.3). We *
*say
that two objects X and Y are Bousfield equivalent if = .
(c) The collection of Bousfield classes then defines a partially ordered class*
* under
inclusion. Write qfor , and ^for . (We shall
see later that this is well-defined. It is easy to see that ^ \<*
*Y >,
but in general they are not equal_see [Bou79a , Lemma 2.5].)
It is more common to define the Bousfield class of X to be the localizing ide*
*al of
X-acyclics. We have chosen to use X-locals instead so that the ordering
just means that the category of X-locals contains the category of Y -locals.
The partially ordered class of Bousfield classes is contained in an apparently
more fundamental lattice called the Bousfield lattice, which we now define.
Let C be a stable homotopy category. Given two objects X and Y , we write
X ? Y if F (X; Y ) = 0. More generally, if D is a class of objects, we write X *
*? D
if X ? Y for all Y 2 D, and so on. We define the left and right annihilators of*
* a
class D as follows:
? D = {X | X ? D};
D? = {X | D ? X}:
Definition 3.6.2.A class D C is a closed localizing ideal if D = ?E for some E.
(It is easy to check that such a class is indeed a localizing ideal.) Dually, D*
* is a
closed colocalizing coideal if D = E? for some E.
The purely formal theory of Galois correspondences tells us that the closed l*
*o-
calizing ideals form a lattice under inclusion, antiisomorphic to the lattice o*
*f closed
colocalizing coideals. (We make the lattice operations explicit in the definiti*
*on be-
low.) Moreover, the smallest closed localizing ideal containing a class D is ?(*
*D? ).
Definition 3.6.3.The Bousfield lattice of C is the lattice L of closed colocali*
*zing
coideals. The meet operation is just intersection, and the join operation is
D q E = (? D \ ?E)? :
We will refer to an element of the Bousfield lattice as a generalized Bousfield
class. Note that Bousfield classes are generalized Bousfield classes, because <*
*X> =
AXIOMATIC STABLE HOMOTOPY THEORY 47
{Z | X ^Z = 0}? . Note also that if we think of L as the lattice of closed loca*
*lizing
ideals ordered by reverse inclusion, then the join operation is just intersecti*
*on.
We do not know whether every generalized Bousfield class is in fact a Bousfie*
*ld
class, although there are a number of generalized Bousfield classes that can on*
*ly be
proved to be Bousfield classes by quite subtle arguments. We do not know whether
the collection of Bousfield classes is closed under intersections.
The Bousfield lattice is analogous to the lattice of torsion theories in an A*
*belian
category [Gol86]. However, the lattice of torsion theories has many good proper*
*ties
which we have been unable to prove in our context.
Lemma 3.6.4. If L: C -! C is a localization functor, then {X | LX = 0} is a
closed localizing ideal, and CL is the corresponding closed colocalizing coidea*
*l.
Proof.See part (b)(iv) of Lemma 3.1.6. |___|
We do not know in general whether the converse of the above lemma holds, nor
do we know whether all localizing ideals are closed. We also do not know whether
the closed localizing ideals form a set or a proper class.
We have shown how to associate a Bousfield class to an object X of a stable
homotopy category C, and a generalized Bousfield class to a localization functor
L. We can also associate a generalized Bousfield class to a homology functor H.
Indeed, recall the localizing ideal D of H-acyclics:
D = {X : H(X ^ Y ) = 0 for allY }:
Then define = D? , the closed colocalizing coideal of H-local objects.
Another way to say this, when C is algebraic, is that is equal to the gen*
*er-
alized Bousfield class of the localization functor LH . In this case, there is *
*also a
way to associate a homology functor to an object X. Recall that we defined the
category (X) in Definition 2.3.7.
Definition 3.6.5.Let C be an algebraic stable homotopy category. Write
bss0(X) = lim-!(X)[S; Xff]:
This is a homology functor by Corollary 2.3.11. If C is unital algebraic, then
bss0(X) = ss0(X). Given an object X 2 C, we define
HX (Y ) = bss0(X ^ Y ):
This is again a homology functor. There is an obvious natural map HX (Y ) =
HY (X) -!ss0(X ^ Y ) = X0Y , which is an isomorphism when C is unital algebraic.
Thus, in an algebraic stable homotopy category, there are two generalized Bou*
*s-
field classes associated to an object X. The following lemma shows they are in *
*fact
equal.
Lemma 3.6.6. Let X be an object of an algebraic stable homotopy category C.
Then for any Y 2 C we have X ^ Y = 0 if and only if Y is HX -acyclic, and thus
= . The category of such Y is a closed localizing subcategory.
Proof.If Y is X-acyclic, then X ^Y = 0, so HX (Y ^Z) = bss0(X ^Y ^Z) = 0 for all
Z. Thus Y is HX -acyclic. Conversely, suppose Y is HX -acyclic. By Lemma 4.1.2,
we find that ss0(X ^ Y ^ Z) = 0 for all small Z. By Spanier-Whitehead duality,
we find that [Z; X ^ Y ] = 0 for all small Z, and in particular for all Z 2 G. *
*Thus,
by Lemma 1.4.5, X ^ Y = 0 and so Y is X-acyclic. The localizing category of
48 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
HX -acyclics is closed by Theorem 3.2.2 and Lemma 3.6.4. The categories and
are by definition the right annihilators of the X-acyclics and the HX_-acy*
*clics,
so they are the same. |__|
Because of this lemma, we denote the localization functor LHX simply by LX .
An annoying feature of this lemma is that we have to define the acyclics of HX
by requiring that HX (Y ^ Z) = 0 for all Z. It is sometimes useful to avoid thi*
*s.
Definition 3.6.7.An object X in an algebraic stable homotopy category C is
monoidal if HX* (Y ) = 0 implies HX* (Y ^ Z) = 0 for all Z. (Note the grading on
HX .)
Lemma 3.6.8. Suppose`C is an algebraic stable homotopy category, and X is an
object of C. Let X0= Z2GX ^ Z. Then = and X0 is monoidal.
Proof.This proof is very similar to that of the preceding lemma. Clearly
X ^ U = 0 ) X0^ U = 0 ) HX0*(U) = 0:
Both claims in the lemma will follow if we prove that HX0*(U) = 0 implies X ^U =
0. To see this, we note that
M M
HX0*(U) = bss*(X ^ U ^ DZ) = HDZ* (X ^ U):
Z2G Z2G
Furthermore, HDZ* (X ^ U) = ss*(DZ ^ X ^ U) = [Z; X ^ U]* by Lemma 4.1.2.__
Lemma 1.4.5 completes the proof. |__|
The following proposition is very similar to results in [Bou79a ] and [Rav84 *
*].
Proposition 3.6.9.Suppose that C is an algebraic stable homotopy category.
(a) if and only if X ^ Z = 0 ) Y ^ Z = 0. Hence **~~ is the largest
Bousfield class, and <0>is the smallest.
(b) For any X and Y we have = q and \ .
(c) If X -! Y -! Z is a cofiber sequence, then q .
(d) If f :dX -f! X is a self map with cofiber X=f and telescope f-1 X (see
Definition 2.2.3), then = q .
(e) Given two objects X and Y , we have LX ' LY if and only if = .
(f) If = then = (so we can define ~~__^ = ____
unambiguously).
(g) The class D of Bousfield classes for which ^ = forms a
distributive lattice with operations q and ^ (but we do not know whether
\ = ^).
Proof.(a): Suppose that , so that Y -local objects are X-local. Suppose
that X ^ Z = 0. As LX Z is initial among X-local objects under Z, we find that
the map Z -! LY Z factors through LX Z, which is zero. As Z -! LY Z becomes
the identity after applying LY , we see that LY Z = 0. This means that Y ^ Z = 0
as claimed. The converse is easy.
(b): This is clear using (a).
(c): Suppose that U ^ (X q Y ) = 0, so that U ^ X = 0 = U ^ Z. From the
cofiber sequence U ^ X -! U ^ Y -! U ^ Z, we see that U ^ Y = 0 also. The claim
follows using (a).
AXIOMATIC STABLE HOMOTOPY THEORY 49
(d): Suppose that U ^ X=f = 0 and U ^ f-1 X = 0. It is easy to see that
(1 ^ f)-1(U ^ X) = U ^ f-1 X = 0. On the other hand, using the cofibration
dU ^ X -1^f-!U ^ X -! U ^ X=f;
we see that 1 ^ f is an isomorphism. Thus, by Lemma 2.2.6, we have U ^ X =
(1 ^ f)-1(U ^ X) = 0. It follows using (a) that q . The
opposite inequality is easy (again using (a)).
(e): The category is the image of LX , and LX is the left adjoint of the
inclusion -!C. Thus and LX determine each other.
(f): This follows easily from (a).
(g): Clearly D is a partially-ordered class, closed under the stated operatio*
*ns.
Clearly q is the join operation; in other words, if and only if
and . Similarly, if and then = *
* __
^ ^ . The converse is trivial, so ^ is the meet operation in D. *
*|__|
Note that, if C is not algebraic, part (a) of Proposition 3.6.9 might fail. W*
*e do
not have examples which tell us whether to worry about this. Note also that in *
*the
homotopy category of spectra, the Brown-Comenetz dual of the sphere (written I)
has I ^ I = 0 but I 6= 0, showing that the inequality in (b) can be strict.
3.7. Rings, fields and minimal Bousfield classes. In this section we consider
some criteria which guarantee that a Bousfield class is minimal. We first need *
*to
define rings, modules and fields.
Definition 3.7.1.Let C be a stable homotopy category.
(a) A ring object in C is an object R equipped with an associative multiplicat*
*ion
map : R ^ R -! R and a unit j :S -! R. Note that this makes ss*(R)
into a graded ring. If R is commutative in the evident sense, then ss*(R) *
*is a
graded-commutative ring. For any object X, the object F (X; X) is a (usual*
*ly
noncommutative) ring under composition.
(b) If R is a ring object, there is an obvious notion of R-module objects. If M
and N are left R-module objects and X is arbitrary then M q N, M ^ X
and F (X; M) are left R-module objects. Moreover, F (M; X) is a right R-
module object. However, the cofiber of a map of R-modules need not admit
an R-module structure.
(c) We say that an R-module M is free if it is isomorphic as an R-module to a
coproduct of suspensions of R.
(d) A skew field object in C is a ring object R such that every R-module is fr*
*ee.
A field object is a skew field object that is commutative.
(e) An object X 2 C is smash-complemented if there is an object Y such that
X ^ Y = 0 and = __~~.
We next give a convenient (and well-known) criterion for recognizing fields.
Proposition 3.7.2.Let R be a ring object in a monogenic stable homotopy cate-
gory C, and suppose that every homogeneous element in ss*(R) is invertible. Then
R is a skew field object.
Proof.Let M be a left R-module object, so that ss*M is a left module over ss*R.
Recall that every left module over a division ring is free. A simple modificati*
*on of
50 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
the proof of this shows that every left module over ss*R is a direct sum of sus*
*pensions
of ss*R. Choose a basis {ei} for ss*M, where ei has degree di. Each ei gives a *
*map
diR = R ^ Sdi-1^ei--!R ^ M -! M:
Putting these together, we get a map
a
f : diR -!M
i
L
which induces an isomorphism iss*-diR ' ss*M. It follows that f is an_equiva-_
lence. |__|
It follows that if R is a commutative ring and K is an R-algebra and a skew f*
*ield
then K defines a skew field object in the derived category of R-modules. It also
follows that the Morava K-theory spectrum K(n) (which depends on a prime p not
written explicitly in the notation) is a skew field object in the category of s*
*pectra.
Indeed, it is known [HS ] that every skew field object in this category is a fr*
*ee module
over some K(n), provided that we allow K(0) = HQ and K(1) = HFp.
An important property of skew field objects is given by the next result.
Proposition 3.7.3.Let R be a smash-complemented skew field object in an al-
gebraic stable homotopy category C. Then locid is minimal among nonzero
localizing ideals of C, and is minimal among nontrivial Bousfield classes.
Proof.Let C be a complement for R, so that R ^ C = 0 and = ~~~~.
Suppose that . As R ^ C = 0, we also have X ^ C = 0. Thus =
^ = . On the other hand, X^R is a (possibly empty) coproduct
of copies of R, so is either 0 or . Thus is minimal.
Now suppose that 0 6= Y 2 locid. As {Z | } is a localizing ideal,
we see that 0 < . As above, we deduce that R is a retract of R ^ Y ,
so R 2 locid. It follows that locid is minimal among nontrivial_localiz*
*ing_
ideals. |__|
It seems likely that the Bousfield class of the Brown-Comenetz dual of the sp*
*here
is minimal, and it is certainly not represented by a field. This also seems lik*
*ely for
where F is a finite complex of type n; note that this is nonzero by
the failure of the telescope conjecture. Conversely, the Eilenberg-MacLane spe*
*c-
trum HFp is a skew field object (that is not smash-complemented), yet neither
locid nor is minimal.
We next relate the two kinds of minimality that we have considered.
Proposition 3.7.4.Let {K(n) | n 2 I} be a family of objects in an algebraic
stable homotopy category C. Suppose that
a
~~~~ =
n2I
and
K(n) ^ K(m) = 0 when n 6= m:
Suppose also that locid is minimal among nontrivial localizing ideals. Th*
*en
is minimal among nontrivial Bousfield classes.
AXIOMATIC STABLE HOMOTOPY THEORY 51
Proof.Suppose that < . This means that there exists an X-acyclic
object U such that K(n) ^ U 6= 0. Note that K(n) ^ U 2 locid, and that
locid is minimal; this implies that
locid = locid locid~~__:
On the other hand, locid____ is clearly contained in the category of X-acyclics.*
* Thus
K(n) ^ X = 0. Moreover, if m 6= n then K(n)`^ K(m) = 0 and so
X ^K(m) = 0. Using the decomposition __~~= , we see that X = 0. The_
claim follows. |__|
We finish this section with a very simple observation.
Proposition 3.7.5.Let {K(n) | n 2 I} be a family of objects in an algebraic
stable homotopy category C. Suppose that
a
~~~~ =
n2I
and
K(n) ^ K(m) = 0 when n 6= m:
Suppose also that for each n, is minimal among nontrivial Bousfield class*
*es.
For any X 2 C, define
supp(X) = {n 2 I | K(n) ^ X 6= 0}:
Then
a
= :
n2supp(X)
Proof.First, we have
a
= ~~~~= :
n2I
Next, observe that K(n) ^ X = 0 unless n 2 supp(X). If n 2 supp(X) then __
6= 0, so = by minimality. The claim follows. |__|
3.8. Bousfield classes of smashing localizations. Let C be an algebraic stable
homotopy category, and write L for the associated Bousfield lattice. Let Ls be
the subclass of colocalizing subcategories of the form CL, where L is a smashing
localization. Note that CL = in this context, and that CL = CL0 if and only
if L is isomorphic to L0. In this section, we shall prove various good properti*
*es of
Ls. We start with a simple lemma.
Lemma 3.8.1. Let L be a smashing localization, and X an arbitrary object of C.
Then = \ .
Proof.It is a general fact that = \ . Suppose that
Y 2 \ , so that Y = LY 2 CL and F (Z; Y ) = 0 whenever X ^ Z = 0.
We need to show that Y is LX-local, in other words that F (W; Y ) = 0 whenever
LS ^ X ^ W = 0. By taking Z = LS ^ W = LW , we see that F (LW; Y ) = 0, but *
* __
F (LW; Y ) = F (W; Y ) as Y is L-local, so F (W; Y ) = 0 as required. *
* |__|
We can now prove the main properties of Ls.
52 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Proposition 3.8.2.The class Ls is a sublattice of L, closed under countable
meets. Moreover, Ls is distributive.
Proof.Let L and L0be smashing localization functors, with corresponding colocal-
izations C and C0. We first show that CL \ CL0 and CL q CL0 lie in Ls. Define
L0X = LL0X = L0LX = LS ^ L0S ^ X:
Clearly L0X 2 CL \ CL0. Moreover, suppose that Y 2 CL \ CL0. We then have
[X; Y ] = [LX; Y ] = [L0LX; Y ];
the first equality because Y 2 CL and the second because Y 2 CL0. Thus L0: C -!
CL \ CL0 is left adjoint to the inclusion. It follows easily that L0 is a smas*
*hing
localization functor with CL0 = CL \ CL0, as required. 0
We next show that CL q CL0 lies in Ls, or equivalently that CC \ CC is the
category of colocal objects for a smashing colocalization functor. We define
C1X = CC0X = C0CX = CS ^ C0S ^ X:
We write L1X = L1S ^ X for the cofiber of the natural map C1X -! X; note that
this is not the same as L0X. An argument similar0to the above shows that C1 is
right adjoint0to the inclusion of CC \ CC in C, so it is a colocalization func*
*tor with
CC1 = CC \CC . It follows that L1 is a (smashing) localization with CL1 = CL qC*
*L0
as required.
It follows from Lemma 3.8.1 that the operations ^ and \ agree on Ls, and ^
clearly distributes over q, so Ls is distributive.
Now consider a countable family of smashing localizations {Lk}, whose meet we
wish to construct. We may replace Lk by the meet of L0; : :;:Lk and thus assume
that we have a descending sequence
: : :
By Lemma 3.1.5, there is unique morphism Lk -! Lk+1 for each k. We define
L1 S to be the sequential limit of the objects LkS, so there is an obvious map
i1 = (S -i0!L0S -! L1 S). Write L1 X = L1 S ^ X, which is clearly a functor of
X. We also see that L1 X is the sequential colimit of the sequence LkX.
For each k, we have a map of cofiber sequences
CkS ----! S ----! LkS
?? ? ?
y =?y ?y
Ck+1S ----! S ----! Lk+1S
We would like to take the sequential colimit of these maps to get a cofiber seq*
*uence,
but the sequential colimit is not always exact. Nonetheless, we get a commutati*
*ve
diagram
` ` `kik `
kCkS ----! kS? ----! k?LkS
f?y g?y
` ` `kik `
kCkS ----! kS? ----! k?LkS
p?y q?y
S L1 S
AXIOMATIC STABLE HOMOTOPY THEORY 53
Here the maps f and g are the usual ones, whose cofibers are the respective se-
quential colimits.` By the analysis in Lemma 2.2.6, we see that pj0 = 1, where
j0: S -! kS includes the first factor. The 3 x 3 Lemma A.1.2 now gives us a
diagram
` ` `kik `
kCkS? ----! kS? ----! k?LkS
?y p?y ?yq
C1 S ----! S ----! L1 S
i
`
Here the rows are exact, and C1 S is a the cofiber of some self-map of kCkS a*
*nd
i is some map S -! L1 S. In fact, we have
ia j
i = ipj0 = q ik j0 = qi0 = i1 :
We can restate this as follows: the fiber C1 S of i1 :S -! L1 S lies in the loc*
*alizing
subcategory generated by the objects CkS.
Note that CkS ^ Lm S = 0 for m k, so CkS ^ L1 S = lim-!mCkS ^ Lm S = 0. It
follows that L1 S ^ C1 S = 0.
We write C1 X = C1 S ^ X, and D = {X | C1 X = 0}. ByTthe above, L1 can
be consideredTas a functor C -! D. We claim that D = kCLk. Indeed, suppose
that X 2 k CLk, so that CkS ^X = 0 for all k < 1. As C1 S lies in the localizi*
*ng
subcategory generated by the CkS, we see that C1 S ^ X = 0 and thus X 2 D.
Conversely, suppose that X 2 D. Then X = L1 X = lim-!mLm X. We may start
the colimit at the kth stage;Tas CkLm X = 0 for k m < 1, we conclude that
CkX = CkL1 X = 0. Thus X 2 k CLk.
Next, we claim that [C1 X; L1 Y ] = 0 for all X and Y . Indeed, we have just
seen that L1 Y is Lk-local for all k, so [CkX; L1 Y ] = 0 for all k. As C1 X 2
loc, we see that [C1 X; L1 Y ] = 0.
Next, we claim that L1 is left adjoint to the inclusion of D in C. Indeed, su*
*ppose
that Y 2 D, so Y = L1 Y , so [C1 X; Y ] = 0 for all Y . Thus the cofibration
C1 X -! X -! L1 X shows that [X; Y ] = [L1TX; Y ] as required. This implies tha*
*t_
L1 is a smashing localization with CL1 = k CLk as required. |__|
The sublattice Ls of the Bousfield lattice is not closed under countable join*
*s.
Indeed, in the category of spectra, the join of the smashing Bousfield classes *
*
is the harmonic Bousfield class, which is not smashing.
The situation is even better for finite localizations, as expressed by the fo*
*llowing
result.
Proposition 3.8.3.Let C be an algebraic stable homotopy category, and let Lf be
the collection of Bousfield classes of the form , where LfAis a finite loc*
*alization
functor. Then Lf is closed under finite joins and arbitrary meets. It is a dist*
*ributive
lattice, antiisomorphic to the lattice of G-ideals of small objects (which is *
*a set
rather than a proper class).
Proof.Let F be the category of small objects in C, and L0fthe collection of G-i*
*deals
of F. This is clearly a lattice with arbitrary meets (given by intersection) an*
*d joins
(given by taking the G-ideal generated by the union). As C is algebraic, there *
*is
only a set of small objects up to isomorphism, so L0fis a set. Suppose that A 2*
* L0f.
54 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Because A is a G-ideal, it is easy to see that loc~~~~ = locid. Theorem 3.3.3
gives a smashing localization functor LfAwith
= CLfA= loc?
and
ker(LfA) = loc:
Moreover, it tells us that loc \ F = A. It follows directly that the map j :*
*A 7!
is an order-reversing map from L0fto the Bousfield lattice L. If we view L
as the opposite of the lattice of closed localizing subcategories, then j becom*
*es the
map A 7! loc, and this makes it easy to see that j sends arbitrary joins to
meets. Also, as A = loc \ F, we see that j is a monomorphism. The image is
by definition Lf, so Lf is closed under arbitrary meets.
Now consider two elements A0; A1 of L0f, and write A2 = A0\A1. We claim that
j(A2) is the join of j(A0) and j(A1) in the Bousfield lattice. It will suffice *
*to show
that loc = loc\loc. It is immediate that loc loc\loc.
For the converse, we first recall that small objects in an algebraic stable hom*
*otopy
category are G-finite, which implies that W ^ X 2 A2 whenever W 2 A0 and
X 2 A1 (or in other words, A0 ^ A1 A2). By considering {Z | Z ^ A1
loc}, we conclude that loc ^ A1 loc. A similar argument then
shows that loc ^ loc loc. In particular (if we write Ci for CfAi) *
*we
have C0S ^ C1S 2 loc. If X 2 loc \ loc then X = C0X = C1X so
X = X ^ (C0S ^ C1S), so X 2 loc as required.
This shows that Lf is closed under the lattice operations in Ls, so it is_a d*
*is-
tributive lattice. |__|
4. Brown representability
In this section we discuss the representability of homology functors and rela*
*ted
issues. Throughout this section, C will be an algebraic stable homotopy categor*
*y.
4.1. Brown categories. We begin with the definition of a representable homology
functor and of a Brown category. The first part of the definition below appeared
as Definition 3.6.5, but we repeat it here for convenience.
Definition 4.1.1.Write
bss0(X) = lim-!(X)[S; Xff]:
This is a homology functor by Corollary 2.3.11. If C is unital algebraic, then
bss0(X) = ss0(X). Given an object X 2 C, we define
HX (Y ) = bss0(X ^ Y ):
This is again a homology functor. There is an obvious natural map HX (Y ) =
HY (X) -!ss0(X ^ Y ).
A homology functor H on C is representable if there is an object Y of C and
an isomorphism of homology functors HY ' H. Note that this definition is incon-
sistent with the usual categorical terminology, in which a covariant functor is*
* said
to be representable if and only if it is equivalent to a functor of the form [Y*
*; -].
Nonetheless, it is close to the standard usage in stable homotopy theory.
Lemma 4.1.2. If X is small then the natural map HX (Y ) -! ss0(X ^ Y ) is an
isomorphism.
AXIOMATIC STABLE HOMOTOPY THEORY 55
Proof.Recall from Theorem 2.1.3 that the subcategory of small objects is closed
under smash products and the duality functor DX = F (X; S).
Suppose that X is small. Then ss0(X ^ Y ) = [DX; Y ] is a homology functor of
Y , by the smallness of DX. When Y is also small, then so is X ^ Y , so HX (Y )*
* =
ss0(X ^ Y ). Thus the natural map HX (Y ) -! ss0(X ^ Y ) is a map of homology *
* __
functors which is an isomorphism for small Y , and thus for all Y . *
* |__|
Lemma 4.1.3. There is a natural isomorphism
HX (Y ) ' lim-!(Ys)s0(X ^ Yff):
Proof.Both sides are homology functors of Y (by Corollary 2.3.11). They are
isomorphic for small Y by Lemma 4.1.2 and the fact that HX Y = HY X. Thus, __
they are isomorphic for all Y . |__|
We can now define a Brown category.
Definition 4.1.4.A Brown category is an algebraic stable homotopy category C
such that every homology functor is representable and every natural transformat*
*ion
HX -! HY of homology functors is induced by a (typically nonunique) map X -! Y .
Naturally one would like to be able to tell when an algebraic stable homotopy
category is a Brown category. Neeman shows in [Nee95] that the derived category
of modules over C[x; y] is not a Brown category, although it is a monogenic sta*
*ble
homotopy category, so it seems that the Brown condition is a genuinely subtle o*
*ne.
Recall, from Definition 2.3.3, that c(C) denotes the (necessarily infinite) c*
*ardi-
nality of the disjoint union of the sets [W; Z]n for all W; Z 2 G and n 2 Z.
Theorem 4.1.5. If C is an algebraic stable homotopy category with c(C) = @0,
then C is a Brown category.
Proof.Suppose that C is algebraic and c(C) = @0. Let H :Fop -!Ab be an exact
functor. We claim that this can be extended to give a cohomology functor defined
on all of C. The proof of this fact is complicated, but it is also the same, mu*
*tatis
mutandis, as the proof in [Mar83 , Chapter 4] (which in turn follows [Ada71 ]).*
* We
give a brief outline. First, we define a functor bH:Cop -!Ab by
bH(Y ) = lim H(Y ):
- (Y ) ff
This converts coproducts to products, minimal weak colimits to limits, and with
considerable work one can show that is exact on a restricted class of cofibrati*
*ons.
This uses the that fact a countable filtered diagram of nonempty sets and surje*
*ctions
has nonempty inverse limit; it is here that the countability hypothesis is used*
*. (The
analogous statement is false for an inverse system indexed by the first uncount*
*able
ordinal.) In any case, this restricted exactness suffices to carry through most*
* of the
proof of representability as in Theorem 2.3.2. This gives an object X and a nat*
*ural
map [Y; X] -!Hb(Y ) which is an equivalence when Y 2 *G and thus when Y 2 F.
Of course, when Y 2 F we also have bH(Y ) = H(Y ). We may therefore take [-; X]
as an extension of H to all of C.
Now suppose that we have two contravariant exact functors H0; H00:Fop -!Ab
and a natural map f :H0 -!H00. Choose objects X0; X00representing H0 and H00
as above. Margolis also proves that f arises from a map g :X0-! X00.
Now let H :C -! Ab be a homology functor. By applying the above to H0 =
H O D (where D is the Spanier-Whitehead duality functor), we obtain an object X
56 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
and an equivalence H(Z) = H0(DZ) = [DZ; X] = X0Z = HX (Z) for small objects
Z. Using Corollary 2.3.11, we see that H(Z) = HX (Z) for all Z, so that H is *
* __
representable. The previous paragraph shows that maps are also representable. *
*|__|
In a Brown category, every natural transformation of homology functors is in-
duced by a map of the representing objects, but there may be more than one such
map inducing the same natural transformation.
To investigate this nonuniqueness, we adopt a somewhat abstract approach.
Suppose that C is an algebraic stable homotopy category, and let F be the full
subcategory of small objects. We write Fo and Fo for the categories of covariant
and contravariant exact functors F -! Ab. Similarly, we write Co and Co for the
categories of homology and cohomology functors from C to Ab. Composition with
the Spanier-Whitehead duality functor D gives an equivalence Fo ' Fo, which we
also call D. Restriction gives functors Uo: Co -! Fo and Uo :Co -! Fo. We also
have functors Vo: C -!Co and V o:C -!Co, sending X to HX and X0 respectively.
Finally, we define a functor L: Fo -! Co by LH = bHF. Recall from Section 2.3
that LH(X) is the colimit of H(Y ) for all small objects Y over X.
Note that all of these functors except Vo and L are defined on any stable hom*
*o-
topy category.
Definition 4.1.6.A map f :X -! Y in a stable homotopy category C is phantom
if, for all small objects Z and maps g :Z -! X, the composite f O g is trivial.
Equivalently, f is phantom if and only if UoV of = 0.
Remark 4.1.7. Margolis [Mar83 ] calls these maps f-phantom maps; his definition
of phantoms is slightly different.
The phantom maps clearly form an ideal, in the following sense: if f; g :X -!*
* Y
are phantom, and u: W -! X and v :Y -! Z are arbitrary, then f + g, fu and vf
are phantom. We denote the subgroup of [X; Y ] consisting of phantom maps by
P(X; Y ).
The above functors fit into a commutative diagram as follows:
Uo
Co_____________Fo-_oe
Vo L |6
||
||
C |||D
@ ||
V @@Ro ||?
Co_____________Fo-o
U
The following theorem is mainly a compendium of results and definitions that
we have already seen.
Theorem 4.1.8. Suppose that C is a Brown category. Then the functors D, Uo
and V oare equivalences, and L is inverse to Uo. The functors Uo and Vo are full
and essentially surjective, and reflect isomorphisms. The kernel of Vo (or Uo) *
*is a
square-zero ideal. To make some of these claims more explicit:
(a) Any homology functor H on F extends to a homology functor LH on C, unique
up to canonical isomorphism.
AXIOMATIC STABLE HOMOTOPY THEORY 57
(b) The group of natural maps f :HX -! HY is isomorphic to [X; Y ]=P(X; Y ). A
map f :X -! Y is an isomorphism if and only if the induced map f0: HX -!
HY is an isomorphism.
(c) Any homology functor H on C is equivalent to HX for some X 2 C. The
representing object X is unique up to isomorphism, but the isomorphism is
only canonical up to the addition of a phantom map.
(d) The composite of any two phantom maps is zero.
Proof.The functor V ois an equivalence in any stable homotopy category. Indeed,
Yoneda's lemma tells us V ois full and faithful, and since every cohomology fun*
*c-
tor is representable, it is essentially surjective. Thus V ois an equivalence.*
* As
D :Fop -! F satisfies D2 ' 1, we see that D :Fo -! Fo is an equivalence in any
stable homotopy category. It follows easily from Propositions 2.3.9 and 2.3.1 t*
*hat
Uo and L are inverse equivalences in any algebraic stable homotopy category.
In view of the above and our commutative diagram of functors, our claims about
Uo are equivalent to the corresponding claims about Vo. Now Vo is full and esse*
*n-
tially surjective by the definition of a Brown category. It is also clear that *
*the kernel
of Vo is the same as that of UoV o, which is the ideal of phantoms. In other wo*
*rds,
a map f :X -! Y induces the zero map HX -! HY if and only if f is phantom.
We next observe that Vo reflects isomorphisms. In other words, if a map f :X *
*-!
Y is such that f0: HX -! HY is an isomorphism, then f is an isomorphism. Indeed,
for Z 2 *G we observe that f gives an isomorphism between [Z; X] = HX DZ and
[Z; Y ] = HY DZ, so the claim follows by Lemma 1.4.5.
This leaves only the claim that the composite of two phantom maps is zero._We
will prove this as Theorem 4.2.5 below. |__|
4.2. Minimal weak colimits. The goal of this section is to show that an algebra*
*ic
stable homotopy category is a Brown category if and only if all filtered minimal
weak colimits of small objects exist. We use this, following Christensen [Chr],*
* to
show that the composite of any two phantom maps in a Brown category is trivial.
We begin by restating and proving Proposition 2.2.2, which gives an easier cr*
*i-
terion for a weak colimit to be minimal in a Brown category.
Proposition 4.2.1.Let C be a Brown category. Suppose that I is a small filtered
category, i 7! Xi is a functor from I to C, and (oi:Xi -! X) is a weak colimit.
Then X is the minimal weak colimit if and only if the induced map
lim-![Z; Xi]* -![Z; X]*
is an isomorphism for all Z 2 G.
Proof.Suppose that X is the minimal weak colimit, and that Z 2 G. Then Z is
small, so [kZ; -] is a homology theory. It follows immediately that
lim-![Z; Xi]* -![Z; X]*
is an isomorphism.
Conversely, suppose that the above map is an isomorphism for all Z 2 G. We
need to prove that lim-!iH(Xi) = H(X) for all homology functors H on C. As C is
a Brown category, we need only show that
HU (X) = lim-!iHU (Xi)
for all U 2 C. The left hand side is the same as HX (U), and the right hand side
is lim-!iHXi(U). These are both homology functors of U (using the exactness of
58 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
filtered colimits). By hypothesis, they agree when U 2 G, so they agree_for all
U. |__|
We also recall the following lemma, whose proof we have given in Remark 2.3.1*
*8.
Lemma 4.2.2. Suppose that C is an algebraic stable homotopy category, and X is
an object of C. If the diagram (X) of small objects over X has a minimal weak_
colimit Y , then X ' Y . |__|
We can now prove the main theorem of this section.
Theorem 4.2.3. Let C be an algebraic stable homotopy category, and F C the
subcategory of small objects. Then C is a Brown category if and only if any fun*
*ctor
i 7! Xi from a small filtered category I to F has a minimal weak colimit.
Proof.First suppose that C is a Brown category. Just as in [Mar83 , Theorem 5.1*
*3],
we define a functor H :C -!Ab by
H(U) = lim-!IHXi(U) = lim-!Iss0(U ^ Xi):
Here the latter isomorphism holds since Xiis small. Then H is a homology functo*
*r,
since filtered colimits of exact sequences of Abelian groups are exact. Let X b*
*e the
representing object for H. There are maps Xi fi-!X induced by including one
factor into the colimit. This map is uniquely determined up to a phantom map,
but since Xiis small, there are no phantom maps out of Xi. By the uniqueness, t*
*he
maps fi are compatible. That is, given a map s: i -!j in I, we have fjO Xs = fi.
Now if we have a compatible family of maps gi:Xi -! Y , we get a natural
transformation of homology functors HX -! HY since HX is just the colimit of the
HXi. This map is induced by a map X -h!Y , which is unique up to a phantom
map. Again, since the Xi are small, we must have hfi = gi. Thus X is a weak
colimit.
To see that X is the minimal weak colimit, suppose that Z 2 *G. It suffices to
show that lim-![Z; Xi] = [Z; X] by Proposition 4.2.1. But
lim-![Z; Xi] = lim-!HXi(DZ) = HX (DZ) = [Z; X]
as required.
Now suppose that C is algebraic, and every filtered diagram of small objects *
*has
a minimal weak colimit. To show that C is a Brown category, we will work with
contravariant exact functors on F (the class of small objects) rather than homo*
*logy
functors. This is equivalent to working with homology functors by the argument *
*of
Theorem 4.1.5 or Theorem 4.1.8. So, suppose that we have a contravariant exact
functor H :Fop -!Ab . We will show that H is representable. Define a category IH
whose objects are pairs (Z; z) where Z 2 F and z 2 H(Z). A map (Z; z) -!(Z0; z0)
is a map f :Z -! Z0 such that H(f)(z0) = z. Since the class of small objects is
essentially small, and H(Z) is a set for each Z, IH is also essentially small.*
* We
claim that IH is filtered. Indeed, given two objects (Z; z) and (Z0; z0), we *
*have
the obvious morphisms (Z; z) -! (Z q Z0; (z; z0))- (Z0; z0). Also, if we have
two morphisms f; g :(Z; z) -! (Z0; z0), we let h: Z0 -! W denote the cofiber of
f - g. Since H(f - g)(z0) = 0, there is a w 2 H(W ) such that H(h)(w) = z0, so a
morphism h: (Z0; z0) -!(W; w) coequalizing f and g.
We have an evident functor IH -! F that takes (Z; z) to Z. Let X denote the
minimal weak colimit of this functor. Then we have compatible maps i(Z;z):Z -! X
AXIOMATIC STABLE HOMOTOPY THEORY 59
for all objects (Z; z) of IH . We will construct a natural equivalence X0 -!H o*
*n F.
To do so, recall that we can extend H to C by defining
bH(Y ) = lim H(Y ):
- (Y ) ff
(This is not generally a cohomology functor on C.) We define a canonical class
x 2 bH(X) as follows. Since X is a minimal weak colimit, a map W -g!X in (X)
factors as g = i(Z;z)O g0 for some object (Z; z) of IH and some map g0:W -! Z.
We can then define x(W;g)= H(g0)(z) 2 H(W ). Then x(W;g)is well-defined and
defines a class x 2 bH(X).
We then have a natural transformation X0 -! Hb that takes a map W -g!X
to bH(g)(x). To see that this natural transformation is always surjective, supp*
*ose
w 2 bH(W ). Then for each (Zff-! W ) 2 (W ), we have a class wff2 H(Zff), and
the wffare compatible. This gives us compatible maps i(Zff;wff):Z -! X. Since W
is a weak colimit of (W ), by Lemma 4.2.2, there is a map W -g!X extending the
i(Zff;wff). It is then easy to see that bH(g)(x) = w.
Now suppose that W is small, and we have a map g :W -! X such that
bH(g)(x) = 0. Then there is an object (Z; z) of IH and a map g0:W -! Z such that
g = i(Z;z)O g0. In particular, H(g0)(z) = 0, so g0 is a morphism in the categor*
*y IH
from (W; 0) to (Z; z). Thus we have i(W;0)= i(Z;z)O g0. Thus g = i(W;0). On the
other hand, the zero map is a morphism in IH from (W; 0) to itself, and from th*
*is
it follows that i(W;0)is trivial.
Therefore, the natural transformation X0 -! bHis an isomorphism on F. Thus
every exact contravariant functor on F is representable. To complete the proof
that C is a Brown category, we must show that any natural transformation of
contravariant exact functors on F is representable. So suppose that H and K are
exact contravariant functors on F, and o :H -! K is a natural transformation. We
then get a functor IH -! IK that takes (Z; z) to (Z; o(z)). If we let X deno*
*te
the minimal weak colimit of IH -! F, and Y denote the minimal weak colimit of
IK -! F, then we get a (non-unique) induced map X -! Y . This is the required_
representation of o. |__|
Theorem 4.2.4. Suppose that C is a Brown category. Then any object X 2 C is
the minimal weak colimit of the diagram (X) (defined in Definition 2.3.7).
Proof.This is immediate from Lemma 4.2.2 and Theorem 4.2.3. |___|
The following result is well-known in the case of spectra with countable homo*
*topy
groups, or with a slightly different notion of phantom maps defined in terms of*
* finite-
dimensional spectra rather than finite spectra. However, it seems less well-kno*
*wn
in full generality. It is probably due to Boardman, but we learned how to prove*
* it
from Dan Christensen [Chr]. It is also proved in [Nee95].
Theorem 4.2.5. In a Brown category C, the composite of two phantom maps is
trivial.
Proof.We have seen that X is the minimal weak colimit of (X). Let us write Xff
for a generic object of (X), and u: Xff-! Xfifor a generic morphism.
60 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
As in the proof of Proposition 2.2.4, we have a non-minimal weak colimit C,
defined as the cofiber in a sequence
a a
Xff-! Xff-! C:
u ff `
Let F denote the fiber of the evident map ffXff-! X. Because X is the
minimal`weak colimit, we get a split monomorphism a: X -! C of objects under
` ffXff(usingapart (c) of 2.2.4). Applying the octahedral axiom to the morphisms
ffXff-! X -! C, we get a diagram as follows.
___-` Xff___
u J
J] J
Jc J
J J
` AE J J
Xff_________-C
ff J
OE J OE J J
J a Jc JJ
J J J
J J^ J^ AE
F ___________oec___________oeXE
0
Because X -! C is`a split monomorphism, we see that E -! X is zero, so`E -! F
is`zero, so F -! uXffis a split monomorphism, so F is a retract of uXff. As
uXffis a coproduct of finite objects, we see that there can be no phantom maps
out of F .
We have just put an arbitrary object X in a cofiber sequence
a v w
F -u! Xff-! X -! F
ff `
where there are no phantom maps out of F or ffXff. Now if g :X -! Y is a
phantom map, then gv = 0 so g = f0w, say. If h: Y -! Z is another phantom map,
then hf0 is a phantom map out of F , so is trivial. Thus hg = hf0w is trivial_as
well. |__|
4.3. Smashing localizations of Brown categories. In this section we show
that a smashing localization of a Brown category is again a Brown category, and
we also prove some slightly sharper statements.
Let C be a Brown category, and L a smashing localization functor on C. As usu*
*al,
we write C for the corresponding colocalization functor, and J for the inclusio*
*n of CL
in C. We know by Theorem 3.5.2 that CL is an algebraic stable homotopy category,
and that L: C -!CL is a geometric morphism that preserves small objects.
Proposition 4.3.1.Let X be an object of CL. Then X is the minimal weak colimit
of the objects LYff, where Yffruns over (JX). Moreover, X is small in CL if and
only if it is a retract of LY for some small object Y in C.
Proof.Suppose that X is small in CL. By Theorem 4.2.4, we see that JX is
the minimal weak colimit of the Yff. As L preserves minimal weak colimits (by
Theorem 3.5.1), we see that X = LJX is the minimal weak colimit of the LYff.
AXIOMATIC STABLE HOMOTOPY THEORY 61
Now suppose that X is small. As [X; -] is a homology theory on CL, we see that
[X; X] = lim-!ff[X; LYff]. This means that the identity map of X factors throu*
*gh
some LYff, so that X is a retract of LYff, and Yffis small in C.
Conversely, we know that L preserves smallness (by Theorem 3.5.2), so any *
* __
retract of a localization of a small object is small. *
*|__|
We now write PL for the ideal of phantoms in CL. We let C=P be the quo-
tient category of C in which the phantoms are sent to zero, so that Vo induces *
*an
equivalence C=P ' Co.
Proposition 4.3.2.Consider objects X 2 C and U 2 CL. Then a map f :X -!
JU is phantom in C if and only if the adjoint map g :LX -! U is phantom in CL.
Thus, the functors L and J induce an adjoint pair of functors between C=P and
CL=PL.
Proof.Suppose that f is phantom. Consider a map u: W -! LX, where W is small
in CL. We can write X as the minimal weak colimit of a diagram of small objects
Xffin C, so LX is the minimal weak colimit of the LXff, so u factors through so*
*me
LXff. However, LXff-! LX -g!U is adjoint to Xff-! X -f!JU, which is zero as
f is phantom. Thus gu = 0 for all such u, which means that g is phantom.
Conversely, suppose that g is phantom. For any small Z in C and any map
Z -! X, the composite Z -! X -f!JU is adjoint to LZ -! LX -g!U, which is zero
as LZ is small in CL. This means that f is phantom. __
The second statement of the proposition follows easily. |_*
*_|
Given an object X 2 C, we write HX for the represented homology functor
C -! Ab, defined using small objects in C. Given an object U 2 CL, we write HLU
for the represented homology functor CL -! Ab, defined using small objects in C*
*L.
Proposition 4.3.3.For objects X 2 C and U 2 CL, there are natural isomor-
phisms HX O J = HLLXand HLUO L = HJU .
Proof.We first show that HX O J = HLLX. Consider a small object Z in C. Then
LZ is small in CL, so HLLXLZ = ss0(LX ^ LZ) = ss0(LX ^ Z) (because LX ^ LZ =
LX ^ LS ^ Z = LLX ^ Z = LX ^ Z). On the other hand,
HX JLZ = bss0(X ^ LS ^ Z) = bss0(LX ^ Z) = ss0(LX ^ Z)
(using the smallness of Z). There is thus an isomorphism HX JLZ = HLLXLZ,
natural in Z. Moreover, we can write any W 2 CL as the minimal weak colimit
over (JW ) of the objects LZff, in a functorial way; it follows that there is an
isomorphism
HX JW = HLLXW = lim-!(JW)ss0(LX ^ Zff);
natural in W . Thus HX O J = HLLXas claimed.
We now show that HLUO L = HJU . It is enough to check this on small objects
of C; let Z be such an object. Then HJU Z = ss0(U ^ Z). Moreover, LZ is small in
CL, so HLULZ = ss0(U ^ LZ). As U ^ LZ = U ^ LS ^ Z = U ^ Z, this is the same_
as HJU Z, as required. |__|
Theorem 4.3.4. Let C be a Brown category and L a smashing localization functor.
Then CL is a Brown category. Moreover, there is a commutative diagram of functo*
*rs
62 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
as follows:
CL=PL --J--! C=P ---L-! CL=PL
? ? ?
VoL?y Vo?y ?yVoL
CLo ----! Co ----! CLo
L* J*
Here VoL is the functor U 7! HLU, L* is the functor H 7! H O L, and everything
else should be clear. The vertical functors are equivalences, the functors L* *
*and
J are full and faithful, and the functors L and J* are essentially surjective. *
*The
horizontal composites are identity functors.
Proof.Proposition 4.3.2 tells us that the functors marked J and L are well-defi*
*ned.
Proposition 4.3.3 says that the squares commute up to natural isomorphism. As
LJ = 1, the horizontal composites are identity functors. It follows that L* and
J are full and faithful, and that L and J* are essentially surjective. As C is*
* a
Brown category, the functor Vo is an equivalence. As three sides of the left ha*
*nd
square are full and faithful, it follows that the fourth side VoL is full and f*
*aithful.
As VoLO L = J* O Vo is essentially surjective, the same is true of VoL. Thus_Vo*
*L is
an equivalence, as claimed. |__|
4.4. A topology on [X; Y ]. In this brief section, we point out that there is a
natural topology on the morphisms in an algebraic stable homotopy category C,
which enriches the category over topological Abelian groups.
Recall that a linear topology on an Abelian group A is a topology such that
the cosets of open subgroups form a basis of open sets. Given a family {Ai} of
subgroups of A, there is a unique linear topology on A such that subgroups Ai a*
*re
open and form a basis of neighborhoods of 0. We write A0O A to indicate that
A0is an open subgroup of A.
There is a natural map ff: A -!lim-A0OA A=A0. We shall say that A is complete
if ff is surjective. Moreover, A is Hausdorff if and only if ff is injective, i*
*f and only
if the intersection of the open subgroups is zero, if and only if {0} is closed.
Fix two objects X and Y of C. For any map F -f!X from a small object to X,
let Uf = Uf(X; Y ) denote the kernel of f* :[X; Y ] -! [F; Y ]. We give [X; Y ]*
* the
linear topology determined by the subgroups Uf, and refer to this as the natural
topology.
Proposition 4.4.1.Let C be an algebraic stable homotopy category.
(a) The composition map [X; Y ] x [Y; Z] -![X; Z] is continuous.
(b) Any pair of maps X0-! X and Y -! Y 0induces a continuous map [X; Y ] -!
[X0; Y 0].
(c) If X is small then [X; Y ] is discrete.
(d) The closure of 0 in [X; Y ] is the set of phantom maps, so [X; Y ] is Haus*
*dorff
if and only if P(X; Y ) = 0.
(e) If`C is a Brown category then [X;QY ] is always complete.
(f) [ iXi; Y ] is homeomorphicQto i[Xi; Y ] with the product topology, but *
*the
natural topology on [X; Yi] is strictly finer than the product topology *
*in
general.
Proof.(a): Call the composition map fl. Suppose that we have maps X -u!Y -v!Z,
and a neighborhood vu + Uf of fl(u; v) in [X; Z] (so f :F -! X for some small F*
* ).
AXIOMATIC STABLE HOMOTOPY THEORY 63
Then one sees easily that fl((u+Uf)x(v +Uuf)) vu+Uf, so that fl is continuous
at (u; v).
(b): This follows immediately from (a).
(c): Immediate from the definitions.
(d): Immediate from the definitions.
(e): This is equivalent to the statement that X is a weak colimit of (X), whi*
*ch
is Theorem 4.2.4.
(f): The first part is easy to see using the fact that any map from a small o*
*bject
to a coproduct factors through a finite sub-coproduct. For the second part, sup*
*pose
that we have a map f :F -! X from a small object F to X. Then
Y Y
Uf(X; Yi) = Uf(X; Yi):
i i
This product is therefore open in the natural topology, but rarely in the_produ*
*ct
topology. |__|
This construction gives an enrichment of C over topological Abelian groups. T*
*his
becomes very important in the K(n)-local category, where even homotopy groups
can have interesting topology [HSS ]. If S is small then many of the groups whi*
*ch
arise are discrete.
We should point out that Brown representability is not compatible with this
enrichment. That is, there are cohomology functors to the category of topologic*
*al
Abelian groups which are not representable. Indeed, given an infinite family {Y*
*i}
of objectsQin a stable homotopy category C, define a cohomology functor H by
H(X) = [X; Yi] with the product topology. This functor cannot be representabl*
*e,
since if it were, H(X) would have to have the discrete topology for all small X*
*. In
general, a cohomology functor H is representable if and only if, for all X, a s*
*ubset
U of H(X) is a neighborhood of 0 if and only if there exists a small object F a*
*nd
a map f :F -! X such that U H(f)-1(0).
5. Nilpotence and thick subcategories
In this section we present analogues of the nilpotence theorems of Devinatz,
Hopkins and Smith [DHS88 ], and the thick subcategory theorems of Hopkins and
Smith [HS ]. None of our theorems imply the theorems just mentioned; we require
stronger finiteness conditions than are available in their context. Our nilpot*
*ence
theorems require a unital algebraic stable homotopy category C, but our thick s*
*ub-
category theorems will hold in an arbitrary algebraic stable homotopy category.*
* We
allow C to be multigraded, and we will often consider graded maps_see Section 1*
*.3.
5.1. A na"ive nilpotence theorem. In this section we prove a nilpotence theorem.
We first present some miscellaneous definitions.
Definition 5.1.1.
(a) We write X(m) = X ^ . .^.X (with m factors), and similarly for maps.
(b) We say that a graded map f :X -! Y is smash nilpotent if f(m): X(m) -!
Y (m)is null for m 0.
(c) Suppose that we have a map f :S -! -dX. We then get a sequence
(5.1.1) S = X(0)-f!-dX(1)-f^1-!-2dX(2): :::
64 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
We write X(1) for the sequential colimit, and f(1) for the evident map
S -! X(1).
(d) We say that a graded self-map f :X -! X is composition nilpotent or just
nilpotent if the mth composition power fm :X -! X is null for m 0.
Here is a rather generic nilpotence theorem. See Definitions 2.1.1 and 3.6.1 *
*for
the relevant definitions.
Theorem 5.1.2 (Nilpotence theorem I).Let C be a unital algebraic stable homo-
topy category. Suppose that we have objects {K(n) | n 2 I} (for some indexing
set I), so that
a
(5.1.2) ~~~~= :
n2I
Then the objects K(n) detect nilpotence:
(a) Let F be small, and X arbitrary. A graded map f :F -! X is smash nilpotent
if 1K(n)^ f = 0 for all n.
(b) Let X be a small object. A graded map f :X -! X is nilpotent if 1K(n)^f = 0
for all n.
(c) Suppose that each K(n) is monoidal, and let X be a small object. A graded
map f :X -! X is nilpotent if and only if for each n, the map K(n)*(f) is
nilpotent.
Equation (5.1.2)simply means that whenever K(n) ^ X = 0 for all n, we have
X = 0. For part (c), we can always replace K(n) by a Bousfield-equivalent monoi*
*dal
object, using Lemma 3.6.8. If the objects K(n) are ring objects (Definition 3.7*
*.1),
then we have results more like those in [HS ].
Theorem 5.1.3 (Nilpotence theorem II).Suppose, in addition, that each K(n) is
a monoidal ring object.
(a) Let R be a ring object. An element ff 2 ss*R is nilpotent if and only if
K(n)*(ff) is nilpotent for all n.
(b) Let X be a small object. A graded map f :S -! X is smash nilpotent if
K(n)*(f) = 0 for all n.
Remark 5.1.4.
(a) Neither of these results implies the nilpotence theorems of [DHS88 ] and [*
*HS ],
because Equation (5.1.2)does not hold in the homotopy category of spectra,
with MU, BP , or the wedge of the Morava K-theories on the right hand
side. Theorems 5.1.2 and 5.1.3 seem to be useful mainly in a stable homoto*
*py
category satisfying certain strong finiteness conditions.
(b) Note that Theorem 5.1.3(b) is weaker than the smash nilpotence result in [*
*HS ].
Hopkins and Smith's theorem applies to maps f :F -! X where F is small.
They reduce to the case F = S using Spanier-Whitehead duality to convert
f to a map ^f:S -! DF ^ X, and the K"unneth isomorphism for K(n)* to
see that K(n)*(f) = 0 ) K(n)*(f^) = 0. This application of the K"unneth
isomorphism seems to be necessary, and one cannot expect it to hold in an
arbitrary stable homotopy category.
We need three lemmas before we begin the proofs.
AXIOMATIC STABLE HOMOTOPY THEORY 65
Lemma 5.1.5. (a) A graded map f :S -! X is smash nilpotent if and only if
X(1) is contractible.
(b) If f :S -! X and E are such that 1E ^ f = 0, then E ^ X(1) = 0.
Proof.(a): Certainly if f is smash nilpotent, then X(1) is contractible. Conver*
*sely,
we have
X(1) ' 0 ) [S; X(1)]* = 0
) S -! X -! X(2)-! : :-:!X(1) is null
) S -! X -! X ^ X -! : :-:!X(n)is null for n 0
) f is smash nilpotent:
(b): Smash the diagram (5.1.1)with E. The sequential colimit is E ^ X(1),_and
each map in the diagram is null. |__|
Lemma 5.1.6. Suppose that f :dX -! X is a graded self-map of a small ob-
ject X. Recall (Definition 2.2.3) that f-1 X denotes the sequential colimit of *
*the
sequence X -f!-dX -! -2dX -! : :.:Let E 2 C be any object.
(a) f is nilpotent if and only if f-1 X = 0.
(b) If E*(f) is nilpotent, then E*(f-1 X) = 0.
Proof.(a): Certainly if f is nilpotent, then f-1 X is trivial. On the other han*
*d,
since X is small we have [X; f-1 X] = lim-![X; X], where the maps in the colimi*
*t are
composition with f. Hence if f-1 X = 0, the identity map of X must be 0 at a
finite stage of the colimit. In other words, f must be nilpotent.
(b): If E*(f) is nilpotent, say E*(fr) = 0, then the maps in the sequence
r 1^fr
E ^ X -1^f--!E ^ X ---! : : :
all induce zero on the homology functor [S; -]*. Hence the minimal weak colimit_
is S*-acyclic. In other words, E*(f-1 X) = 0. |__|
Proof of Theorem 5.1.2.For part (a), we use Lemma 5.1.5. First, using Spanier-
Whitehead duality, we can reduce to the case where F = S, so assume we have a
map f :S -! X. Let X(1) be as above. By`assumption, 1K(n)^ f = 0 for all n,
so K(n) ^ X(1) = 0 for all n. As ~~~~= n, we conclude that X(1) = 0.
Thus f is smash nilpotent.
Part (b) follows from (a), as in [HS ].
For part (c), we use Lemma 5.1.6, so we have to show that f-1 X = 0. The
hypothesis implies immediately that K(n)*f-1 X = 0; as K(n) is monoidal, we see*
*__
that K(n) ^ f-1 X = 0. As this holds for all n, we have f-1 X = 0. |*
*__|
Suppose that R is a ring object. Fix ff 2 ss*(R), and let ^ffdenote the "mult*
*ipli-
cation by ff" self-map O (ff ^ 1) of R.
Lemma 5.1.7. With notation as above, ff is nilpotent in ss*(R) if and only if
^ff-1R = 0.
Proof.Because C is unital algebraic, we know that ^ff-1R = 0 if and only if
ss*(^ff-1R ^ DZ) = 0
66 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
for all Z 2 G. The right hand side is just the direct limit of the R-module ss**
*(R^DZ)
under left multiplication by ff. This vanishes if ff is nilpotent, and the_con*
*verse
holds by the special case Z = S. |__|
Proof of Theorem 5.1.3.Part (a) follows from Lemma 5.1.7_if fi = K(n)*(ff) is
nilpotent, then
K(n) ^ ^ff-1R = ^fi-1(K(n) ^ R) = 0:
If this holds for all n, then by the decomposition (5.1.2), we see that ^ff-1R *
*= 0, so
ff is nilpotent.
(b): Let j :S -! K(n) denote the unit map, and : K(n) ^ K(n) -! K(n) the
product. Suppose that f :S -! X induces zero on K(n)*; then the composite S -j!
K(n) 1^f--!K(n)^X is null. But K(n) 1^f--!K(n)^X factors as K(n)^S -(1^f)Oj----!
K(n) ^ K(n) ^ X -^1-!K(n) ^ X, and so is null. Now apply Theorem 5.1.2(b). |__*
*_|
5.2. A thick subcategory theorem. In this section we present a classification
of the G-ideals of small objects in an algebraic stable homotopy category; the *
*basic
argument is of course inspired by [HS ], and a few of the details are drawn fro*
*m [Ric].
Definition 5.2.1.Suppose that C is a stable homotopy category. Fix a collection
of objects {K(n) | n 2 I} (for some indexing set I). Given an object X, we
define the support of X (with respect to the K(n)'s) to be the set supp(X) =
{n | K(n) ^ X 6= 0}. Similarly,Sgiven a replete subcategory D, define the suppo*
*rt
of D to be the set supp(D) = X2D suppX. We say that the K(n)'s determine
G-ideals if whenever D is a G-ideal of small objects, we have
D = {X | X finite; supp(X) supp(D)}:
Theorem 5.2.2. Suppose that C is an algebraic stable homotopy category, and that
we have objects {K(n)} such that
1. If R is a nontrivial ring object, then there is some n such that K(n) ^ R *
*is
nontrivial (in other words, the objects K(n) detect ring objects).
2. If X is finite and K(n) ^ X 6= 0, then = .
Then the objects K(n) determine G-ideals.
Note that we always have . If C is monogenic, then every
thick subcategory is a G-ideal, and vice versa, so we get a classification of t*
*hick
subcategories in this setting.
Note as well that any collection of ring objects that detect nilpotence, as in
Theorem 5.1.3, automatically detects ring objects. Theorem 5.2.2 tells us what *
*else
we need to know to get a thick subcategory theorem from a nilpotence theorem. In
particular, we recover the Hopkins-Smith thick subcategory theorem [HS ] from t*
*he
Devinatz-Hopkins-Smith nilpotence theorem [DHS88 ].
Corollary 5.2.3.Suppose that C is monogenic, so that thick subcategories are the
same as G-ideals. If the family {K(n)} detects ring objects and each K(n) satis*
*fies
one of the following conditions, then the K(n)'s determine thick subcategories:
(i)For X and Y arbitrary objects, K(n)*(X^Y ) = 0 if and only if K(n)*(X) = 0
or K(n)*(Y ) = 0.
(ii)K(n)* satisfies a K"unneth isomorphism: K(n)*(X ^ Y ) ' K(n)*(X) K(n)*
K(n)*(Y ).
AXIOMATIC STABLE HOMOTOPY THEORY 67
(iii)K(n) is a skew field object (Definition 3.7.1). __
(iv)is a minimal nonzero Bousfield class. |__|
Proof of Theorem 5.2.2.Suppose that D is an G-ideal, and supp(Y ) supp(D).
We need to show that Y 2 D. Note first that Y and F (Y; Y ) = Y ^ DY generate
the same thick subcategory (by Lemma A.2.6), so we can replace Y by the ring
object F (Y; Y ). We therefore assume that Y is a ring object.
Since C is algebraic, every G-ideal of small objects is essentially small, so*
* we can
use the finite localization functors of Definition 3.3.4. Thus, it suffices to *
*show that
Y ^ LfDS = LfDY = 0. Fix n. If K(n) ^ Y = 0, then certainly K(n) ^ Y ^ LfDS = 0.
If K(n) ^ Y 6= 0, then (because supp(Y ) supp(D)) there is some X 2 D such
that K(n) ^ X 6= 0. Since X ^ LfDS = 0, we have (K(n) ^ X) ^ (Y ^ LfDS) = 0;
and since = , we have K(n) ^ (Y ^ LfDS) = 0. So for all n, we
have K(n) ^ (Y ^ LfDS) = 0; hence, since Y ^ LfDS is a ring object, it is trivi*
*al. |___|
6. Noetherian stable homotopy categories
In this section we consider a multigraded stable homotopy category C_see Sec-
tion 1.3. We shall assume that C is monogenic in the multigraded sense.
Definition 6.0.1.If C is a monogenic stable homotopy category such that ss*(S) =
[S; S]* is Noetherian (as a multigraded-commutative ring), then we say that C i*
*s a
Noetherian stable homotopy category.
We shall use a number of standard theorems that are proved in the literature
for ungraded commutative rings. These will all apply to multigraded-commutative
rings, when suitably interpreted. On the one hand, one has to keep track of the
grading, but this is trivial if we insist that everything in sight be homogeneo*
*us.
On the other hand, we need to think about the fact that odd-dimensional elements
anticommute instead of commuting. If 2 is invertible in R, then all odd-dimensi*
*onal
elements square to zero and thus lie in every prime ideal, and this makes every*
*thing
work as expected. If we work modulo 2, then R is strictly commutative. By
combining these observations, we see that everything works as expected integral*
*ly.
The very cautious reader may wish to assume that R is concentrated in even degr*
*ees,
as we could not honestly claim to have checked every detail otherwise.
The derived category of a Noetherian ring is a Noetherian stable homotopy
category. If B is a finite-dimensional commutative Hopf algebra, then C(B) is o*
*ften
a Noetherian stable homotopy category (see Section 9.5). In particular, this ho*
*lds
if B = (kG)* with G a finite p-group and char(k) = p, or if B is graded and
connected.
For the rest of this section, we assume that C is a Noetherian stable homotopy
category. We write R for the graded ring ss*S = [S; S]*. We also let SpecR
(respectively, Max R) denote the space of prime (maximal) homogeneous ideals of*
* R,
under the Zariski topology. Given an R-module M and a prime ideal p 2 SpecR we
write the localization, completion and completed localization of M at p as foll*
*ows:
Mp = (R \ p)-1M
M^p = lim-kM=pkM
Mp = (Mp)^p
68 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
(We need Mp because M^pneed not be p-local in general. There does not seem to
be a standard notation for this, so we have invented one.)
We also write E(R=p) for the injective hull of R=p, which is well-defined up *
*to
non-canonical isomorphism.
We record some basic facts from commutative algebra.
Proposition 6.0.2.Rp and Rp are flat over R, and Rp is faithfully flat over Rp.
For any M, we have Mp = Rp R M; hence M 7! Mp is an exact functor. If M_is
finitely generated, we have M^p= R^pR M and Mp = Rp R M. |__|
Proposition 6.0.3.A module M is zero if and only if Mm = 0 for all m 2 Max (R),_
if and only if Mp = 0 for all p 2 SpecR. |__|
The following less well-known result is proved in [Mat89 , Section 18].
Proposition 6.0.4.E(R=p) is a p-torsion module; more precisely, for all x 2
E(R=p), there is a t 1 so that ptx = 0. The localizations of E(R=p) are
(
E(R=p)q = E(R=p) p q;
0 otherwise:
Moreover, there is a natural isomorphism Rp = Hom R(E(R=p); E(R=p)) |___|
The following fact is standard, but seldom stated explicitly.
Proposition 6.0.5.In a Noetherian ring, any nonempty collection of prime ideals
has a minimal element.
Proof.If R is a Noetherian local ring, then it has finite Krull dimension (boun*
*ded
by dimR=m m=m2). Given an arbitrary Noetherian ring R, and a collection T of
prime ideals, choose p 2 T such that the Krull dimension of Rp (also called the
height of p) is minimal. It is easy to see that any q < p has strictly smaller_*
*height,
so p is minimal in T . |__|
We now start to apply these results to stable homotopy theory.
Proposition 6.0.6.If X and Y are small objects in a Noetherian stable homotopy
category C, then [X; Y ]* is finitely generated as a module over R = ss*S.
Proof.By Spanier-Whitehead duality it suffices to show that if Y is small, then
ss*Y is a finitely generated R-module. This is proved by showing that the categ*
*ory
of all Y such that ss*Y is a finitely generated R-module is thick, which follow*
*s from_
well-known properties of finitely-generated modules over a Noetherian ring. *
* |__|
For example, if X is small, then the noncommutative ring [X; X]* is finitely
generated as a module over the image of R -![X; X]*.
A basic technique in Noetherian ring theory is to work one prime at a time
by localizing. Fortunately, we have an analogous procedure in Noetherian stable
homotopy theory.
Proposition 6.0.7.For each p 2 Spec R, there is a ring object Sp such that
ss*(Sp ^ X) = ss*(X)p. Moreover, the functor Lp: X 7! Xp = Sp ^ X is an al-
gebraic localization.
Proof.This is an immediate consequence of Theorem 3.3.7 and Proposition_3.1.8.
|__|
AXIOMATIC STABLE HOMOTOPY THEORY 69
It follows that the category of local objects, which we denote Cp, is again a
Noetherian stable homotopy category. We call it the p-localization of C.
Now we define a number of other objects associated to a prime ideal p R = ss*
**S.
Definition 6.0.8.Fix a prime ideal p R.
(a) S=p: Write the prime ideal p as p = (y1; : :;:yn). Define S=yi as the cofi*
*ber
of the (graded) map S -yi!S, and let
S=p = S=y1 ^ . .^.S=yn:
Note that S=p depends on the choice of generators {yi}. We shall show in
Lemma 6.0.9 that any two choices generate the same thick subcategory, and
thus have the same Bousfield class.
(b) K(p): Let K(p) = Sp ^ S=p = (S=p)p. Using Lemma A.2.6, we see that this
generates the same thick subcategory as the ring object F (K(p); K(p)) =
Sp ^ D(S=p) ^ S=p.
(c) Ip: As E(R=p) is an injective module, the functor
X 7! Hom R(ss*X; E(R=p))
is a cohomology functor. We let Ip denote the representing object. Note th*
*at
ss*Ip = E(R=p).
(d) Sp: Define Sp = F (Ip; Ip). Note that ss*Sp = Rp, by Proposition 6.0.4.
(e) MpS: Let Lp be the p-localization functor; this is a finite localization, *
*and
the corresponding subcategory of finite acyclics is
{Z 2 F | ss*(Z)p= 0} = {Z 2 F | ss*(Z)q= 0 for allq p}:
We let L~~.
Proof.Suppose thatVwe chose generators (y1; : :;:yn) for p, so the resulting mo*
*del
of S=p is Y = iS=yi. One can check that y2i= 0 as a self-map of S=(yi), and
thus that every element of p acts nilpotently on Y . Now choose a different set*
* of
generators (z1; : :;:zm ). For large N we see that each zNiacts trivially on Y *
*, and
thus that Y is a retract of Y ^ S=(zN1; : :;:zNm). It follows easily that Y *
*lies in
the thick subcategory generated by Z = S=(z1; : :;:zm ), as required. An eviden*
*t_
extension of this argument gives the second claim. |__|
6.1. Monochromatic subcategories. Let C be a Noetherian stable homotopy
category. We will need to investigate certain subcategories that are strongly c*
*on-
centrated at a single prime ideal p of R = ss*(S).
Definition 6.1.1.The monochromatic category Mp is the localizing subcategory
generated by K(p).
70 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Theorem 6.1.8 gives a number of other descriptions of this category.
We make the following conjectures:
Conjecture 6.1.2.For each prime p, the category Mp is minimal among nonzero
localizing subcategories of C.
Conjecture 6.1.3.For each p, the Bousfield class is minimal among non-
trivial Bousfield classes.
We shall show in Proposition 6.1.7 and Theorem 6.1.9 that
~~ = qp; and
K(p) ^ K(q) = 0 forp 6= q:
This means that each K(p) is smash-complemented. Thus, if each K(p) is (or gene*
*r-
ates the same thick subcategory as) a skew field object, then both conjectures *
*follow
from Proposition 3.7.3. We shall show in Proposition 6.1.11 that Conjecture 6.1*
*.2
implies Conjecture 6.1.3.
Example 6.1.4. Both conjectures_hold in the derived category D(R) of a Noether-
ian ring R. Indeed, let kp denote (the representative in D(R) of) the residue_f*
*ield
kp = (R=p)p_at p. We will see in Proposition_9.3.2 that loc = loc and
= . We will show also that kpis a smash-complemented field object, so
both conjectures follow from Proposition 3.7.3. Hence we recover the thick subc*
*at-
egory theorem of [Hop87 , Nee92a] from Theorem 5.2.2. The telescope conjecture
also holds in this setting. This was proved in [Nee92a] for the derived categor*
*y, and
will be proved in a more general context in Theorem 6.3.7.
The thick subcategory theorem is also known to be true for C((kG)*) when G
is a p-group, as proved in [BCR ]. Their method is to verify that the objects M*
*pS
(defined in Definition 6.0.8) satisfy
MpS ^ (X ^ Y ) = 0 , MpS ^ X = 0 orMpS ^ Y = 0:
In this context, this also implies that is minimal. (They write (p) for
MpS, essentially, and [BCR , Theorem 10.8], is precisely the above statement. S*
*ee
also [BCR , Lemma 10.3].) We shall show in Theorem 6.1.8 that = .
Assuming Conjecture 6.1.2, we can state our results quite simply. Recall that*
* a
subset T SpecR is said to be closed under specialization if whenever p 2 T and
p q, we also have q 2 T . This is (easily) equivalent to T being a union of Za*
*riski
closed sets. The following theorem is a summary of Theorem 6.2.3, Corollary 6.1*
*.10,
Corollary 6.3.4 and Theorem 6.3.7.
Theorem 6.1.5. Suppose that Conjecture 6.1.2 holds for C. Then the Bousfield
lattice is isomorphic to the lattice of subsets of SpecR. Moreover, every gener*
*alized
Bousfield class is a Bousfield class, and \= for all X and Y . E*
*very
smashing localization is a finite localization, and the lattice of such is isom*
*orphic
to the lattice of thick subcategories of small objects, or to the lattice of su*
*bsets of
SpecR that are closed under specialization. The objects K(p) detect nilpotence_*
*and
determine thick subcategories. |__|
We still have strong results in the absence of Conjecture 6.1.2, but they can*
*not
be stated so succinctly. A major r^ole is played by the following definition.
AXIOMATIC STABLE HOMOTOPY THEORY 71
Definition 6.1.6.For any object X, we define
supp(X) = {p | K(p)*(X) 6= 0} SpecR:
Similarly, if D is a thick subcategory of C, we define
[
supp(D) = supp(X):
X2D
For a general object X, this can be an arbitrary subset of SpecR, but for sma*
*ll
objects it is constrained by the following result.
Proposition 6.1.7.
(a) If p 6= q, then K(p) ^ K(q) = 0.
(b) Fix a small object X and a prime ideal q. Then K(q)*X = 0 if and only if
Xq = 0. In particular (taking X = S), the K(q) are all nontrivial.
(c) If X is small then supp(X) is Zariski closed (and thus closed under specia*
*l-
ization).
(d) supp(S=p) = V (p) = {q | p q}.
Proof.(a): Without loss of generality p 6 q, so there exists y 2 p \ q. As y 2 *
*p,
it is nilpotent as a self-map of K(p), while it is an equivalence on K(q). It i*
*s both
nilpotent and an equivalence on K(p) ^ K(q), so this object must be zero.
(b): Certainly, if Xq = 0 then K(q) ^ X = S=q ^ Xq = 0. Conversely, suppose
that S=q ^ Xq = 0. Choose a set of generators (y1; : :;:yn) for q. We will show*
* by
downward induction on i that Yi= S=(y1; : :;:yi) ^ Xq is trivial. Suppose that *
*Yk
is trivial. By considering the cofibration sequence
Yk-1 yk-!Yk-1 -!Yk;
we find that multiplication by yk on ss*Yk-1 is an isomorphism. Now Yk-1 is the*
* q-
localization of a small object. Hence ss*Yk-1 is a finitely generated module ov*
*er the
Noetherian local ring Rq by Proposition 6.0.6. The element yk is in the Jacobson
radical, and ykss*Yk-1 = ss*Yk-1. Hence ss*Yk-1 = 0 by Nakayama's lemma.
(c): By (b), we have supp(X) = {p | ss*(X)p6= 0}. As ss*(X) is finitely
generated, a well-known algebraic lemma identifies this with the Zariski closed*
* set
V (ann(ss*X)).
(d): If q 6 p then (S=p)q = 0 by the argument of (a). If q p then K(p) = (S=*
*p)p
is a further localization of (S=p)q, and K(p) 6= 0 by (b), so (S=p)q 6= 0._The *
*claim
follows, using (b). |__|
We can now give a number of new characterizations of the category Mp =
loc.
Theorem 6.1.8. We have loc = loc = Mp, and = .
Moreover, for any object X 2 C, the following are equivalent.
(a) X = Xp and Xq = 0 for all q < p.
(b) ss*(X) is p-local and p-torsion.
(c) X = MpX.
(d) = .
(e) X 2 loc = loc = Mp.
Proof.Consider the following auxiliary statements:
(d0) .
72 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
(e0)X 2 loc.
(e00)X 2 loc.
We shall prove that
(e00) ) (e0) ) (c) ) (d0) ) (a) ) (b) ) (e00):
This implies everything except that = . However, for any U, the
category {V | ~~__} is localizing. Using this and the equality loc =
loc, we easily deduce that = .
Before we start, note that any of (a)-(e) (or the primed versions) implies th*
*at
X is p-local. Nothing changes if we replace C by the p-local category Cp, so we
may assume that R = ss*(S) is local with maximal ideal p. This implies that
K(p) = S=p, and that Mp is just the finite colocalization functor C__

__.
(e0) ) (c): If X 2 loc then L). Thus, if , then Xq = Sq^X = 0.
(a) ) (b): If (a) holds, then ss*(X)q= 0 for q < p. A well-known piece of
algebra implies that ss*(X) is then p-torsion (use the following fact: an eleme*
*nt
x 2 N is nonzero in Nq if and only if the radical of the annihilator of x is co*
*ntained
in q).
(b) ) (e00): Suppose that ss*(X) is p-torsion. The localizing category
D = {Y | ss*(Y ) is p-torsion}
contains X, so it also contains Y = LfS=pS ^ X = LfS=pX. Choose generators
(y0; : :;:yn-1) for p, and set Yk = F (S=(y0; : :;:yk-1); Y ) = D(S=(y0; : :;:y*
*k-1)) ^
Y . Note that Yn = F (S=p; LfS=pX) = 0 by the definition of LfS=p. If Yk =
Yk-1=yk-1 = 0 then yk-1 acts isomorphically on ss*(Yk-1), but it also acts nilp*
*o-
tently as yk-1 2 p and Yk-1 2 D. It follows that ss*(Yk-1) = 0, and thus
that Yk-1 = 0. By downwards induction, we conclude that Y = Y0 = 0, so
X = CfS=pX 2 loc__~~ as required. |___|
Theorem 6.1.9. We have an equality of Bousfield classes
a
~~~~ = :
p2SpecR
(Note also that ^ = 0 for p 6= q, by Proposition 6.1.7.)
Proof.Suppose that K(p) ^ X = 0 for all p; we need to show that X = 0. By
Theorem 6.1.8, we have = , so MpX = 0 for all p. We claim that
Xp = 0 for all p; we prove this by induction on p. Fix p and suppose that Xq = 0
for all q < p. Then by the implication (a))(c) of Theorem 6.1.8 (applied to Xp),
we find that Xp = MpX = 0. Thus Xp = 0 for all p; using Lemma 6.0.3, we __
conclude that ss*(X) = 0 and thus X = 0. |__|
AXIOMATIC STABLE HOMOTOPY THEORY 73
It follows that the nilpotence results of Section 5 apply:
Corollary 6.1.10.Let C be a Noetherian stable homotopy category. Then the
objects {K(p) | p 2 SpecR} detect nilpotence, in the sense that Theorem 5.1.3_
applies. Moreover, Theorem 5.1.2 applies with K(p) replaced by Mp. |__|
We pause briefly to deduce the claimed relation between our two conjectures.
Proposition 6.1.11.If Mp is minimal among nontrivial localizing subcategories,
then is a minimal Bousfield class.
Proof.This now follows from Proposition 3.7.4, Theorem 6.1.9, and part (a)_of
Proposition 6.1.7. |__|
6.2. Thick subcategories. We next attempt to classify thick subcategories of
small objects in a Noetherian stable homotopy category C; we succeed completely
if each is a minimal Bousfield class. We first recall some convenient ter*
*mi-
nology.
Definition 6.2.1.Let A -f!B -g!A be maps of partially ordered sets. We say
that g is left adjoint to f (and write g ` f) if g(b) a is equivalent to b f(*
*a).
(It is equivalent to say that g is left adjoint to f when A and B are regarded *
*as
categories, and f and g as functors, in the usual way.)
The particular lattices of interest are as follows.
Definition 6.2.2.We define
Lopf= { thick subcategories of small objects};
Lt= { subsetsT SpecR that are closed under specialization}:
Note that Lopfis antiisomorphic to the lattice of finite localization functors,*
* by
Proposition 3.8.3. We define f :Lt-! Lopfby
f(T ) = thick~~~~ = {Z 2 F | supp(Z) T }:
(We shall verify below that these two definitions are equivalent.) We also def*
*ine
maps g; g0:Lopf-!Lt by
g(A) = supp(A); g0(A) = {p | S=p 2 A}:
(We shall verify below that these sets are closed under specialization).
Theorem 6.2.3. The two definitions of f given above are the same, and the sets
g(A) and g0(A) are closed under specialization. Moreover, we have
g ` f ` g0; g g0;
gf = 1 = g0f; fg0 1 fg:
If each is a minimal Bousfield class, then f; g and g0are isomorphisms wi*
*th
g = g0= f-1 .
Proof.We first show that the two definitions of f are equivalent. Consider a set
T 2 Lt, and write A = thick~~~~ and B = {Z 2 F | supp(Z) T }.
If p 2 T then supp(S=p) = V (p) T (because T is closed under specialization).
It follows that A B. For the converse, write C = CfAand L = LfA. Suppose
that Z 2 B. If p 62 T we have Z ^ K(p) = 0 (by the definition of B) and thus
LZ ^ K(p) = LS ^ Z ^ K(p) = 0. On the other hand, if p 2 T then we have
74 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
S=p 2 A, so LS=p = 0 (by the definition of L), so LZ ^ K(p) = Z ^ Sp ^ LS=p = 0.
It follows that LZ ^K(p) = 0 for all p, so LZ = 0 by Theorem 6.1.9. Theorem 3.3*
*.3
now tells us that Z 2 A. Thus A = B as claimed.
We next show that g(A) and g0(A) are closed under specialization. For g(A) =
supp(A), this is immediate from Proposition 6.1.7. For g0(A), it is immediate f*
*rom
Lemma 6.0.9.
Using our first definition of f, we see that T g0(A) if and only if f(T ) A,
so that f ` g0. Using the second definition, it is clear that g(A) T if and on*
*ly if
A f(T ), so that g ` f. It is clear that g0 g. By setting T = g(A) or T = g0(A)
or A = f(T ) in the adjunctions above, we obtain (co)unit inequalities
fg0(A) A fg(A)
gf(T ) T g0f(T )
By combining the latter with the inequality g0 g, we conclude that gf(T ) = T =
g0f(T ).
Now suppose that all the Bousfield classes are minimal. We can then
apply Theorem 6.1.9 and Corollary 5.2.3 to show that the collection of K(p)'s
determines thick subcategories, in other words that the map g is injective. As
gf = 1, we conclude that g is also surjective, in fact that f and g are inverse_
isomorphisms. As g0f = 1, we see that g0= f-1 = g. |__|
6.3. Localizing subcategories. We next study localizing subcategories of C. We
start with the following lemma.
Lemma 6.3.1. Suppose that T SpecR is closed under specialization, and let
A = f(T ) = thick~~~~. Then
(
CfAK(p) = K(p) ifp 2 T;
0 otherwise.
Proof.If p 2 T then S=p 2 A, so K(p) 2 loc~~~~ loc~~~~, so CfAK(p) = K(p).
On the other hand, if p 62 T = supp(A) then for all Z 2 A we have Zp = 0, so
[Z; K(p)] = [Zp; K(p)] = 0. It follows that CfAK(p) = 0. |_*
*__|
The key fact for our understanding of localizing subcategories is as follows.
Proposition 6.3.2.For any object X 2 C we have
X 2 loc:
Proof.Write D = loc, which is the same as loc, because loc = loc. Write T = {p 2 SpecR | S=p ^ X 2
D}; this is closed under specialization by Lemma 6.0.9. Define A = f(T ) =
thick~~~~ and CT = CfA. Note that CTX 2 loc~~~~ D,
by the definition of T .
We claim that T = SpecR. If not, then as R is Noetherian, we can choose a
maximal element p of SpecR \ T . Maximality means that T 0= T [ {p} is closed
under specialization. There is a morphism CT -! CT0 of colocalization functors;
call the cofiber M. By Lemma 6.3.1, we have CTK(q) = CT0K(q) (and thus
MS ^ K(q) = MK(q) = 0) unless q = p. It follows from Theorem 6.1.9 that
, and thus from Theorem 6.1.8 that MS 2 loc, and thus that
MX 2 loc D (by the definition of D). We observed above that CTX 2 D,
AXIOMATIC STABLE HOMOTOPY THEORY 75
so we conclude from the cofibration that CT0X 2 D. As S=p ^ CT0S = CT0S=p =
S=p, we conclude that S=p ^ X 2 D, contradicting our assumption that p 62 T .
Thus T = SpecR as claimed.
We thus have A = f(SpecR) = {Z | supp(Z) SpecR} = F, so X = CTX 2 __
D, as required. |__|
We can deduce the following splitting of the Bousfield lattice.
Corollary 6.3.3.Let Bp be the lattice of localizing subcategories that are cont*
*ained
in Mp, and B theQlattice of all localizing subcategories. Then there is a natu*
*ral
isomorphism B = pBp.
Q
Proof.The map B -! pBp sends D to the collection of subcategories Dp =
D \ Mp.S The map the other way sends a collection of subcategories Ep to E =
loc< pEp>. It is clear that Ep E \ Mp. Conversely, if X 2 E \ Mp then
[
X = MpS ^ X 2 loc< MpS ^ Eq> = MpS ^ Ep = Ep;
q
S
so E \ Mp = Ep. It is also clear that loc< pDp> D, and Proposition 6.3.2 impl*
*ies
the opposite inequality. It follows that these two constructions are mutually_i*
*nverse,
as required. |__|
Corollary 6.3.4.If Conjecture 6.1.2 holds for C, then every localizing subcateg*
*ory
of C is closed, and the lattice of such subcategories is isomorphic to the latt*
*ice of
subsets of SpecR (and antiisomorphic to the Bousfield lattice).
Proof.Conjecture 6.1.2 says that each Mp is minimal, so thatQeach lattice Bp is
isomorphic to the two-element lattice {0; 1}, so that B = p2SpecRBp is isomor*
*phic
to the lattice of subsets of SpecR. We`can describe the maps more explicitly. L*
*et
T be a subset of SpecR, and write X = q62TK(q). Then the corresponding
localizing subcategory is
loc = {Y | X ^ Y = 0}:
It follows from Lemma 3.6.6 that this is a closed localizing subcategory; thus *
*all
localizing subcategories are closed. In any stable homotopy category, the Bousf*
*ield __
lattice is antiisomorphic to the category of closed localizing subcategories. *
* |__|
We now turn to the telescope conjecture, in other words, the classification of
smashing localization functors. For this, we need to study the objects Sp.
Lemma 6.3.5.
(a) There is a natural isomorphism ss*(Sp ^ X) = RpRp ss*(Xp), and this is the
same as ss*(X)pif`ss*(Xp) is finitely generated over Rp.
(b) = = qp .
(c) LK(p)S = Sp.
Proof.(a): There is an obvious pairing ss*(Y ) R ss*(X) -! ss*(Y ^ X). Taking
Y = Sp gives a map Rp R ss*(X) -! ss*(Sp ^ X). As Sp is p-local, the left hand
side is the same as RpRp ss*(Xp). This is a homology functor of X, because Rp is
flat over Rp, and ss*(Sp ^ X) is also a homology functor for more obvious reaso*
*ns.
We thus have a map of graded homology functors that is an isomorphism when
76 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
X = S, and thus for all X. It is well-known that Rp Rp N = Np for finitely
generated modules over Rp, which gives the last statement.
(b): Using (a) and the fact that Rp is`faithfully flat over Rp, we see that <*
*Sp>=
. By smashing`the equivalence ~~~~ = qwith Sp, we see that this is
the same as qp .
(c): Observe that ss*K(p) is a p-torsion module, finitely generated over Rp, *
*so
that ss*(Sp^K(p)) = ss*(K(p))p= ss*(K(p)) by (a). Thus the obvious map S -! Sp
gives an equivalence S ^ K(p) = Sp ^ K(p) and thus LK(p)S = LK(p)Sp. It will
therefore be enough to show that Sp is K(p)-local. Suppose that K(p) ^ X = 0,
so we need to prove that [X; Sp] = 0. Recall that Sp = F (Ip; Ip), so it is eno*
*ugh
to show that X ^ Ip = 0. However, ss*Ip is p-local and p-torsion, so Theorem 6.*
*1.8_
tells us that , and K(p) ^ X = 0, so Ip ^ X = 0 as required. |_*
*_|
Corollary`6.3.6.If L is a smashing localization functor and , then
qp .
Proof.If , then LK(p)S is LLS-local, which is the same as being
L-local, as L is smashing; in other words LS ^ LK(p)S = LK(p)S. It follows that
a
= = :
qp
|___|
We can now prove the promised classification theorem.
Theorem 6.3.7. Suppose that each Bousfield class is minimal. Then the
telescope conjecture holds for C_every smashing localization is a finite locali*
*zation.
The lattice of finite localizations is antiisomorphic to the lattice of thick s*
*ubcate-
gories of small objects, which is isomorphic to the lattice of subsets of SpecR*
* that
are closed under specialization.
Proof.Let L be a smashing localization functor, and write
T = {p 2 SpecR | LK(p) = 0}:
If LS ^ K(p) 6= 0 then by minimality, and thus LS ^ K(q) 6= 0 for
q p by Corollary 6.3.6. It follows that T is closed under specialization. Let *
*LT
be the corresponding localization functor. By Lemma 6.3.1, we have LTK(p) = 0 if
and only`if p 2 T , and LTK(p) = K(p) otherwise. By smashing the decomposition
~~~~= pwith LS and LTS, we see that = . As a localization
functor is determined by the corresponding category of local objects, we have L*
* =
LT. Thus, every smashing localization is a finite localization. The rest follow*
*s_from
Proposition 6.2.3 and Proposition 3.8.3. |__|
7. Connective stable homotopy theory
In this section, we discuss stable homotopy categories with a good notion of
connectivity. These share many features with the homotopy category of spectra.
Definition 7.1.1.Suppose that C is a monogenic stable homotopy category. We
say that C is connective if ssnS = 0 for n < 0.
AXIOMATIC STABLE HOMOTOPY THEORY 77
Of course, this is the case for ordinary stable homotopy theory. In fact, it *
*is the
case for many of our examples in Section 1.2 (including D(R) and C(B)). However,
Bousfield localizations of connective categories are rarely connective. Here we*
* do
little more than summarize the properties of connective categories and briefly *
*sketch
a proof, referring to the work of Margolis for details.
Proposition 7.1.2.Let C be a connective monogenic stable homotopy category.
(a) Suppose that X 2 C has ssm X = 0 for m < k (in this case, we say X is
bounded below). Then there is a cellular tower
Xk fk-!Xk+1 fk+1---!Xk+2 fk+2---!: : :
whose minimal weak colimit is X, such that Xk is a coproduct of copies of
Sk, and such that the cofiber of fn is a coproduct of copies of Sn+1.
(b) Suppose that R = ss0S is a Noetherian ring of finite global dimension, and
that every projective R-module is free. Then every small object of C has a
finite cellular tower as in (a).
(c) Given an object X 2 C and an integer k, there is a diagram X[k; 1] f-!X -g!
X[-1; k - 1] such that ssm X[k; 1] = 0 for m < k, ssm X[-1; k - 1] = 0
for m k, ssm f is an isomorphism for m k, and ssm g is an isomorphism
for m < k. Furthermore, [X[k; 1]; X[-1; k - 1]] = 0. In the terminology
of [BBD82 ], C admits a t-structure.
(d) Using the diagram in part (c), we can construct a Postnikov tower for any
object X:
: :-:---! X[-1; r] ----! X[-1; r - 1]----! X[-1; r - 2]- ---! : : :
x? x x
? ?? ??
X[r] X[r - 1] X[r - 2]
The sequential colimit of X[-1; r] is 0 and the sequential limit of X[-1; *
*r]
is X. (See [Mar83 , Chapter 5].)
(e) Let R denote the ring ss0S. Let A denote the full subcategory of C consist*
*ing
of objects X such that ssm X = 0 unless m = 0. Then A is a closed symmetric
monoidal Abelian category, and is equivalent as such to the category of R-
modules. In the terminology of [BBD82 ], the category of R-modules forms t*
*he
heart of C. We call elements of A Eilenberg-MacLane objects.
(f) Let H = S[0] denote the object of A corresponding to R. We call H*X =
ss*(H ^ X) the ordinary homology of X. Then H is a ring object, and if
ssm X = 0 for all m < k, then the natural (Hurewicz) map sskX -! HkX is
an isomorphism.
Proof.Given an object X and an integer k, we construct X[-1; k] as follows. We
set X0 = X, choose a system of generators {xi} for ssk+1X0 as a module over R,
let B1 be a coproduct of copies of Sk+1 (one for each i), and consider the evid*
*ent
map B1 -! X0. We define X1 to be the cofiber of this map. One can check that
ssm X1 = ssm X for m k, and sskX1 = 0. In a similar way, one can construct
a coproduct B2 of copies of Sk+2 and a cofibration B2 -! X1 -! X2, such that
ss*X2 = ss*X below degree k, and ssk+1X2 = ssk+2X2 = 0. Continuing in this
manner and passing to the sequential colimit, we get X1 = X[-1; k]. This comes
equipped with a natural map X -! X[-1; k], and we define X[k + 1; 1] to be the
fiber. We also define X[k; l] = X[k; 1][-1; l] and X[k] = X[k; k].
78 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
The proof of (a) is essentially identical to the construction above. We refer
to [Mar83 , Chapter 5] for the proofs of (c), (d) and (f). For (e), observe tha*
*t ss0
is a functor from A to the category M of R-modules.LThisLis essentially surject*
*ive:
for any module M, we can choose a`presentation` iR -! jR -! M, construct
a corresponding cofiber sequence iS -! jS -! X, and then X[0] 2 A and
ss0X[0] = M. Small modifications of the arguments of Margolis show that this is
also full and faithful.
For (b), suppose that R is Noetherian with finite global dimension, and that *
*all
projective R-modules are free. Let X be a small object in C. It is easy to see *
*that
H*X is then a finitely generated graded module over R, and that sskX = 0 for
k 0. We may assume without loss of generality that sskX = 0 for k < 0. Write
X0 = X. Much as above, we let B0 be a finite coproduct of copies of S0 and choo*
*se
a map B0 -!X that is surjective on ss0 (which agrees with H0 by (f)). We let X1
be the cofiber, and note that H*X1 is the same as H*X0 except in dimension zero
(where H0X1 = 0) and dimension one. Continuing in this way, we get a sequence of
cofibrations Bk -!Xk -! Xk+1, where H*Xk is zero below dimension k and agrees
with H*Xk-1 above dimension k. It follows that for large k, the groups H*Xk are
concentrated in a single degree. After that point, the projective dimension of *
*the
single group in question decreases by one at each stage, until it becomes zero,*
* so the
group is free. At that point, we can choose the map Bk -! Xk so that it induces
an isomorphism in homology, so that H*Xk+1 = 0. As ss*Xk+1 = 0 for k 0,
part (f) implies that ss*Xk+1 = 0, and thus Xk+1 = 0. Now let Xj be the fiber o*
*f __
the evident map X -! Xj, to get a finite cellular tower of the required type. *
* |__|
8.Semisimple stable homotopy theory
In this section we give conditions under which a stable homotopy category is
actually an Abelian category. The most familiar example is the category of rati*
*onal
spectra, which is equivalent to the category of graded rational vector spaces. *
*We
will allow C to be multigraded as in Section 1.3.
Definition 8.1.1.Suppose that C is an algebraic stable homotopy category. We
say that C is semisimple if, for every pair Y; Z 2 G, we have
(a) If Y 6= Z, then [Y; Z]* = 0; and
(b) [Z; Z]* is a (multigraded) division algebra kZ, where the multiplication is
given by composition.
Example 8.1.2. Consider the category C(kG ) of chain complexes of projective kG-
modules where G is a finite group and p = char(k) does not divide |G|. Then kG
is semisimple, so every kG-module is a direct sum of simple modules. Also, every
simple module appears as a summand of kG; hence every kG-module is projective.
Schur's lemma says that if S and T are non-isomorphic simple kG-modules, then
Hom kG(S; T ) = 0, while kS = Hom kG(S; S) is a division algebra.
In this case we are doing something just a bit more complicated than rational
stable homotopy (equivalently, graded rational linear algebra).
Proposition 8.1.3.If C is a semisimple stable homotopy category , then every
object X of C is equivalent to a coproduct of suspensions of elements of G.
AXIOMATIC STABLE HOMOTOPY THEORY 79
Proof.Given an object X of C and Z 2 G, note that [Z; X]* is a right module over
the division algebra kZ; let BZ denote a basis. We have a map F given by
a a F
nZ --! X:
Z2G f2BZ
|f|=n
Here n could be a multi-index. We claim that F is an equivalence. It suffices to
show that [Y; F ]* is an isomorphism for each Y 2 G. This is clear, though:
a a a
(8.1.1)[Y; |f|Z]* = [Y; |f|Y ]* =
Z f2BZ f2BY
M M
-|f|[Y; Y ]* = -|f|kY = [Y; X]*:
f2BY f2BY
|___|
We can extend this proposition a little as follows.
Definition 8.1.4.Given a semisimple stable homotopy category C, a G-module is
an assignment of a graded right kZ-module to each Z 2 G. The class of G-modules
forms an Abelian category G-Mod in the obvious way, where a sequence is exact if
and only if it is exact for each Z.
There is a natural functor F from C to G-Mod that assigns to X and Z the
kZ-module [Z; X]*.
Proposition 8.1.5.The functor F is an equivalence of categories. |___|
The proof is clear. Note that the induced triangulation on G-Mod is a very si*
*mple
one. Given a f :M -! N and a generator Z, the cofiber of f at Z is the direct s*
*um
of the suspension of the kernel of f and the cokernel of f as a kZ-vector space.
So, for example, the set of maps between finite objects W and X is in one-to-*
*one
correspondence with matrices of the appropriate shape_if rZW is the kZ-rank of
[Z; W ]* (and similarly for rZX ), then
M
[W; X]* = MrZXxrZW (kZ)
Z2G
(where Mrxs(k) is the set of r x s matrices with coefficients in k).
Note that one can define the product of two stable homotopy categories C and
C0in the obvious way, and this again gives a stable homotopy category: the obje*
*cts
are pairs (X; X0), and all of the structure in the axioms is defined coordinate*
*-wise.
For example (X; X0) ^ (Y; Y 0) is defined to be (X ^ Y; X0^ Y 0). We point out *
*that
a semisimple stable homotopy category will not in general be decomposable into a
product of stable homotopy categories, one for each element of G. In the case of
C(kG ) where p does not divide |G|, if Z is a simple kG-module, then Z Z does
not necessarily decompose as a direct sum of copies of Z (as it would if the ca*
*tegory
split). Similarly, the function object in C(kG ) will not behave "coordinate-wi*
*se,"
as it would in a product stable homotopy category. Thus, the induced symmetric
monoidal structure on G-Mod might be complicated. Indeed, this is the content of
classical representation theory.
One can describe the G-ideals in a semisimple stable homotopy category in the
following way. Draw a graph with one vertex for each generator, and an edge joi*
*ning
Z to W if and only if Z ^ W 6= 0, and let T be the set of components of this gr*
*aph.
80 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
One can check that the G-ideals biject with the subsets of T . In the case of a*
* group
algebra, T bijects with the set of blocks of kG.
9.Examples of stable homotopy categories
9.1. A general method. Suppose that one wants to do homotopy theory in a
category C, which has a natural notion of homotopy on its morphism sets. The fi*
*rst
step is to consider the category hC in which the morphisms are homotopy classes
of maps. It is by now familiar that this is insufficient to give a good theory.*
* One
must instead_formally invert a suitable class of weak equivalences to get a cat*
*egory
called hC. This procedure cures such pathologies as the long line, which is a n*
*on-
contractible topological space whose homotopy_groups_all vanish. However, there*
* is
no guarantee a priori that the morphism sets hC(X; Y_) are actually sets rather*
* than
proper classes. In many cases it can be shown that hC is equivalent to some full
subcategory D hC. If C is a closed model category [Qui67, DS95] we can take D
to be the subcategory of objects which are both cofibrant and fibrant. In some *
*other
cases one can use a category of cell objects. This general approach has been mu*
*ch
used by May and his coauthors [LMS86 ] in more recent work [EKMM95 , KM95 ]
they have also given closed model structures.
We now state a theorem which codifies these ideas. See Definitions 1.1.1, 1.1*
*.4,
1.1.6, and 3.4.1 for the relevant terms.
Theorem 9.1.1 (Cellular approximation).Let C be an enriched triangulated cat-
egory , and G a set of small strongly dualizable objects containing the unit S.*
* Let
D denote the localizing subcategory generated by G. Suppose in addition that if
X; Y 2 G, then X ^ Y and DX are in D. Then:
(a) D is a unital algebraic stable homotopy category with generating set G.
(b) The inclusion functor J :D -!C is a geometric morphism, with right adjoint
C say.
(c) The functor C preserves coproducts and the unit.
(d) Let S be the class of morphisms f :X -! Y in C such that f*: [Z; X]* '
[Z; Y ]* for all Z 2 G. Then C induces an equivalence C[S-1] ' D.
Note that if C is an enriched triangulated category in which S is small, then*
* we
can take G = {S} and the hypotheses of the theorem are satisfied. In particular,
in any unital algebraic stable homotopy category, the localizing subcategory lo*
*c~~~~
is a monogenic stable homotopy category. One might wonder why we need more
general sets G at all. A good example of this is provided by the homotopy categ*
*ory
of G-spectra discussed in Section 9.4. There, if we just take G = {S} we only g*
*et
spectra on which G acts trivially, which is equivalent to the homotopy category*
* of
non-equivariant spectra. Therefore, in that case, to get anything new we must t*
*ake
a larger set G.
Proof of Theorem 9.1.1.Let X be an object of C. As in the proof of Theorem 2.3.*
*2,
we construct a cofibration CX -q!X -i!LX with CX 2 D, and [Z; LX]* = 0 for
all Z 2 G (and therefore all Z 2 D). It follows as in Lemma 3.1.6 that C and L *
*are
functorial, and indeed are exact functors of X. Clearly, if Y 2 D then [Y; LX] *
*= 0
so [Y; X] = [Y; CX]. It follows that C is right adjoint to J.
We can now show that D is a unital algebraic stable homotopy category. Since
D is localizing, it is certainly a cocomplete triangulated category. By defini*
*tion,
the localizing subcategory generated by G is D, and the objects of G are small.*
* It
AXIOMATIC STABLE HOMOTOPY THEORY 81
remains to show that D has a compatible closed symmetric monoidal structure, and
that the generators are strongly dualizable. It will then follow from Theorem 1*
*.2.1
that cohomology functors are representable.
We will first show that D is closed under the smash product in C (clearly this
smash product is compatible with the triangulation and the coproduct). We begin
by showing that if X 2 D and Y 2 G, then X ^ Y 2 D. This is immediate since the
subcategory of such X is localizing and contains G. Similarly, given an arbitra*
*ry
X 2 D, we consider the set of all Y such that X ^ Y 2 D. This is again localizi*
*ng,
and we have just seen that it contains G, so it is all of D as claimed. Moreove*
*r, the
unit S lies in G D.
We still need to construct function objects and show that elements of G are
strongly dualizable. Define FD (X; Y ) to be CF (X; Y ). If X 2 D we have
[X; FD (Y; Z)] = [X; F (Y; Z)] = [X ^ Y; Z]:
Therefore FD (X; Y ) is adjoint to the smash product on D. Since C is exact,
FD (-; -) is exact as well.
Each object Z 2 G is strongly dualizable in C, so that the map S -j!F (Z; Z)
factors through an isomorphism F (Z; S) ^ Z ' F (Z; Z). It follows from our as-
sumptions on G and the closure of D under smash products that all the objects j*
*ust
discussed lie in D. Thus
FD (Z; Z) = F (Z; Z) = F (Z; S) ^ Z = FD (Z; S) ^ Z;
which means that Z is strongly dualizable in D.
This proves that D is a unital algebraic stable homotopy category. It is cle*
*ar
that J :D -!C is a geometric morphism, with right adjoint C, and that C preserv*
*es
the unit.
Consider a family {Xi} of objects of C. We then have a cofibration
a a a
CXi-! Xi-! LXi:
i i i
The first term lies in D. Moreover, for any Z 2 G we have
a M
[Z; LXi] = [Z; LXi] = 0:
i i
` `
Thus iCXihas the universal property characterizing C iXi, which means that
C preserves coproducts.
Recall that the category of fractions C[S-1] has the same objects as C, and t*
*hat
there is a functor Q: C -! C[S-1] which sends the maps in S to isomorphisms.
Moreover, Q is the initial example of such a functor: given any functor F :C -!*
* E
which inverts S, there is a factorization F ' F 0Q, unique up to natural isomor*
*phism.
Clearly C :C -! D inverts S, so C ' C0Q for some functor C0:C[S-1] -! D. We
also write J0 = QJ :D -!C[S-1]. It follows that
C0J0 ' C0QJ ' CJ ' 1: D -!D:
On the other hand, for any object X = QX 2 C[S-1], we have a map J0C0QX =
CX -q!X = QX which lies in S. Thus q is an isomorphism in C[S-1], so J0C0'_1.
Thus D ' C[S-1] as claimed. |__|
82 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
9.2. Chain complexes. Most of the non-topological examples of stable homotopy
categories we will consider involve chain complexes of objects in additive cate*
*gories.
If an additive category A is sufficiently nice, the category of chain complexes*
* of
objects of A and chain homotopy classes of maps will satisfy the hypotheses of
Theorem 9.1.1. Since all of our algebraic examples fit this description, we fol*
*low a
somewhat abstract approach, just as we did for triangulated categories.
For this section we assume that A is an enriched additive category (Defini-
tion 1.1.6). If A happens to be graded, we assume that the closed symmetric
monoidal structure has the usual sign conventions. In particular, the symmetric
monoidal structure is graded-commutative rather than commutative.
Given any additive category A, we can form the category Ch(A) of (Z-graded)
chain complexes and chain maps. As usual, if X is such a chain complex, we
denote its nth component by Xn and its differential by d. It is essentially irr*
*elevant
whether the differential raises or lowers degree, but for concreteness we assum*
*e it
lowers degrees.
Proposition 9.2.1.If A is an enriched additive category, then Ch (A) is an en-
riched additive category. The obvious inclusion functor A -! Ch(A) (sending an
object M to a complex concentrated in degree zero) is full and faithful, and pr*
*eserves
all structure in sight. If A is Abelian, then so is Ch(A).
Proof.This is all well-known. We can define products and coproducts dimension-
wise, so Ch(A) isLcomplete and cocomplete. If X and Y are chain complexes, we d*
*e-
fine (X^Y )n = kXkYn-k. The differential on XkYn-k is dX 1+(-1)kdY .
It is easy to check, and standard, that this gives a symmetric monoidal structu*
*re
on Ch(A). The component of the twist map X ^ Y -! Y ^ X sending Xn Ym to
Ym Xn involves a sign (-1)nm as usual.Q
Similarly, we define F (X; Y )n = kHom (Xk; Yn+k). The component of the
differential landing in Hom (Xk; Yn-1+k) is the composite
Y
Hom (Xl; Yn+l) -!Hom (Xk-1; Yn-1+k) Hom (Xk; Yn+k)
-(-1)n+1Hom(dX;1)Hom(1;dY)--------------------!Hom(X
*
*k; Yn-1+k):
It is then straightforward to check that this structure makes Ch (A) an enriched
additive category. If A is Abelian then we can define kernels and cokernels_dim*
*en-
sionwise, and Ch(A) becomes an Abelian category. |__|
There is an evident notion of chain homotopy in Ch(A). For later use, we will
describe this somewhat differently than usual. Define I to be the chain complex
consisting of R in dimension 1, R R in dimension 0, and 0 everywhere else. The
differential is given by R (1;-1)----!RR. There are two evident chain maps R i0*
*;i1---!I.
Two chain maps f0; f1: X -! Y are then said to be chain homotopic if there is a
map H :X ^ I -! Y such that H(1 ^ ik) = fk for k = 0; 1.
It is easy to see that a homotopy X ^ I -! Y is the same thing as a map
X -! F (I; Y ). This is just a formalization of the standard equivalent notions*
* of
chain homotopy.
The resulting category of chain complexes and chain homotopy classes of maps
will be denoted K(A).
AXIOMATIC STABLE HOMOTOPY THEORY 83
Proposition 9.2.2.If A is an enriched additive category , then K(A) is an en-
riched triangulated category. Furthermore, every small (resp., strongly dualiza*
*ble)
object of A is small (resp., strongly dualizable) in K(A).
Proof.The coproduct in Ch(A) descends to K(A), as is easy to check. The trian-
gulation on K(A) is of course well-known. That is, we define the suspension X
by (X)n = Xn-1 with dX = -dX . An exact triangle is a sequence isomorphic
in K(A) to one of the form
X -f!Y -g!Z -h!X
i j
where Zn = Yn Xn-1, dZ = dY0-fdX , g is the evident inclusion, and h is the
evident projection.
There is also a slightly more flexible way to define triangles. Suppose that *
*X -f!
Y -g!Z is a sequence of chain complexes, such that in each degree Xn f-!Yn -g!Zn
is a split short exact sequence. (Note that this makes sense even if A is not A*
*belian.)
This means that we can choose maps X -r Y -s Z (usually not chain maps) such
that
rf = 1; rs = 0; gf = 0; gs = 1; sg + fr = 1:
It turns out that h = rdY s defines a chain map Z -! X, which is independent of
the choice of r and s up to homotopy. Moreover, the sequence X -f!Y -g!Z -h!X
is a triangle. For more details, see [Ive86], for example.
It is straightforward to check that the symmetric monoidal structure defined *
*on
Ch(A) descends to K(A). Although we have assumed nothing about the exact-
ness of the symmetric monoidal structure on A, cofiber sequences in K(A) split
dimensionwise, so the smash product will automatically be exact on K(A).
Using the alternative description of a chain homotopy as a map X -! F (I; Y ),
it is easy to check that the function object construction F (X; Y ) descends to*
* the
homotopy category and is exact there. One must check that the adjointness also
descends of course.
It is immediate that strongly dualizable objects of A give strongly dualizable
objects of K(A), since the relevant function objects and smash product are the
same in both categories. To see that a small object M of A remains small in K(A*
*),
one need only check that M ^ I is small in Ch(A). This is easy to do using the_
standard definition of chain homotopy. |__|
There is also a natural notion of weak equivalences in Ch(A) (or K(A)):
Definition 9.2.3.A quasi-isomorphism is a chain map f :X -! Y which induces
an isomorphism H*X -! H*Y .
9.3. The derived category of a ring. Let R be a commutative ring, and M(R)
the enriched Abelian category of R-modules. We write K(R) for K(M(R) ). Note
that the unit in this category is R, so that
ss*X = [R; X]* = H*X:
Theorem 9.3.1. Let R be a commutative ring. Let D(R) be the category of frac-
tions obtained from K(R) by inverting quasi-isomorphisms, so that we have a fun*
*c-
tor Q: K(R) -! D(R). Then D(R) can be identified with a subcategory of K(R),
and Q with the right adjoint of the inclusion functor. Moreover:
84 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
(a) D(R) is a monogenic stable homotopy theory (with small Hom sets).
(b) There is a geometric morphism J :D(R) -!K(R), with right adjoint Q, with
the property that Q(X JY ) = (QX) Y .
(c) Let X be a bounded-below complex of projectives, and Y an arbitrary comple*
*x.
Then [QX; QY ] = [X; Y ].
(d) Let X be an arbitrary complex, and Y a bounded-above complex of injectives.
Then [QX; QY ] = [X; Y ].
(e) For any short exact sequence X -! Y -! Z of bounded-below complexes in
Ch(R), there is a natural exact triangle QX -! QY -! QZ -! QX.
(f) A map R -!R0induces a stable morphism D(R) -!D(R0).
We refer to D(R) as the derived category of R. Our approach to the derived
category was inspired by [BN93 ].
Note that in this category S = R, so ss*S = R. Thus, if R is Noetherian, then
the results of Section 6 apply. Note also that if R is countable, then D(R) is*
* a
Brown category.
Of course, most of the axioms are well-known for the derived category. See for
example [Ver77, Har66, Wei94].
Proof.Parts (a) and (b) are an immediate corollary of Theorem 9.1.1 (with G =
{R}) and Proposition 9.2.2. The only point to check is that Q(JXY ) = (QX)Y .
This really means that if Y is a cell object then the functor (-) Y preserves *
*weak
equivalences, or equivalently, preserves acyclic complexes. The category of tho*
*se Y
for which this is true is clearly a localizing subcategory, and it clearly cont*
*ains S;
it is therefore the whole of D(R).
(c): Suppose that X is a bounded-below complex of projectives. Let Z be an
exact complex; it is well-known that any chain map X -! Z is null-homotopic. The
null-homotopy is constructed as usual by induction on the dimension, and we hav*
*e a
place to start because X is bounded below. In particular, if QX = CX -! X -! LX
is the usual cofibration (as in the proof of Theorem 9.1.1), we find that the m*
*ap
X -! LX is zero. As L is idempotent, we find that 0 = 1: LX -! L2X = LX, so
that LX = 0 and X = CX = QX. It follows that [X; Y ] = [QX; QY ] as claimed.
(d): Suppose that Y is a bounded-above complex of injectives, and X is arbitr*
*ary.
By the adjoint property of C, we have [CX; CY ] = [CX; Y ]. Any map from an
exact complex (such as LX) to Y is null-homotopic, so [CX; Y ] = [X; Y ].
(e): It is well-known (see [Ive86, Proposition 6.10], for example) that such*
* a
sequence can be replaced by a quasi-isomorphic sequence X0-! Y 0-!Z0, which is
a dimensionwise-split short exact sequence of bounded-below complexes of projec-
tives, and thus a cofiber triangle. Thus, CX = CX0 = X0 (using (c)). The claim
follows.
(f): Given a map R -! R0, we get a functor T = R0R (-): D(R) -! D(R0).
This clearly preserves the tensor product and the unit, and sends G = {R} to
G0= {R0}. In the other direction, we can start with an object of D(R0), regard *
*it
as a complex of modules over R, and apply Q; it is not hard to see that this gi*
*ves_
a right adjoint to T , so that T is indeed a stable morphism. *
*|__|
For the interested reader, we describe the closed model structure [Qui67] on
the category of chain complexes and chain maps that gives rise to D(R); this is,
essentially, in [Wei94]. We need to identify the weak equivalences, the fibrati*
*ons,
and the cofibrations. A morphism is a weak equivalence if and only if it is a
AXIOMATIC STABLE HOMOTOPY THEORY 85
homology isomorphism, a fibration if and only if a dimensionwise surjection, an*
*d a
cofibration if and only if a dimensionwise injection where the dimensionwise co*
*kernel
is cofibrant. A complex is cofibrant if it can be written as an increasing unio*
*n of
complexes so that the associated quotients are complexes of projectives with ze*
*ro
differential.
Notice as well that we can do this same construction if R is a graded ring, u*
*sing
chain complexes of graded R-modules. In that case, we will have two orthogonal
directions in which to suspend, so we should consider D(R) as a multigraded sta*
*ble
homotopy category as in Section 1.3.
We record the following result for use in Section 6.
Let p R be_a prime ideal, and let kp denote the residue field (R=p)p of p. We
also write kp= Q(kp) 2 D(R).
Proposition 9.3.2.Suppose that R is Noetherian. Then we have
__
loc = loc = loc
and
__
= = :
Proof.First note that, by [Nee92a, Lemma 2.12], we have
a __
~~~~ = :
__
The homotopy_groups of kpare_p-local_and p-torsion. It follows from Theorem_6.1*
*.8
that , so that kp^ kq= 0 when p 6= q. This means that each kp is
smash-complemented. __
Next, we can use the_pairing QX QY -! Q(X Y ) to make kp into a ring
object,_such that ss*kp = kp is a_field. It follows from Propositions 3.7.2 and*
*_3.7.3
that kpis a field object, that is a minimal Bousfield class, and that loc is
a minimal localizing subcategory._
We next claim that K(p) 2 loc. It is easy to reduce to the case in which
R is a local ring with maximal ideal p, so we shall assume this.S Let M be a p-
torsion R-module. Write Mk = {m 2 M | pkm = 0}, so that M = kMk and
Mk=Mk-1 is a vector_space over kp. It follows (using part (e) of Theorem 9.3.1)
that Q(Mk) 2 loc. The map from the sequential colimit of the objects Q(Mk)
to QM is a quasi-isomorphism of bounded-below complexes_of projectives, hence
a homotopy equivalence, which shows that QM 2 loc. Now suppose that
X 2 D(R) is such that ssk(X) = 0 for k < 0 and sskX is p-torsion for all k. We
define X0 = X, and observe that there is a natural map Q(ss0X0) -!X0; we write
X1 for the cofiber. Note that sskX1 = 0 for k < 1, and ss*X1 is p-torsion. We d*
*efine
objects Xk and cofibrations Q(sskXk) -!Xk -!Xk+1 in the evident way. We_write
Xk for the fiber of the map X -! Xk. It is easy to see that Xk 2 loc, and t*
*hat
ssm Xk = ssm X for k > m + 1. By arguments similar to those of Proposition_2.3.*
*1,_
we see that X is the sequential colimit of the Xk, and thus_X lies in loc.
In particular, this shows that Ip and K(p) lie in loc (recall that ss*S_=*
* R
is concentrated in degree_zero_in this context). As K(p) 6= 0 6= Ip and loc*
* is
minimal,_we see that loc = loc = loc as claimed. It follows that
= = . |___|
86 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Example 9.3.3. Let k be a field, and R = k[[x; y]]=(xy). Let p = (x; y) be the
maximal ideal. Then K(p) is the finite complex
(y-x)
R (x;y)---R2 ---- R- 0- : : :
__
and kpis the infinite complex
iy0j ix0j iy0j
R (x;y)---R2 -0x---R2 -0y---R2 -0x---R2- : : :
__
As R is not regular, k has infinite projective dimension as an R-module, and kp*
*is
not a small object.
9.4. Homotopy categories of equivariant spectra. In this section, we will
recall enough equivariant stable homotopy theory from [LMS86 ] to conclude that
the homotopy category of G-spectra (based on a complete G-universe, where G is
a compact Lie group) satisfies our axioms. We will not adopt the most modern
approach [EKMM95 , Smi] because it is not necessary for the results of this s*
*ection.
We should also point out that the homotopy category of non-equivariant spectra
(the case G = 1) has been known for a long time to satisfy the axioms for a
monogenic stable homotopy category [Vog70, Ada74, Mar83, EKMM95 , Smi].
Fix a compact Lie group G.
Definition 9.4.1.Let U be an inner product space isomorphic to R1 , with an or-
thogonal action of G. U is a G-universe if every finite-dimensional subrepresen*
*tation
of U occurs infinitely often in U and if the trivial representation is a subrep*
*resen-
tation of U. A G-universe is complete if every finite-dimensional representatio*
*n of
G is a subrepresentation of U.
If G is trivial, all universes are isomorphic to each other, though non-canon*
*ically.
However, if G is nontrivial, each isomorphism class of universes will give rise*
* to a
distinct stable homotopy category. It turns out to be important to consider the*
* set
of isotropy groups of points of U, or equivalently
Isotropy(U) = {H | G=H embeds inU}:
This is clearly closed under conjugation, and it is also closed under intersect*
*ion.
Indeed, the isotropy subgroup of (x; y) 2 U x U is just the intersection of the
isotropy subgroups of x and y, and U x U is G-isomorphic to U.
To discuss, even briefly, G-spectra, we need first to recall G-spaces. A G-sp*
*ace
is a compactly generated weak Hausdorff space with a continuous action of G. A
G-map between two G-spaces is just a G-equivariant continuous map. A based G-
space is a G-space with a distinguished point which is fixed by G. A based G-ma*
*p is
a G-map that preserves the basepoints. Given two G-spaces X and Y , X xY is the
G-space on which G acts diagonally. This allows us to define the smash product
X ^ Y of two based G-spaces as the quotient of X x Y by the one-point union
X _ Y . We can also define F (X; Y ), the space of based maps from X to Y , with
G acting by conjugation. Given a representation V of G, we have the associated
one-point compactification SV ; we consider this as a based G-space, with basep*
*oint
at infinity. We denote SV ^ X by V X, and call this the V th suspension of X. As
usual, suspension has an adjoint V X = F (SV ; X).
Given a G-space X and a subgroup H of G, we denote the H-fixed point space
by XH . A weak equivalence of based G-spaces is a based G-map f :X -! Y that
AXIOMATIC STABLE HOMOTOPY THEORY 87
induces isomorphisms ss*(XH ) -! ss*(Y H) of all homotopy groups (relative to t*
*he
basepoint) of all fixed point sets by closed subgroups H.
Definition 9.4.2.Suppose that U is a G-universe. A prespectrum X is a collec-
tion of based G-spaces X(V ) for each finite-dimensional subrepresentation V of*
* U,
together with G-equivariant maps oeV;W :W-V X(V ) -! X(W ) for each V W .
Here W -V denotes the orthogonal complement of V in W . The maps oeV;W are re-
quired to satisfy the transitivity conditions: oeV;V is the identity, and if U *
* V W
then oeU;W = oeV;W O (W-V oeU;V). A spectrum is a prespectrum such that the ad-
joints X(V ) -!W-V X(W ) of the structure maps are homeomorphisms.
Note that the notions of prespectrum and spectrum depend on both the group
G and the universe U. Note as well that there is an evident notion of maps of
prespectra and spectra, which are simply maps that commute with the structure
maps. This makes the category of spectra GSU a full subcategory of the category
of prespectra. One of the most important points in this theory is that the incl*
*usion
functor from spectra to prespectra has a left adjoint L. This makes the categor*
*y of
spectra have both arbitrary limits and arbitrary colimits, by taking colimits i*
*n the
category of prespectra and then applying L. One can also use L to define the sm*
*ash
product of a G-space and a spectrum, by smashing space-wise and then applying
L.
The category of spectra is enriched over the category of G-spaces, so we can
define X ^Y when X is a G-space and Y is a spectrum. This gives a natural notion
of homotopy in the category of spectra. Indeed, by allowing G to act trivially *
*on
the unit interval I, we have cylinder objects I+ ^ X. We can then define homoto*
*py
in the usual way.
For an integer n 0, we define the spectrum Sn by applying L to the prespec-
trum which is SV ^ Sn at the representation V , where we are thinking of Sn as
a based G-space with trivial G-action, and where the structure maps are isomor-
phisms. For n < 0, we define the spectrum Sn by applying L to the prespectrum
which is SV -nfor all representations V which contain n, the sum of n copies of*
* the
trivial representation, and which is the basepoint otherwise, with the evident *
*struc-
ture maps. We then define SnHfor all closed subgroups H G by SnH= G=H+ ^Sn.
Given a spectrum X, we then define its homotopy group ssnH(X) as the homotopy
classes of maps from SnHto X. A map f : X -! Y of spectra is then defined to
be a weak equivalence if it induces an isomorphism on all homotopy_groups._We
denote the category of spectra with weak equivalences inverted by hGSU . This is
an honest category with small Hom sets, by [LMS86 , Section I.6].
The next theorem follows from the first three chapters of [LMS86 ].
__
Theorem 9.4.3. The category hGSU is an enriched triangulated category (Def-
inition 1.1.6). For any closed subgroup H of G, the object S0H = G=H+ ^_S_
is small, and the localizing subcategory generated by the S0H is all of hGSU . *
* If
H 2 Isotropy(U), then G=H+ ^ S is strongly dualizable.
Given the last statement of the theorem, it is natural to define
G = {G=H+ ^ S | H 2 Isotropy(U)}:
Proof.This is all contained in the first three chapters of [LMS86 ], but we rec*
*all
some of the structure. It is easy to see that the coproduct in spectra descends*
* to
88 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
__
the homotopy category. A cofiber sequence is any sequence isomorphic in hGSU to
a sequence
X -f!Y -g!Z -h!X
where Z = Y [f CX is defined as the evident pushout in the category of spectra
(as is CX), g is the inclusion, and h is the map obtained by collapsing Y . Then
the results of [LMS86 , III, Section 2] show that this gives a triangulation on*
* KUG,
and that fiber sequences, defined analogously using pullbacks and function spac*
*es,
are equivalent to cofiber sequences (up to a sign).
The symmetric monoidal structure is a little tricky in [LMS86 ] because it do*
*es
not come from a symmetric monoidal structure on the category of spectra. This
problem has been fixed more recently [EKMM95 , Smi]. The problem is that the
smash product of two prespectra indexed on U is naturally indexed on UxU, rather
than on U. But one can choose an equivariant linear isometry between U x U and
U and use this,_together with L, to make a structure that becomes symmetric
monoidal on hGSU . This, and the analogous function spectrum construction, is a*
*ll
contained in [LMS86 , II,Section 3].
We learn from [LMS86 , Lemma I.5.3] that G=H+ ^ S is always small. The_CW
approximation theorem [LMS86 , Chapter I] implies that any object in hGSU is in
the localizing subcategory generated by the S0H. Moreover, when H 2 Isotropy(U)*
*,__
we see from [LMS86 , Theorem III.2.7] that G=H+ ^ S is strongly dualizable. *
*|__|
__
Unfortunately, the category hGSU is not a stable homotopy category as we have
defined it, unless the universe is complete, since otherwise some of the genera*
*tors SH
need not be strongly dualizable. In fact, Lewis has shown (personal communicati*
*on)
that SH is never strongly dualizable unless H 2 Isotropy(U). This may be a
flaw with our axiom system, or one could interpret it as an unpleasant feature *
*of
incomplete universes. __
One can always take the "stable hull" of the category hGSU to obtain a stable
homotopy category.
Corollary 9.4.4.Let SUGbe the localizing subcategory generated by the SH such
that H 2 Isotropy(U). Then SUGis a unital algebraic Brown category.
Proof.First, we claim that
G=H+ ^ G=K+ ^ S = (G=H x G=K)+ ^ S 2 SUG
whenever H and K are in Isotropy(U). From [LMS86 ], we find that G=H x G=K
is a finite G-CW complex. If there is a cell of type G=L x en in G=H x G=K, then
there must be a point with isotropy group L. But then L is an intersection of a
conjugate of H with a conjugate of K, so L 2 Isotropy(U). Thus G=H+ ^G=K+ ^S
is a finite G-CW spectrum built from the G=L+ ^ Sn where L 2 Isotropy(U), and
in particular is in SUG.
Similarly, the Wirthm"uller isomorphism [LMS86 , Chapter II] shows that
D(G=H+ ^ S) 2 SUG:
Thus Theorem 9.1.1 applies, showing that SUGis a unital algebraic stable homo-
topy category. To see that SUGis actually a Brown category, we use Theorem 4.1.*
*5.
The maps between the generators are given by the values of the Burnside ring
Mackey functor, which is countable, by the results of [LMS86 , Chapter 5], in_p*
*ar-
ticular [LMS86 , Corollary V.9.4]. |__|
AXIOMATIC STABLE HOMOTOPY THEORY 89
There is a closed model structure on the category of spectra whose associated
homotopy category is SUG, at least when U is complete. The fibrations are space*
*wise
fibrations, and the weak equivalences are spacewise weak equivalences of G-spac*
*es.
(A weak equivalence of G-spaces is a weak equivalence on each fixed point set.)
The cofibrations are defined by the left lifting property. Hopkins [Hop ] has g*
*iven a
proof that this does indeed give a closed model structure.
9.5. Cochain complexes of B-comodules. Let B be a commutative Hopf al-
gebra over a field k. Write Comod (B) for the category of left B-comodules, and
K(B) for the homotopy category of chain complexes in Comod (B). As with the
derived category of modules over a ring, this needs a little modification befor*
*e it
becomes a stable homotopy category. In this case the right thing to consider is*
* the
homotopy category C(B) of chain complexes of injective comodules.
We work with chain complexes (so that the differential decreases degrees), for
consistency with previous sections. Thus, an injective resolution of a comodule*
* will
be concentrated in negative degrees. Of course, everything can be translated by
the usual prescription Ci = C-i.
Recall that a comodule M is simple if it is nontrivial, but has no proper non*
*trivial
subcomodules.
Theorem 9.5.1. Let B be a commutative Hopf algebra over a field k. Then C(B) is
a unital algebraic stable homotopy category, with a geometric morphism L: K(B) *
*-!
C(B), whose right adjoint is the inclusion functor. We may take
G = {LM | M is a simple comodule}:
Moreover, L sends complexes of finite total dimension over k to strongly dualiz*
*able
objects.
If B = (kG)* where G is a p-group and p = char(k), then C(B) is monogenic.
The same applies if B is graded and connected (and we consider only graded co-
modules).
If M and N are comodules then
[LM; LN]* = Ext*B(M; N):
A map B -! B0 of Hopf algebras gives rise to a stable morphism (see Defini-
tion 3.4.1) C(B) -!C(B0).
This theorem will be proved after a number of auxiliary results.
Note that in C(B) we have L(k) = S and thus
ss*S = [S; S]* = Ext*B(k; k):
If B is finite-dimensional, then Friedlander and Suslin have shown that this is
Noetherian [FS ]. (This was known in special cases earlier: for B = (kG)* where*
* G
is a finite group [Eve61], for B a finite-dimensional graded connected commutat*
*ive
Hopf algebra [Wil81], and for B = V (L)* where L is a finite-dimensional restri*
*cted
Lie algebra [FP87 ]). If in addition, k (along with its suspensions, in the gr*
*aded
case) is the only simple comodule, then the results of Section 6 apply. For ins*
*tance,
the nilpotence theorem (Corollary 6.1.10) provides a new way to detect nilpoten*
*ce
in Ext*kG(M; M) for G a finite p-group and M a finitely generated kG-module.
We begin with some results about the category of B-comodules. It is well-
known that the category of modules over a commutative ring is enriched, but it *
*is
90 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
less well-known that the category of comodules over a commutative Hopf algebra
is also enriched.
Note that the dual vector space B* is a (typically non-commutative) k-algebra.
Definition 9.5.2.A B*-module M is tame if for every m 2 M, the generated
submodule B*m has finite dimension over k.
Lemma 9.5.3. Let B be a Hopf algebra over a field k.
(a) Given a left B-comodule M and m 2 M, then the subcomodule generated by
m is finite-dimensional over k.
(b) There is an isomorphism of categories (which is the identity on objects) b*
*e-
tween (left) B-comodules and the full subcategory of tame (left) B*-module*
*s.
(c) The resulting inclusion functor J from the category of (left) B-comodules *
*to
the category of (left) B*-modules has a right adjoint R.
(d) The category of (left) B-comodules is complete and cocomplete.
(e) If B is commutative, then the category of (left) B-comodules is enriched.
P
Proof.(a): Choose a basis {bi} for B. Relative to this basis, write (m) = bi
mi, where all but finitely many of the miare 0. Let M0denote the vector space s*
*pan
of the mi; we claim that M0 is a comodule. This is a consequence of coassociati*
*vity,
and we leave it to the reader.
(b): Given a left B-comodule M, define a B*-module structure by the composite
B* M -1--!B* B M -ev1--!M
where ev:B*B -! k is the evaluation map. One can check that the subcomodule
generated by m coincides with the sub B*-module generated by m, so by part (a)
these are all finite-dimensional. This correspondence clearly defines a functor.
We now define the inverse functor. Let N be a B*-module, with action map
B* N -! N. By adjunction we get a map N -! Hom k(B*; N). There is an
inclusion B N -i!Hom k(B*; N) whose adjoint is the evaluation map tensored
with N. The image of i is precisely the set of maps f which factor through a
finite-dimensional quotient of B*. In particular, if N is tame then the map N -!
Hom k(B*; N) will factor through i and give a B-comodule structure on N.
(c): Given an arbitrary B*-module N, define RN to be the set of all n 2 N such
that the submodule generated by n is finite-dimensional. Then RN is clearly a
submodule, and R is both a left inverse and a right adjoint to the inclusion fu*
*nctor.
(d): Coproducts of comodules can be defined in the usual way, by just taking
the direct sum of the underlying vector spaces and giving it the evident comodu*
*le
structure.Q To defineQproducts, we use R. Given a family of comodules {Mi},
define Mi = R( JMi). Then adjointness guarantees that this is a product in
the category of B-comodules. Note however that the product of infinitely many
nontrivial comodules could easily be trivial. Since the category of B-comodules*
* is
Abelian, it has arbitrary (co)limits whenever it has arbitrary (co)products.
(e): The symmetric monoidal structure on the category of B-comodules is well-
known. Given B-comodules M and N, we define the comodule structure on M kN
by the composite
M N ---! B M B N -1T1---!B B M N -11---!B M N:
AXIOMATIC STABLE HOMOTOPY THEORY 91
To make the category of B-comodules enriched, we also need a notion of function
object. It is easier to do this for B*-modules, so suppose that M and N are B*-
modules such that every principal submodule is finite-dimensional. Dual to the
multiplication and conjugation on B we have maps *: B* -! Hom k(B; B*) and
O*: B* -! B*. We are going to define a B*-module structure on Hom k(M; N)
by a horrendous formula, which is necessary since we are not assuming that B is
finite-dimensional. Recall we have chosen a basis {bi} for B; we let {b*i} den*
*ote
the dual basis for B*. Given f 2 Hom k(M; N) and u 2 B*, we define uf by the
formula
X
uf(x) = b*if[O*((*u)(bi))x]:
i
Since M and N are both tame, this sum is in fact finite. Indeed, B*x is finite-
dimensional, so f(B*x) is as well; thus b*if(B*x) is zero for almost all i. We *
*leave
it to the reader to check that this gives a B*-module structure on Hom k(M; N).
Now, given B-comodules M and N, let us define the function comodule F (M; N)
to be R Hom k(JM; JN). Then one can verify that the function comodule is right_
adjoint to the tensor product as required. |__|
We next investigate the structure of injective comodules.
Lemma 9.5.4.
(a) The forgetful functor U from comodules to vector spaces has a right adjoint
V 7! B V , with coaction map = 1. We refer to B V as the cofree
or extended comodule on V .
(b) For any comodule M we have B M ' B UM. Here the left hand side has
the structure described in part (e) of Lemma 9.5.3, and the left hand side*
* has
the comodule structure described in (a).
(c) A comodule I is injective if and only if it is a retract`of a cofreeQcomod*
*ule.
(d) If {Iff} is a collection of injective comodules, then Iffand Iffare in*
*jective.
(e) If I is an injective comodule and M is arbitrary, then I M and F (M; I) a*
*re
injective.
Proof.(a): This is well-known, and easy to check.
(b): The isomorphism is as follows (cf. [Mar83 , Proposition 12.4]):
B M ! BX UM
b m 7-! bbi mi
X i
bO(bi) mi - | b m
i
(c): It is clear from the adjunction Comod (B)(M; B V ) ' Hom k(M; V ) that
cofree comodules (and thus their retracts) are injective. Conversely, suppose t*
*hat I
is injective. We have an embedding of comodules k -! B, and thus an embedding
I = k I -! B I ' B UI:
Because I is injective, this splits, so I is a retract of the cofree comodule B*
* UI.
(d): It is formal (in any category) that products of injectives are injective*
*. For
coproducts, use (c).
92 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
(e): Let M be an arbitrary comodule. Using (b) and (c), we find that B M is
injective. Using (b) again, we conclude that I M is injective whenever I is. We
also have
Comod (B)(N; F (M; I)) = Comod (B)(N M; I):
This is clearly an exact functor of N, so F (M; I) is injective. *
* |___|
Lemma 9.5.5. Every simple comodule is finite-dimensional, and the collection of
isomorphism classes of simple comodules forms a set.
Proof.Let S be a simple comodule, and m 2 S a nonzero element. We know by
part (a) of Lemma 9.5.3 that m lies in a finite-dimensional subcomodule of S. T*
*his
must be all of S, because S is simple. It follows that S is a cyclic B*-module;*
*_there
is clearly only a set of these (up to isomorphism). |_*
*_|
Proof of Theorem 9.5.1.The strategy of the proof is very similar to that of The*
*o-
rem 9.3.1, except we have to localize rather than colocalize. That is, we begin*
* with
the category K(B) of chain complexes of B-comodules and chain homotopy classes
of chain maps. By Proposition 9.2.2, K(B) is an enriched triangulated category.
Note as well that any finite-dimensional comodule is small and strongly dualiza*
*ble
in the category of comodules, so will also be in K(B). Indeed, if M and N are
finite-dimensional, so is Hom (jM; jN), so F (M; N) = Hom (jM; jN). In partic-
ular, since any simple comodule is necessarily finite-dimensional by Lemma 9.5.*
*5,
simple comodules are small and strongly dualizable.
But, of course, the simple comodules do not generate K(B). Ordinarily in this
situation we would look at the localizing subcategory generated by the simple c*
*o-
modules, but if we did that in this case we would not get ExtB(k; k) as the hom*
*otopy
of S. So instead we look at C(B), the full subcategory of K(B) consisting of (a*
*ll
complexes chain homotopy equivalent to) complexes of injectives. (This is in fa*
*ct
a Bousfield localization.)
Using Lemma 9.5.4, we see that C(B) is a localizing ideal and a colocalizing
coideal in K(B). In particular, it is closed under products, coproducts, tensor
products and function objects. However, it does not contain the unit k.
Let k -! L be the cobar resolution of the comodule k. (Any other resolution
would do, but we take the cobar resolution to be definite.) Thus L is a complex
of injectives, whose homology is k, concentrated in degree zero. The complex L
itself is concentrated in degrees less than or equal to zero. We also write C f*
*or the
fiber of the map k -! L, so that H*C = 0. This means that C is contractible as a
complex of vector spaces, so that H*(C X) = 0 for any complex X. Finally, we
write CX = C X and LX = L X.
Note L is a functor K(B) -! C(B). We next show that it is left adjoint to the
inclusion J :C(B) -!K(B).
To do so, we first recall the well-known fact that if Y is a bounded-above ch*
*ain
complex (of objects in any Abelian category) with no homology, and U is a compl*
*ex
of injectives, then every map from Y to U is chain homotopic to the zero map. T*
*his
is proved, as usual, by induction, and since X is bounded above there is a plac*
*e to
start.
Now suppose that X 2 K(B) and U 2 C(B). Then C is bounded above and
acyclic, and F (X; U) is a complex of injectives, so
[CX; U] = [C; F (X; U)] = 0:
AXIOMATIC STABLE HOMOTOPY THEORY 93
It follows from the fibration C -! k -! L that
[LX; U] = [X; U] = [X; JU]:
In other words, L is left adjoint to J, as claimed.
It follows immediately that L ' 1 on C(B), in other words that L U ' U
whenever U 2 C(B). Thus L is the unit of the smash product on C(B), and C(B)
becomes an enriched triangulated category. This also means that L is a geometric
morphism.
It follows by juggling adjunctions that we have an internal version of the ad*
*junc-
tion: F (LX; U) ' F (X; U) whenever U is a complex of injectives.
Now let X be a complex of finite total dimension over k, so that X is strongly
dualizable in K(B). We now show that LX is strongly dualizable in C(B). The
dual of LX in C(B) is
DC(B)(LX) = F (LX; L) ' F (X; L) = F (X; k) L:
More generally, for any U 2 C(B) we have
F (LX; U) ' F (X; U) = F (X; k) U:
On the other hand, L U ' U so
DC(B)(LX) U ' F (X; k) L U ' F (X; k) U ' F (LX; U):
Thus LX is strongly dualizable as claimed. In particular, this applies to LM wh*
*en
M is a simple comodule.
Now we must show that injective resolutions of simple comodules form a set of
weak generators for C(B). Suppose that
X = (: : :dn-1---Xn-1 -dn-Xn dn+1---:): :
is a complex of injectives, and [LM; X]* = [M; X]* = 0 for every simple comodule
M. Let ZXn denote the cocycles in dimension n and let BXn denote the boundaries
in dimension n. The exact sequence
0 -!ZXn+1 -i!Xn+1 p-!BXn -! 0
(where p is just the coboundary d) gives rise to an exact sequence
0 -!Hom B (M; ZXn+1) f-!Hom B(M; Xn+1)
g-!Hom 1
B (M; BXn) -!Ext B(M; ZXn+1 ) -!0
since Xn is injective. On the other hand, any map M -ff!ZXn is a chain map
from M (as a complex in degree 0) to X of degree n. Thus, since [M; X]* = 0, it
must be null-homotopic. That is, there must be a lift of ff to fi :M -! Xn-1. T*
*he
composite Hom B(M; Xn+1) -g!Hom B(M; BXn) -h!Hom B (M; ZXn) is therefore
surjective. Since h is monic, it follows that h is an isomorphism and g is surj*
*ective.
In particular, Ext1B(M; ZXn+1 ) = 0 for every simple comodule M.
Lemma 9.5.7 then shows that ZXn-1 is injective. There are then splittings
r :Xn-1 -! ZXn-1 of i and q :BXn -! Xn-1 of p. In particular, BXn is also
injective. By considering the exact sequence
0 -!BXn -! ZXn -! HnX -! 0
94 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
we find an exact sequence
0 -!Hom B (M; BXn) h-!Hom B(M; ZXn) -!Hom B (M; HnX) -!0:
Since h is an isomorphism for any simple comodule M, we find that
Hom B(M; HnX) = 0
for every such M. By Lemma 9.5.6, HnX = 0, so ZXn = BXn.
We can now define a chain homotopy D :Xn -! Xn-1 by the composite Xn -r!
ZXn ' BXn -q!Xn-1. It is then easy to see that dD + Dd is the identity of X, so
X is a contractible chain complex.
It follows that C(B) is a unital algebraic stable homotopy category, with
G = {LM | M is a simple comodule}:
If k is the only simple comodule, then C(B) is monogenic. It is well known that
this is the case when G is a p-group with p = char(k), and B = (kG)*. Similarly*
* if
B is graded and connected.
Let M and N be comodules. Then LN is a complex of injectives with homology
H*(LN) = H*(L)N = N, in other words an injective resolution of N. Moreover,
we know that [LM; LN]* = [M; LN]*. It follows that
[LM; LN]* = Ext*B(M; N)
as claimed.
Finally, suppose we have a map f :B -! B0of Hopf algebras. Given a complex of
injective B-comodules, we can think of it as a complex of B0-comodules through *
*f.
We then apply L to get a complex of injective B0-comodules. This gives a functor
C(B) -! C(B0), which we claim is a stable morphism; we leave the details_to the
reader. |__|
We still owe the reader Lemmas 9.5.6 and 9.5.7.
Lemma 9.5.6. The set {M} of simple comodules weakly generates Comod (B). In
other words, for every nonzero comodule N, there is an inclusion M ,! N of a
simple comodule into N.
Proof.By Lemma 9.5.3, every B-comodule N 6= 0 has a finite-dimensional subco-
module N0 6= 0. Clearly every finite-dimensional B-comodule N0 6= 0 has a simpl*
*e_
sub-comodule (by induction on dimension). |__|
Lemma 9.5.7. Suppose that J is a comodule such that Ext1B(M; J) = 0 for all
simple comodules M. Then J is injective.
Proof.We first note that Ext1B(F; J) = 0 for all finite-dimensional (over k) co*
*mod-
ules F . Indeed, we prove this by induction on the dimension. Any comodule of
dimension 1 is simple. Given a nonzero finite-dimensional comodule F , there is*
* a
nonzero simple comodule M F . By considering the short exact sequence
0 -!M -! F -! F=M -! 0
we find an exact sequence
Ext 1B(F=M ; J) -!Ext 1B(F; J) -!Ext 1B(M; J)
and so, by induction, Ext1B(F; J) = 0.
Now, suppose that we have an arbitrary inclusion of comodules M N, and
a map f :M -! J. We must extend f to N. Consider the set of pairs (P; g)
AXIOMATIC STABLE HOMOTOPY THEORY 95
where M P N and g :P -! J is an extension of f, ordered in the evident
way. By applying Zorn's lemma, we find a maximal such extension (N0; g0). We
claim that N0 = N. Indeed, suppose not. Then choose an element n 2 N but
not in N0. Let N00denote the subcomodule generated by N0 and n. Then N00=N0
is generated by n, and hence, by Lemma 9.5.3, N00=N is finite-dimensional. Thus,
Ext1B(N00=N0; J) = 0, and so the map Hom B(N00; J) -!Hom B (N0; J) is onto. Thus
there is an extension of g0to N00, violating the fact that (N0; g0) is maximal.*
*_Hence
N0 = N, as required. |__|
Note that the homotopy groups in C(B) of a complex of injectives X are
ss*X = [L; X]* = H*(P X);
where P denotes the primitive functor. If k is the only simple comodule, as will
occur for example when B is graded connected, then H*(P X) = 0 , X = 0.
As usual, C(B) is the homotopy category of a closed model structure on the
Abelian category of chain complexes of comodules. We will describe the structure
without giving proofs. The cofibrations are dimensionwise inclusions, and the f*
*ibra-
tions are dimensionwise surjections with kernel a complex of injectives. The we*
*ak
equivalences are generalized homotopy isomorphisms_that is, a map f :X -! Y is
a weak equivalence if and only if it induces isomorphisms [Z; L ^ X] -! [Z; L ^*
* Y ]
for all finite-dimensional comodules Z.
Remark 9.5.8. Suppose that A is a finite-dimensional graded connected cocom-
mutative Hopf algebra over a field k; suppose also that A is a Koszul algebra [*
*Pri70,
BGS ] with Koszul dual A!. Then by [BGS , Theorem 16] there is an equivalence of
triangulated categories between FC(A*)and FD(A!). Therefore we have a classific*
*a-
tion of the thick subcategories of these, by Example 6.1.4. Note that this agre*
*es
with the classification of thick subcategories of FC((kG)*), for G a p-group.
9.6. The stable category of B-modules. In this section, we let B be a finite-
dimensional commutative Hopf algebra over a field k, and we study comodules over
B. It would be equivalent to consider modules over a finite-dimensional cocommu-
tative Hopf algebra, in view of the following result.
Proposition 9.6.1.There is an isomorphism (which is the identity on objects)
between the categories of B-comodules and B*-modules.
Proof.Any B*-comodule is clearly tame (Definition 9.5.2), given that B* has fin*
*ite_
dimension. The claim therefore follows from part (b) of Lemma 9.5.3. |_*
*_|
We can now define the stable categories which we wish to study.
Definition 9.6.2.Given two comodules M and N over B, write
Hom B(M; N)0 = {f 2 Hom B(M; N) | f factors through an injective comodule}:
This is a subspace of Hom B(M; N), so we can define
Hom__B(M; N) = Hom B(M; N)= Hom B(M; N)0:
The composition map Hom B(L; M) Hom B(M; N) -! Hom B(L; N) descends to
give a well-defined composition
Hom__B(L; M) Hom__B(M; N) -!Hom__B(L; N):
We can therefore define a category StComod (B) whose objects are B-comodules,
and whose morphisms are the sets Hom__B(M; N).
96 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
In this section we prove the following theorems, and we also make a few remar*
*ks
about classifying thick subcategories and detecting nilpotence.
Theorem 9.6.3. Suppose that B is a finite-dimensional commutative Hopf algebra
over a field k; then StComod (B) is a unital algebraic stable homotopy category*
*. If
k is countable then it is a Brown category. If k is the only simple B-comodule,*
* then
StComod (B)is monogenic.
It turns out that StComod (B) is equivalent to a Bousfield localization (in f*
*act,
a finite localization_Definition 3.3.4) of C(B). Because of this, there is a st*
*rong
relation between the two categories; see Lemma 3.5.6 and Proposition 9.6.8, for
example.
Theorem 9.6.4. Let B be a finite-dimensional commutative Hopf algebra over a
field k. Let LfB:C(B) -! C(B) denote finite localization with respect to the th*
*ick
subcategory generated by B (as a complex of injectives concentrated in degree 0*
*).
Then we have an equivalence of stable homotopy categories
LfBC(B) ' StComod (B):
Before proving Theorems 9.6.3 and 9.6.4, we need three lemmas.
Lemma 9.6.5. There is an isomorphism B ' Hom (B; k) of B-comodules. A co-
module is projective if and only if it is injective.
Proof.See [LS69, p. 85]. |___|
Lemma 9.6.6. A chain complex X in C(B) is LfB-local if and only if H*(X) = 0.
Proof.X is LfB-local if and only if [B; X]* = 0. But H*(X) = ss*(B ^ X) = __
[DB; X]* = [B; X]*, since B is self-dual. |__|
Recall that L denotes the cobar resolution of the ground field k (which has f*
*inite
dimension in each degree). Given any comodule M, we have both an injective
resolution
M -! L M
and a projective resolution
Hom (L; k) M = Hom (L; M) -!M:
We can then splice these resolutions together to form the Tate complex tB (M):
: :-:!Hom (L-1; M) -!Hom (L0; M) -!L0 M -! L-1 M -! : :::
Here L0 M is in degree zero, so Hom (L-k; M) is in degree k + 1.
Because projectives and injectives are the same, tB (M) is an element of C(B).
Since projective and injective resolutions of M are unique up to chain equivale*
*nce,
tB (M) is independent of our choice of L. Since tB (M) clearly has no homology,*
* it
is LfB-local by the preceding lemma.
Lemma 9.6.7. For any comodule M, tB (M) is the LfB-localization of LM.
Proof.There is certainly a map LM -f!tB (M), and we have already seen that
tB (M) is LfB-local. The cofiber of f is the projective resolution Hom (L; M) o*
*f M,
and it suffices to show that Hom (L; M) is in the localizing subcategory D = lo*
*c~~**
AXIOMATIC STABLE HOMOTOPY THEORY 97
generated by B. We prove more generally that any bounded below complex of
injectives lies in D.
First consider an injective comodule J thought of as a complex concentrated in
a single degree. Then J is a retract of a coproduct of copies of B, so it lies *
*in D.
Next, let I be a bounded-below complex of injectives. Let I(k) be the truncat*
*ed
complex
: :-:!0 -!Ik -!Ik-1 -!: :::
Note that I(k) = 0 for k 0, and there are cofibrations I(k) -! I(k + 1) -!
I(k + 1)=I(k). By the previous paragraph, I(k + 1)=I(k) 2 D, so I(k) 2 D for al*
*l k.
Let I(1) be the sequential colimit of the I(k)'s, so that we have a map I(1) -!I
with cofiber I0 say. It is not hard to see that I0 is a bounded below complex *
*of
injectives with H*I0= 0, so I0 is contractible and I(1) ' I. It follows_that_I *
*2 D
as claimed. |__|
We now prove Theorems 9.6.3 and 9.6.4 simultaneously.
Proof of Theorems 9.6.3 and 9.6.4.The strategy of the proof is first to define *
*struc-
tures on StComod (B), then to construct an equivalence of categories between
LfBC(B) and StComod (B)that preserves these structures. Since LfBis smashing, as
finite localizations always are, this shows that with these structures StComod *
*(B)is
a stable homotopy category. We leave it to the reader to check that the structu*
*res
we define on StComod (B)are equivalent to similar structures on StMod (B* ).
We begin by defining the standard structures on StComod (B)that will make it
into a stable homotopy category. First, the coproduct of stable comodules is the
same as the coproduct of comodules. The suspension of a stable comodule M is
defined by embedding M into the injective module B M and taking the cokernel.
A cofiber sequence in StComod (B)is any sequence isomorphic to
M -f!N -g!P -h!M
where
0 -!M -f!N -g!P -! 0
is a short exact sequence of comodules, and where h is any map such that the
following diagram commutes:
0 ----! M ----! B M ----! M ----! 0
? ? ?
=?y ?y -h?y
0 ----! M --f--! N --g--! P ----! 0:
The smash product of two stable comodules is just the tensor product M N
and the function object is just Hom k(M; N). (Since B is finite, the subtleties*
* in
Lemma 9.5.3 do not arise.) We leave it to the reader to check that these do def*
*ine
functors in the stable category and that they remain adjoint.
We still have to prove that with these structures and with generating set con*
*sist-
ing of the simple comodules, StComod (B) is a stable homotopy category. Rather
than proving this directly, we will construct an equivalence of categories betw*
*een
StComod (B) and LfBC(B) that preserves coproducts, cofiber sequences, smash
98 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
products, and function objects. Since LfBis a finite localization, it is in pa*
*rtic-
ular smashing, so LfBC(B) is a unital algebraic stable homotopy category, and t*
*he
result follows.
We begin by considering the functor G: Comod (B) -! C(B) that takes M to
L M. This functor preserves coproducts and takes the tensor product to the
smash product. Also, if
0 -!M -f!N -g!P -! 0
is a short exact sequence, then the cofiber of G(f) is a complex of injectives *
*begin-
ning in degree 0 whose only homology group is P , so it is equivalent to G(P ).
The composite functor LfBG is equivalent to tB by Lemma 9.6.7. Since any
injective module lies in loc****, the functor tB factors through the stable cate*
*gory
to give a functor
tB :StComod (B)-! LfBC(B):
Since LfBpreserves the coproduct and the smash product, so does tB . By consid-
ering the short exact sequence
0 -!M -f!B M -! M -! 0
we find that the cofiber of tB (f) is tB (M). But tB (B M) is trivial, so this
cofiber is tB (M). Thus tB preserves the suspension, and it is then easy to see
that it preserves cofiber sequences as well.
We now construct an inverse to tB . Consider an object X = (: :-:!X1 -!
X0 -! X-1 -! : :):which is LfB-local, i.e., which is an acyclic chain complex of
injectives. Define
u(X) = ker(X0 -!X-1) = image(X1 -!X0):
This is clearly functorial for chain maps. Moreover, if f :X -! Y is null-homot*
*opic
then u(f) factors through the injective comodule Y1. It follows that u gives a *
*functor
LfBC(B) -!StComod (B). It is easy to check that tB and u define an equivalence *
*of
categories. It follows by adjointness that tB preserves function objects, so in*
*_fact
tB is an equivalence of stable homotopy categories. |__|
The proof of Theorem 9.6.4 makes it clear that the homotopy of S in the stable
module category of kG is just the Tate cohomology of G, hence the notation tB .
For a general commutative Hopf algebra B, the category of complexes of injectiv*
*es
with no homology will not in general be colocalizing, so there is no way one co*
*uld
perform Bousfield localization and land in it. On the other hand, one can always
perform Bousfield localization LH with respect to the ordinary homology functor*
* H
on C(B). Denote the fiber of the localization map X -! LH X by CH X. Then one
could define the Tate cohomology of a general Hopf algebra B to be [CH S; CH S].
This agrees with the usual Tate cohomology of a finite group. Mislin [Mis] has
recently given an extension of Tate cohomology to arbitrary groups. We have not
checked whether his definition agrees with ours.
Theorem 9.6.4 implies that there is a strong tie between the two categories C*
*(B)
and StComod (B). Here is one example, motivated by Rickard's classification [Ri*
*c]
of thick subcategories of small objects in StMod (kG ), for G a p-group.
Proposition 9.6.8.There is a one-to-one correspondence between the thick sub-
categories of FStComod(B)and the nonzero thick subcategories of FC(B).
AXIOMATIC STABLE HOMOTOPY THEORY 99
Before proving this proposition, we need a lemma.
Lemma 9.6.9. A comodule M is small in StComod (B) if and only if M is iso-
morphic in StComod (B) to a finite-dimensional comodule.
Proof.First suppose M is a finite-dimensional comodule. Then tB (M) = LfBLM is
small in LfBC(B), since LM is small in C(B). Thus M is small in StComod (B). Co*
*n-
versely, any comodule M which is small in StComod (B)is in the thick subcategory
generated by the simple comodules, since StComod (B) is a unital algebraic stab*
*le
homotopy category. Any simple comodule is finite-dimensional, and the property *
*of
being isomorphic to a finite-dimensional comodule is preserved under suspension*
*s,
cofibrations, and retracts. The only one of these claims that is not immediate*
*ly
clear is the closure under retracts. To see this, we use [Mar83 , Proposition 1*
*3.13]
(and the comments immediately following it) to write any comodule M uniquely as
I M0, where I is injective and M0 has no injective summands. Of course, M and
M0 are isomorphic in StComod (B), and conversely, if M and N are isomorphic in
StComod (B), then M0 and N0 are isomorphic as comodules [Mar83 , Proposition
14.1]. Thus, if N is a retract of M in StComod (B), then N0 is a retract of M0
as comodules. In particular, if M is isomorphic to a finite-dimensional comodul*
*e,_
then M0 must be finite-dimensional, and so N0 is as well. |_*
*_|
Proof of Proposition 9.6.8.Let L = LfB, so that StComod (B) is equivalent to
C(B)L. We claim that there is a correspondence
ae oe ae oe
nonzero thick subcats thick subcats
in FC(B) ! in FC(B)L
D 7-! thick
L-1D0\ FC(B) - | D0
To see that these maps give a one-to-one correspondence, note first that the pr*
*e-
ceding lemma shows that LfB:FC(B)-! FStComod(B)is surjective on objects. From
this, it is straightforward to check that, if D0is a thick subcategory of FStCo*
*mod(B),
then thick = D0. __
Conversely, suppose that D is a thick subcategory of FC(B), and let D = L-1LD
be the L-replete_thick subcategory generated by D. By Lemma 3.5.6 it suffices to
show that D \ FC(B)= D.
For all X 2 D, we have LfDX = 0. Note that if Y -f!Z is an L-equivalence,
then it is also an LfD-equivalence; it follows that the full_category of all ob*
*jects X
satisfying LfDX = 0 is thick_and_L-replete, so it contains D . By Theorem 3.3.3,
then, if X is finite and in D, then X 2 D. |___|
We finish by pointing out a familiar result which is an application of the fa*
*ct
that StMod (kG ) is a stable homotopy category.
Theorem 9.6.10 ([QV72 , Car81]).Fix a finite group G, a field k of characterist*
*ic
p > 0, and a finitely-generated kG-module M. Then an element z 2 Ext*kG(M; M)
is nilpotent if and only if resG;E(z) 2 Ext*kE(M; M) is nilpotent for every ele*
*men-
tary Abelian p-subgroup E of G.
Proof.Chouinard's theorem [Cho76 ] says that a kG-module M is projective (i.e.,
zero in StMod(kG )) if and only if M#E is a projective kE-module for all elemen*
*tary
100 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Abelian p-subgroups E of G (here, M#E denotes M restricted to kE). Now, one
can easily show that M#E is a projective kE-module if and only if k[G=E]*M is a
projective kG-module. In other words, we have the following equality of Bousfie*
*ld
classes:
a
**~~ = :
EG
elem.ab.
Hence Theorem 5.1.2 applies. On the other hand, these modules k[G=E]* represent
the homology functors
M 7! Ext*kE(k; M):
This finishes the proof. |___|
Remark 9.6.11.
(a) One can also give a proof of Chouinard's theorem in the language of stable
homotopy theory. Of course one needs to use Serre's characterization of el-
ementary Abelian p-groups via products of Bocksteins [Ser65]; given that,
however, it is straightforward.
(b) We have shown that Theorem 9.6.10 is a formal consequence of Chouinard's
theorem. This is perhaps different from the usual point of view on this re*
*sult.
(c) A similar result (with the same proof) holds in the category of chain comp*
*lexes
of projective kG-modules_one can imitate the arguments in Section 9.5 to
show that this is a stable homotopy category. We leave the details to the
reader. If G is a p-group, then this category is monogenic; so since Ext*k*
*G(k; k)
is Noetherian, then the results of Section 6 apply, giving a different nil*
*potence
theorem in this setting.
(d) One can also use the same arguments to recover the nilpotence theorem for a
finite-dimensional cocommutative Hopf algebra B*_see [Wil81, Palb]. Again,
one can work either in StMod (B* ) or C(B* ).
10.Future directions
We close the paper by briefly discussing some topics we feel merit further st*
*udy.
One such topic is the Adams spectral sequence. Given a ring object E in the
ordinary stable homotopy category, we can construct a spectral sequence that at-
tempts to calculate [X; Y ] in terms of E-homology information [Ada74 ]. One ne*
*eds
some hypotheses on E to determine the E2-term of this spectral sequence, and
some hypotheses on E, X, and Y to guarantee convergence. This construction of
the Adams spectral sequence will certainly work in a monogenic Brown category,
but one might hope to be able to construct it more generally. Convergence will *
*cer-
tainly be delicate, as it is even with spectra. One might hope for a constructi*
*on in
an algebraic stable homotopy category, and for some convergence results analogo*
*us
to those in [Bou79b ], but we have not investigated this question. It is certai*
*nly an
important one. It also seems likely that many familiar spectral sequences (such*
* as
the Cartan-Eilenberg spectral sequence associated to an extension of Hopf algeb*
*ras)
can be presented in these terms.
AXIOMATIC STABLE HOMOTOPY THEORY 101
10.1. Grading systems on stable homotopy categories. There are times
when we would like to allow a stable homotopy category to have a more gen-
eral grading than the Z-grading enjoyed by any triangulated category, or even t*
*han
the multigrading discussed in Section 1.3. In this section we briefly discuss *
*such
grading systems. The basic examples are the possibility of grading over the Pic*
*ard
group (Definition A.2.7), and, in the setting of G-equivariant stable homotopy *
*the-
ory, grading so that [X; Y ]* is a Z-graded Mackey functor. The main advantage
of this more complicated grading is that one can make any unital algebraic sta-
ble homotopy category monogenic, if one is willing to modify the grading of the
category.
Suppose that we have a stable homotopy category C. Consider (Z; +) as a
symmetric monoidal category with the only maps being the identity map 1m and
(-1)m on each object m, with the evident composition. The symmetric monoidal
structure is just addition on the objects and multiplication on the maps. It is
strictly associative and unital, but not strictly commutative. The commutativity
natural transformation is (-1)mn on m ^ n.
Recall that a strict symmetric monoidal functor F :C -! D between symmetric
monoidal categories is a functor equipped with natural isomorphisms F X ^ F Y '
F (X ^ Y ) and F S ' S, which are compatible in the evident sense with the com-
mutativity, associativity and unity maps in C and D. We shall simply refer to s*
*uch
a thing as a monoidal functor.
If C is an enriched triangulated category, then we can define a monoidal func*
*tor
(Z; +) S-!C, which takes m to Sm .
Definition 10.1.1.Let C be an enriched triangulated category.
(a) A pointed symmetric monoidal category is a symmetric monoidal category
C equipped with a commutative monoidal functor (Z; +) -! C. There is an
evident notion of a pointed functor of pointed symmetric monoidal categori*
*es.
(b) A grading system on an closed symmetric monoidal triangulated category C
is a small pointed symmetric monoidal category I and a pointed commutative
monoidal functor G: I -!C such that Ga is strongly dualizable for all a 2 *
*I.
(c) Given a grading system G on an closed symmetric monoidal triangulated
category, we write Sa for Ga, and we write aX for Sa ^ X. Furthermore,
we define [X; Y ]a = [aX; Y ]. If H is a homology functor, we define HaX =
H(DSa^X) and if H is a cohomology functor we define HaX = H(DSa^X).
(d) Given a grading system G on an closed symmetric monoidal triangulated
category, we modify the definitions of thick, localizing, and colocalizing*
* sub-
categories so as to require them to be closed under smashing with all the *
*Sa,
which we think of as a generalized suspension.
(e) An I-graded stable homotopy category is a category C equipped with
(i)Arbitrary coproducts
(ii)A triangulation
(iii)A closed symmetric monoidal structure compatible with the triangulation
(iv)A grading system G: I -!C such that the localizing subcategory (in the
above sense) generated by S is all of C.
such that all cohomology functors are representable.
Example 10.1.2. (a)Every closed symmetric monoidal triangulated category
admits a grading system where I = Z.
102 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
(b) If C is a unital algebraic stable homotopy category, then there is always a
choice of grading system on C so that C becomes an I-graded stable homotopy
category. Indeed, we can take for I a small equivalent subcategory of F and
for G the inclusion functor.
(c) If the Picard category (see Definition A.2.7) of an closed symmetric monoi*
*dal
triangulated category C is small, then the inclusion functor Pic-! C defin*
*es
a grading system on C. Grading over the Picard group is not always possibl*
*e,
but can be done if and only if one can choose a monoidal section of the map
from the Picard category to the Picard group. In particular, one can always
grade over free Abelian subgroups of the Picard group. A bigraded category
is an example of this, as is the RO(G) grading on the homotopy category
of G-spectra defined over a complete universe [LMS86 ]. Note, however, that
the category of G-spectra is not an RO(G)-graded stable homotopy category,
because the localizing subcategory generated by the "representation sphere*
*s"
is not usually the whole category.
(e) Let G be a finite group. The homotopy category of G-spectra (based on a
complete universe) admits a grading system over (Z; +) x L where L is the
Lindner category of finite G-sets. Here objects are finite G-sets, but map*
*s are
arbitrary maps of G-sets together with transfers. The symmetric monoidal
structure is given by the product. In this case [X; Y ]*, as a contravari*
*ant
functor from (Z; +)xL to Abelian groups, is a graded Mackey functor, and t*
*he
category of G-spectra becomes a (Z; +)xL-graded stable homotopy category.
See [LMS86 ] for details.
(f) In any I-graded stable homotopy category, there is a symmetric monoidal
structure on the category of contravariant functors from I to Ab so that s*
*s*S
becomes a commutative ring object. In the equivariant case, one recovers t*
*he
notion of Green functors.
The approach outlined here is a useful one in equivariant stable homotopy the*
*ory,
but we do not know if it is useful in other stable homotopy categories.
10.2. Other examples. To conclude the paper, we provide a partial list of coho-
mology theories that we think are associated with stable homotopy categories. We
have not considered these, for reasons of space, time, and energy.
(a) Ext of modules over a commutative S-algebra in the sense of [EKMM95 ],
possibly in a G-equivariant setting. These certainly form a stable homotopy
category, but we have not studied any examples in depth.
(b) Ext of differential graded modules over a commutative differential graded
algebra. This example should be very similar to the derived category of an
ordinary ring. It is likely that the Adams spectral sequence in this categ*
*ory
is the spectral sequence of Eilenberg-Moore that goes from ordinary Ext to
differential Ext.
(c) Extof sheaves of OX -modules, where X is a scheme.
(d) Continuous cohomology of a topological group, especially of a profinite gr*
*oup.
(e) Extof comodules over a commutative Hopf algebroid, such as BP*BP .
(f) Extof Mackey functors which are modules over a Green functor. This example
might help one to understand what to expect for a thick subcategory theorem
in equivariant stable homotopy theory.
(g) Motivic cohomology in any of its various guises.
AXIOMATIC STABLE HOMOTOPY THEORY 103
We have also considered generalizing the definition of a stable homotopy cate*
*gory
to allow for a non-commutative smash product. This would cover the derived
category of bimodules over a non-commutative ring, for example. One might hope
to be able to put the following things in this context:
(a) Cyclic cohomology.
(b) Hochschild cohomology.
(c) Extof bimodules over a noncommutative S-algebra.
It is possible that Hochschild cohomology should be thought of in the same
way as Tate cohomology of spectra [GM95 ]. That is, given an algebra A over
a commutative ring B, we should construct the Hochschild cohomology of A as
a ring object in the derived category of B-modules. We do not yet understand
the relationship between this approach and the idea of making a stable homotopy
category of bimodules.
Appendix A. Background from category theory
A.1. Triangulated categories. We recall the notion of a triangulated category.
We give the definition from [Mar83 ]; this definition is equivalent to the more*
* usual
one as in [Ver77], and see [Nee92b] for yet another set of axioms.
Definition A.1.1.A triangulation on an additive category C is an additive (sus-
pension) functor : C -!C giving an automorphism of C, together with a collection
4 of diagrams, called exact triangles or cofiber sequences, of the form
X -! Y -! Z -! X
such that
1. Any diagram isomorphic to a diagram in 4 is in 4 .
2. Any diagram of the following form is in 4 :
0 -!X -1!X -! 0
3. If the first of the following diagrams is in 4 , then so is the second:
X -f!Y -g!Z -h!X
Y -g!Z -h!X --f--!Y:
4. For any map f :X -! Y , there is a diagram of the following form in 4 .
X -f!Y -! Z -! X
5. Suppose we have a diagram as shown below (with h missing), in which the
rows lie in 4 and the rectangles commute. Then there exists a (nonunique)
map h making the whole diagram commutative.
U________-V ________-W ________U-
| | || |
| | | |
|f | |h |f
| | | |
|? |? ||? |?
X ________Y-________-Z ________X-
104 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
6. Verdier's octahedral axiom holds: Suppose we have maps X -v!Y -u!Z, and
cofiber triangles (X; Y; U), (X; Z; V ) and (Y; Z; W ) as shown in the dia*
*gram.
(A circled arrow U -!O X means a map U -! X.) Then there exist maps
r and s as shown, making (U; V; W ) into a cofiber triangle, such that the
following commutativities hold:
au = rd es = (v)b sa = f br = c
_____-V ______
J
J] J
b c J a J
r J J s
AE J J
X ___________-Zuv
J
OE J OE J J
c c vJ u J f JJ
J J J
J J^ J^ AE
U ___________oeY___________oecW
d e
(If u and v are inclusions of CW spectra, this essentially just says that
(Z=X)=(Y=X) = Z=Y . The diagram can be turned into an octahedron by
lifting the outer vertices and drawing an extra line from W to U.)
A category equipped with a triangulation is called a triangulated category. G*
*iven
an exact sequence -1Z -! X -f!Y -! Z, we say that Z is the cofiber and -1Z
the fiber of f. The cofiber of f is only determined up to unnatural isomorphism*
* in
the category of objects under Y and over X. If Z is the cofiber of f :X -! Y ,
we will often abuse notation by referring to the sequence X -f!Y -! Z as a cofi*
*ber
sequence.
An exact functor between triangulated categories is a functor L which is equi*
*pped
with an equivalence L ' L, and which preserves cofiber sequences. More pre-
cisely, suppose that X -! Y -! Z -! X is a cofiber sequence. We can apply L
and use the given equivalence LX = LX to get a sequence LX -! LY -! LZ -!
LX. The requirement is that this should again be a cofiber sequence.
A natural transformation of exact functors is required to commute with the gi*
*ven
suspension equivalences in the obvious sense.
We will use a number of well-known properties of triangulated categories with-
out proof, and usually without explicit mention; see [Mar83 ] and [Ver77] for m*
*ore
complete references. In particular, in any triangulated category, coproducts a*
*nd
products, when they exist, preserve cofiber sequences.
We will state the (perhaps poorly named) 3 x 3 lemma here, however. A proof
can be found in [BBD82 ] and an interesting discussion of related matters can be
found in [Nee92b].
AXIOMATIC STABLE HOMOTOPY THEORY 105
Lemma A.1.2 (3 x 3 lemma). Let C be a triangulated category. Consider a com-
mutative square as shown, in which the rows and columns are cofiber sequences.
-1X0 -1Y 0
?? ?
y ?y
-1Z00 ----! X00 - ---! Y 00 ----! Z00
?? ?
y ?y
-1Z ----! X - ---! Y ----! Z
?? ?
y ?y
X0 Y 0
Then there exists an object Z0 and maps Y 0-! Z0- Z, such that the following
diagram commutes (except that the top left square anticommutes) and the rows and
columns are exact.
-2Z0 - ---! -1X0 ----! -1Y 0 ----! -1Z0
?? ? ? ?
y ?y ?y ?y
-1Z00 - ---! X00 ----! Y 00 ----! Z00
?? ? ? ?
y ?y ?y ?y
-1Z - ---! X ----! Y ----! Z
?? ? ? ?
y ?y ?y ?y
-1Z0 - ---! X0 ----! Y 0 ----! Z0
It is understood that the map X0 -! Y 0is the suspension of the map -1X0 -!
-1Y 0, and so on.
A.2. Closed symmetric monoidal categories.
Definition A.2.1.A closed symmetric monoidal category is a category C equipped
with:
1. A unit object S.
2. A functor (X; Y ) 7! X ^ Y from C x C to C, which is associative and com-
mutative up to coherent natural isomorphism, such that S ^ X = X up to
coherent natural isomorphism. We shall call this functor the smash product,
by analogy with the category of spectra.
3. Function objects F (X; Y ), which are functorial contravariantly in X and *
*co-
variantly in Y , such that [X; F (Y; Z)] ' [X ^ Y; Z], naturally in all th*
*ree
variables.
We shall say that this structure is compatible with a given triangulation on *
*C if:
1. The smash product preserves suspensions. That is, there is a natural equiv-
alence eX;Y :X ^ Y -! (X ^ Y ). Furthermore, if we let rX denote the
unital equivalence X ^S -! X, then we have rX OeX;S = rX , and if aX;Y;Z
denotes the associativity isomorphism (X ^ Y ) ^ Z -! X ^ (Y ^ Z), then the
106 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
following diagram commutes:
(X ^ Y ) ^ Z
j3 Q
eX;Y ^ 1Zj j Q Q eX^Y;Z
j j Q Q
j j QQs
(X ^ Y ) ^ Z ((X ^ Y ) ^ Z)
B
B
B
aX;Y;Z B aX;Y;Z
B
B
BBN fl
X ^ (Y ^ Z) _________(X-^ (Y ^ Z))
eX;Y ^Z
One can easily construct from e isomorphisms F (X; Y ) ' -1F (X; Y ) and
F (X; Y ) ' F (X; Y ).
2. The smash product is exact. More precisely, suppose that X -f!Y -g!Z -h!
X is an exact triangle, and that W is an object of C. If we use eX;W to
identify (X) ^ W with (X ^ W ), then the following triangle is required to
be exact:
X ^ W -f^1-!Y ^ W -g^1-!Z ^ W -h^1-!(X ^ W )
3. The functor F (X; Y ) is exact in the second variable in a similar sense, *
*and is
exact in the first variable up to sign. That is, suppose X -f!Y -g!Z -h!X
is an exact triangle, and that W is an object of C. If we use the adjoint *
*of e to
identify F (X; W ) with -1F (X; W ), then the following triangle is requir*
*ed
to be exact:
-1F (X; W ) -F(h;1)-----!F (Z; W ) F(g;1)----!F (Y; W ) F(f;1)----!F (X;*
* W )
4. The smash product interacts with the suspension in a graded-commutative
manner. That is, the following diagram is commutative for all integers r
and s, where T is the twist map (i.e., the commutativity equivalence for t*
*he
smash product), Sr = rS, and the horizontal equivalences come from the
equivalence e above together with the symmetric monoidal structure.
Sr ^ Ss --'--! Sr+s
? ?
T?y ?y(-1)rs
Ss ^ Sr --'--! Sr+s
We really need only require that the last diagram commute with r = s = 1;
it then follows that each transposition in the symmetric group n acts as -1 on
Sn = S1 ^ : :^:S1, so every permutation acts as its signature. One can deduce
from this that the diagram commutes for all r and s.
AXIOMATIC STABLE HOMOTOPY THEORY 107
There are a number of theorems which say approximately the following: if dia-
grams of a certain type commute in the category of (possibly infinite-dimension*
*al)
vector spaces over C, then they commute in any closed symmetric monoidal cate-
gory. See for example [Sol95].
One way to interpret the sign that prevents F (X; Y ) from being exact in the*
* first
variable is that mapping out of a cofiber sequence should produce a fiber seque*
*nce,
not a cofiber sequence, and in the stable case, cofiber sequences differ from f*
*iber
sequences only by a sign. This sign can usually be ignored, since most of our
arguments do not rely on the features of the maps in an exact triangle, but only
on the existence of the exact triangle.
Remark A.2.2. Note that, in a closed symmetric monoidal category C, the smash
product is always compatible`with coproducts. That is, given a family {Xi} of
objects`of C such that Xiexists, and`given another`object Y of C, the coprodu*
*ct
(Xi^Y ) exists, and the natural map (Xi^Y ) -!( Xi)^Y is an isomorphism.
Indeed, we have, for any object Z of C,
a a Y Y
[( Xi) ^ Y; Z] = [ Xi; F (Y; Z)] = [Xi; F (Y; Z)] = [Xi^ Y; Z];
as required.
Proposition A.2.3. Suppose that C is a closed symmetric monoidal category.
(a) There is an associative composition map
F (X; Y ) ^ F (Y; Z) O-!F (X; Z):
For all X, there is a map S -j!F (X; X) which is a two-sided unit for the
composition. This makes C a category enriched over itself, as in [Kel82].
(b) Both the smash product and the function object functor are (canonically) e*
*n-
riched functors, and they are adjoint as enriched functors. That is, there*
* is a
natural isomorphism
F (X; F (Y; Z)) ' F (X ^ Y; Z):
(c) Coproducts and products in C are enriched coproducts and products as well.
That is, we have equivalences
a Y
F ( Xff; Y ) -! F (Xff; Y )
and
Y Y
F (X; Yff) -! F (X; Yff):
Proof.The unit map S -! F (X; X) is adjoint to the unit equivalence S ^ X -! X.
Because of the adjunction, we have evaluation maps
F (X; Y ) ^ X -ev!Y
and we use these to define the composition map as the adjoint of the composite
F (X; Y ) ^ X ^ F (Y; Z) ev^1---!Y ^ F (Y; Z) ev-!Z:
We leave it to the reader to check that this composition is associative and uni*
*tal.
To say that the smash product is an enriched functor means that we have a natur*
*al
map
F (X; X0) ^ F (Y; Y 0) -!F (X ^ Y; X0^ Y 0)
108 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
compatible with the composition and the unit. To construct this map, we take the
adjoint to the map
F (X; X0) ^ X ^ F (Y; Y 0) ^ Y -ev^ev--!X0^ Y 0:
We leave it to the reader to check that this is compatible with composition and*
* the
unit and to construct analogous maps for the function object functor.
One way to see that F (X ^ Y; Z) is naturally equivalent to F (X; F (Y; Z)) i*
*s to
show that they represent the same functor. That is, we have
[W; F (X ^ Y; Z)] = [W ^ X ^ Y; Z] = [W ^ X; F (Y; Z)] = [W; F (X; F (Y; Z))]:
One can use a similar method to show that coproducts and products behave_as
expected. |__|
Recall from [LMS86 , Chapter III] the definition of strongly dualizable objec*
*ts in
a closed symmetric monoidal triangulated category: Z is strongly dualizable if *
*the
natural map
F (Z; S) ^ X -! F (Z; X)
is an isomorphism for all X. We can now see that this natural map is nothing
more than composition, if we interpret X as F (S; X). This motivates the follow*
*ing
definition.
Definition A.2.4.In a closed symmetric monoidal category C, we denote F (X; S)
by DX and refer to it as the Spanier-Whitehead dual of X. Note that D is an exa*
*ct
contravariant functor that takes coproducts to products.
Strongly dualizable objects were studied in [LMS86 , Chapter III], and are th*
*e ba-
sis for Spanier-Whitehead duality. We recall the results of [LMS86 ] in the fol*
*lowing
theorem.
Theorem A.2.5. Let C be a category with a triangulation and a closed symmetric
monoidal structure compatible with the triangulation.
(a) The full subcategory of strongly dualizable objects is thick and closed un*
*der
smash products and function objects. In particular, if X is strongly duali*
*zable,
so is DX.
(b) If X is strongly dualizable, the natural map X -! D2X adjoint to the evalu-
ation map
X ^ DX -! S
is an isomorphism.
(c) If Y is strongly dualizable and X and Z are arbitrary objects of C, there *
*is a
natural isomorphism
F (X ^ Y; Z) -!F (X; DY ^ Z):
(d) The natural map
F (X; Y ) ^ F (X0; Y 0) -!F (X ^ X0; Y ^ Y 0)
is an equivalence when X and X0 are strongly dualizable, and also when X is
strongly dualizable and Y = S. In particular, if X is strongly dualizable,*
* the
map
DX ^ DY -! D(X ^ Y )
is an isomorphism for all Y .
AXIOMATIC STABLE HOMOTOPY THEORY 109
(e) If X or Z is strongly dualizable, the natural map
F (X; Y ) ^ Z -! F (X; Y ^ Z)
is an isomorphism.
(f) If X is strongly dualizable and {Yff} is a family of objects, then the nat*
*ural
map
Y Y
X ^ Yff-! (X ^ Yff)
is an isomorphism.
With the exception of the fact that strongly dualizable objects form a thick
subcategory, this theorem holds in an arbitrary closed symmetric monoidal categ*
*ory.
Proof.This is all proved in [LMS86 ] except for two things: the fact that stron*
*gly
dualizable objects form a thick subcategory, and part (f). The former follows e*
*asily
from the exactness of F (-; Y ). For the latter, we have the equivalences
Y Y Y Y
X ^ Yff' F (DX; Yff) ' F (DX; Yff) ' (X ^ Yff);
completing the proof. |___|
Another useful lemma proved in [LMS86 ] (see Proposition III.1.3 and its proo*
*f)
is the following.
Lemma A.2.6. Suppose that X is a strongly dualizable object in a closed symmet*
*-_
ric monoidal category. Then X is a retract of X ^ DX ^ X. |__|
Of course, S is always strongly dualizable in any closed symmetric monoidal
category. The Picard category, first introduced into stable homotopy theory by
Hopkins [HMS94 , Str92], provides another source of strongly dualizable objects.
Definition A.2.7.Let C be a closed symmetric monoidal category. We say that
an object X 2 C is invertible if there is an object Z and an isomorphism X ^Z -*
*! S.
Define the Picard category to be the full subcategory of invertible objects. We*
* refer
to the isomorphism classes of this category with the operation ^ as the Picard
group, though in general it may be a proper class rather than a set. We will de*
*note
the Picard group by Pic.
Proposition A.2.8. Let C be a closed symmetric monoidal category. Then any
object X of the Picard category is strongly dualizable. Furthermore, the invers*
*e of
X is DX.
Proof.Let Y denote an inverse of X. Then smashing with Y is an equivalence of
C with itself. Therefore, for all Z, we have
[Z; DX] ' [Z ^ X; S] ' [Z ^ X ^ Y; Y ] ' [Z; Y ]:
Hence Y = DX. Now for all W and Z we also have
[W; F (X; Z)] ' [W ^ X; Z] ' [W; Z ^ Y ] ' [W; DX ^ Z];
Thus F (X; Z) ' DX ^ Z and so X is strongly dualizable. |___|
110 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
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Index
acyclic, 29, 35, 46 injective, 91
Adams spectral sequence, 100 complete, 12, 86
additive category, 8 connective, 76, 77
enriched, 82, 83 Cp, 69
algebraic, 7
algebraic localization, 40, 68 derived category, 8, 41, 55, 67, 70, 84
detect nilpotence, 64, 73
bounded below, 77 determine G-ideals, 66
Bousfield class division algebra, 78
generalized, 46
Bousfield class, 46-48, 50, 51, 64 Eilenberg swindle, 13
Bousfield equivalent, 46 Eilenberg-MacLane object, 77
Bousfield lattice, 46, 75 enriched, 8, 21, 83
Brown category, 7, 42, 54, 55, 57-62 equivalence, 29
E(R=p), 68
Co; Co, 56, 61, 62 essentially small, 10, 23
C , 22, 23 exact functor, 6, 104
C(B), 70, 78, 89, 95, 96, 98 exact triangle, 103
c(C), 22, 23, 55 extended, 91
c(X), 22
cardinality, 22 Fo; Fo, 56
CC, 30 F(X; Y ), 5
cellular approximation, 80 f-phantom, 56
cellular tower, 21, 23, 28, 77 F-small, 14
chain complex, 82 fiber, 104
chain homotopy, 82 field object, 49
CL, 29, 42, 44, 45, 47 skew, 49
closed model category, 5, 16, 18, 19, 35,f36,iltered category, 24, 26, 27, 57
80, 84, 89, 95 finite localization, 37, 38, 40, 44, 96
closed symmetric monoidal category, 105
closed under specialization, 70, 73 GSU, 87
cocomplete, 12 G-coideal, 11, 12
cofiber, 103, 104 G-finite, 25
cofinal functor, 24 G-ideal, 11, 12, 14, 23
cofree, 91 G-map, 86
cohomology functor, 6 G-Mod, 79
graded, 10 G-module, 79
representable, 6 G-space, 86
coideal, 11 G-spectrum, 86
colocalizing, 11 generated, 11
closed, 46 generators, 7
colimit geometric morphism, 41, 42, 80
homotopy, 18 lax, 41
minimal weak, 16, 57-59 grading system, 101
sequential, 16-18
weak, 16, 17 bH, 24, 25
coloc~~~~, 11 bHA, 24, 27, 28
colocal, 30 heart,_77
colocalization hGSU, 87, 88
functor, 29, 31 Hom_B(M; N), 95
colocalizing homology functor, 6
coideal, 11 graded, 10
subcategory, 11 representable, 54
Comod(B), 89, 94 homotopy colimit, 18
comodule, 9 homotopy group, 10
cofree, 91 Hopf algebra, 9, 67, 89, 90, 95, 96, 98, *
*100
extended, 91 HX , 47, 54, 55
113
114 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
ideal, 11 nilpotence, 73
localizing, 11 nilpotence theorem, 63, 64, 100
closed, 46 nilpotent, 63, 64, 99
idempotent, 12 Noetherian, 67, 68, 73, 89
injective comodule, 94
injective hull, 68 octahedral axiom, 21, 60, 104
inverse limit, 27 ordinary homology, 77
invertible, 109 P(X; Y ), 56
Ip, 69 phantom, 56, 59, 61, 62
Isotropy(U), 86, 87 bss, 47, 54
K(B), 89 Picard category, 102, 109
Koszul algebra, 95 Picard group, 102, 109
K(p),_69-71, 73 ssk, 10
kp, 70 Postnikov tower, 77
kZ, 78 prespectrum, 87
(X), 24, 59 quasi-isomorphism, 83
A(X), 23, 24 replete, 45
lax, 41 representable, 6, 54, 55, 63
LfB, 96 ring object, 49, 64
Lie group, 8, 86
limit S-finite, 14
sequential, 20 semisimple, 78, 79
weak, 19 sequential colimit, 16-19, 65
Lindner category, 102 sequential limit, 20
linear topology, 26, 62 SUG, 88, 89
linearly compact, 26, 27 simplekcomodule, 89, 92, 94
loc~~~~, 11 S , 10
local, 29, 35, 46 small, 6, 14, 25, 27
localization, 10, 42, 44, 45 smash nilpotent, 64, 65
algebraic, 40, 68 smash-complemented, 49
at a prime ideal, 69 smashing, 36, 42, 51, 60, 61, 76
away from, 37, 38, 40 Sp, 69, 75
finite, 37, 38, 40, 44, 53, 96 S=p, 69
functor, 29, 31, 35 Sp, 68
smashing, 36, 42, 51, 60, 61, 76 Spanier-Whitehead dual, 108
with respect to homology, 35, 36 SpecR, 67, 71
localizing, 11 spectrum, 87
ideal, 11 stable homotopy category, 6
subcategory, 5, 28, 75 I-graded, 101
L~~, 11
triangulated category, 104
cocomplete, 12
complete, 12
enriched, 8, 21, 80, 83
triangulation, 103
Uo; Uo, 56
Uf, 62
unital algebraic, 7
universe, 86
complete, 86
incomplete, 88
Vo; V o, 56, 61, 62
weak colimit, 16, 17
weak limit, 19
, 46-48, 50, 51, 64
X(1), 64
X(n), 63
X ^ Y , 5
116 MARK HOVEY, JOHN H. PALMIERI, AND NEIL P. STRICKLAND
Department of Mathematics, M. I. T. 2-388, 77 Massachusetts Ave., Cambridge, *
*MA
02139, USA
E-mail address: hovey@math.mit.edu
Department of Mathematics, M. I. T. 2-388, 77 Massachusetts Ave., Cambridge, *
*MA
02139, USA
E-mail address: palmieri@math.mit.edu
Trinity College, Cambridge CB2 1TQ, England
E-mail address: neil@pmms.cam.ac.uk