Series Logo
Volume 00, Number 00, Xxxx 19xx
MORAVA K-THEORIES AND LOCALISATION
MARK HOVEY AND NEIL P. STRICKLAND
Abstract.We study the structure of the categories of K(n)-local and E(n)-
local spectra, using the axiomatic framework developed in earlier work of*
* the
authors with John Palmieri. We classify localising and colocalising subca*
*t-
egories, and give characterisations of small, dualisable, and K(n)-nilpot*
*ent
spectra. We give a number of useful extensions to the theory of vn self m*
*aps
of finite spectra, and to the theory of Landweber exactness. We show that
certain rings of cohomology operations are left Noetherian, and deduce so*
*me
powerful finiteness results. We study the Picard group of invertible K(n)*
*-local
spectra, and the problem of grading homotopy groups over it. We prove (as
announced by Hopkins and Gross) that the Brown-Comenetz dual of MnS lies
in the Picard group. We give a detailed analysis of some examples when n *
*= 1
or 2, and a list of open problems.
Introduction
The stable homotopy category S is extraordinarily complicated. However, there
is a set of approximations to it that are much simpler and closer to algebra. T*
*he
stable homotopy category is somewhat analogous to the derived category of a rin*
*g R,
except that R is replaced by the stable sphere S0. In practice, we always consi*
*der the
p-local stable homotopy category and the p-local sphere for some prime p, witho*
*ut
changing the notation. It is very common to study R-modules using the fields ov*
*er
R, and in recent years there has been much work on studying the stable homotopy
category via its fields. These fields are referred to as Morava K-theories, den*
*oted
by K(n), and were introduced by Morava in the early 1970's. See [Mor85 ] for a
description of Morava's earlier work.
Associated to the Morava K-theories are various (homotopy) categories of local
spectra that are the approximations to the stable homotopy category mentioned
above. There is the category L of spectra local with respect to K(0) _ . ._.K(n)
and the category K of spectra local with respect to K(n). (We will always have a
fixed n with 0 < n < 1 in mind in this paper). These categories are themselves
stable homotopy categories, in the sense of [HPS95 ]. The purpose of this paper*
* is
to study the structure of these categories. We will show that the category K is*
* in
a certain sense irreducible; it has no nontrivial further localisations. On the*
* other
____________
Received by the editors April 20, 1997.
1991 Mathematics Subject Classification. 55P42, 55P60, 55N22,55T15.
Key words and phrases. Morava K-theory, stable homotopy category, Bousfield*
* localisation,
Picard group, phantom maps, Landweber exact homology theories, Adams spectral s*
*equence,
spectrum, Brown-Comenetz duality, thick subcategory.
The first author was partially supported by an NSF Postdoctoral Fellowship.
The second author was partially supported by an NSF Grant.
Oc0000 American Mathematical Society
0000-0000/00 $1.00 + $.25 per page
1
2 M. HOVEY AND N. P. STRICKLAND
hand, there are a number of results (such as the Chromatic Convergence Theorem
of Hopkins and Ravenel [Rav92a], or the proof by the same authors that suspensi*
*on
spectra are harmonic [HR92 ]) which indicate that an understanding of K for all
n and p will give complete information about S. Hopkins' Chromatic Splitting
Conjecture [Hov95a], if true, would be a still stronger result in this directio*
*n.
The first half of the paper is mostly concerned with issues we need to resolve
before beginning our study of K. In Section 1, we define the basic objects of s*
*tudy:
the ring spectra E(n), E = [E(n)and K = K(n), the categories L and K and so
on. In Section 2, we study E-cohomology. We prove that E*X is complete, using a
slightly modified notion of completeness which turns out to be more appropriate*
* than
the traditional one. We show that a well-known quotient of the ring of operatio*
*ns in
E-cohomology is a non-commutative Noetherian local ring. We also show that there
are no even degree phantom maps between evenly graded Landweber exact spectra
such as E, a result that has been speculated about for a long time. In Section *
*3, we
prove some basic results about the (very simple) category of K-injective spectr*
*a. We
then turn in Section 4 to the study of generalised Moore spectra, and prove a n*
*umber
of convenient and enlightening extensions to the theory developed by Hopkins and
Smith [HS , Rav92a] and Devinatz [Dev92]. In Section 5, we assemble some (mostly
well-known) results about the Bousfield classes of a number of spectra that we *
*will
need to study. In Section 6, we study the E(n)-local category L. We prove some
results about nilpotence, and we classify the thick subcategories of small obje*
*cts,
the localising subcategories and the colocalising subcategories.
In the second half of the paper, we concentrate on K. In Section 7, we prove *
*our
most basic results about the K(n)-local category K. In particular, we prove that
it is irreducible, in the sense that the only localising or colocalising subcat*
*egories
are {0} and K itself. We also study the localisation functor bL= LK and some
related functors, describing them in terms of towers of generalised Moore spect*
*ra.
In Section 8 we study two different notions of finiteness in K, called smallnes*
*s and
dualisability. The (local) sphere is dualisable but not small; some rather unex*
*pected
spectra are dualisable, such as the localisation of BG for a finite group G. We*
* prove
a number of different characterisations of smallness and dualisability; in part*
*icular,
we give convenient tests in terms of computable cohomology theories. We also sh*
*ow
that dualisable spectra lie in the thick subcategory generated by E, and that K-
small spectra lie in the thick subcategory generated by K. In Section 9 we study
homology and cohomology theories on K, and prove that both are representable
in a suitable sense. In Section 10 we study a version of Brown-Comenetz duality
appropriate to the K(n)-local setting, and we prove that the Brown-Comenetz dual
of the monochromatic sphere is an element of the Picard group. This result was
stated in [HG94 ]. In Section 11, we introduce a natural topology on the groups*
* [X; Y ]
for X and Y in K, and prove a number of properties. In Section 12, we return to*
* the
study of the category D of dualisable spectra. We prove a nilpotence theorem an*
*d an
analogue of the Krull-Schmidt theorem, saying that every dualisable spectrum can
be written as a wedge of indecomposables in an essentially unique way. We make
some remarks about ideals in D, but we have not been able to prove the obvious
conjectures about them. In Section 13 we study K-nilpotent spectra, proving a
number of interesting characterisations of them. In Section 14 we study the pro*
*blem
of grading homotopy groups over the Picard group of invertible spectra, rather *
*than
just over the integers. We have a satisfactory theory for homotopy groups of K-*
*small
MORAVA K-THEORIES AND LOCALISATION 3
spectra, but the general case seems less pleasant. We show that the Picard group
is profinite in a precise sense, but we do not know if it is finitely generated*
* over the
p-adics. Section 15 is devoted to the study of the simplest examples. Even when
n = 1 and p is odd, there are simple counterexamples to plausible conjectures. *
*We
also consider the case n = 2 and p > 3, showing that the Picard graded homotopy
groups of the Moore spectrum are mostly infinite. We conclude the main body of
the paper with Section 16, which contains a list of interesting questions that *
*we have
been unable to answer, some of them old and others new.
We also have two appendices: the first addresses the sense in which the E-
cohomology of a K-local spectrum is complete, as mentioned above, and the second
shows that some other interesting localisations of the category of spectra have*
* a
rather different behaviour, in that they contain no nonzero small objects at al*
*l.
We have chosen to rely on a minimum of algebraic prerequisites; in particular,
we say almost nothing about formal groups or the Morava stabiliser groups. We
have preferred to use thick subcategory arguments rather than spectral sequences
where possible. We have chosen our methods very carefully to avoid having to say
anything special when p = 2. For this reason we have generally used E rather th*
*an
K, as E is commutative at all primes (for example).
Our debt to Mike Hopkins will be very obvious to anyone familiar with the sub-
ject. We have been heavily influenced by his point of view and a large number o*
*f our
results were previously known to him. We also thank Matthew Ando, Dan Chris-
tensen, Chun-Nip Lee, John Palmieri and Hal Sadofsky for helpful conversations
about the subject matter of this paper.
Contents
Introduction 1
1. Basic definitions 4
2. E theory 7
3. K-injective spectra 19
4. Generalised Moore spectra 21
5. Bousfield classes 32
6. The E(n)-local category 33
7. General properties of the K(n)-local category 41
8. Smallness and duality 45
9. Homology and cohomology functors 50
10. Brown-Comenetz duality 52
11. The natural topology 55
12. Dualisable spectra 57
13. K-nilpotent spectra 64
14. Grading over the Picard group 67
15. Examples 74
16. Questions and conjectures 81
Appendix A. Completion 82
Appendix B. Small objects in other categories 90
References 95
Index 97
4 M. HOVEY AND N. P. STRICKLAND
1. Basic definitions
Fix a prime p and an integer n > 0. We shall localise all spectra at p; in
particular, we shall write MU for what would normally be called MU(p). We write
S for the category of p-local spectra.
1.1. The spectra E(n), Ed(n)and K(n). We next want to define the spectra
E = [E(n)and K = K(n). It is traditional to do this using the Landweber
exact functor theorem and Baas-Sullivan theory. Here we will use the more recent
techniques of [EKMM96 ] instead of the Baas-Sullivan construction.
It is well-known that the integral version of MU has a natural structure as an
E1 ring spectrum or (essentially equivalently) a commutative S-algebra in the s*
*ense
of [EKMM96 ]. It follows from [EKMM96 , Theorem VIII.2.2] that the same appli*
*es
to our p-local version. We can thus use the framework of [EKMM96 , Chapters
II-III] to define a topological closed model category MMU of MU-modules. The
associated homotopy category (obtained by inverting weak equivalences) is called
the derived category of MU-modules and written DMU . It is a stable homotopy
category in the sense of [HPS95 ]. There is a "forgetful" functor DMU -!S and a
left adjoint MU ^ (-): S -!DMU .
Let wk 2 ss2(pk-1)MU be the coefficient of xpkin the p-series of the universa*
*l for-
mal group law over MU* (so w0 = p) and write In = (w0; : :;:wn-1). We can con-
struct an object w-1nMU=In of DMU by the methods of [EKMM96 , Chapter V] (see
also [Str96]). Using [EKMM96 , Theorem VIII.2.2] again, we can Bousfield-local*
*ise
MU with respect to w-1nMU=In to get a strictly commutative MU-algebra which
we call dMU. As explained in [GM95a ], the homotopy ring of dMU is (w-1nMU*)^In.
Next, consider the graded ring
E(n)* = Z(p)[v1; : :;:vn][v-1n] |vk| = 2(pk - 1):
There is a unique p-typical formal group law over this ring with the property t*
*hat
XF k
[p]F(x) = expF(px) +F vkxp :
0 2 there is a unique commutative product on E(n) 2 DMU , but
we avoid using this here so that we can handle all primes uniformly.
We now define
E = [E(n)= E(n) ^MU dMU:
MORAVA K-THEORIES AND LOCALISATION 5
This is clearly a module over dMU with a given map dMU -! E, and one can check
that this gives an isomorphism
E* = (E(n)*)^In= Zp[v1; v2; : :;:vn-1; v1n]^In:
It is again well-defined up to non-canonical isomorphism under dMU, and it admi*
*ts a
non-canonical associative ring structure. If p > 2 then there is a unique commu*
*tative
product on E as an object of DMdU.
Because E(n) is an MU-module under MU, there is a canonical map
E(n)* MU* MU*X -! E(n)*X:
This is of course an isomorphism, by the Landweber Exact Functor Theorem [Lan76*
*].
Similarly, we have an isomorphism
E* MU* MU*X -! E*X:
It follows that the homology theory represented by E(n) is independent of the c*
*hoice
of object E(n) 2 DMU up to canonical isomorphism, and thus the underlying spec-
trum of E(n) is well-defined up to an isomorphism that is canonical mod phantoms
(see [HPS95 , Section 4] for a discussion of phantoms and representability). We
shall show later that the relevant group of phantoms is zero, so as a spectrum *
*E(n)
is well-defined up to canonical isomorphism. We shall also show that there is a
canonical commutative ring structure on this underlying spectrum. Similar remar*
*ks
apply to E.
We can also define MU-modules MU=Ik 2 DMU for 0 k n in the evident
way, and then define
E(n)=Ik = MU=Ik ^MU E(n)
E=Ik = MU=Ik ^MU E
K = K(n) = E(n)=In = E=In:
It is clear that ss*(E(n)=Ik) = E(n)*=Ik and so on. In particular we have K(n)**
* =
E(n)*=In = E*=In = Fp[v1n]. These MU-modules admit (non-unique) associative
products, so (E(n)=Ik)*X is canonically a module over E(n)*=Ik. Similar remarks
apply to E=Ik.
There are evident cofibrations
k-1) vk
2(p E=Ik -! E=Ik -!E=Ik+1;
and similarly for E(n)=Ik.
We also know from [Bak91] that there is an essentially unique A1 structure
on the spectrum E. It is widely believed that this can be improved to an E1
structure, and that the maps MU -! E characterised by Ando [And95 ] (which do
not include the map MU -! E considered above) can be improved to E1 maps.
Unfortunately, proofs of these things have not yet appeared. Nonetheless, just *
*using
the A1 structure we can still use the methods of [EKMM96 ] to define a derived
category DE of left E-modules. This is a complete and cocomplete triangulated
category, with a smash product functor ^: S x DE -! DE .
1.2. Categories of localised spectra. We use the following notation.
Notation 1.1.
1.S is the (homotopy) category of p-local spectra. We write S for S0.
6 M. HOVEY AND N. P. STRICKLAND
2.L = Ln is the category of E(n)-local spectra, and L = Ln : S -! L is the
localisation functor. The corresponding acyclisation functor is written C, *
*so
there is a natural cofibre sequence CX -! X -! LX.
3.M = Mn is the monochromatic category. This is defined to be the image of
the functor Mn = Cn-1Ln: S -! S. Note that there is a natural fibration
MnX -! LnX -! Ln-1X = Ln-1LnX.
4.K = Kn is the category of K-local spectra, and bL= bLn: S -! K is the
localisation functor. The corresponding acyclisation functor is written bC.
5.F (m) denotes a finite spectrum of type m, and T (m) = v-1mF (m) is its tel*
*e-
scope. Recall from [Rav92a, Chapter V] or [HS ] that any two F (m)'s genera*
*te
the same thick subcategory and have the same Bousfield class, so it usually
does not matter which one we use. Note also that the Spanier-Whitehead dual
of an F (m) is again an F (m).
There are topological closed model categories whose homotopy categories are S,
L and K [EKMM96 , Chapter VIII].
1.3. Stable homotopy categories. In this section we collect some basic defini-
tions from the theory of stable homotopy categories developed in [HPS95 ]. The
reader will be familiar with most of these: we assemble them here as a convenie*
*nt
reference. We will not recall the definition of a stable homotopy category, exc*
*ept to
say that a stable homotopy category is a triangulated category with a closed sy*
*m-
metric monoidal structure which is compatible with the triangulation and which *
*has
a set of generators in an appropriate sense. The symmetric monoidal structure is
written X ^ Y and the closed structure is written F (X; Y ). The unit for the s*
*mash
product is written S.
Definition 1.2.A full subcategory D of any triangulated category is thick if it*
* is
closed under retracts, cofibres, and suspensions. That is, D is thick if both o*
*f the
following conditions hold.
(a)If X _ Y 2 D, then both X and Y are in D; and
(b)If
X -! Y -! Z -! X
is a cofibre sequence and two of X, Y , and Z are in D, then so is the thir*
*d.
Certain kinds of thick subcategories come up frequently.
Definition 1.3.Let C be a thick subcategory of a stable homotopy category D.
(a)C is a localising subcategory if it is closed under arbitrary coproducts.
(b)C is a colocalising subcategory if it is closed under arbitrary products.
(c)C is an ideal if, whenever X 2 D and Y 2 C we have X ^ Y 2 C.
(d)C is a coideal if, whenever X 2 D and Y 2 C we have F (X; Y ) 2 C.
Whenever the localising subcategory generated by S is all of D, every (co)loc*
*alising
subcategory is a (co)ideal [HPS95 , Lemma 1.4.6]. This is true in S, L, and K (*
*and
any other localisation of S).
The ideal generated by a ring object is particularly important.
Definition 1.4.Let C be a stable homotopy category, and R a ring object in C.
We say that an object X 2 C is R-nilpotent if it lies in the ideal generated by*
* R.
MORAVA K-THEORIES AND LOCALISATION 7
We now recall some different notions of finiteness in a stable homotopy categ*
*ory.
We have replaced "strongly dualisable" by "dualisable" for brevity. Also recall*
* that
DZ = F (Z; S) is the usual duality functor.
Definition 1.5.Let C be a stable homotopy category, and Z an object of C. We
say that Z is
L
(a)small`if for any collection of objects {Xi}, the natural map [Z; Xi] -!
[Z; Xi] is an isomorphism. `
(b)F -small`if for any collection of objects {Xi}, the natural map F (Z; Xi)*
* -!
F (Z; Xi) is an isomorphism.
(c)A-finite (for any family A of objects of C) if Z lies in the thick subcateg*
*ory
generated by A.
(d)dualisable if for any X, the natural map DZ ^X -! F (Z; X) is an equivalenc*
*e.
A triangulated category with a compatible closed symmetric monoidal structure
is an algebraic stable homotopy category if there is a set G of small objects s*
*uch that
the localising subcategory generated by G is the whole category. An algebraic s*
*table
homotopy category is called monogenic if we can take G = {S}.
Finally, we recall that limits and colimits generally do not exist in triangu*
*lated
categories, but sometimes suitable weak versions do exist. In particular, give*
*n a
sequence X0 -! X1 -! :W:,:we can form the sequential colimit as the cofibre of
the usual self-map of Xi. We denote this by holim-!Xi , as it is the homotopy
colimit of a suitable lift of the sequence to a model category. Similarly, we d*
*enote
the sequential limit of a sequence : :-:!X1 -!X0 by holim-Xi.
In case we have a more complicated functor F : I -! C to a stable homotopy
category, we say that a weak colimit X of F is a minimal weak colimit if the in*
*duced
map lim-!H O F -! HX is an isomorphism for all homology functors H. We write
X = mwlim-!F . Minimal weak colimits are unique in algebraic stable homotopy
categories when they exist, and are extremely useful. See [HPS95 , Section 2] *
*for
details.
2.E theory
In this section we assemble some basic results about E theory. We begin with
the fact that E*X is L-complete in the sense of Appendix A. In Section 2.1 we s*
*how
that the ring structure on E is canonical by showing there are no phantom maps
between evenly graded Landweber exact spectra. We recall the modified Adams
spectral sequence briefly in Section 2.2. Finally, we briefly discuss operatio*
*ns in
E-theory in Section 2.3.
Proposition 2.1.If X is a finite spectrum and R is one of E, E(n), E=Ik,
E(n)=Ik or K(n) then R*X is finitely generated over R*.
Proof.We first recall that in each case R has an associative ring structure, so*
* that
R*X is a module over R*. The ring structure is not canonical but the module
structure is induced by the MU*-module structure so it is canonical. In each ca*
*se __
R* is Noetherian and the claim follows easily by induction on the number of cel*
*ls. |__|
Proposition 2.2.For any spectrum X, there is a natural topology on E0X making
it into a profinite (and thus compact Hausdorff ) Abelian group. Moreover, if (*
*X)
8 M. HOVEY AND N. P. STRICKLAND
is the category of pairs (Y; u) where Y is finite and u: Y -! X, then E0X is
homeomorphic to lim- E0Y .
(Y;u)2(X)
Proof.If X is a finite spectrum then E*X is a finitely generated module over E*,
and it follows easily that the In-adic topology on E0X is profinite. For an arb*
*itrary
spectrum X, define F 0X = lim- E0Y , with the inverse limit topology.
(Y;u)2(X)
By [HPS95 , Proposition 2.3.16], F is a cohomology theory with values in the ca*
*te-
gory of profinite groups and continuous homomorphisms. There is an evident map
E0X -! F 0X which is an isomorphism when X is finite (because (X; 1X ) is a
terminal object of (X) in that case). It follows easily that E0X = F_0X for all
X. |__|
Corollary 2.3.For any spectrum X, the module E*X is L-complete in the sense
of Definition A.5.
Proof.This is immediate when X is finite. It thus follows for general X because*
* the_
category of L-complete modules is closed under inverse limits (by Theorem A.6).*
* |__|
Proposition 2.4.For any spectrum X, the module E*X is finitely generated over
E* if and only if K*X is finitely generated over K*.
Proof.The cofibration 2(pk-1)E=Ik -vk!E=Ik -! E=Ik+1 gives a short exact se-
quence
k+1
(E=Ik)*(X)=vk ae (E=Ik+1)*(X) i ann(vk; (E=Ik)*-2p (X)):
It follows that if (E=Ik)*X is finitely generated then the same is true of (E=I*
*k+1)*X.
Conversely, suppose that (E=Ik+1)*X is finitely generated over E*, and write M =
(E=Ik)*X. The above sequence shows that M=vkM is finitely generated. This
means that there is a finitely generated free module F over E*=Ik and a map f :*
*F -!
M such that the induced map F=vkF -! M=vkM is surjective. If we let N be the
cokernel of f, we conclude that N is an L-complete module over E* with N = vkN.
It follows from Proposition A.8 that N = 0, so f is surjective and M is finitely
generated.
It follows that K*X = (E=In)*X is finitely generated if and only if E*X_=_
(E=I0)*X is finitely generated. |__|
Proposition 2.5.Let X be a spectrum. Suppose E*X is pro-free (in the sense of
Definition A.10). Then K*X = (E*X)=In. Conversely, if K*X is concentrated in
even degrees, then E*X is pro-free and concentrated in even degrees.
Proof.The first statement follows from the fact that the sequence (v0; : :;:vn-*
*1)
is regular on E*X, by Theorem A.9. Conversely, suppose that (E=Ik+1)*X is
concentrated in even degrees. Consider the short exact sequence
k+1
(E=Ik)*(X)=vk ae (E=Ik+1)*(X) i ann(vk; (E=Ik)*-2p (X)):
It follows that (E=Ik)odd(X)=vk = 0. As (E=Ik)odd(X) is L-complete, we conclude
from Proposition A.8 that it must be zero. It also follows from the sequence th*
*at
ann(vk; (E=Ik)even(X)) = 0, so that vk acts injectively on (E=Ik)*(X). Finally,*
* we
also see from the sequence that (E=Ik+1)*(X) = (E=Ik)*(X)=vk.
By an evident induction we conclude that E*X is concentrated in even de-
grees, that the sequence {v0; : :;:vn-1} is regular on E*X and that K*(X) =
MORAVA K-THEORIES AND LOCALISATION 9
E*(X)=(v0; : :;:vn-1) = K* E* E*X. It follows from Theorem A.9 that E*X_
is pro-free. |__|
2.1. Landweber exactness. We next recall the theory of Landweber exact homol-
ogy theories, and prove some convenient extensions. Many of the theorems we pro*
*ve
were proved by Franke [Fra92] in the case where ss*M is countable; our methods
are a generalisation of his.
Definition 2.6.An MU*-module M* is said to be Landweber exact if the sequence
(w0; w1; : :):is regular on M*. We write E* for the category of Landweber exact
modules that are concentrated in even degrees. We also write E for the category*
* of
MU-module spectra M such that ss*(M) 2 E*. Maps in E are MU-module maps.
Finally, we write EF for the category of finite spectra X such that H*X is free*
* and
concentrated in even degrees. We refer to such an X as an even finite spectrum.
Note that any even finite spectrum has a finite filtration where the filtration*
* quotients
are finite wedges of even spheres.
The basic result is as follows.
Theorem 2.7 (Landweber).If M* 2 E* then the functor M* MU* MU*X is a
homology theory. Thus (by Brown representability), there is a spectrum M equipp*
*ed
with a natural isomorphism M*X ' M* MU* MU*X. This M is unique up to
isomorphism, and the isomorphism is canonical modulo phantoms.
Proof.See [Lan76, Theorem 2.6]. |___|
The following result summarises Proposition 2.20 and Proposition 2.19, which
are proved below. It justifies the statements made in Section 1 about the uniqu*
*eness
of E and E(n) and their ring structures.
Theorem 2.8. The functor ss*: E -! E* is an equivalence of categories. The in-
verse functor sends commutative MU*-algebras to commutative MU-algebra_spec-
tra. |__|
It is convenient to introduce a new category E0at this point; it will follow *
*from
the theorem that E0= E.
Definition 2.9.E0 is the category of spectra M such that M* is concentrated in
even degrees, with a given MU*-module structure on M* and a stable natural iso-
morphism M*MU* MU*X -! M*X which is the identity when X = S. Morphisms
of E0must preserve the module structure.
The converse of the Landweber exact functor theorem [Rud86] shows that if
M 2 E0, then M* 2 E*. Theorem 2.7 says that ss*:E0-! E* is essentially surjecti*
*ve
on objects.
In order to show that ss* is an equivalence of categories, we introduce the f*
*ollowing
definition.
Definition 2.10.A spectrum X is evenly generated if and only if, for every fini*
*te
spectrum Z and every map Z -f!X, there is an even finite spectrum W and a
factorisation Z -g!W -h!X of f.
10 M. HOVEY AND N. P. STRICKLAND
Every even finite spectrum is evenly generated. The collection of evenly gene*
*rated
spectra is closed under even suspensions, coproducts, and retracts, but does no*
*t form
a thick subcategory. We will see in Proposition 2.18 that evenly generated spec*
*tra
are closed under the smash product.
Lemma 2.11. MU is evenly generated.
Proof.Any map from a finite spectrum to MU factors though a skeleton of MU._
Any skeleton of MU is an even finite. |__|
The following result is essentially due to Hopkins.
Proposition 2.12.Suppose M 2 E0. Then M is evenly generated.
Proof.Suppose f : Z -! M is a map from a finite spectrum to M. Then f is a
class in M0Z. Spanier-WhiteheadPduality implies that M*Z = M* MU* MU*Z.
We can thus write f = mi=1bi cisay. As M* is concentrated in even degrees we
see that |ci| = -|bi| is even. Each map ci thus has a factorisation
ci= (Z -gi!Wi-ei!|bi|MU);
where Wiis a skeleton of |bi|MU, and so is an even finite. Write W = W1_ : :_:
Wm , let g :Z -! W be the map with components giand let h: W -! M be the map
with components bi ei 2 M* MU* MU*Wi = [Wi; M]*. This gives the desired__
factorisation. |__|
There are several different characterisations of evenly generated spectra.
Proposition 2.13.The spectrum X is evenly generated if and only if X can be
written as the minimal weak colimit [HPS95 , Section 2.2] of a filtered system *
*{Mff}
of even finite spectra.
Proof.First suppose that X can be written as such a minimal weak colimit. Then,
by smallness, any map from a finite to X will factor through one of the terms i*
*n the
minimal weak colimit, and so through an even finite. Thus X is evenly generated.
Conversely, suppose X is evenly generated. We replace EF by a small skeleton *
*of
EF without change of notation. Let EF(X) be the category of pairs (U; u), where
U 2 EF and u: U -! X. Let (X) be the category of pairs (W; w), where W is any
finite spectrum and w :W -! X. We know from [HPS95 , Theorem 4.2.4] that X
is the minimal weak colimit of (X). It will therefore be enough to show that the
obvious inclusion EF(X) -!(X) is cofinal.
We first show that EF(X), like (X), is filtered. Consider two objects (U; u)
and (V; v) of EF(X). We need to show that there is an object (W; w) and maps
(U; u) -! (W; w)- (V; v) in (X). Clearly we can take W = U _ V , and let
w :W -! X be the map with components u and v. We also need to show that
when f; g :(U; u) -!(V; v) are two maps in EF(X), there is an object (W; w) and
a map h: (V; v) -! (W; w) with hf = hg. To see this, let W 0be the cofibre of
f - g and h0:V -! W 0the evident map. We have vf = u = vg so v(f - g) = 0
so v = w0h0 for some w0:W 0-! X. Because X is evenly generated, the map w0
factors as W 0k-!W -w!X for some even finite W . We can evidently take h = kh0.
It is now easy to check that the inclusion EF(X) -!(X) is cofinal,_as require*
*d.
|__|
MORAVA K-THEORIES AND LOCALISATION 11
Corollary 2.14.A spectrum X is evenly generated if and only if there is a cofib*
*re
sequence
P -! Q -!X -ffi!P;
where P and Q are retracts of wedges of even finite spectra and ffi is a phantom
map.
Proof.If there is such a cofibre sequence, then there is a short exact sequence
[Z; P ] ae [Z; Q] i [Z; X]
for all finite Z, since ffi is phantom. It follows easily that X is evenly gene*
*rated. The
converse follows from the Proposition and the construction in [CS96 ,_Propositi*
*on_
4.6]. |__|
Corollary 2.15.Suppose X is evenly generated and Y is a spectrum such that Y*
is concentrated in even degrees. Let P*(X; Y ) be the graded group of phantom m*
*aps
from X to Y . Then
P2k(X; Y )= 0
P2k-1(X; Y )= [X; Y ]2k-1:
In particular, this holds if X; Y 2 E0.
Proof.Proceeding by induction, one proves easily that Y *W is concentrated in e*
*ven
degrees for all even finite W . In particular, in the cofibre sequence of Corol*
*lary 2.14,
Y *Q and Y *P are concentrated in even degrees. Any phantom map in Y 2kX would
be in the image of Y 2k-1P = 0, so there are no even phantoms. On the other han*
*d,_
every class in Y 2k+1X is in the image of Y 2kP , so is phantom. *
*|__|
Although this characterisation of phantoms is the main fact that we need, we *
*will
prove something rather sharper.
Proposition 2.16.Suppose R is a ring spectrum, S is an R-module spectrum, and
X is evenly generated. Suppose as well that R* and S* are concentrated in even
degrees. Then R*X is flat, has projective dimension at most 1, and is concentra*
*ted
in even degrees. Furthermore, we have
S*X = S* R* R*X
S2kX = Hom 2kR*(R*X; S*)
S2k-1X = Ext1;2kR*(R*X; S*):
Proof.Choose a cofibre sequence P -! Q -! X -ffi!P as in Proposition 2.14. If
W is an even finite, it is easy to see by induction on the (even) cells that R**
*W is
free over R* and concentrated in even degrees. Since R*X = lim-! R*W , we
EF(X)
see that R*X is also concentrated in even degrees and is a filtered colimit of *
*free
modules, so is flat. One can also check by induction on the cells, using the fa*
*ct that
S* is evenly graded, that the natural map S*R* R*W -! S*W is an isomorphism
for W an even finite. Taking colimits, we find that S* R* R*X -! S*X is also an
isomorphism.
Similarly, one can check by induction on the cells that when W is an even fin*
*ite,
the natural map [W; S]* -!Hom R*(R*W; S*) is an isomorphism. As P and Q are
retracts of wedges of such W , we see that R*P and R*Q are projective over R*
12 M. HOVEY AND N. P. STRICKLAND
and that [Q; S]* = Hom R*(R*Q; S*) and [P; S]* = Hom R*(R*P; S*). In particular,
these groups are concentrated in even degrees. Because ffi is phantom, we have a
short exact sequence
R*P ae R*Q i R*X;
which is a projective resolution of R*X. We now apply the functor [-; S]* to the
cofibration P -! Q -!X to get an exact sequence
Hom *+1R*(R*Q; S*) -!Hom *+1R*(R*P; S*) -![X; S]* -!
Hom*R*(R*Q; S*) -!Hom *R*(R*P; S*);
and thus a short exact sequence
Ext1;*+1R*(R*X; S*) ae [X; S]* i Hom *R*(R*X; S*):
The first term is concentrated in odd degrees and the last one in even degrees,*
*_so
the sequence splits uniquely. |__|
Note that this proposition implies that spectra such as P (n) and K(n) are not
evenly generated for n > 0. Indeed, P (n)*P (n) and K(n)*K(n) both contain a
Bockstein element in degree 1. Similarly, HZ and HFp are not evenly generated. *
*In
fact, the only MU-module spectra that are evenly generated are the even Landweb*
*er
exact MU-module spectra. One can prove this by using the fact that X ^ F (n) is
a retract of (MU ^ F (n)) ^ X and is therefore evenly graded. Applying this to *
*the
spectra S=I of Section 4 shows that vn acts injectively on X*=I.
Corollary 2.17.Let M and N be MU-module spectra in E0. Then
[M; N]2k= Hom 2kMU*MU(MU*M; MU*N)
= Hom 2kMU*(MU*M; N*)
[M; N]2k-1= Ext1;2kMU*MU(MU*M; MU*N)
= Ext1;2kMU*(MU*M; N*):
If s > 1 then Exts;*MU*MU(MU*M; MU*N) = 0.
Proof.By Landweber exactness we have X*N = X*MU MU* N* and in particular
MU*N = MU*MU MU* N*. This is an extended comodule, so for any comodule
C* we have
Hom MU*MU (C*; MU*N) = Hom MU*(C*; N*):
More generally, if we resolve N* by injective MU*-modules and apply the functor
MU*MU MU* (-) we get a resolution of MU*N by injective comodules. Using
this it is not hard to check that
Exts;*MU*MU(C*; MU*N) = Exts;*MU*(C*; N*)
for all s. The rest of the corollary is proved in Proposition 2.16. *
* |___|
Proposition 2.18.Suppose X and Y are evenly generated. Then X ^ Y is evenly
generated.
Proof.It is clear that if W and Z are even finites, so is W ^ Z. Now suppose X
is evenly generated and W is an even finite. Then X ^ W is the minimal weak
colimit of the functor on EF(X) which takes (U; u) to U ^ W . Thus X ^ W
is evenly generated. Now suppose Y is also evenly generated. Choose a cofibre
MORAVA K-THEORIES AND LOCALISATION 13
sequence P -! Q -! Y ffi-!P as in Corollary 2.14. Then X ^ P and X ^ Q
are evenly generated, as they are retracts of wedges of terms of the form X ^ W*
* .
Furthermore, 1X ^ ffi is still a phantom map (using the characterisation of pha*
*ntom
maps as those maps not seen by any homology theory). It follows that X_^_Y is
evenly generated. |__|
In fact, it is also true that E0is closed under the smash product, though we *
*do
not need this.
Proposition 2.19.If M 2 E0 then M admits a canonical structure as an MU-
module spectrum (in the traditional homotopical sense). If N is another spectrum
in E0then the degree-zero MU-module maps M -! N biject with the MU*-module
maps M* -!N*.
Proof.For any spectrum X we have a natural map
ffl: MU*(MU ^ X) = MU*MU MU* MU*X -! MU*X
of left MU*-modules, and thus a natural map
(M ^ MU)*X = M*(MU ^ X) = M* MU* MU*(MU ^ X) -!
M* MU* MU*X = M*X:
By Brown representability, we get a map :M ^ MU -! M which is unique mod
phantoms. It is not hard to check that this is associative and unital mod phant*
*oms.
However, Proposition 2.18 shows that M ^ MU and M ^ MU ^ MU are evenly
generated, so Corollary 2.15 tells us that there are no degree-zero phantom maps
M ^ MU -! M or M ^ MU ^ MU -! M. Thus is unique, associative and unital,
and M 2 E.
Now let N be another spectrum in E0, and consider the following diagram.
ss*
[M; N]MU _________wHom MU*(M*; N*)
v v
| |f
| |
|u |u
[M; N]_____wHom'MU*MUM(MU*M;UMU*N):*
Here [M; N]MU denotes the group of MU-module maps, and all the groups are
groups of degree-zero maps. The bottom map is an isomorphism by Proposi-
tion 2.17. The map f sends a map u: M* -!N* to
1 u: MU*MU MU* M* = MU*M -! MU*N = MU*MU MU* N*:
It is easy to check that this is injective and that the diagram commutes. It fo*
*llows
that ss* is injective. Now suppose we have a map v :M* -! N* of MU*-modules.
We then have a unique map u: M -! N such that MU*u = f(v). A diagram chase
shows that u is a map of MU-module spectra (up to a phantom term which is zero *
* __
as usual), and f(ss*(u)) = f(v) so ss*(u) = v. Thus ss* is an isomorphism. *
* |__|
Proposition 2.20.If A 2 E and ss*(A) is a commutative MU*-algebra then there
is a unique product on A making it into a commutative MU-algebra spectrum.
14 M. HOVEY AND N. P. STRICKLAND
Proof.For any X and Y we have a pairing MU*X MU* MU*Y -! MU*(X ^ Y )
and thus a pairing
A*(X) A*(Y ) = A* MU* MU*X A* MU* MU*Y -!
A* MU*(X ^ Y ) = A*(X ^ Y ):
This is easily seen to be commutative, associative and unital. By writing A as
a minimal weak colimit of finite spectra Affand taking X = DAff, Y = DAfi
we construct a map A ^ A -! A which is well-defined, commutative, associative
and unital up to phantoms. However, all the relevant phantom groups vanish_by_
Proposition 2.18 and Corollary 2.15. |__|
Proposition 2.21.If A is a Landweber exact ring spectrum and M is an A-module
spectrum then there are universal coefficient spectral sequences of A*-modules
Exts;tA*(A*X; M*)=) Mt+sX
TorA*s;t(M*; A*X)=) Mt+sX:
Proof.The first spectral sequence follows from Proposition 2.12, Proposition 2.*
*13_
and [Ada74, Theorem 13.6]. The second is constructed by the same methods. |__|
2.2. The E-based Adams spectral sequence. We next briefly recall an ap-
proach to the E-based Adams spectral sequence that we learnt from Mike Hop-
kins, which is explained in more detail in [Dev96b]. This approach is also used
by Franke [Fra96], who attributes it to Brinkmann. Let A be an even Landweber
exact commutative ring spectrum; in our applications, A will be E or E(n). By
Proposition 2.16, A*A is flat as a left module over A*. Similarly, it is flat a*
*s a right
module. It follows in the usual way that it is a Hopf algebroid, so we can thin*
*k about
comodules over A*A. If I* is an injective module over A* then the extended como*
*d-
ule A*A A* I* is injective. It follows that there are enough injective comodule*
*s,
and that they can be used to define Extgroups. If J* is an injective comodule t*
*hen
Brown representability gives a spectrum W such that [X; W ] = Hom A*A(A*X; J*)
for all X. The identity map of W corresponds to a map A*W -! J*, which we
claim is an isomorphism. Indeed, when X 2 EF we know that A*X is free over A*
and using duality we find that
X*W = [DX; W ] = Hom A*A(A*DX; J*) = Hom A*A(A*; A*X A* J*):
By Proposition 2.13, we can write A = mwlim-!Afffor Aff2 EF. By taking X = Aff
and passing to the limit we find that
A*W = Hom A*A(A*; A*A A* J*) = J*:
There is some inconsistency in the literature about what to call spectra such*
* as
W . We will use the following terminology.
Definition 2.22.Let A be a ring spectrum. We say that a spectrum X is A-
injective if it is a retract of A ^ Y for some Y . Suppose in addition that A*A*
* is flat
as a module over A*. We say that X is strongly A-injective if A*X is an injecti*
*ve
comodule and the natural map [Z; X] -! Hom A*A(A*Z; A*X) is an isomorphism
for all Z. Note that if X is A-injective then A*X is injective relative to A*-s*
*plit
exact sequences, but not necessarily absolutely injective.
MORAVA K-THEORIES AND LOCALISATION 15
If X = X0 is any spectrum then we can embed A*X0 in an injective comodule
J* and define W = W0 as above. We then have a map X0 -!W0, and we let X1 be
the fibre. Continuing in the obvious way, we get a tower
X = X0- X1- X2- : :::
For any spectrum Y we can apply the functor [Y; -] to get a spectral sequence
Es;t2= Exts;tA*A(A*Y; A*X) =) [Y; LAX]t-s:
We call this the modified Adams spectral sequence (MASS) . It need not converge
without additional assumptions. We will prove a convergence result in Proposi-
tion 6.5.
We will also have occasion to use the (unmodified) Adams spectral sequence.
For this, we choose a complex X -! I0 -! I1 -! : :o:f E-injective spectra which
becomes a split exact sequence after applying the functor E ^ (-). Such a compl*
*ex
is unique up to chain homotopy equivalence under X. It can be converted into
a tower X = X0- X1- : :j:ust as above, and by applying [Y; -] we get a
spectral sequence, called the Adams spectral sequence. If X is E-nilpotent then*
* this
converges to [Y; X]. If E*Y is projective over E* then the modified and unmodif*
*ied
Adams spectral sequences coincide from the E2 page onwards [Dev96b], and in
particular the E2 page can be identified as an Ext group.
2.3. Operations in E theory. We now turn to the study of operations in E-
theory and K-theory. There is a well-known connection between this and the study
of the Morava stabiliser group, but no really adequate account of this. For thi*
*s and
a variety of technical reasons we have chosen to use a more traditional approac*
*h.
Write b* = E*E for the (non-commutative) ring of operations in E-cohomology.
Note that K*E = Hom (K*E; K*) (by Proposition 2.16 or by the general theory of
modules over a field spectrum) and K*E = K*BP BP*E* because E* is Landweber
exact. Because In is an invariant ideal, it is easy to check that K*E is the sa*
*me as
the ring * = (n)* studied in [Rav86, Chapter VI]. We thus have
n pk-1
K*E = * = Fp[tk | k > 0]=(tpk- vn tk):
Here |tk| = 2(pk - 1), so K*E and * = K*E are in even degrees. It follows from
Proposition 2.5 that b* = E*E is pro-free, and that * = b*=Inb*. Moreover,
because In is an invariant ideal in BP* one can check that Inb* = b*In, and thus
that * is a quotient ring of b*.
Our main result is as follows.
Theorem 2.23. The ring * is left Noetherian in the graded sense.
It is also true that b* and K*K are Noetherian, but we do not need this so we
will not prove it.
As the theory of non-commutative Noetherian rings is less familiar than the c*
*om-
mutative version, we start with some elementary remarks. Let R be a possibly
non-commutative ring. Unless otherwise specified, we shall take "ideal" to mean
"left ideal" and "module" to mean "left module". As in the commutative case, one
checks easily that the following are equivalent:
(a)Every ideal J R is finitely generated.
(b)Every ascending chain J0 J1 : :o:f ideals is eventually constant.
(c)Any submodule of a finitely generated module over R is finitely generated.
16 M. HOVEY AND N. P. STRICKLAND
(d)Every ascending chain of submodules of a finitely generated module over R is
eventually constant.
If so, we say that R is (left) Noetherian. If R is a graded ring and all ideals*
* and
modules are required to be homogeneous, then the corresponding conditions are
again equivalent; if they hold, we say that R is Noetherian in the graded sense.
Lemma 2.24. Let R -!S be a map of rings such that S is finitely generated and
free both as a left R-module and a right R-module. Then S is left Noetherian if*
* and
only if R is left Noetherian. Similarly for the graded case.
Proof.Suppose that R is Noetherian. Then any ascending chain of ideals in S
is a chain of R-submodules of a finitely generated R-module, and thus eventually
constant. L
Conversely, suppose that S is Noetherian, and that S = d-1i=0aiR. Then the
map
M M
J 7! SJ = aiJ ' J
i i
embeds the lattice of ideals in R into the Noetherian lattice of ideals_in S, s*
*o R is
Noetherian. |__|
Lemma 2.25._ Let R* be a graded algebra_over K* (so that K* is_central_in R*).
Write R = R*=(vn-1) and ss :R* -!R for the projection map. If R is Noetherian,
then R* is Noetherian in the graded sense.
Proof.It is enough to show that the map J 7! ssJ embeds_the lattice of homogene*
*ous
ideals of R* into the Noetherian lattice of ideals in R , and thus enough to sh*
*ow
that the set of homogeneous elements in ss-1ssJ is just J. As R*=J is a graded
module over the graded field K*, it is free, generated by elementsPei of degree*
* di
say. Suppose that a 2 R* is homogeneous of degree d; then a = iaiv(d-di)=|vn|*
*nei
(mod J) where ai 2 Fp, and aiPis zero if the indicated exponent of vn is not an
integer._If ss(a) 2 J then iaiss(ei) = 0, but it is clear that {ss(ei)} is a *
*basis for
R=ssJ, so that ai= 0 for all i and thus a 2 J. |___|
Lemma 2.26. Let R be ring, and {Is} a decreasing filtration such that I0 = R a*
*nd
IsIt Is+t. Suppose that R=Is is a finite set for all s, that R = lim-R=Is, and
Q s
that the associated graded ring R0= E0R = sIs=Is+1 is Noetherian. Then R is
Noetherian.
Q
Proof.Let J be a left ideal in R. Then J0= s(J \ Is)=(J \ Is+1) is a left ide*
*al in
R0. It is thus finitely generated, so there are elements ai2 J \ Idi(for i = 1;*
* : :;:m
say) whose images generate J0.PThis means that for any element a 2 J \ Is there
are elements bi such that a = ibiai (mod J \ Is+1). In other words, if K J
is the ideal generated by {a1; : :;:am },Tthen J \ Is K + J \ Is+1. It follows
easily that J = J \ I0 is contained in s(K + Is), which is the closure of K in*
* the
evident topology given by the ideals Is. On the other hand, as R=Is is finite, *
*we
see that R is I-adically compact and Hausdorff. As K is the image of an evident
continuous map Rm -! R, we see that K is compact and thus closed. It follows *
* __
that J = K = (a1; : :;:am ), which is finitely generated as required. *
* |__|
We next recall that for each k > 0, the ideal (tj | 0 < j < k) . * is a Hopf
ideal, so that (k)* = *=(tj | 0 < j < k) is a Hopf algebra, and (k)* is a quoti*
*ent
MORAVA K-THEORIES AND LOCALISATION 17
Hopf algebra of *. We also write S = *=(vn - 1) and S(k) = (k)*=(vn - 1). We
write S* = Hom (S; Fp) and S(k)* = Hom (S(k); Fp). Note that Ravenel [Rav86]
calls these objects (n; k)*, S(n; k) and so on.
Proposition 2.27.If k > pn=(p-1) then the Hopf algebra S(k)* can be filtered so
that the associated graded ring is a commutative formal power series algebra ov*
*er
Fp on n2 generators.
Proof.In this proof, all theorem numbers and so on refer to Ravenel's book [Rav*
*86].
Our proposition is essentially Theorem 6.3.3. That theorem appears to apply to S
rather than S*, but this is a typo; this becomes clear if we read the preceding
paragraph. Some modifications are necessary to replace S* by S(k)*, and anyway
Ravenel does not give an explicit proof of his theorem, so we will fill in some*
* details.
The Hopf algebra filtration of S given by Theorem 6.3.1 clearly induces a fil*
*tration
on S(k). It is easy to see that
E0S(k) = T [tij| i k; j 2 Z=n]
j
as rings, where T [t] = Fp[t]=tp and tijcorresponds to tpi. There is therefore*
* an
automorphism F on E0S(k) that takes ti;jto ti;j+1which has order n. Moreover,
this is a connected graded Hopf algebra (using the grading coming from the filt*
*ration,
so that the degree of tijis the integer dn;iof Theorem 6.3.1). The coproduct is*
* given
by Theorem 4.3.34,_which says that (tij) is the sum of the elements in a certain
unordered list ij. These lists are used in such a way that we may ignore any t*
*erms
which have filtration less than that of tij. We start with the list
___
M ij= {tij 1; 1 tij}:
(This comes from Lemma 4.3.32, using the fact that k > pn=(p-1).) We also recall
the Witt polynomialskwJJdefinedkin-Lemma|4.3.8.J|We are working modulo p so
we have wJ = wp|J| . In E0S(k) we must interpret this as wJ = F kJk-|J|w|J|.
We have set vn = 1 and killed all other v's, so vJ = 0 unless J has the form
J = Jr = (n; : :;:n) (with r terms), in which case vJ = 1. With these observati*
*ons,
Lemma 4.3.33 becomes
__ ___ -r __
ij= M ij[ {F wr( i-nr;j) | r > 0}:
___ __
As M ijis invariant under the twist map, we see that the same holds for ij, a*
*nd
thus that E0S(k) is cocommutative. We also see that
(tij) = tij 1 + 1 tij+ w1(ti-n;j-1 1; 1 ti-n;j-1) (mod trs| r < i - n):
Here w1 is given by
X p X
w1(x1; x2; : :):= ( xt - ( xt)p)=p:
t t
It is not hard to conclude that the Verschiebung is
V (tij) = ti-n;j-1 (mod trs| r < i - n):
We also observe that the degree of tijis less than that of tklwhenever i < k (t*
*his
follows easily from the definition in Theorem 6.3.1). It follows by induction *
*on i
that each tijlies in the image of V , so that V is surjective.
We now dualise, and conclude that E0S(k)* is a bicommutative connected graded
Hopf algebra for which the Frobenius map is injective. We can thus apply the Bo*
*rel
structure theory [Spa66, Section 5.8][Bor53] to conclude that E0S(k)* is a form*
*al
18 M. HOVEY AND N. P. STRICKLAND
power series algebra. One can check that there must be n2 generators, correspon*
*ding_
to the tk+i;jfor 0 i < n. |__|
We can now prove as promised that * is Noetherian.
Proof of Theorem 2.23.According to Lemma 2.25, it is enough to check that S* is
Noetherian. By Lemma 2.26, it is enough to check that E0S* is Noetherian. We
have an extension of Hopf algebras
n
S0(k) = Fp[t1; : :;:tk-1]=(tpj- tj) ae S i S(k);
and thus an injective map of connected graded Hopf algebras E0S(k)* ae E0S*.
It follows from the Milnor-Moore theorem (the dual of [Rav86, Corollary A1.1.20*
*])
that E0S* is a free (as a left or right module) over E0S(k)*. The rank is just
dim(S0(k)) < 1. Thus, by Lemma 2.24, it is enough to check that E0S(k)* is __
Noetherian, and this follows from Proposition 2.27. |__|
We next show that * is local in a suitable sense. Let I be the kernel of the
augmentation map * -!K*.
Proposition 2.28.* = lim-*=Ik.
k
Proof.This follows easily from the fact [Rav86, Theorem 6.3.3] that S* has a fi*
*l-
tration whose associated graded ring is a graded connected Hopf algebra_of fini*
*te
type. |__|
This gives us a version of Nakayama's lemma.
Corollary 2.29.Let M be a finitely generated graded module over *. If IM = M
then M = 0.
Proof.If M = *m is cyclic and IM = M,Pthen m = am for some a 2 I. This
means that (1 - a)m = 0, but the sum kak converges to an inverse for (1 - a),
so M = 0 as required.
More generally, suppose M = *{m1; : :;:mk} and IM = M. Write M0 =
*{m1; : :;:mk-1}, so that N = M=M0 is cyclic and IN = N. It follows that __
N = 0, so M = M0, and the claim follows by induction on k. |__|
This means that the usual theory of minimal projective resolutions applies.
Corollary 2.30.Let M be a finitely generated graded module over *. Then M
has a resolution P* -!M by finitely generated free modules over *, such that the
maps Pk=IPk -! Pk-1=IPk-1 are zero. This is a retract of any other projective
resolution, and is unique up to non-canonical isomorphism.
Proof.Suppose that the first k stages
M- P0- : :-: Pk-1
have been constructed. Let N be the kernel of the differential Pk-1 -! Pk-2. As
Pk-1 is finitely generated and * is Noetherian, we see that N is finitely gener*
*ated.
We can thus choose finitely many elements n1; : :;:nr in N giving a basis for N*
*=IN
over K*. Let Pk be a direct sum of r copies of * (suitably shifted in degree), *
*and
let Pk -! N -! Pk-1 be the obvious map. We leave it to the reader to check_that
this does the job. |__|
MORAVA K-THEORIES AND LOCALISATION 19
3.K-injective spectra
We say that a spectrum X is K-injective if it is equivalent to a wedge of sus*
*pen-
sions of copies of K. We recall a number of facts about such spectra, and prove*
* a
few more.
Remark 3.1.It will follow from the results below that this is consistent with D*
*efi-
nition 2.22.
We start by considering K-module spectra. If p > 2 then K is commutative, and
if p = 2 it has a canonical antiautomorphism O with O2 = 1 (see [Nas96]), so we
can convert freely between left and right modules.
Proposition 3.2.Suppose that R is a ring spectrum, M is an R-module spectrum,
and X is arbitrary. Then there are natural R-module structures on M ^X, F (X; M)
and F (M; X). Moreover, if R = K, then M is K-injective.
Proof.For the first part, let : R ^ M -! M be the R-module structure map. We
can make M ^ X into a R-module with structure map ^ 1X . We can use the
following composite as a structure map for F (X; M):
R ^ F (X; M) = F (S; R) ^ F (X; M) ^-!F (X; R ^ M) *-!F (X; M):
Finally, the structure map R ^ F (M; X) -! F (M; X) is adjoint to the following
composite:
M ^ R ^ F (M; X) o^1--!R ^ M ^ F (M; X) ^1--!M ^ F (M; X) eval--!X:
For the last part, note that K* is a graded field, so M* is necessarily a free *
*module
over K*, say M* = K*{ei | i 2 I}.WEach map ei:S|ei|-!M gives a K-module
map |ei|K -! M, so we have a map i|ei|K -! M which is an equivalence_by_
construction. |__|
The following proposition (most of which appears in [HS ]) summarises the main
facts that we need.
Proposition 3.3.
(a)For any spectrum X, the smash product K ^ X is K-injective.
(b)A spectrum X is K-injective if and only if it admits a K-module structure.
(c)Any retract of a K-injective spectrum is K-injective.
(d)If X is K-injective, then ss*X has a natural structure as a K*-module. Any
map f : X -! Y of K-injective spectra induces a K*-linear map f* : ss*X -!
ss*Y .
(e)If f : X -! Y is a map of K-injective spectra such that f* is a split monom*
*or-
phism of K*-modules, then f is a split monomorphism and the cofibre of f
is K-injective. If f is compatible with given module structures on X and Y ,
then the splitting may also be chosen compatibly. Similarly for epimorphism*
*s.W
(f)IfQ{Xi} is a family of K-injective spectra, then the natural map iXi -!
iXi is a split monomorphism.
Proof. (a)K ^ X is clearly a K-module, and thus K-injective by Proposition 3.2.
(b)It is clear that a K-injective spectrum admits a module structure; the conv*
*erse
is just Proposition 3.2 again.
(c)This is proved after (d).
20 M. HOVEY AND N. P. STRICKLAND
(d)Any choice of K-module structure on X gives a K*-module structure on ss*X.
Suppose that we have two different module structures, given by maps ff; fi *
*:K^
X -! X. We then have a map
fl = (K ^ K ^ X -1^ff-!K ^ X -fi!X):
The two different vn-multiplications are given by composing fl with the two
maps
ff0= j ^ vn ^ 1: |vn|X -! K ^ K ^ X
fi0= vn ^ j ^ 1: |vn|X -! K ^ K ^ X
and applying ss*. If X = K then a well-known calculation shows that ss*(ff0*
*) =
ss*(fi0), so this remains true for any K-injective spectrum X. It follows t*
*hat
the two K*-module structures coincide.
Before proving the rest of (d), we need to prove a part of (f).WNamely, g*
*iven
aQset I and integers di for i 2 I, we show that the natural map idiK -!
idiK is a split monomorphism. Note that this map is a mapLof K-module
spectra,Qso induces the usual monomorphismLof K*-modules idiK*Q-!
idiK*. Extend the usual basis {ei}of idiK* to a basis of idiK* by
adding new basis elementsWfjWfor j 2 J. JustQas in the proof of Proposition*
* 3.2,
we get an equivalence idiK_ j|fj|K -! idiK. This gives an obvious
left inverse to the inclusion of the wedge, as required.
Now consider a map f :X -! Y of K-injective spectra. We claim that
f*: ss*X -! ss*Y is K*-linear. Note that X is a wedge of K's, and we have
just shown that Y is a retract of a product of K's; this reduces us easily
to the case X = Y = K (up to suspension). The usual theory of flat ring
spectra shows that the graded map r*: E* -! E* induced by a graded map
r :E -!E (that is, an element r 2 E*E = Hom E*(E*E; E*)) is the composite
E* jR-!E*E -r!E*. Normally, jR is a homomorphism of right E*-modules,
but for E = K we have jR = jL. Thus, r* is K*-linear.
(c)Let X be K-injective, and e: X -! X an idempotent. By (d) we know that
e* is K-linear, and thus ss*X splits as a direct sum ss*eX ss*(1 - e)X of *
*K*-
modules. By choosing a basis for ss*X adapted to this splitting and proceed*
*ing
as in the proof of Proposition 3.2, we see that eX is K-injective.
(e)Let f :X -! Y be a map of K-injective spectra such that f*: X* -! Y* is
a monomorphism. Note that the image is a K*-submodule by (d). Choose
elements {ei 2 Y* | i 2 I} giving a basis for Y*=f*X*. Choose a K-module
structure on Y , and use it to convertWthe maps ei:S|ei|-!Y into module
maps |ei|K -! Y , and thus a map Z = i|ei|K -! Y . By construction,
the evident map X _ Z -! Y is an equivalence, so that X -! Y is a split
monomorphism with cofibre Z. Clearly, if f is compatible with given module
structures on X and Y , then the splitting will also be compatible. The pro*
*of
for epimorphisms is similar.
(f)This follows immediately from (e). __
|__|
Recall that * = K*E = (E*E)=In, using the invariance of In. There is thus a
natural coaction of * on (E*X)=In induced from the coaction of E*E. In particul*
*ar,
if X is such that E*X is annihilated by In then E*X is a comodule over *.
MORAVA K-THEORIES AND LOCALISATION 21
Proposition 3.4.Let M be a K-module spectrum, and X a spectrum. Then there
are natural isomorphisms
M*X ' M* K* K*X
[X; M]*' Hom K*(K*X; M*)
E*M ' * K* M*
M* ' Prim* E*M
Proof.The map
M ^ K ^ X -mult^1---!M ^ X
induces a natural map M* K* K*X -! M*X, which is visibly an isomorphism
when X is a sphere. Both sides are homology theories because M* is free over K*,
so the map is an isomorphism for all X. The second part is similar.
The third part follows from the first part, since we have
E*M = M*E = M* K* K*E = M* K* *:
A diagram chase shows that this is actually an isomorphism of comodules (where *
*__
M* has trivial coaction) from which the fourth part is immediate. |*
*__|
4.Generalised Moore spectra
We now discuss generalised Moore spectra. The most important ideas are due
to Hopkins and Smith [HS ], and are also explained in [Rav92a, Chapter 6]. Here
we offer some convenient technical improvements.
In this section, if we write w :X -! Y then we always allow the possibility t*
*hat
w has nonzero degree, but we always insist that the degree should be even.
We shall consider vn self maps with two extra properties. The first is strong
centrality in the following sense:
Definition 4.1.Let X be a finite spectrum. We say that a self map w :X -! X
is strongly central if 1X ^ w is central in End(X ^ X)*.
It seems that this condition was first considered by Ethan Devinatz [Dev92]. *
*It
turns out to have surprisingly strong implications. To explain this, we need t*
*he
following definition.
Definition 4.2.If X is a finite spectrum, we let JX be the full subcategory of
(possibly infinite) spectra Y such that Y may be written as a retract of some s*
*pec-
trum of the form X ^ Z. (Note that we do not take such a retraction as part of *
*the
structure of Y .)
Remark 4.3.Let R be the finite ring spectrum End(X) = F (X; X) = DX ^ X.
If Y is an R-injective spectrum then Y is a retract of R ^ Y = X ^ (DX ^ Y ),
so Y 2 JX . Conversely, if Y 2 JX then Y is a retract of some spectrum X ^ Z,
which is an R-module (because X is). It follows that JX is precisely the catego*
*ry
of R-injective spectra, and thus that it is closed under products, coproducts, *
*and
retracts. Moreover, if Y 2 JX and U is an arbitrary spectrum then Y ^ U, F (U; *
*Y )
and F (Y; U) are all in JX (by Proposition 3.2).
Later in this section we will prove the following result.
22 M. HOVEY AND N. P. STRICKLAND
Proposition 4.4.Let v be a strongly central self-map of a finite spectrum X. Th*
*en
there is a unique natural transformation vY :Y -! Y for Y 2 JX , such that vX^Z*
* =_
v ^ 1Z for all spectra Z. Furthermore, vY ^Z= vY ^ 1Z for all spectra Z. *
*|__|
This means that when we work with spectra in JX we can pretend that v is an
element of ss*S. We will often write v instead of vY .
Our second extra condition involves the group of MU-module self maps of MU ^
X. This can be interpreted in two different ways. On the one hand, we can work
entirely in the homotopy category of spectra. The spectrum MU is a ring object
in this category, so we can consider the category of modules over it; we shall *
*call
this CMU . On the other hand, we can work in the derived category DMU of strict
MU-modules, as explained in Section 1. In the present context it does not matter
which interpretation we use, because
CMU (MU ^ X; M)* = DMU (MU ^ X; M)* = [X; M]*:
In particular,
CMU (MU ^ X; MU ^ X)* = [X; X ^ MU]* = MU*(DX ^ X):
Definitionk4.5.Let X be a p-local finite spectrum of type at least n > 0. We say
that w :p |vn|X -! X is a good vn self map if
1.1 ^ w = vpkn^ 1 in CMU (MU ^ X; MU ^ X).
2.w is strongly central.
Every good vn self map is a vn self map in the sense of Hopkins and Smith [HS*
* ].
Indeed, we shall eventually prove the following result:
Theorem 4.6. Any finite spectrum X of type atileastjn > 0 admits a good vn self
map, and if v and w are two such maps then vp = wp for some i and j. Moreover,
for any MU-module spectrum M 2 CMU we have
k
1 ^ w = vpn^ 1: M ^ X -! M ^ X:
If M is an object of DMU , then the above equation also holds when interpreted
in DMU . In any case,kit follows that the induced map w*: M*X -! M*X is just
multiplication by vpn. |___|
We now start work on the proofs.
Lemma 4.7. Let w be a strongly central self map of a finite spectrum X. Then
w ^ 1X = 1X ^ w in End(X ^ X)*. Moreover, for any spectra Y; Z and any map
u: X ^ Y -! X ^ Z , the following diagram commutes:
X ^ Y _____wXu^ Z
| |
w^1 | |w^1
|u |u
X ^ Y _____wXu^ Z
(In particular, by taking Z = Y we see that w ^ 1 is central in End(X ^ Y )*; in
particular, by taking Y = S we see that w is central in End(X)*.)
Proof.Let w be a strongly central self map of a finite spectrum X. Let o 2
End(X ^ X) be the twist map. As 1X ^ w is central, we have
1X ^ w = o O (1X ^ w) O o = w ^ 1X :
MORAVA K-THEORIES AND LOCALISATION 23
Now consider a map u: X ^ Y -! X ^ Z, and write
g = u O (w ^ 1Y ) - (w ^ 1Z) O u;
so the claim is that g = 0. By rewriting 1X ^w as w^1X , we see that both compo*
*sites
in the following diagram are w ^ u, and thus that the diagram commutes.
X ^ X ^ Y _____wX1^^Xu^ Z
| |
1^w^1 | |1^w^1
|u |u
X ^ X ^ Y _____wX1^^Xu^ Z
This means that 1X ^ g = 0; a fortiori we have 1DX^X ^ g = 0. On the other han*
*d,
X ^ Z is a module-spectrum over DX ^ X = End(X), so the map
j ^ 1X^Z :X ^ Z -! DX ^ X ^ X ^ Z
is a split monomorphism. By considering the following diagram, we conclude that
g = 0 as claimed.
g
X ^ Y ______________Xw^ Z
v v
j^1| |j^1
| |
|u |u
DX ^ X ^ X ^ Y _____wDX1^^Xg^=X0^ Z
|___|
Proof of Proposition 4.4.Let Y be a spectrum in JX . We can choose a spectrum
Z and maps Y -j!X ^ Z -q!Y such that qj = 1Y . By naturality, we must define
vY = q(v ^ 1)j :Y -! Y . Suppose that we have a different spectrum Z0 and maps
0 q0
Y j-!X ^ Z0 -! Y with q0j0 = 1Y . We claim that q0(v ^ 1)j0 = q(v ^ 1)j, so
that our definition is independent of the choice of Z, j and q. To see this, no*
*te that
Lemma 4.7 gives j0q(v ^ 1Z) = (v ^ 1Z0)j0q. We thus have
q(v ^ 1)j = q0j0q(v ^ 1)j = q0(v ^ 1)j0qj = q0(v ^ 1)j0
as required. It follows that vY ^Z= vY ^ 1Z as well.
Now suppose that we have a morphism f :Y0 -! Y1 in JX . We need to show
that v is natural with respect to f, or in other words that the central square *
*in the
following diagram commutes:
q0 f j1
X ^ Z0 _____Y0ww______wY1 v____X_^wZ1
| | | |
| v | |v |
v^1| Y0| | Y1 |v^1
| | | |
|u |u |u |u
X ^ Z0 _____Y0wwq0____wY1fv____X_^wZ1j1
Because Y0; Y1 are in JX , we can choose spectra Z0, Z1 and a split monomorphism
j1 and split epimorphism q0 as shown. By the definition of v given above, the o*
*uter
squares commute. The total rectangle commutes by Lemma 4.7. It follows easily_
that the middle square commutes, as required. |__|
24 M. HOVEY AND N. P. STRICKLAND
Definition 4.8.A -spectrum is a spectrum X equipped with maps S -j!X -
X ^ X such that
O (j ^ 1) = 1 : X -! X:
(We reserve the term ring spectrum for examples in which is associative and j *
*is
a two-sided unit.) If X is a -spectrum, then ss*X is a (possibly non-associativ*
*e)
graded ring, with possibly only a left unit. A module spectrum over X is a spec*
*trum
Y equipped with a map : X ^ Y -! Y such that O (j ^ 1) = 1 : Y -! Y . Note
that this is independent of . If X -! Y is a map of -spectra, then Y has an
obvious structure as an X-module; in particular, X is an X-module. We shall say
that a -spectrum X is atomic if every map f :X -! X such that fj = j is an
isomorphism.
Definition 4.9.Given a finite -spectrum X and an element v 2 ssdX, we define
(v) = (dX -v^1-!X ^ X -! X) 2 [X; X]d:
If (v) is a strongly central self map of X, we say that v is a strongly central*
* element
of ss*X. If in addition the Hurewicz image of v in MU*X is (the image under the
unit map of) vpknfor some k, we say that v is a good vn element . Given a self *
*map
w :dX -! X, we define j*(w) = w O j 2 ssdX.
Proposition 4.10.Let X be a finite, atomic -spectrum. Then every strongly
central self map of X is a map of X-modules. Moreover, the set of such maps
forms a commutative and associative ring under composition, which is isomorphic
to the ring of strongly central elements of ss*X via and j*. This also induces*
* a
bijection between good vn self maps and good vn elements.
Proof.Write A* for the set of strongly central self maps of X; this is clearly a
commutative and associative graded ring under composition. Write B* for the set
of strongly central elements of ss*X, so we have a map : B* -!A*.
Consider w 2 A*. We first claim that w is a map of X-modules, in other words
that the following diagram commutes.
X ^ X _____wX1^^Xw
| |
| |
|u |u
X _________wXw
This is just Lemma 4.7 (with Y = X; Z = S; u = ).
Next, we claim that w = (j*w). To see this, write v = j*w and consider the
following diagram.
X|[_______________Xww|[
| [ | [
| |
| [ | [
| [ | [
| [ | [1
j^1 | v^1=j^w |j^1
| [ | [
| [ | [
| [ | [
| [] | []
|u |u
X ^ X ____________Xw^wX^1=1^w_________wX:
MORAVA K-THEORIES AND LOCALISATION 25
We remarked earlier that w^1 commutes with the twist map and thus w^1 = 1^w.
By thinking of this map as 1 ^ w we see that the square commutes and that the
diagonal map is j ^ w. By thinking of it as w ^ 1 instead we see that the diago*
*nal
can also be described as v ^ 1. The right hand triangle commutes by the axiom
for a -spectrum. By looking at the long composite in the diagram we see that
w = (j*w) as claimed.
This means that v 2 B*, so we have maps
*
A* j-!B* -!A*
with j* = 1.
Next, consider two strongly central self maps w; w0, and write v = j*w and
v0= j*w0. We claim that j*(ww0) = vv0 (which means O (v ^ v0)). This follows
by inspecting the following diagram:
0
S [____________Xw^vX^v
| [ 0 aeo|
j| v v^1 |
| [] ae |
|u ae |u
X _____wXw0____wX:w
Thus, j* is a ring map from A* to B*.
Next, consider the map = O (1 ^ j): X -! X. As we do not assume that j is
a two-sided unit for , this need not be the identity. However, it is easy to se*
*e that
j = j; as X is atomic, we conclude that is an isomorphism. For any v 2 B*, we
have a commutative diagram as follows:
S _______wXv[
| | [
| |
j| 1^j| [
| | []
|u |u
X _____wXv^^X1_____wX
This shows that j*(v) = O v, so that : B* -!A* is injective. As j* = 1, it is
not hard to see that and j* are mutually inverse isomorphisms.
We leave it to the reader to check that good vn self maps go to good_vn_eleme*
*nts
and vice versa. |__|
In view of the above proposition, we allow ourselves to write v for (v), when*
* v
is a strongly central homotopy element for a finite, atomic -spectrum.
Proposition 4.11.Let X be a finite atomic -spectrum, and v 2 ss*X a strongly
central element. Suppose that k > 1, and let X=vk be the cofibre of vk: X -! X.
Then X=vk can be made into a -spectrum in such a way that the map q :X -!
X=vk is a map of -spectra and the maps
k q d
X -v!X -! X=vk -!X
are maps of X-modules.
Proof.This is essentially due to Ethan Devinatz [Dev92]. His Theorem 1 states
that if w :X -! X is a self map of a finite -spectrum and
(i)w is strongly central
(ii)w2 is a map of X-modules
26 M. HOVEY AND N. P. STRICKLAND
then X=w2 can be made into a -spectrum, with properties essentially above. Our
Proposition 4.10 shows that hypothesis (ii) is redundant. We assume that X is
atomic so that we can translate cleanly between (powers of) strongly central se*
*lf
maps and homotopy elements. This proves the case k = 2 of our claim. The general
case is essentially the same; in Devinatz' Lemma 5, one simply has to apply Ver*
*dier's
k-1 v v v __
axiom to the maps X -v--!X -! X rather than X -! X -! X. |__|
Definition 4.12.Consider an ideal I MU* of the form
I = (va00; : :;:van-1n-1);
where each aiis a power of p. We call n the height of I. We give a recursive de*
*finition
of "generalised Moore spectra of type S=I"; such a thing will be a certain kind*
* of
finite, atomic -spectrum. The only spectrum of type S=0 is S, equipped with the
obvious structure. Consider an ideal I as above, and write J = (va00; : :;:van-*
*2n-2).
A -spectrum X has type S=I if there is a -spectrum Y of type S=J with a good
vn-1 self map v of degree |van-1n-1| and a cofibration X -v!X -q!Y -d!X such th*
*at
q is a map of -spectra and q and d are maps of X-modules.
If a spectrum of type S=I exists, we shall refer to it as S=I; this is an abu*
*se,
as there may be many non-isomorphic spectra of type S=I. It is easy to see that
the unit map S -! S=I induces an isomorphism MU*=I ' MU*(S=I). It follows
that any map f :S=I -! S=I with f O j = j satisfies MU*f = 1 and thus is an
isomorphism; this means that S=I is atomic. We can also construct an object
MU=I 2 DMU by the methods of [EKMM96 , Chapter V] or [Str96], and it is easy
to check that MU ^ S=I is isomorphic in DMU to MU=I.
Proposition 4.13.Suppose that there is a spectrum X of type S=I; then it has a
good vn element, and thus S=(I; vpjn) exists for j 0.
Proof.This is based on results in [HS ]; we refer to Ravenel's account [Rav92a]*
* as
it is more readily available.
By [Rav92a, Theorem 1.5.4], there exists k 0 and a map w 2 End(X)* such
that K(m)*w = 0 for m 6= n, and K*w = K(n)*w is an isomorphism. By [Rav92a,
Lemma 6.1.1], we may assume that K*w = vrnfor some r. By examining the
proofs, we see that r may be assumed to have the form pk. It followskthat 1 ^ w*
* 2
End(X ^ X)* also has K(m)*(1 ^ w) = 0 for m 6= n and K*(1 ^ w) = vpn. Thus,
by [Rav92a, Lemma 6.1.2], the map 1 ^ wpiis central in End(X ^ X)* for large i.
After replacing w by wpi, we may assume that w is a strongly central self map of
X. Write v = j*w 2 ss*X, so that v is a strongly central element. Note that v
maps to a primitive element v0 in the MU*MU-comodule MU*X = MU*=I. As
Prim(MU*=In) = Fp[vn], we conclude that v0= ffvpkn(mod In) for some ff 2 Fp.
Recall that K*v = vpkn; by comparing the unit maps, we conclude that ff = 1.
Using [HS , Lemma 3.4], we see that (v0)pi = vpk+infor i 0; we may replace v
by vpi and thus assume that v0 = vpkn. This means that v is a good vn element,
as required. Using Proposition 4.11, we see that S=(I; vpjn) = X=vpj-kexists_wh*
*en
j > k. |__|
Corollary 4.14.For any ideal J MU* with radical In, there exists an ideal *
* __
I = (va00; : :;:van-1n-1) such that I J and S=I exists. *
*|__|
MORAVA K-THEORIES AND LOCALISATION 27
Corollary 4.15.Any finite spectrum X of type at least n admits a good vn self
map.
Proof.Let C be the category of those X that admit a good vn self map. Spectra of
type S=I (where I has height n) lie in C, so it will suffice to check that C is*
* thick.
It is clearly closed under suspensions. Suppose that X 2 C and that e: X -! X is
idempotent. Write Y = eX and Z = (1 - e)X, so that X = Y _ Z. Choose a good
vn self map w :X -! X. As w is central, it commutes with e, so it must have the
form u _ v where u: Y -! Y and v :Z -! Z. It is easy to check that u and v are
good vn self maps, so C is closed under retracts.
Now consider a cofibration X -f!Y -g!Z with X; Z 2 C. For k 0, we cank
choose good vn self maps u: X -! X and w :Z -! Z with |u| = |w| = |vpn|. We
may also choose a compatible fill-in map v :Y -! Y . After replacing u; v and w
by suitable iterates, we may assume that v is strongly central. Indeed, u and w
are strongly central, so 1 ^ v - v ^ 1 is nilpotent. It follows from [HS , Lemm*
*a 3.4]
that vpjkis strongly central for large j. For typographical convenience, we wr*
*ite
x = vpn:MU -! MU. We thus have a commutative diagram
1^f 1^g
MU ^ X _____wMU ^ Y _____wMU ^ Z
| | |
x^1| 1^v| |x^1
| | |
|u |u |u
MU ^ X _____wMU1^^Yf _____wMU1^^Zg
Write d = 1^v-x^1 2 End(MU ^Y )*. Note that dO(1^f) = 0 and (1^g)Od = 0,
so that d can be written in the form (1 ^ g) O r and also in the form s O (1 ^ *
*f). Thus
d2 factors through (1 ^ f)(1 ^ g) = 0, so d2 = 0. As x ^ 1 commutes with 1 ^ v,*
* the
same is true of d. Thus we have
i pi i pi-1
1 ^ vp = x ^ 1 + p (x ^ 1)d:
Moreover, we may assume that n > 0 so that End(MU ^ Y )* is a torsion group. It
follows that
i pi pk+i
1 ^ vp = x ^ 1 = vn ^ 1
for i 0, as required. |___|
Proof of Theorem 4.6.Let X be a finite spectrum of type at least n. Wekknow
from Corollary 4.15 that X admits a good vn self map v, with |v| = |vpn| say. L*
*et
w be another such map. By the asymptotic uniqueness of vn self maps [Rav92a,
Lemma 6.1.3], we know that vN = wM for some N and M. By inspecting the
proof (bearing in mind that MU*v and MU*w are powers of vn and not merely unit
multiples of powers of vn) we see that M and N may be taken to be powers of p.
Now let M be a module-spectrum over MU in the classical sense, so M 2 CMU ;
let :M ^MU -! M be the structure map. Using the bijection CMU (MU ^X; MU ^
X) = [X; MU ^ X], we find that
k
j ^ v = vpn^ 1: X -! MU ^ X:
By applying M ^ (-) and composing with ^ 1X , we find that
k
1 ^ v = vpn^ 1: M ^ X -! M ^ X
as claimed.
28 M. HOVEY AND N. P. STRICKLAND
On the other hand, let N be an MU-module in the strict sense, so N 2 DMU .
We know that
k
1 ^ v = vpn^ 1 2 CMU (MU ^ X; MU ^ X) = DMU (MU ^ X; MU ^ X):
We can apply the functor N ^MU (-): DMU -!DMU to this equation and conclude
that
k
1 ^ v = vpn^ 1 2 DMU (N ^ X; N ^ X)
as claimed. |___|
Proposition 4.16.If S=I is a generalised Moore spectrum and X is an S=I-
module then S=I ^ X is a wedge of finitely many suspended copies of X.
Proof.We may assume that I has nonzero height, so S=I = S=(J; v) for some
generalised Moore spectrum S=J and some good vn element v. Note that there is a
map S=J -! S=I of -spectra so X is a module over S=J. Thus X 2 JS=J and vX is
defined. By induction we may assume that W = X ^ S=J is a wedge of suspended
copies of X, and thus that X ^ S=I is the cofibre of v :W -! W . However, it is
clear that vS=I= 0 and X is a retract of S=I ^ X so vX = 0 so vW = 0. It follo*
*ws_
that X ^ S=I is a wedge of suspended copies of X, as claimed. |__|
Proposition 4.17.Let X be a spectrum. Then the following are equivalent:
(a)X is a retract of some spectrum of the form Y ^Z, where Z is a finite spect*
*rum
of type at least n.
(b)There is a generalised Moore spectrum S=I of type n such that X is a module
over S=I.
Moreover, the category of such X is an ideal.
Proof.(a))(b): It is enough to show that Z is a module over some S=I of type
n. By induction, we may assume that it is a module over some S=J of type n - 1.
Let v be a good vn-1 self map of S=J. As K(n - 1)*Z = 0 and K(m)*v = 0 for
all m 6= n - 1, the Nilpotence Theorem tells us that v ^ 1: S=J ^ Z -! S=J ^ Z *
*is
nilpotent, say vpN ^ 1 = 0. It follows that Z is a module over S=(J; vpN).
(b))(a): Clear.
Suppose that X -! Y -! Z is a cofibration and that X and Z satisfy (b). We
claim that Y satisfies (b) as well. By induction, we may assume that X, Y and Z
are all modules over some S=J of type n - 1. Let v be a good vn-1 self map of S*
*=J.
By the argument above we seeNthat vX and vZ are nilpotent, and it follows easily
that vY is nilpotent, say vpY = 0. It follows that Y is a module over S=(J; vpN*
*).
This shows that the category in question is thick. Using (a) it is trivial_to s*
*ee that
it is an ideal. |__|
It is sometimes convenient to know that S=I is self-dual.
Proposition 4.18.Let X be a spectrum of type S=I, and let d be the dimension
of its top cell. Then there is an isomorphism dDX = X.
Proof.Write I = (J; vpkn), so there is a generalised Moore spectrum Y of type S*
*=J
and a good vn self map v of Y such that X = Y=v. We can assume by induction
that Y is self-dual. Given any map f :U -! V , Df is the composite
DV -1^j-!DV ^ U ^ DU -1^f^1---!DV ^ V ^ DU -ffl^1-!DU
MORAVA K-THEORIES AND LOCALISATION 29
In particular, 1DY = D(1Y ) = (ffl ^ 1) O (1 ^ j), so DY is in the category JY*
* and
we have a self map vDY of DY . Using the naturality of v with respect to 1 ^ j *
*and
ffl ^ 1 we see that vDY = Dv. Now let t : Y -! DY be a self-duality isomorphism,
where we have omitted the suspension. By naturality, we have t O v = Dv O t. He*
*nce
t induces an isomorphism from Y=v to (a suspension of) DY=Dv = D(Y=v),_as
required. |__|
4.1. Towers of generalised Moore spectra. We next try to build a tower of
spectra S=I(j), where the I(j) all have height n, their intersection is trivial*
*, and
the maps are compatible with the unit maps S -! S=I(j). We start with some
generalities about towers (or more general pro-systems) of finite spectra. The*
*se
ideas also appear in [CS96 ].
Let F be the category of finite spectra. Its pro-completion is the dual of t*
*he
category of those functors F :F -!Setsthat can be written as a small filtered c*
*olimit
of representable functors. Consider two such functors, say F Z = lim-![Xi; Z] a*
*nd
i2I
GZ = lim-![Yj; Z]. Using the Yoneda lemma, we see that
j2J
Pro(F)(F; G) = [F; Sets](G; F ) = [F; Sets](lim-![Yj; -]; lim-![Xi; -]) = lim-l*
*im-![Xi; Yj]:
j i j i
An alternate definition of Pro(F) is as the category of pairs (I; X), where I i*
*s a small
filtered category, {Xi} is an Iop-indexed system of finite spectra, and the mor*
*phisms
from (I; X) to (J; Y ) are given by the above formula. This clearly gives a cat*
*egory
equivalent to the one described previously.
Note that the Yoneda functor X -! [X; -] gives a full and faithful covariant
embedding of F in Pro(F), which we think of as an inclusion. Every object in
Pro(F) is an inverse limit of objects of F (because inverse limits in Pro(F) ar*
*e the
same as direct limits in the functor category Pro(F)op).
There is a popular full subcategory Tow (F) of Pro(F) consisting of limits of
towers (or equivalently, countable filtered systems with countably many morphis*
*ms)
of representable functors. In this case we can be more explicit about the morph*
*isms.
Let Seq(F) be the category of inverse sequences of finite spectra, like
X0- X1- X2- : :::
The maps from Xo to Yo are just systems of maps fk: Xk -!Yk making the evident
diagrams commute. Let U be the directed set of strictly increasing maps N -!N.
If u 2 U and Xo 2 Seq(F) then we define u*Xo 2 Seq(F) by (u*Xo)k = Xu(k). The
maps Xu(k)- Xu(k+1)are the evident composites of the maps Xj- Xj+1. It is
not hard to check that
Tow (F)(lim-Xi; lim-Yj) = lim-!Seq(F)(u*Xo; Yo):
i j U
In particular, a map f :Xo -! Yo becomes zero in the Pro category if and only if
there exists u 2 U such that the composite
fu(k)
Xu(k+1)-!Xu(k)---! Yu(k)
is zero for all k. It is equivalent to say that for all i there exists j > i su*
*ch that
the composite Xj -!Xi fi-!Yi vanishes. In particular (taking f = 1) we see that
lim-Xi = 0 in Pro(F) if and only if for all i there exists j > i such that the *
*map
i
30 M. HOVEY AND N. P. STRICKLAND
Xj -!Xi is zero. Note that this certainly implies that holim-Xi= 0 in S, but is*
* a
much stronger statement.
It turns out that Pro(F) is a rather familiar category in disguise.
Proposition 4.19.Pro(F) is the dual of the category of homology theories (or
equivalently, spectra mod phantoms). Moreover, Tow(F) corresponds to the subcat-
egory of homology theories with countable coefficients.
Proof.Any homology theory defined on finite spectra has an essentially unique
extension to the category of all spectra, so it doesn't matter which we meant.
As filtered colimits are exact, it is clear that any filtered colimit of repr*
*esentable
functors is a homology theory. Conversely, let H be a homology theory. It is not
hard to check that the category I of pairs (X; u) (where X 2 F and u 2 HX)
is filtered, and it is formal that H = lim-![X; -]. Moreover, I is essentially *
*small
I
because F is.
If H has countable coefficients, then I is easily seen to be countable, so th*
*ere_is
a cofinal functor N -!I, so H 2 Tow(F). The converse is also clear. |_*
*_|
Remark 4.20.As the smash product of a phantom map and any other map is phan-
tom, there is an induced smash product on the category of spectra mod phantoms.
We can transfer this to the equivalent category of homology theories, or the du*
*al
category Pro(F). One can easily check that with this definition we have
(lim-Xi) ^ (lim-Yj) = lim-Xi^ Yj = lim-lim-Xi^ Yj:
i j i;j i j
Remark 4.21.One can see either directly from the definitions or from Proposi-
tion 4.19 that an object X 2 Pro(F) is zero if and only if [X; W ] = 0 for all *
*W 2 F,
and that a map f :X -! Y in Pro(F) is an isomorphism if and only if the induced
map [X; W ]- [Y; W ] is an isomorphism for all W 2 F.
It follows from the above proposition that Pro(F) is not a triangulated cate-
gory. Indeed, in a triangulated category every monomorphism is split, but any
map of spectra with phantom fibre gives a non-split monomorphism of repre-
sented homology theories. Nonetheless, some features of triangulated categories
remain. In particular, suppose that we have an inverse system of cofibre sequen*
*ces
Xi -fi!Yi -gi!Zi -hi!Xi. Write X = lim-Xi 2 Pro(F) and so on, so we
i
have maps X -f!Y g-!Z -h!X. For any W 2 F we have an exact sequence
[Xi; W ]- [Yi; W ]- [Zi;*W ]- [Xi;*W ], and passage to direct limits gives a*
*n ex-
*
act sequence [X; W ] -f- [Y; W ] -g- [Z; W ] -h- [X; W ]. In view of Remark 4.2*
*1,
we see that f is an isomorphism if and only if Z = 0.
We next construct some special objects of Pro(F). Fix an integer n and let Fn
be the category of finite spectra of type at least n. Let n be the category of *
*pairs
(X; u) where X 2 Fn and u: S -!X. It is easy to see that opnis a filtered categ*
*ory,
so we have an object Tn = lim- X of Pro(Fn). It is formal to check that th*
*is
(X;u)2n
is the initial object in S # Pro(Fn).
Proposition 4.22.Fix an integer n. Then there exists a tower
: :g:3-!X2 g2-!X1 g1-!X0
such that
MORAVA K-THEORIES AND LOCALISATION 31
(a)Xiis a generalised Moore spectrum of type S=J(i) for some ideal J(i) of hei*
*ght
n.
(b)We have giO ji = ji-1 for all i, where ji:S -! Xi is the unit map of the
-spectrumTXi.
(c) iJ(i) = 0.
(d)For any Z 2 Fn we have Z = lim-Z ^ Xi in Pro(Fn).
i
Note that the unit maps ji give a map from S to lim-Xi in Pro(F), which is
i
implicitly used in (d). For any such tower, the corresponding object of S # Pro*
*(Fn)
is isomorphic to Tn.
Remark 4.23.We shall often talk about towers of the form
: :-:!S=J(3) -!S=J(2) -!S=J(1) -!S=J(0);
by which we mean a tower with the properties mentioned in the above proposition.
We will also use notation like holim-S=J to refer to the homotopy inverse limit*
* of
J
any such tower. This depends only on the isomorphism class of the tower in Pro(*
*F)
and thus is independent of the choices made.
Proof.We assume that n > 1 and that the result holds for n-1; small modificatio*
*ns
are required for n = 1, but we leave them to the reader. By induction, there is*
* a
tower {Wi} of the required type for the n - 1 case. Choose a spectrum Y of type*
* n.
Write Wi0for the cofibre of the map S -!Wi. These cofibres can be assembled into
a tower. Because Y -! lim-Y ^ Wiis an isomorphism, we see that limY ^ Wi0= 0.
i - i
After replacing {Wi} and {Wi0} by cofinal subtowers, we may assume that the maps
Y ^ Wi0-!Y ^ Wi0-1are all zero.
We shall recursively build a tower {Xi} and cofibrations as shown below, such
that wi is a good vn-1 self map, and 1Y ^ wi= 0: Y ^ Wi-! Y ^ Wi.
Wi ________Wiwwi______wXi _______Wiw
| | | |
| | | |
wpk-1Of| fi| |gi |wpk-1Of
i-1 |i | | | i-1 i
| | | |
|u |u |u |u
Wi-1 ______Wi-1wwi-1__wXi-1 _____Wi-1w
Suppose we have constructed all this up to thej(i - 1)'th level. Choose a good *
*vn
self map wi of Wi. After replacing wi by wpi for some j > 0, we may assume that
k
wiis compatible with wi-1, say fiO wi= wpi-1O fifor some k. We may also assume
that k > 0 and 1Y ^ wi= 0. We let Xibe the cofibre of wi; the usual comparison *
*of
cofibrations gives a map gias shown. Moreover, Xican be made into a -spectrum
of type S=J(i) for a suitable ideal J(i), as in Proposition 4.11. Thus (a) hol*
*ds.
The maps Wi -!Xi are maps of -spectra and thus carry units to units. By the
induction hypothesis, fi carries the unit of Wi to that of Wi-1. It follows tha*
*t gi
carries the unit of Xi to that of Xi-1, so (b)Tholds. The assumption k > 0 above
ensures that |wi| -!1 as i -!1, and thus that iJ(i) = 0, so (c) holds.
Clearly, when we smash the towers at either end of the above diagram with Y ,
all the maps become zero. It follows that lim-Y ^ Xi= limY ^ Wi= Y in Pro(F),
i - i
as claimed.
32 M. HOVEY AND N. P. STRICKLAND
It is not hard to check that {Z | Z = lim-Z ^ Xi} is a thick subcategory of F*
*, so
i
it contains all of Fn. This proves (d).
Similarly, for Z 2 Fn we have
Pro(F)(lim-Xi; Z) = Pro(F)(lim-DZ ^ Xi; S) = Pro(F)(DZ; S) = [S; Z]:
i i
It follows that for any object Y = lim-Zj 2 Pro(Fn) we have
j
Pro(F)(lim-Xi; Y ) = lim-[S; Yj] = Pro(F)(S; Y ):
i j
This means that lim-Xi is initial in S # Pro(Fn), so limXi= Tn. |__*
*_|
i - i
Corollary 4.24.We have Tn^Tn = Tn in Pro(F) (because Tn^Tn = lim-S=I ^Tn
I __
and S=I ^ Tn = S=I). |__|
Remark 4.25.In any tower as above, the spectrum Xi= S=J(i) can be made into
a module over Xj for j > i. Indeed, as Xi is a -spectrum we know that the map
Xi ji^1---!Xi^ Xi is a split monomorphism. This can be factored as Xi jj^1---!
Xj^ Xi-g^1-!Xi^ Xi, so jj^ 1 is a split monomorphism, as required.
5.Bousfield classes
In this section, we recall some known results about Bousfield classes that we*
* need
in the rest of this paper. Recall that in a stable homotopy category C, the Bou*
*sfield
class of an object X is the collection of X-local objects. We order Bousfie*
*ld
classes by inclusion. (It is more traditional to define to be the category *
*of X-
acyclic objects, and to order Bousfield classes by reverse inclusion. Either ap*
*proach
gives the same lattice of Bousfield classes, but the former has some conceptual
advantages.) We begin with the crucial result of Hopkins and Ravenel.
Theorem 5.1 ([Rav92a, Theorem 7.5.6]).The localisation functor L is smashing._
That is, the natural map LS ^ X -! LX is an isomorphism. |__|
Note that the Bousfield class in L of an L-local spectrum X is the same as the
Bousfield class in S, since any X-local spectrum is L-local. Indeed, since L i*
*s a
smashing localisation functor, if W is L-acyclic, then X ^ W = X ^ LS ^ W =
X ^ LW = 0.
If X and Y have the same Bousfield class, we write X ~ Y .
Recall that Cj is the acyclisation functor corresponding to Lj, and Mj = Cj-1*
*Lj
is the fibre of the natural map Lj -!Lj-1= Lj-1Lj.
Lemma 5.2. We have bLE(n) = E.
Proof.Let F be the fibre of the evident map E(n) -! E. Choose a generalised
Moore spectrum S=I of type n. By Landweber exactness we have E(n)*(S=I) =
E(n)*=I and E*(S=I) = E*=I. But E(n)*=I = E*=I so F ^ S=I = 0. Thus
K*(F ) K* K*(S=I) = 0 and K*(S=I) 6= 0 so K*F = 0. Thus, the map E(n) -!E
is a K-equivalence. On the other hand, one can use the techniques of [EKMM96 ,
Chapter V] to build a tower of spectra E=I 2 DMU with ss*(E=I) = E*=I. There
is a natural map from E to the sequential limit of this tower, which is easily *
*seen
to be an isomorphism. The spectra E=I lie in the thick subcategory generated by
MORAVA K-THEORIES AND LOCALISATION 33
E=In = K, so the sequential limit is K-local. Thus E is a K-local spectrum that*
* is
K*-equivalent to E(n), so E = bLE(n). |___|
Proposition 5.3.We have the following equalities of Bousfield classes:
= = = = _ : :_:
= _ : :_:
= = _ : :_:
= =
Proof.It is proved in [Rav84, Theorem 2.1] that = _ : :_:.
By Theorem 5.1, LX = LS ^ X. It is immediate from this that = .
By Landweber exactness we have E*X = E* E(n)*E(n)*X, and E* is faithfully
flat over E(n)* so = . We have ring maps LS -! bLS -! bLE(n) = E so
but we have seen that = so = also.
Next, we claim that = _ : :_:. Indeed, because Lj is
smashing, we know that
CjLS = CjS ^ LS ~ CjK(0) _ : :_:CjK(n)
If i j then K(i) is E(j)-local. Indeed, K(i) is self-local, and ,*
* so
K(i) is also E(j)-local. Hence CjK(i) = 0. If j < i n then
LjK(i) = LjS ^ K(i) ~ (K(0) ^ K(i)) _ : :_:(K(j) ^ K(i)):
We know from [Rav84, Theorem 2.1] that K(k) ^ K(i) = 0 unless i = k. Thus,
if j < i n, then LjK(i) = 0 and CjK(i) = K(i). It follows that CjLS ~
K(j + 1) _ : :_:K(n) as claimed. Clearly MjS = Cj-1LjS, so = .
By the definition of a type j spectrum, we have K(i) ^ F (j) = 0 if and only *
*if
i < j. Moreover, if i j then K(i) ^ F (j) is a nontrivial wedge of suspensions*
* of
K(i), hence Bousfield equivalent to K(i). It follows that
LF (j) ~ (K(0) ^ F (j)) _ : :_:(K(n) ^ F (j)) ~ K(j) _ : :_:K(n)
Now let v : dF (j) -! F (j) be a vj-self map, so that v-1F (j) = T (j). By
definition, v induces a nilpotent endomorphism of K(i)*F (j) when i 6= j, so th*
*at
K(i)*T (j) = 0. It follows easily that . On the other hand, the
Telescope Lemma [Rav84, Lemma 1.34] says that LF (j) ~ LT (j) _ LF (j + 1).
By smashing with K(j) we deduce that and thus that K(j)_~
LT (j). |__|
6. The E(n)-local category
In this section, we discuss the general properties of L. Some of this materia*
*l also
appears in [Dev96a] or [HS95 ].
Proposition 6.1.L is a monogenic stable homotopy category in the sense of
[HPS95 ]. In particular, it is a triangulated category with coproducts and comp*
*atible
smash products and function spectra, giving a closed symmetric monoidal struc-
ture. Coproducts, minimal weak colimits, smash products and function spectra are
the same as in S. The unit for the smash product is LS, which is a small graded
34 M. HOVEY AND N. P. STRICKLAND
weak generator. This last statement means that for any family of spectra Xi 2 L
we have
_ M
[LS; Xi] = [LS; Xi];
and that if X 2 L satisfies [LS; X]* = 0 then X = 0.
Proof.Apply [HPS95 , Theorems 3.5.1 and 3.5.2]. |___|
Theorem 6.2.
(a)Homology and cohomology functors L -!Ab are representable.
(b)Any spectrum X 2 L is the minimal weak colimit of L(X), where (X) is
the category of finite ordinary spectra over X.
(c)Suppose that X 2 L. The following are equivalent:
1. X is small.
2. X is F -small.
3. X is LS-finite.
4. X is dualisable.
5. X is a retract of LY for some finite spectrum Y 2 S.
See Definition 1.5 for definitions.
(d)Every spectrum in L is E(n)-nilpotent (Definition 1.4).
Proof.Part (a) follows from [HPS95 , Theorem 3.5.2] and [HPS95 , Theorem 4.1.5].
For part (b), recall that X is the minimal weak colimit in S of (X), by [HPS95 ,
Theorem 4.2.4] or [Mar83 , Chapter 5]. It follows from part (e) of [HPS95 , Pro*
*posi-
tion 2.2.4] that X = LS^X is the minimal weak colimit in S of LS^(X) = L(X).
Moreover, minimal weak colimits are the same in S or L. For part (c), Theorem 2*
*.1.3
of [HPS95 ] implies that the first four conditions are equivalent. Moreover, if*
* Y 2 S
is finite, then it is easy to see that LY (and hence, any retract of LY ) is LS*
*-finite.
Conversely, suppose that X 2 L is small, so that [X; -] is a homology theory on
L. It follows from part (b) that [X; X] = lim-! [X; LY ], so the identity map
Y 2(X)
1X 2 [X; X] factors through LY , in other words, X is a retract of LY ._Part_(d*
*) is
proved in [HS95 ]. |__|
Proposition 6.3.Suppose that X; Y 2 L are small. Then [X; Y ]* is countable.
Proof.By an evident thick subcategory argument, we need only show that ss*LS is
countable. This is proved in [Hov95a, Lemma 5.5] (by showing that the E1 term
of the Adams-Novikov spectral sequence converging to ss*LS is countable in each_
bidegree, and that the E1 term has only finitely many lines). |_*
*_|
For calculations when n = 1, see [Bou79] or [Rav84]. For the case n = 2, see *
*the
many papers by Katsumi Shimomura and his coworkers, for example [SY95 ]. See
also Section 15.
We can now prove a convergence theorem for the modified Adams spectral se-
quence described in Section 2.2.
Lemma 6.4. Let C be a stable homotopy category, and R a ring object in C. Then
the ideal of R-nilpotent objects is the same as the thick subcategory generated*
* by the
R-injective objects.
MORAVA K-THEORIES AND LOCALISATION 35
Proof.Let N be the category of R-nilpotent spectra, which is by definition the *
*ideal
generated by R. Let I be the category of R-injectives, and N0the thick subcateg*
*ory
generated by I. Write
N00= {X | X ^ Y 2 N0for allY }:
Clearly I N and N is thick so N0 N. Clearly R 2 N00and N00is an ideal
so N N00. By putting Y = S in the definition we see that N00 N0. Thus_
N0= N = N00as required. |__|
Proposition 6.5.There are constants r0; s0 such that for all spectra X and Y , *
*in
the modified Adams spectral sequence
Exts;tE(n)*E(n)(E(n)*X; E(n)*Y ) =) [X; LY ]t-s
(see Section 2.2) we have Es;tr= 0 whenever r r0 and s s0. Moreover, the
spectral sequence converges.
_____
Proof.Let i: E(n)-!_LS_be_the fibre of the unit map LS -! E(n). It is easy to
see that i ^ 1E(n):E(n) ^ E(n) -! E(n) is null, and thus that i ^ 1I = 0 for any
E(n)-injective spectrum I. It follows using Lemma 6.4 that for any E(n)-nilpote*
*nt
_____(N)
spectrum X we have i(N)^ 1X = 0: E(n) ^ X -! X for some N. However, we
_____(N)
know that LS is E(n)-nilpotent, so that i(N):E(n) -!LS is null for some N.
Now suppose that X is such that E(n)*X is a projective module over E(n)*. We
then have an ordinary Adams spectral sequence for [X; LY ]* obtained by mapping
X to the tower
_____ _____(2)
LY- E(n)^ LY- E(n) ^ LY- : :::
As every composite in the tower of length at least N vanishes, one can show tha*
*t the
spectral sequence converges, and that it stops at the EN page with a flat vanis*
*hing
line. One also knows from [Dev96b, Proposition 1.9 and Remark 1.10] that this
spectral sequence is isomorphic to the modified Adams spectral sequence from the
E2 page onwards, and that the resulting filtrations of [X; LY ]* are also the s*
*ame.
In particular, in the filtration arising from the MASS we have F s= 0 for s N.
Now choose a tower LY = Y 0- Y 1- Y 2- : :o:f the type used in the MASS.
By the above, we see that when E(n)*X is projective, the map [X; Y s]* -![X; Y *
*0]*
is null when s N. Moreover, if we truncate the tower below Y twe get a tower of
the type that computes the MASS for [X; Y t]* so we can apply the same logic and
deduce that the map [X; Y t+s]* -![X; Y t]* is null for s N.
Now consider a general spectrum X. By the methods of [Ada74, Theorem 13.6]
(taking account of Proposition 2.13) we can choose a tower
X = X0 -!X1 -!: :-:!Xn
and triangles -1Xk+1 -! Pk -! Xk such that E(n)*Pk is projective and the
map E(n)*Pk -! E(n)*Xk is surjective. As E(n)* = Z(p)[v1; : :;:v1n] has global
dimension n (in the graded sense) we see that E(n)*Xn is projective. It is not
hard to show by induction that [Xn-j; Y t+(j+1)N]* -![Xn-j; Y t]* is null and t*
*hus
that [X; Y t+(n+1)N]* -! [X; Y t]* is null. It follows that the MASS for [X; L*
*Y ]*_
converges, and that it stops with a flat vanishing line at the E(n+1)Npage. *
* |__|
Jens Franke has shown [Fra96, Theorem 10] that when 2p - 2 > n2 + n, the
category L is equivalent to a purely algebraically defined derived category of *
*periodic
36 M. HOVEY AND N. P. STRICKLAND
complexes of comodules. He shows that (in contrast to Quillen's algebraicisation
of rational unstable homotopy) the equivalence does not capture all the higher-
order homotopical phenomena of the model category underlying L, but that the
approximation improves as p increases. His result implies for example that ther*
*e is
an L-small spectrum X with E(n)*X = K(n)*. There is some reason to suspect
that this is not of the form LY for any finite spectrum Y (although it must be a
retract of some such LY ) but the situation is rather unclear at the moment. Et*
*han
Devinatz has also constructed the spectrum X by a more direct argument.
6.1. Nilpotence and thick subcategories. In this situation, we have the simple
nilpotence theorem proved in [HPS95 , Section 5]. Recall that a map f : X -! Y *
*is
called smash nilpotent if f^m : X^m -! Y ^m is null for large enough m.
Corollary 6.6 (Nilpotence theorem).
(a)Let F be an L-small spectrum, and X an arbitrary spectrum in L. A map
f : F -! X is smash nilpotent if and only if K(i)*f = 0 for all i n.
(b)Let X be an L-small spectrum. A map f : X -! X is nilpotent if and only if
K(i)*f is nilpotent for all i n.
(c)Let R be a ring spectrum in L. An element ff 2 ss*R is nilpotent if and only
if K(i)*ff is nilpotent in the ring K(i)*R for all i n.
Proof.Most of this is proved in [HPS95 , Section 5]. The only thing left to pro*
*ve
is that if f : F -! X is a smash nilpotent map from an L-small spectrum F to X,
then K(i)*f = 0. This follows easily from the K"unneth theorem since K(i)*(f^N_*
*) =
(K(i)*f)N . |__|
Note that this nilpotence theorem follows directly from the Bousfield decompo*
*si-
tion of LS, but that Bousfield decomposition depends on the smashing theorem of
Hopkins-Ravenel, whose proof requires the much more difficult nilpotence theorem
of Devinatz-Hopkins-Smith [DHS88 ].
The nilpotence theorem can be used to classify thick subcategories (Defini-
tion 1.2) of L-small spectra. For this, we need a definition of the support of*
* a
spectrum or full subcategory of L.
Definition 6.7.If X is any class of spectra (in particular if X L), we write
K(m) ^ X = {K(m) ^ X | X 2 X}
supp(X) = {m | K(m) ^ X 6= {0}}:
We refer to supp(X) as the support of X. Similarly, the cosupport of X is
cosupp(X) = {m | F (K(m); X) 6= {0}}:
We have to classify the possible supports of L-small spectra. Since not every*
* such
spectrum is the localisation of an ordinary finite spectrum, this does not imme*
*diately
follow from knowledge of possible supports in S. However, the same result does *
*hold.
Proposition 6.8.If X is small in L and j < n, then dimK(j)*X dimK(j +
1)*X. Thus supp(X) = {m; m + 1; : :;:n} for some m with 0 m n + 1 (where
the case m = n + 1 means supp(X) = 0 and thus X = 0).
Proof.The proof is the same as that of [Rav84, Theorem 2.11]. |__*
*_|
MORAVA K-THEORIES AND LOCALISATION 37
Now, given an integer j such that 0 j n + 1, let Cj denote the thick sub-
category of L consisting of all small spectra X such that K(i)*X = 0 for all i *
*< j.
Then we have
C0 C1 . . .Cn+1 = {0}:
Note that C0 consists of all L-small spectra. All of the inclusions are strict,*
* since
LF (j) is in Cj but not Cj+1.
We now have the following theorem, which is a corollary of the general thick
subcategory theorem proved in [HPS95 , Section 5] and the classification of sup*
*ports
proved above.
Theorem 6.9 (Thick Subcategory Theorem).If C is a thick subcategory of L,_then
C = Cj for some j such that 0 j n + 1. |__|
The telescope conjecture also holds in L. More explicitly, consider the follo*
*wing
three functors:
Li: L -!L = localisation with respectKto(0) _ : :_:K(i)
Llfi: L -!L = finite localisation awayLfromF (i + 1)
Lfi: S -!S = finite localisation awayFfrom(i + 1)
(See [HPS95 , Section 3.3] for discussion of finite localisation). The followin*
*g corol-
lary is the telescope conjecture for L.
Corollary 6.10.If L0 is a smashing localisation functor on L, then L0= Li for
some i. Moreover, Li= Llfi= LLfi.
Proof.Since L0is smashing, L0is Bousfield localisation with respect to L0S. Note
that if L0S ^ K(j) is nonzero then it is a wedge of suspensions of K(j) and thu*
*s has
the same Bousfield class as K(j). We thus have
_n _
= ^ = :
j=0 j2supp(L0S)
Now, if j 2 supp(L0S), then LK(j)S is a module over L0S so we have
= _ : :_:
(using Proposition 5.3). Thus L0= Li for the largest i in the support of L0S.
Next, recall that there is a cofibre sequence CfiS -! S -! LfiS, in which CfiS
has a cellular tower built from F (i + 1) (see [HPS95 , Theorem 3.3.3]). Moreov*
*er,
[F (i + 1); LfiS]* = 0, so DF (i + 1) ^ LfiS = 0. It follows that LCfiS has a c*
*ellular
tower built from LF (i + 1), so it is Llfi-acyclic. Moreover,
[LF (i + 1); LLfiS]* = [LS; L(DF (i + 1) ^ LfiS)]* = 0:
It follows that LLfiS is Llfi-local, so LLfi= Llfi.
We know that Llfiis again a smashing localisation. Thus, to see that Llfi= Li,
it is enough to check that supp(LlfiS) = {0; : :;:i}. We know from Proposition *
*5.3
that LfiS ~ T (0) _ : :_:T (i), so
LlfiS ~ LT (0) _ : :_:LT (i) ~ K(0) _ : :_:K(i)
The claim follows. |___|
38 M. HOVEY AND N. P. STRICKLAND
Corollary 6.11.If X 2 Cj then X is a retract of LY for some finite spectrum Y
of type at least j.
Proof.We have seen that X lies in the thick subcategory generated by LF (i), so
X = ClfiX, but we have also seen that this is the same as LCfiX, which is a min*
*imal
weak colimit of terms of the form LY . As X is small, we conclude that it_is a *
*retract
of one of these terms. |__|
We conclude this section by showing that the theory of vj self-maps (defined *
*by
the evident generalisation of Definition 4.5) works as expected in L.
Theorem 6.12. Every spectrum X 2 Cj admits a good vj self map .
Proof.Choose a finite spectrum Y of type at least j such that X is a retract of
LY = LS ^ Y , so that X lies in the category JY of Definition 4.2. Choose a good
vj self map v :dY -! Y . Proposition 4.4 gives an induced map vX :dX -! X, *
*__
and it is easy to check that this is a good vj self map. |*
*__|
6.2. Localising and colocalising subcategories. In this section, we classify the
localising and colocalising subcategories (Definition 1.3) of L . Every such ca*
*tegory
is determined by the Morava K-theories which it contains.
We write loc and coloc for the localising and colocalising subcategories
generated by a class X of spectra. We also write
X? = {Y | [X; Y ]* = 0}
?X = {Y | [Y; X]* = 0}
(where [X; Y ]* = {[X; Y ]* | X 2 X} and so on). Note that X? is a colocalising
subcategory and ?X is a localising subcategory. Also note that every (co)locali*
*sing
subcategory is a (co)ideal, since L is monogenic.
Definition 6.13.
(a)We write P for the lattice of subsets of {0; : :;:n}. For any S 2 P we write
Sc = {0;W: :;:n} \ S.
(b)K(S) = m2S K(m)
(c)CS = {X 2 L | K(Sc)*X = 0} = {K(Sc)-acyclic spectraLin}
(d)DS = {K(S)-local spectra} L
Theorem 6.14.
(a)The lattice of localising subcategories is isomorphic to P, via the maps S *
*7! CS
and C 7! supp(C).
(b)The lattice of colocalising subcategories is isomorphic to P, via the maps *
*S 7!
DS and D 7! cosupp(D).
(c)CS = loc = {X 2 L | }.
(d)DS = coloc = {X 2 L | = }.
(e)CSc = ?DS and DSc = C?S.
The notation denotes the cohomological Bousfield class of X, the collecti*
*on
of X*-local spectra. A spectrum Y is said to be X*-local if [Z; Y ]* = 0 for a*
*ll
spectra Z such that [Z; X]* = 0.
MORAVA K-THEORIES AND LOCALISATION 39
This theorem is similar to the analogous theorem proven as [HPS95 , Corol-
lary 6.3.4] for a Noetherian stable homotopy category (with some hypotheses). H*
*ow-
ever, we have been unable to find a worthwhile common generalisation of the two
theorems. We shall prove the theorem after some intermediate results.
Proposition 6.15.If X is a finite spectrum of type at least n, then LX is K-
nilpotent.
Proof.By a thick subcategory argument, it is sufficient to verify this for X = *
*S=I.
We know from [HS95 ] that LS is E(n)-nilpotent, and {Y | Y ^S=I 2 ideal}
is an ideal containing E(n), so it contains LS. Thus, LS=I lies in the ideal ge*
*nerated_
by E(n)=I, which is easily seen to be the same as the ideal generated by K. *
*|__|
Remark 6.16.We shall show in Corollary 8.12 that the localisation of a finite s*
*pec-
trum of type n actually lies in the thick subcategory generated by K.
Proposition 6.17.For any X 2 S, the spectrum Mm X = Mm S ^ X lies in the
localising subcategory generated by K(m), and F (Mm S; X) lies in the colocalis*
*ing
subcategory generated by K(m).
Proof.Observe that Mm X = Cm-1 Lm X, which is Lm-1 -acyclic. Recall that the
functor Lm-1 : Lm -! Lm-1 is simply finite localisation away from Lm F (m) (Cor*
*ol-
lary 6.10), so that the Lm-1 -acyclics in Lm are precisely loc. Becau*
*se
Lm F (m) is K(m)-nilpotent (Proposition 6.15), this is contained in loc, *
*so
Mm X 2 loc.
The spectrum F (K(m); X) is a K(m)-module, hence a wedge of suspensions of
K(m). By part (f) of Proposition 3.3, the wedge splits off as a retract of the *
*product,
so it lies in coloc. The subcategory {Y | F (Y; X) 2 coloc} is loca*
*lis-_
ing and contains K(m), so it contains Mm S. Thus F (Mm S; X) 2 coloc. |_*
*_|
Proposition 6.18.For any spectrum X 2 L we have
X 2 loc = loc
and
X 2 coloc = coloc:
Proof.We have a tower of spectra as follows, in which the vertical maps are the
fibres of the horizontal maps.
MnS Mn-1S M1S M0S
?? ? ? ?
y ?y ?y ?y
LnS ----! Ln-1S : : : L1S ----! L0S - ---! 0
After smashing this with X, we see that X 2 loc. Because
= (by Proposition 5.3), we see that Mm X = 0 unless m 2 supp(X).
On the other hand, when m 2 supp(X) we have Mm X 2 loc by Proposi-
tion 6.17. It follows easily that X 2 loc = loc,
so loc loc. Moreover, if m 2 supp(S) then loc contains
K(m) ^ X, which is a nonzero wedge of suspensions of K(m), so loc contains
K(m). It follows that loc = loc. For the colocalising case, we*
*__
apply the functor F (-; X) to the above tower and argue in the same way. |*
*__|
40 M. HOVEY AND N. P. STRICKLAND
Proof of Theorem 6.14.It is clear that S 7! CS and S 7! DS are order preserving
maps from P to the lattices of localising and colocalising subcategories. Next,*
* recall
from Proposition 5.3 that K(m)*K(n) 6= 0 if and only if m = n. Using this,
we see that K(S) 2 CS, and that supp(CS) = S. Moreover, because K(m)*X =
HomK(m)*(K(m)*X; K(m)*), we see that K(m)*K(n) 6= 0 if and only if m = n.
Using this and the fact that K(S) 2 DS, we see that cosupp(DS) = S.
It is immediate from Proposition 6.18 that any localising subcategory C L
satisfies C = loc, and similarly for colocalising subcategories. In*
* par-
ticular, CS = loc. Thus, for any C, we have C = Csupp(C). This proves
part (a) of the theorem, and part (b) is similar.
We have already proved the first equality in part (c). For the second, write
C = {X 2 L | }. It is easy to see that this is a localising subcatego*
*ry.
Suppose that . As K(S)^K(Sc) = 0, we conclude that X^K(Sc) = 0,
so that X 2 CS. Thus C CS. On the other hand, it is clear that K(S) 2 C, so
CS = loc C. Thus C = CS as claimed. The proof of part (d) is similar.
For the first part of (e), observe that K(S)*X = 0 if and only if K(S)*X = 0,
so CSc = {X | [X; K(S)]* = K(S)*X = 0}. In other words,
CSc = ?{K(S)} = ?coloc = ?DS:
For the second part, recall that by definition
DSc = {K(S)-local spectra} = {K(S)-acyclics}? = C?S:
|___|
Note that Theorem 6.14 implies that we can localise with respect to any local*
*ising
subcategory of L (because we can localise with respect to the homology theory
K(S)*).
6.3. The monochromatic category. As mentioned in the introduction, there are
two different ways to investigate the difference between the E(n)-local categor*
*y and
the E(n-1)-local category. The first is to consider the fibre MX = MnX of the m*
*ap
LnX -! Ln-1X, and the second is to observe that E(n) ~ E(n-1)_K(n) (where ~
means Bousfield equivalence, as in Section 5) and thus consider LK(n)X = bLX. T*
*he
purpose of this section is to demonstrate that these two approaches are essenti*
*ally
equivalent.
First, we observe that Mn (considered as a functor Ln -!Ln-1) is just E(n-1)-
acyclisation, so it is an idempotent exact functor.
Theorem 6.19. For any X 2 S, there are natural equivalences MLbX = MX and
bLMX = bLX. It follows that the functors M -bL!K -M! M are mutually inverse
equivalences between the monochromatic category M and the K-local category K.
Proof.We start by observing that for any X 2 S, we have MX = MLX and
bLX = bLLX.
Next, we know that Ln-1S ~ K(0) _ : :_:K(n - 1), so that K ^ Ln-1S = 0,
so bLLn-1S = 0. By applying bLto the cofibration MX -! LX -! Ln-1X, we
conclude that bLMX = bLX. Next, using [Hov95a, Lemma 4.1] (or part (e) of
Proposition 7.10) and the cofibration MS -!LS -!Ln-1S, we obtain a cofibration
F (Ln-1S; X) -!LX = F (LS; LX) -!LbX = F (MS; LX):
MORAVA K-THEORIES AND LOCALISATION 41
We claim that F (Ln-1S; X) is E(n - 1)-local. Indeed, if Z is E(n - 1)-acyclic *
*then
[Z; F (Ln-1S; X)]* = [Ln-1S ^ Z; X]* = [Ln-1Z; X]* = 0
because Ln-1Z = 0. It follows easily that MF (Ln-1S; X) = 0, and thus that
MX = MLbX.
If X 2 K then bLX = X so bLMX = X. If X 2 M then MX = X so MLbX = X. __
It follows that M and bLare mutually inverse equivalences between M and K. |_*
*_|
7.General properties of the K(n)-local category
This section begins the second part of the paper, in which we will study the *
*K-
local category K. Unlike many other categories of localised spectra (see Append*
*ix B)
this category has a good supply of small objects. These are the K-localisations*
* of
finite type n spectra. We will show that these K-small spectra can be used as
replacements for the sphere in many cases. That is, many of the constructions
commonly used in stable homotopy categories, such as the cellular tower, have
analogues in the K-local category. It is just that the cells are no longer sphe*
*res, but
suspensions of LF (n).
Theorem 7.1. K is an algebraic stable homotopy category in the sense of [HPS95 *
*].
The spectrum LF (n) is a graded weak generator, and loc = K. The co-
product, smash product and function objects in K are
_ _
Xi= bL( Xi)
K S
X ^K Y = bL(X ^S Y )
FK (X; Y ) = FS(X; Y ):
Proof.This all follows from [HPS95 , Theorem 3.5.1], except for the fact that L*
*F (n)
is a graded weak generator, which will be proved as Theorem 7.3. (It follows fr*
*om_
this that loc = K, as in [HPS95 , Theorem 1.2.1]). |_*
*_|
We shall usually not write the subscript K on the symbols for the coproduct a*
*nd
smash product. Instead, we shall say "in K, we have X = Y ^ Z" (rather than
"X = Y ^K Z").
Lemma 7.2. If X is a finite spectrum of type at least n and Y 2 S then L(X^Y )*
* =
bL(X ^ Y ) = X ^ bLY .
Proof.For any finite spectrum X, we have bL(X ^ Y ) = X ^ bLY , and similarly f*
*or
the functor L, as we see easily by induction on the number of cells. To complet*
*e the
proof, it suffices to show that X ^ LY is already K-local. Suppose Z is K-acycl*
*ic.
Then Z ^ DX is E-acyclic, since X, and thus DX, has type at least n. Hence __
[Z; X ^ LY ] = [Z ^ DX; LY ] = 0, as required. |__|
Theorem 7.3. If X is a finite type n spectrum then LX = bLX is a small graded
weak generator in K. It follows that any spectrum Y 2 K is the sequential colim*
*it
of a sequence 0 = Y0 -!Y1 -!Y2 -!: :Y:, in which the cofibre of Yk -!Yk+1 is a
coproduct of suspensions of X (and therefore Y 2 loc).
42 M. HOVEY AND N. P. STRICKLAND
Proof.Suppose that {Yi} is a family of K-local spectra. We have
_ _
[bLX; bL( Yi)]=[X; bL( Yi)]
_
= [S; bL( Yi) ^ DX]
_
= [S; L( Yi^ DX)]
_
= [S; L(Yi^ DX)]
M
= [S; L(Yi^ DX)]
M
= [S; Yi^ DX)]
M
= [bLX; Yi]:
Here DX is the Spanier-Whitehead dual of X. The third and sixth equalities use
Lemma 7.2, and the fact that Yi= bLYi. The fourth equality uses the fact that L*
* is
smashing. We conclude that LX is small.
Now suppose that Y 2 K has [LX; Y ]* = 0. Observe that
[LX; Y ]* = [X; Y ]* = ss*(DX ^ Y ):
It follows that DX ^ Y = 0, so
0 = K*(DX ^ Y ) = Hom K*(K*X; K*Y ):
This means that K*Y = 0, and thus (because Y is K-local) that Y = 0. In other
words, LX is a graded weak generator. __
The remaining claims follow from [HPS95 , Theorem 1.2.1]. |__|
7.1. K-nilpotent spectra. This short section concerns a small technicality. The
class of K-nilpotent spectra is by definition the ideal generated by K. Because
there are different smash products in L and K, it might a priori make a differe*
*nce
whether we interpret this definition in K or L.
Lemma 7.4. The category N of K-nilpotent spectra is the thick subcategory of K
generated by the K-injective spectra. It does not make a difference whether we
interpret the definition in S, L or K. If X 2 N and Y 2 K then X ^K Y = X ^SY .
Moreover, N locK.
Proof.It is easy to see that K ^S X 2 K for all X, and thus that the K-injectiv*
*es
are the same in S, L or K. It follows immediately from Lemma 6.4 (applied in S,*
* L
and K) that the K-nilpotents are the same in all three categories and that they*
* are
always the same as the thick subcategory generated by the K-injectives.
Now write C = locK, and
D = {X 2 S | 8Y 2 S X ^S Y 2 C}:
It is easy to see that D is an ideal in S containing K, so NS D. On the other_
hand D C (take Y = S in the definition of D). |__|
MORAVA K-THEORIES AND LOCALISATION 43
7.2. Localising and colocalising subcategories.
Theorem 7.5. The only localising or colocalising subcategories of K are {0} and
K.
Proof.It is easy to see that locK = K (because the analogous thing holds i*
*n L),
and thus that any (co)localising subcategory is a (co)ideal [HPS95 , Lemma 1.4.*
*6].
Suppose that C K is a nontrivial localising subcategory, say 0 6= X 2 C.
Then K ^K X = K ^S X lies in C and is a nontrivial wedge of suspensions of K.
Thus K 2 C, and therefore every K-nilpotent spectrum lies in C. In particular,
LF (n) 2 C. By Theorem 7.1, we conclude that C = K.
Now suppose that D K is a nontrivial colocalising subcategory, say 0 6= X 2 *
*K.
Because D is a coideal, the K-injective spectrum F (K; X) lies in D. Observe th*
*at
{Y 2 K | [Y; X]* = 0} is a localising subcategory of K which does not contain X,
so it is zero. It follows that [K; X]* 6= 0, so F (K; X) 6= 0. Thus K is a retr*
*act of
F (K; X), so K 2 D. Any K-injective spectrum Z is a wedge of suspensions of K,
and the wedge is a retract of the product by part (f) of Proposition 3.3, so Z *
*2 D.
It follows by Lemma 7.4 that any K-nilpotent spectrum lies in D.
Suppose that Y 2 K. Consider C = {U 2 K | F (U; Y ) 2 D}. This is clearly a
localising subcategory. Because F (LF (n); Y ) = LDF (n)^Y is K-nilpotent, we s*
*ee
that F (n) 2 C. We know that locK = K, so C = K. In particular,_bLS 2 C,
so Y 2 D. Thus D = K. |__|
Corollary 7.6.If X; Y 2 K are both nontrivial, then F (X; Y ) 6= 0 (equivalentl*
*y,
[X; Y ]* 6= 0) and X ^ Y 6= 0.
Proof.C = {Z | [Z; Y ]* = 0} is a localising subcategory of K not containing Y ,
so C = {0}, so X 62 C. Similarly, {Z | Z ^ Y = 0} is a localising subcategory
which does not contain bLS, so it is trivial. More directly, K*X 6=_0_6= K*Y ,*
* so
K*(X ^ Y ) = K*X K* K*Y 6= 0. |__|
As an aside, we point out that from this theorem it is very easy to prove tha*
*t the
Bousfield class of K is minimal among all cohomological Bousfield classes, as w*
*as
conjectured in [Hov95b].
Theorem 7.7. Suppose o : H -! K is a natural transformation of homology or co-
homology functors on K. If there is a nontrivial X such that oX is an isomorphi*
*sm,
then o is a natural equivalence.
Proof.{X | oX is an isomorphism} is a nontrivial localising subcategory,_hence *
*is
all of K. |__|
Corollary 7.8.If H is a homology or cohomology functor on K such that H*(X) =_
0 or H*(X) = 0 for some nontrivial X, then H is trivial. |__|
For cohomology functors, this corollary follows immediately from Corollary 7.6
and the fact that cohomology functors are representable (discussed in Section 9*
*).
We do not yet have a complete classification of ideals in K, but we do have t*
*he
following simple result.
Proposition 7.9.Suppose that D is a nonzero ideal of K. Then D contains the
ideal of K-nilpotent spectra.
44 M. HOVEY AND N. P. STRICKLAND
Proof.Suppose X is a nontrivial element of D. Then X ^ K 2 D, but X ^ K is a
wedge of suspensions of K. Therefore K 2 D, so D contains the ideal of_K-nilpot*
*ent
spectra. |__|
7.3. The localisation functor bL. In this section we recall some facts about the
functor bL: S -!K. Let Lf : S -!S be the finite localisation away from F (n), a*
*nd
Cf the corresponding acyclisation functor [Mil92] [HPS95 , Section 3.3]. Thus, *
*we
have a cofibration
CfX -! X -! LfX
in which CfX has an F (n)-cellular tower and [F (n); LfX]* = 0.
Proposition 7.10.In the claims below, all homotopy colimits are calculated in S.
(The homotopy limits are the same in S, L or K). They are taken over a tower of
generalised Moore spectra, as in Proposition 4.22.
(a)CfX = holim-!F (S=I; X) = holimD(S=I) ^ X.
I -! I
(b)LLfX = Ln-1X = LfLX.
(c)LCfX = MX = CfLX = holim-!D(S=I) ^ LX.
I
(d)LF(n)X = holim-X ^ S=I = F (CfS; X).
I
(e)bLX = LF(n)LX = holim-LX ^ S=I = F (MS; LX).
I
Proof.By the Thick Subcategory Theorem [HS , Theorem 7] [Rav92a, Theorem
3.4.3], we know that D(S=I) lies in the thick subcategory generated by F (n). *
*It
follows that holim-!D(S=I) ^ X 2 loc. On the other hand,
I
[F (n); holim-!D(S=I) ^ X] = lim-![F (n) ^ S=I; X] = [F (n); X]
I I
(the second equality by Proposition 4.22). It follows that holim-!D(S=I) ^ X -!*
* X
I
has universal property of CfX -! X, which gives (a). For part (b), we know that
L and Lf are both smashing, so that LLf = LfL, and Corollary 6.10 tells us that
LfL = Ln-1. Part (c) follows immediately, as MX is by definition the fibre of
LX -! Ln-1X.
We now prove (d) (which is also proved as [Hov95a, Theorem 2.1]). The second
equality is clear from (a). Suppose that Z is F (n)-acyclic. As CfS 2 loc,
we see that [Z; F (CfS; X)]* = [Z ^ CfS; X]* = 0. Thus F (CfS; X) is F (n)-loca*
*l.
We have a cofibration
F (LfS; X) -!X = F (S; X) -!F (CfS; X):
We know that [U; LfS]* = 0 whenever U is a finite type n spectrum. In partic-
ular ss*(F (n) ^ LfS) = [DF (n); LfS]* = 0, so F (n) ^ LfS = 0. It follows that
[F (n); F (LfS; X)]* = [F (n) ^ LfS; X]* = 0, so that X -! F (CfS; X) is an F (*
*n)-
equivalence. Claim (d) follows.
Finally, we prove (e). By part (c), we know that MS = holim-!LD(S=I), which
I
implies that F (MS; LX) = holim-LX ^ S=I, and this is the same as LF(n)LX
I
by (d). Because X -! LX is an LS-equivalence, and LX -! LF(n)LX is an F (n)-
equivalence, we see that X -! LF(n)LX is an LS ^ F (n)-equivalence. Because
LS ^ F (n) = LF (n) is Bousfield equivalent to K, it is a K-equivalence. Thus,
we need only show that LF(n)LX is K-local. This follows immediately from the
MORAVA K-THEORIES AND LOCALISATION 45
equivalence LF(n)LX = F (MS; LX) and the fact that MS is Bousfield equivalent_
to K. |__|
Corollary 7.11.If X 2 K and we compute sequential (co)limits in K then
X = holim-!F (S=I; X) = holim-X ^ S=I
I I
Proof.Note that X = LX = bLX. By part (c) of Proposition 7.10, we see that
MX = holim-!SD(S=I) ^ X:
I
By Theorem 6.19, we have X = bLMX = holim-!KD(S=I) ^ X as claimed. It is
I __
immediate from part (e) of Proposition 7.10 that X = holim-X ^ S=I. |__|
I
8. Smallness and duality
We next consider various notions of smallness in K. To show that things are n*
*ot
as simple as in S or L, we have the following lemma.
Lemma 8.1. The functor bLis not smashing, and bLS is not small in K.
Proof.We know from Proposition 5.3 that = > . On the other
hand, if bLwere smashing we would have
bLS ^ X = 0 , bLX = 0 , K*X = 0 , K ^ X = 0
It would follow that = , a contradiction. If bLS were small then bLwould
preserve smallness, so it would be smashing by [HPS95 , Theorem 3.5.2],_another
contradiction. |__|
In spite of this, we have a good understanding of two different notions of sm*
*all-
ness, as indicated by the theorems below. We start with some definitions.
Definition 8.2.We say that a graded Abelian group A* is finite if Ak is a finite
set for each k. (This is the most natural definition given that most of our gra*
*ded
Abelian groups are periodic.)
Definition 8.3.For any spectrum X we define E_*X = ss*bL(E ^ X).
We will see below that E_*X is a better covariant analogue of E*X than E*X
for X 2 K, though E_*X is not a homology theory. For now we note the following
analogue of some of the results in Section 2.
Proposition 8.4.
(a)The E*-module E_*X is L-complete.
(b)If X is finite in S or in L, then E_*X = E_*bLX = E*X.
(c)If E*X is a free E*-module, then E_*X = (E*X)^In(which is a pro-free E*-
module).
(d)E_*X is finitely generated if and only if K*X is finite-dimensional.
(e)If E_*X is pro-free, then K*X = (E_*X)=In.
(f)If K*X is concentrated in even dimensions, then E_*X is pro-free and con-
centrated in even dimensions.
46 M. HOVEY AND N. P. STRICKLAND
Proof. (a)Proposition 7.10 gives a Milnor exact sequence
lim-1(E=I)*+1(X) ae E_*X i lim(E=I)*X:
I - I
Theorem A.6 then implies that E_*X is L-complete.
(b)If X is finite in S or L then E^X is already K-local, so E_*X = E_*bLX = E**
*X.
(c)This follows from the Milnor sequence and the fact that (E=I)*X = (E*X)=I,
so there is no lim-1term.
(d)This is very similar to the proof of Proposition 2.4, except that we replace
(E=Ik)*X by (E=Ik)_*X = ss*bL(E=Ik^ X). Note that (E=In)_*X = K*X and
that (E=Ik)_*X is L-complete (by induction on k).
(e)If E_*X is pro-free, then the sequence {v0; : :;:vn-1} is regular on E_*X, *
*so
K*X = (E=In)_*X = (E_*X)=In.
(f)Proceed just as in Proposition 2.5 using the theories (E=Ik)_*X._
|__|
Recall that we have different notions of finiteness in K, or any stable homot*
*opy
category (Definition 1.5). Throughout the rest of this paper, we will denote t*
*he
category of small spectra in K by F and the category of dualisable spectra in K*
* by
D. Note that if X is dualisable, we have E*DX = ss*(E ^K X) = E_*X.
Theorem 8.5. Suppose that X 2 K. Consider the following statements:
(a)X is small.
(b)X is LF (n)-finite.
(c)X is a retract of LX0= bLX0 for some finite spectrum X0 of type at least n.
(d)E_*X is finite.
(e)E*X is finite.
(f)X is dualisable and K-nilpotent.
Then (a),(b) and (c) are equivalent, and they imply (d),(e) and (f ). (We shall
show in Corollary 12.16 that (d),(e) and (f ) are also equivalent to (a),(b) an*
*d (c).)
The category F D is thick. Moreover, if X 2 F and Y 2 D then X ^ Y ,
F (X; Y ) and F (Y; X) lie in F. In particular, DX = F (X; bLS) 2 D.
Proof.It follows from part (c) of [HPS95 , Theorem 2.1.3] that (a),(b), and that
small implies dualisable. Because every finite spectrum of type at least n lies*
* in the
thick subcategory generated by F (n), we see that (c))(b).
We now prove that (a))(c). Suppose that X is small. By Corollary 7.11, we see
that X = holim-!D(S=I) ^ X, so [X; X] = lim[X; D(S=I) ^ X], so X is a retract of
I -!I
Y = D(S=I) ^ X for some I. We claim that Y is small in L. To see this, consider
a family {Zi} of spectra in K. By Proposition 5.3 we have L(S=I) ~ K, which
implies that
L_ K_ K_
L(S=I) ^ Zi= L(S=I) ^ Zi= S=I ^ Zi
i i i
It follows that
L_ K_ M M
[Y; Zi]* = [X; S=I ^ Zi]* = [X; S=I ^ Zi]* = [Y; Zi]*
i i i i
as claimed. Since K(i)*Y = 0 for i < n, Corollary 6.11 implies that Y , and hen*
*ce
X, is a retract of LZ = bLZ for some finite spectrum Z of type at least n.
MORAVA K-THEORIES AND LOCALISATION 47
We now show that (c) implies (d),(e), and (f). By an evident thick subcategory
argument, we need only show that E_*S=I and E*S=I are finite, and that S=I is
dualisable and K-nilpotent. As S=I is finite, we see that E_*S=I = E*S=I = (E=I*
*)*,
which is clearly finite. Similarly, E*(S=I) = (E=I)* which is finite. Moreove*
*r,
Proposition 6.15 shows that S=I is K-nilpotent, and it is easy to see that S=I *
*is
dualisable.
From (b) it is clear that F is thick, and from (c) it is clear that F is clos*
*ed under
the Spanier-Whitehead duality functor D. Part (a) of [HPS95 , Theorem 2.1.3]
says that F is an ideal in D. This also means that if X 2 F and Y 2 D then_
F (X; Y ) = DX ^ Y and F (Y; X) = X ^ DY lie in F. |__|
Theorem 8.6. Suppose that X 2 K. The following are equivalent:
(a)X is F -small.
(b)X is dualisable.
(c)K*X is finite.
(d)K*X is finite.
(e)E_*X is finitely generated.
(f)E*X is finitely generated.
The category D of dualisable spectra is thick. Moreover, bLS 2 D and if X; Y 2 D
then X ^ Y 2 D and F (X; Y ) 2 D. In particular, DX = F (X; bLS) 2 D.
Proof.It follows from part (c) of [HPS95 , Theorem 2.1.3] that (a),(b), and also
that the subsidiary claims after (f) hold. Because
K*X = Hom K*(K*X; K*);
we see that (c),(d). Proposition 2.4 shows that (d),(f), and Proposition 8.4 sh*
*ows
that (c),(e).
We now prove that (a))(c). Suppose that X is F -small. We know from Theo-
rem 8.5 that LF (n) is small, and thus that F (n)^X is small, and thus that F (*
*n)^X
is a retract of LY for some finite type spectrum Y of type at least n. For any *
*finite
spectrum Y , it is easy to see that K*Y = K*LY is finite (by induction on the
number of cells). It follows that K*(F (n) ^ X) = K*F (n) K* K*X is finite. As
K*F (n) 6= 0, we conclude that K*X is finite.
Finally, we prove that (c))(b). Suppose that K*X is finite. There is a natural
map
X ^ F (X; U) ^ Y -ev^1--!U ^ Y
with adjoint
aeU;Y : F (X; U) ^ Y -! F (X; U ^ Y ):
Write
C = {U 2 K | 8Y 2 K aeU;Y is an isomorphism}:
we claim that every K-module M lies in C. Indeed, using Proposition 3.4 repeate*
*dly,
we see that
ss*(F (X; M) ^ Y ) = Hom K*(K*X; M*) K* K*Y
ss*(F (X; M ^ Y )) = Hom K*(K*X; M* K* K*Y ):
One can check that the map induced by aeM;Y is the obvious one, which sends f y
to x 7! f(x) y. Given that K*X is finite, we see that this map is an isomorphi*
*sm.
48 M. HOVEY AND N. P. STRICKLAND
As C is clearly thick, we see from Lemma 7.4 that all K-nilpotent spectra lie*
* in
C, in particular LF (n) 2 C. On the other hand, because LF (n) is dualisable, w*
*e can
identify F (X; LF (n)) ^ Y with LF (n) ^ (F (X; bLS) ^ Y ) and F (X; LF (n) ^ Y*
* ) with
LF (n) ^ F (X; Y ). Under these identifications, aeLF(n);Ybecomes 1LF(n)^ aebL*
*S;Y.
It follows that the smash product of LF (n) with the fibre of aebLS;Yis zero, s*
*o (by
Corollary 7.6) aebLS;Yis an isomorphism. This means that X is dualisable. *
*|___|
Corollary 8.7.Let G be a finite group, and X a finite G-complex. Then bL(EG^G
X) is dualisable. Moreover, we have DbLBG+ = bLBG+.
Proof.It is shown in [GS96b , Corollary 5.4] that K*(EG ^G X) is finite, so by
the preceding theorem, bL(EG ^G X) is dualisable. We next use the Greenlees-
May Tate construction [GM95b ] (see also [GS96a ]). We can think of any spectrum
X as a G-equivariant spectrum with trivial action indexed on a G-fixed universe,
apply a change of universe functor to get to a complete universe and perform the
Tate construction to get a G-spectrum tG X. We write PG X for the Lewis-May
fixed-point spectrum (tG X)G . It fits into a natural cofibre sequence
BG+ ^ X -! F (BG+; X) -!PG (X):
If we let X be bLS and localise, we get a cofibre sequence
bLBG+ -! DbLBG+ -! bLPG bLS:
Now in [HS96 ] it is shown that the Bousfield class of PG bLX is the same as th*
*at of
Ln-1X if X is finite. In particular, its K-localisation is trivial. Thus bLBG*
*+ =
DbLBG+. |___|
Corollary 8.8.If X is a connected p-local loop space such that ssk(X) is finite*
* for
all k and zero for almost all k. Then bL1 X is dualisable.
Proof.It is shown in [RW80 ] that K*K(A; q) is finite whenever A is a finite Ab*
*elian_
group, so it follows from the results of [HRW96 ] that K*X is finite. *
* |__|
Another important example of a dualisable spectrum is the Brown-Comenetz dual
of MS; see Section 10.
Theorem 8.9. Every dualisable spectrum in K is E-finite.
The proof depends on the following lemmas.
Lemma 8.10. Let X 2 K be a spectrum such that
(a)E*X is pro-free as a module over E.
(b)K*X is a finitely generated module over *, of projective dimension 0 d <
1.
Then there is a cofibration X -! J -! Y such that
(i)J is a finite wedge of suspensions of E.
(ii)K*J is the *-projective cover of K*X.
(iii)Y satisfies (a) and (b), with projective dimension d - 1 (and Y = 0 if d =*
* 0).
Proof.Since E*X is pro-free, we have K*X = (E*X)=In by Proposition 2.5.
In particular, * = (E*E)=In acts on K*X, so (b) makes sense. Let I be the
augmentation ideal in *. Choose a (finite) basis {__mi} for (K*X)=I over K*.
Since K*X = (E*X)=In, we can lift the __mito get elements mi 2 E*X. Define
MORAVA K-THEORIES AND LOCALISATION 49
W
J = i|mi|E, so the m's give an evident map m: X -! J. Define Y to be the
cofibre. It is immediate, using the arguments of Corollary 2.30, that K*J -! K*X
is the *-projective cover, and thus that the kernel K*Y has projective dimension
d - 1. In the exceptional case d = 0, we see that K*X = K*J and thus K*Y = 0;
as Y is K-local, we conclude that Y = 0. In any case, because * is Noetherian,
we see that K*Y is finitely generated over *.
By construction, the image of E*J in (E*X)=In = K*X is K*X. As E*J and
E*X are L-complete, we conclude from Theorem A.6 that E*J -! E*X is epi. As
E*X is pro-free, we conclude that this epimorphism splits. As E*J is also pro-f*
*ree,_
we see that E*Y is again pro-free. |__|
In the next lemma, we use the derived category DE of E-modules defined as
in [EKMM96 ]; see Section 1 for more discussion.
Lemma 8.11. Let M be an object of DE such that ss*M is a finitely generated
module over E*. Then M is E-finite in DE (and thus also in the category of
spectra).
Proof.This is very similar to the last result. We can recursively construct fin*
*itely
generated free modules Pk over the ring spectrum E and cofibrations Mk+1 -!
Pk -!Mk (with M0 = M), such that the modules ss*Mk form a minimal projective
resolution for ss*M over E*. As E* has finite global dimension n, we see that_
Mn+1 = 0, and thus M is E-finite as claimed. |__|
We can now prove the theorem.
Proof of Theorem 8.9.Jeff Smith has constructed a finite p-local spectrum X with
only even cells such that
Extk*(K*X; K*) = Extk*(K*; K*X) = 0 fork > n2:
See [Rav92a, Section 8.3] for a proof of this. Ravenel actually shows that
ExtkBP*BP(BP*; v-1nBP*X=In) = 0
for large k, but the Miller-Ravenel change of rings theorem [Rav86, Theorem 6.6*
*.1]
and the fact that X has only even cells implies that this is equal to Ext* (K*;*
* K*X).
That we can choose the horizontal vanishing line to be n2 does not appear to be
in the published literature, but is not important for this argument. As X has o*
*nly
even cells, the Atiyah-Hirzebruch spectral sequence collapses to show that E*X *
*is
free of finite rank over E*. It follows a fortiori that K*X = E*X=In is finite*
*ly
generated over *. Now choose a minimal projective resolution P* of K*X as
in Corollary 2.30. We find that the complex Hom * (P*; K*) has zero differentia*
*l,
so the projective dimension of K*X is at most n2. Thus Lemma 8.10 applies to
X0 = bLX to give a sequence of cofibrations Xk -! Jk -! Xk+1, where K*Xk has
projective dimension at most n2-k over R, and Xk = 0 for k > n2. As each Jk is a
finite wedge of E's, we conclude that X0 = bLX is E-finite. By a thick subcateg*
*ory
argument, we even see that bLS is E-finite.
Now let Y 2 K be dualisable. By the above, Y = bLS ^K Y is in the thick
subcategory generated by M = E ^K Y . Moreover, because Y is dualisable, we
know that ss*M is a finitely generated module over ss*E. It follows by Lemma 8.*
*11
that M is E-finite in DE (and thus in the category of spectra). It follows that
everything in the thick subcategory generated by M is E-finite, and thus_that Y*
* is
E-finite. |__|
50 M. HOVEY AND N. P. STRICKLAND
Corollary 8.12.If X 2 K is small, then X is K-finite, and in particular sskX is
finite for all k.
Proof.Because X is small it is easy to see using Theorem 8.5 and Proposition 4.*
*17
that the unit map X -! S=I ^ X is a split monomorphism for some generalised
Moore space S=I of height n. As bLS is E-finite, we see that S=I ^ X lies in the
thick subcategory generated by E ^ S=I ^ X = E=I ^ X. It is easy to see that
this lies in the thick subcategory generated by K ^ X, which is a finite wedge_*
*of
suspensions of K. This implies that X is K-finite. |__|
9. Homology and cohomology functors
In this section, we discuss the representability of (co)homology functors fro*
*m K
to the category Ab of Abelian groups. We write Ko and Ko for the categories of
homology and cohomology functors.
Theorem 9.1. Every cohomology functor H : K -!Ab is uniquely representable.
More precisely, the Yoneda functor X 7! [-; X] is an equivalence K ' Ko.
Proof.Observe that H O bLis a cohomology theory on S, and apply the Brown
representability theorem for S. It is easy to check that the representing_spect*
*rum
automatically lies in K. |__|
Before we discuss the more complicated representability of homology functors,
we discuss the closely related subject of realizing any object of K as a minima*
*l weak
colimit of small spectra. Recall that colimits do not usually exist in triangul*
*ated
categories, but that weak colimits always exist but are not unique. Minimal weak
colimits, discussed in [HPS95 , Section 2.2], often exist and are always unique*
* (in
algebraic stable homotopy categories).
We define
0(X) = {X0-u!X | X02 K is small}
00(X) = {X00-u!X | X002 S is a finite spectrum of type at}least:n
As usual, we make 0(X) and 00(X) into categories, in which the morphisms are
commutative triangles. It is not hard to see that they are filtered (compare [H*
*PS95 ,
Proposition 2.3.9]). Recall that every K-small spectrum is a retract of LY for
some finite spectrum Y of type at least n, and that there are only countably ma*
*ny
isomorphism classes of finite spectra. It follows that 0(X) and 00(X) both have
only a set of isomorphism classes, in other words that they are essentially sma*
*ll.
Proposition 9.2.For any X 2 K we have
X = mwlim-!X0= mwlim-!bLX00
0(X) 00(X)
(where the minimal weak colimits are computed in K). Moreover, the evident func-
tor bL:00(X) -!0(X) is cofinal.
Proof.Let LfX be the finite localisation of X away from F (n) and CfX the cor-
responding acyclisation. We then have a cofibration CfX -! X -! LfX, and
[F (n); LfX] = 0, so F (n) ^ LfX = 0. As K*F (n) 6= 0 and we have a K"unneth
isomorphism, we see that K*LfX = 0, so bLLfX = 0 and bLCfX = bLX = X.
MORAVA K-THEORIES AND LOCALISATION 51
By [HPS95 , Proposition 2.3.17], we know that CfX = mwlim-!S X00. As local-
00(X)
isation functors preserve minimal weak colimits [HPS95 , Theorem 3.5.1], we find
that X = bLCfX = mwlim-!K bLX00.
00(X)
We now prove that bL: 00(X) -! 0(X) is cofinal. Suppose that (X0 u-!X) 2
0(X). Because X0 is small, [X0; -] is a homology functor on K. It follows that
[X0; X] = lim-![X0; bLX00]:
00(X)
We can apply this fact to the element u 2 [X0; X]. The conclusion is that there
is a map X0 -! bLX00over X, for some X002 00(X). Moreover, given two such
maps X0 -! bLX00i(for i = 0; 1), there are maps X000-!X002- X001in 00(X)
such that the two composites X0 -!bLX002are the same. This means precisely that
bL: 00(X) -!0(X) is cofinal. It follows easily that X = mwlim X0. |__*
*_|
-! 0(X)
Recall from [HPS95 , Section 4.1] that a homology functor H : K -! Ab is
representable if there is a spectrum X 2 K and a natural isomorphism H(Y ) '
[S; X ^ Y ] for K-small spectra Y . (It does not matter whether the smash produ*
*ct
is interpreted in S or K, because Y is small). Recall also that for any X and Y*
* in
K we have
M(X ^K Y ) = M(X ^S Y ) = MS ^S X ^S Y:
Lemma 9.3. A natural isomorphism H(Y ) ' [S; X ^ Y ] for small spectra Y gives
rise to a natural isomorphism H(Y ) ' [S; M(X ^ Y )] = MX0Y for all Y 2 K.
Proof.Write {Yi} for the objects of 00(Y ) (as in Proposition 9.2). Thus each Yi
is a finite spectrum of type at least n, so LYi = MYi = bLYi is K-nilpotent by
Proposition 6.15. Moreover, Y = mwlim-!KLYi and thus HY = lim[S; X ^ Yi].
i -!i
Next, because X 2 K we see that X = LX, so there is a cofibration MX -!
X -! Ln-1X. Because Ln-1X ^S Yi= X ^S Ln-1Yi= 0, we see that
X ^ Yi= MX ^ Yi:
We know that mwlim-!SYi= CfY , using the notation of the proof of Proposition 9*
*.2.
i
Moreover [F (n); CfY ]* = [F (n); Y ]*, so DF (n) ^ CfY = DF (n) ^ Y , so MCfY =
MY (since MZ = MS ^ Z and MS ~ LS ^ DF (n)). It follows that
mwlim-!SMX ^ Yi= MX ^ CfY = MX ^ Y:
i
This gives an isomorphism
[S; MX ^ Y ] ' lim-![S; MX ^ Yi] ' lim-![S; X ^ Yi] ' H(Y ): |___|
i i
Before stating the representability theorem for homology functors, it is conv*
*enient
to record the definition of phantom maps.
Definition 9.4.A map f : X -! Y in K is phantom if (Z -u!X -f!Y ) = 0 for all
small Z and all u : Z -! X. We also say that f is cophantom if (X -f!Y -u!Z) = 0
for all small Z and all u : Y -! Z.
52 M. HOVEY AND N. P. STRICKLAND
Theorem 9.5. Any homology functor H :K -! Ab is representable. More pre-
cisely, there is a spectrum X 2 K such that H(Y ) = X0Y = ss0(X ^ Y ) for all
K-small Y , and H(Y ) = MX0Y = ss0(MX ^ Y ) for all Y 2 K. The group of natu-
ral maps MX0 -!MY0 is naturally isomorphic to [X; Y ]=phantoms. Any composite
of a phantom with another map is phantom, and any composite of two phantoms is
zero.
Proof.This follows easily by combining the results of [HPS95 , Section_4.1] with
Lemma 9.3. |__|
10. Brown-Comenetz duality
In this section, we define and study a good notion of Brown-Comenetz duality *
*on
the category K. We have been strongly influenced by the outline in [HG94 ]. Rec*
*all
the usual definition of Brown-Comenetz duality. In keeping with our convention
that everything is p-local, we write Q=Z for Q=Z(p). Because ss0X is a homology
functor on S and Q=Z is an injective Abelian group, we see that Hom (ss0Y; Q=Z)*
* is
a cohomology functor. By the Brown representability theorem, there is a spectrum
I unique up to canonical isomorphism, with a natural isomorphism
Hom (ss0Y; Q=Z) ' [Y; I]:
We also write IX = F (X; I), so there is a natural isomorphism Hom (X0Y; Q=Z) '
[Y; IX]. Recall from Section 9 that any homology functor on K has the form Y 7!
ss0(MX ^ Y ) for some X 2 K. This motivates the following definition:
Definition 10.1.If X 2 K, the Brown-Comenetz dual of X is
bIX = IMX = F (MX; I):
We also write bI= bIS, so that bIX = F (X; bI).
Theorem 10.2. (a) There is a natural isomorphism
[Y; bIX] ' Hom (MX0Y; Q=Z):
(b)The correspondence X 7! bIX is a contravariant exact functor Kop-! K.
(c)bIK ' K.
(d)The natural map X -! bI2X is an isomorphism when ss*(F (n) ^ X) is finite
in each degree (in particular, when X is dualisable, for example if X = S).
(e)The spectrum bIis invertible under the smash product.
(f)The functor X 7! bIX preserves the categories F, D, I and N of small, dual-
isable, K-injective and K-nilpotent spectra.
The isomorphism classes of invertible spectra form the Picard group Pic, stud*
*ied
in Section 14 below and in [HMS94 , Str92]. There is a natural homomorphism from
Picto an algebraically defined Picard group, as explained in [HMS94 ]. It is na*
*tural
to ask where bIgoes under this map. The answer is stated in [HG94 ], but we will
not discuss this issue here.
We will prove Theorem 10.2 after two lemmas.
Lemma 10.3. Let {Ak} be an inverse system of finite-dimensional vector spaces
of dimension at most d. Then lim-Ak has dimension at most d (and in particular
k
is finite-dimensional), and lim-1Ak = 0.
k
MORAVA K-THEORIES AND LOCALISATION 53
Proof.This follows easily from the standard Mittag-Leffler construction. Explic*
*itly,
we define
"
A0k= image(Am -! Ak):
m>k
As Ak has finite dimension we must have A0k= image(Am -! Ak) for m 0. The
groups A0kform a subtower of A, and the quotient A=A0is pro-trivial so lim-sAi=
i
lim-sA0ifor s = 0; 1. The maps A0k-! A0k-1are surjective so lim1A0i= 0. The
i - i
dimension of A0iis a nondecreasing function of i that is bounded by d, so for l*
*arge
i it is constant and thus the map A0i-!A0i-1is an isomorphism. This means that_
lim-A0i= A0mfor large m, so limA0ihas dimension at most d. |__|
i - i
The next lemma is crucial.
Lemma 10.4. Let S=I be a localised generalised Moore spectrum of type n. Then
[E; S=I]* is finite.
Proof.We can form the standard E-based Adams resolution of S=I and apply the
functor [E; -] to get a spectral sequence whose E1 term is the group
__(s) -s__(s)
Cst(I) = Es;t1= [E; E ^ -sE ^ S=I]t-s= [E; E ^ E=I]t-s:
__
Here E is the cofibre of the unit map S -! E. As E*E is not a projective module
over E*, the standard theorems are not enough to identify the E2 page as an Ext
group but this turns out not to be important. What is important is that S=I is
K-nilpotent and thus E-nilpotent, so the resulting spectral sequence converges *
*to
[E; S=I]* (by the argument of Proposition 6.5). It will thus be enough to show *
*that
the E2 page is finite in each total degree._
To do this, we define Cst= [E; E ^ -sE (s)]t-s. We make C** into a module
over E* using the second copy of E. Using Proposition 8.4 (f) we see that ss*bL*
*(E ^
__(s) __(s)
E ) is pro-free and in even degrees,Wso that bL(E ^ E ) is isomorphicQas an E-
module to a spectrumQof the form bL( i2diE). This is a retract of i2diE so
Cs* is a retract of i[E; 2diE]* and thus is pro-free. It is not hard to dedu*
*ce
that C**(I) = C**=IC** and that this admits a finite filtration with quotients
C**=InC** = C**(In). It will thus be enough to show that H*C**(In) is finite in
each total degree. But we have E=In = K, so by Proposition 3.4 we have
__(s) t __s t __s
Es;t1= Hom t-sK*(K*E; -sK*(E )) = Hom K*(*; * ) = Hom * (*; * * )
__
where * is the coaugmentation coideal of *. It follows that
H*C**(In) = Ext***(*; K*):
Consider the subring 0(k)* < * generated by t1; : :;:tk-1. It is easy to check
thatSthis is a sub Hopf algebra, and thus a subcomodule of *. Moreover, we have
* = k 0(k)*, which gives a short exact sequence of comodules
M M
0(k)* ae 0(k)* i *:
k k
This in turn gives a Milnor sequence relating Ext* (*; K*) to lim-iExt*(0(k)*; *
*K*)
for i = 0; 1. As 0(k) is a finite-dimensional cocommutative Hopf algebra, it is
54 M. HOVEY AND N. P. STRICKLAND
self-dual (up to a dimension shift) as a comodule over itself [LS69] and thus a*
*s a
comodule over *. We thus have
Ext* (0(k)*; K*) = Ext* (K*; 0(k)*) = Ext*==0(k)*(K*; K*):
We know from [Rav86, Theorem 6.3.7] that for k 0 this is an exterior algebra
over K* on n2 generators, and thus has dimension 2n2 independently of k. It fol*
*lows
from Lemma 10.3 and the Milnor sequence that Ext***(*; K*) has finite dimension_
over K*, as required. |__|
Corollary 10.5.[K; S]* is finite.
Proof.It follows from the lemma that [E ^ DS=I; S]* is finite, and it is easy_t*
*o_see
that K lies in the thick subcategory generated by E ^ DS=I. |__|
Proof of Theorem 10.2.(a): This is clear.
(b): We first need to show that bIX is K-local. Suppose that Z is K-acyclic. *
*As
MS ~ K and MX = MS ^ X, we see that MX ^ Z = 0, so
[Z; bIX]* = Hom (ss*(MX ^ Z); Q=Z) = 0
as required. Because bIX = F (X; bI), the correspondence X 7! bIX is clearly a
contravariant exact functor.
(c): Note that MK = K. Think of ss0K = Fp as a subgroup of Q=Z in the
obvious way. We know that [X; K]* ' Hom K*(K*X; K*). Looking in degree zero,
we see that
[X; K] ' Hom (MK0X; Fp) = Hom (MK0X; Q=Z)
the first equality because vn is a unit, and the second equality because MK0X i*
*s a
vector space over Fp. It follows that bIK = K.
(d): The evaluation map X ^ F (X; bI) -! bIgives by adjunction a natural map
: X -! F (F (X; bI); bI) = bI2X. Write F = F (n) and Y = F ^X, so that ss*Y is*
* finite
in each degree. Because F is dualisable, we have bIY = DF ^bIX and bI2Y = F ^bI*
*2X.
Under this identification, we have Y = F^X = 1F ^X , so it is enough to show th*
*at
Y is an isomorphism. Because F and DF are K-nilpotent, we see that MY = Y
and MIbY = bIY . It follows that ss*bI2Y = Hom (Hom (ss*Y; Q=Z); Q=Z), which is
the same as ss*Y because ss*Y is finite in each degree. Thus Y is an isomorphis*
*m,
as required.
If X is dualisable then F ^ X is small by Theorem 8.5, so ss*(F ^ X) is finit*
*e in
each degree by Corollary 8.12, so the above applies.
(e): We know from Lemma 10.5 that [K; S]* is finite in each degree. We also
know from (c) and (d) that K and S are isomorphisms, from which it follows that
bI: [K; S]* -![bI; bIK]* = K*bIis an isomorphism. Thus K*bIis finite, which imp*
*lies
by Theorem 8.6 that bIis dualisable. This implies that F (bI; bI) = DbI^ bI, bu*
*t we
also know that F (bI; bI) = bI2S = S. Thus DbI^ bI' S, which means that bI2 Pic.
(f): By part (c), bIK = K. Any K-injective spectrum X is a wedge of sus-
pensions of K, so bIX is a product of suspensions of K, which is still K-inject*
*ive
by Proposition 3.3. Thus X 7! bIX preserves I, and so also N, by Lemma 7.4.
Part (e) implies that bI2 D, so it follows from Theorems 8.6 and 8.5 that the
functor X 7! bIX = DX ^ bIpreserves D and F. |___|
Corollary 10.6.If X is an arbitrary spectrum in K, then bIX ' DX ^ bI.
MORAVA K-THEORIES AND LOCALISATION 55
Proof.Because bIis dualisable, we have F (X; bI) ' F (X; S) ^ bI. *
* |___|
Recall the definition of phantom and cophantom maps from Definition 9.4.
Theorem 10.7. Suppose that X; Y 2 K.
(a)There are no nonzero phantom maps X -! IY .
(b)If ss*(F (n) ^ Y ) is finite in each degree (in particular, if Y is dualisa*
*ble, for
example if Y = S) then there are no nontrivial phantom maps X -! Y .
(c)A map u : X -! Y is phantom if and only if it is cophantom.
Proof.(a): Suppose that u : X -! IY is phantom. This means that the correspond-
ing map u# : ss0(X ^ MY ) = ss0(MX ^ Y ) -! Q=Z vanishes on ss0(Z ^ MY ), for
every small spectrum Z equipped with a map Z -! X. On the other hand, X is
the minimal weak colimit in K of these Z's (Proposition 9.2), and ss0(- ^ MY ) *
*is
a homology functor, so ss0(X ^ MY ) = lim-!ss0(Z ^ MY ). It follows that u# = 0,
Z
and thus u = 0.
(b): By part (d) of Theorem 10.2, we know that Y = bI2Y . It therefore follows
from (a) that there are no nonzero phantoms X -! bI2Y = Y .
(c): In the following statements, Z runs over K-small spectra. Thus MDZ = DZ
and
Hom ([Z; X]; Q=Z) = [X; Z ^ bI]:
u is phantom , 8Z [Z; X] -![Z; Y ] is zero
, 8Z Hom([Z; X]; Q=Z)- Hom ([Z; Y ]; Q=Z) is zero
, 8Z [X; Z ^ bI]- [Y; Z ^ bI] is zero
, 8Z [X; Z] -![Y; Z] is zero
, u is cophantom
(The fourth implication uses the fact that bI2 Pic, so that Z 7! Z ^ bIis an_au*
*to-
morphism of the category of small spectra). |__|
11.The natural topology
As in [HPS95 , Section 4.4], we can give [X; Y ] a natural topology. For any *
*map
f : F -! X with F small, we write Uf = {u : X -! Y | u O f = 0}, and then give
[X; Y ] the linear topology for which the sets Uf form a basis of neighbourhood*
*s of
0. We recall from [HPS95 , Proposition 4.4.1] the basic properties of this topo*
*logy:
Proposition 11.1.
(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 Hausd*
*orff
if and only if there are no phantoms from X to Y .
(e)[X;WY ] is always complete. Q
(f)[ iXi; Y ] is homeomorphicQto i[Xi; Y ] with the product topology, but t*
*he
natural topology on [X; iYi] is finer than the product topology in general.
56 M. HOVEY AND N. P. STRICKLAND
Corollary 11.2.If we give Hom (sskX; sskY ) the compact-open topology, then it
follows from (a) that the evident map
[X; Y ] -!Hom (sskX; sskY )
is continuous.
We next prove some additional properties.
Proposition 11.3.The smash product map [W; X] x [Y; Z] -! [W ^ Y; X ^ Z] is
continuous.
Proof.Suppose we have maps u: W -! X and v :Y -! Z, a K-small spectrum A
and a map A f-!W ^ Y , so that
N = {w :W ^ Y -! X ^ Z | wf = (u ^ v)f}
is a basic neighbourhood of u ^ v in [W ^ Y; X ^ Z]. We know from Proposition 9*
*.2
that W can be written as a minimal weak colimit W = mwlim-!Wffwith Wffsmall,
ff
and similarly Y = mwlim-!Yfi. Note that [A; W ^ (-)] is a homology theory, so t*
*hat
fi
[A; W ^ Y ] = lim-![A; W ^ Yfi], so f factors through a map g :A -!W ^ Yfifor s*
*ome
fi
fi. By a similar argument, g factors through a map h: A -! Wff^ Yfifor some ff.
Write i and j for the given maps Wff-!W and Yfi-!Y . Write
N0 = {(u0; v0) 2 [W; X] x [Y; Z] | u0i = ui andv0j = vj}:
Then N0 is a neighbourhood of (u; v) and if (u0; v0) 2 N0 then u0^ v0 2 N. This*
* __
means that the smash product is continuous at (u; v), as required. *
*|__|
Corollary 11.4.The adjunction map [X; F (Y; Z)] -! [X ^ Y; Z] is a continuous
bijection, and a homeomorphism if Y is dualisable.
Proof.Let ffl: F (Y; Z) ^ Y -! Z be the evaluation map. Then the adjunction map
is the composite
[X; F (Y; Z)] (-)^1Y-----![X ^ Y; F (Y; Z) ^ Y ] ffl*-![X ^ Y; Z];
which is continuous by Proposition 11.3 and Proposition 11.1(a). It is also a b*
*ijec-
tion, by the defining property of F (Y; Z). If Y is dualisable and j :S -!Y ^ D*
*Y =
F (Y; Y ) is the unit map then we can identify the inverse of the adjunction ma*
*p with
the composite
*
[X ^ Y; Z] (-)^1----![X ^ Y ^ DY; Z ^ DY ] (1^j)----![X; Z ^ DY ];
which is again continuous. |___|
Proposition 11.5.If [F (n); Y ]* is finite, then for any X 2 K we have
[X; Y ] = lim-{[X0; Y ] | (X0-! X) 2 0(X)};
and this is a compact Hausdorff group.
Remark 11.6.This applies when Y is E-finite, and thus (by Theorem 8.9) when Y
is dualisable.
Proof.Because X is the minimal weak colimit of 0(X), the map from [X; Y ] to the
inverse limit is surjective. The kernel is the set of phantom maps X -! Y_,_whi*
*ch is
zero by part (b) of Theorem 10.7. |__|
MORAVA K-THEORIES AND LOCALISATION 57
Corollary 11.7.If X is dualisable and Y is small then [X; Y ] is both finite_(b*
*y_
Corollary 8.12) and compact Hausdorff, hence discrete. |__|
Corollary 11.8.If X and Y are dualisable then the map D :[X; Y ] -![DY; DX]
is a homeomorphism.
Proof.As D2 = 1 it is enough to check that D is open. Consider a K-small spectr*
*um
Z and a map f :Z -! X, so that Uf = {u: X -! Y | uf = 0} is a basic open
neighbourhood of 0 in [X; Y ]. The image under D is {v :DY -! DX | (Df)Ov = 0}.*
* __
This is open in [DY; DX] because (Df)* is continuous and [DY; Z] is discrete. *
*|__|
Proposition 11.9.If X is dualisable, then the natural topology on [X; E] is the
In-adic topology. This is also the same as the topology defined by the kernels *
*of the
maps [X; E] -! [X; E=I] as I runs over a tower as in Proposition 4.22. Similar
remarks apply to E_*X.
Proof.The natural topology is profinite by Proposition 11.5. The In-adic topol-
ogy is profinite because E*X is finitely generated over E* by Theorem 8.6 and
Proposition 2.4. Using Theorem 8.6 again and a thick subcategory argument we see
that (E=I)*X is finite. We also know from Corollary 7.11 that E = holim-E=I and
I
there are no lim-1terms because everything is finite so [X; E] = lim-[X; E=I] a*
*nd the
I
topology defined by the kernels is again profinite. As the maps [X; E] -! [X; E*
*=I]
are continuous, we see that the natural topology is at least as large as the to*
*pology
defined by the E=I, and this is at least as large as the In-adic topology becau*
*se
the groups [X; E=I] are In-torsion. By a well-known lemma, comparable compact
Hausdorff topologies are always equal. The result carries over to E_*X_because_*
*of
Corollary 11.4. |__|
12. Dualisable spectra
In this section we prove some more results about the category D of dualisable
spectra in K, and its subcategory F of K-small spectra.
First observe that F is the same as the category Cn L considered in Sec-
tion 6.1. The following three results thus follow from Theorem 6.9, Theorem 6.12
and Corollary 6.6.
Proposition 12.1.The only thick subcategories of F are {0} and F. |___|
Proposition 12.2.Every small spectrum X has a good vn self map v :dX -!_X_
(which is an isomorphism). |__|
Proposition 12.3.Let u: dX -! X be a self map of a K-small spectrum such __
that K*u is nilpotent. Then u is nilpotent. |__|
For self maps of spectra that are dualisable but not small it is natural to c*
*onsider
self maps that are topologically nilpotent rather than nilpotent. We first need*
* to
define topological nilpotence.
Proposition 12.4.Let X be a spectrum in K and u: dX -! X a map. Let u-1X
denote the sequential colimit in K of the sequence
X -u!-dX -u!-2dX -! : :::
Consider the following statements, in which Z and W run over K-small spectra.
(a)u-1X = 0.
58 M. HOVEY AND N. P. STRICKLAND
(b)For all Z and all z :Z -! X we have uN O z = 0 for N 0.
(c)For all Z we have 1Z ^ uN = 0 for N 0.
(d)For all W and all w :W -! S, the following composite vanishes for N 0:
N
NdW ^ X -w^1-!NdX -u-!X:
Equivalently, the following adjoint map vanishes:
#)N
NdW -w!SNd -(u---!F (X; X):
(e)d = 0 and uN -! 0 in the natural topology on [X; X].
Then (a),(b)((c) and (b)((d). If X is dualisable then (b),(c),(d). If d = 0
then (c),(e).
Definition 12.5.If (a) and (b) hold in the above proposition, we say that u is
topologically nilpotent.
Proof of Proposition 12.4.(a),(b): We know that u-1X = 0 if and only if we have
[Z; u-1X] = 0 for all Z, and as Z is small we have [Z; u-1X] = u-1[Z; X]. The
claim follows easily.
(a)((c): If (c) holds then we see that Z ^ u-1X = 0 and thus u-1X = 0, so (a)
holds. Conversely, suppose (a) holds and X is dualisable. Then Z ^ X is small by
Theorem 8.5, and (1 ^ u)-1(Z ^ X) = 0. It follows that (c) holds.
(d))(b): By Corollary 7.11 we have X = holim-!D(S=I)^X, so any map z :Z -!
I
X factors through D(S=I) ^ X -! X for some I. If (d) holds then D(S=I) ^ X -!
N uN
X -u-!-Nd X vanishes for N 0 and thus so does Z -! X --! -Nd X, so (b)
holds.
We next assume that X is dualisable; then if (b) holds we can put Z = W ^ X
and deduce that (d) holds. *
* __
Finally, if d = 0 then (e) is a direct translation of (b). *
* |__|
Lemma 12.6. If u: dX -! X is topologically nilpotent, then the same is true of
u ^ 1Y for any Y .
Proof.The functor (-) ^ Y preserves sequential colimits, so the claim is_obviou*
*s_
from criterion (a). |__|
Recall that a self map u : A -!A of a topological Abelian group A is said to *
*be
topologically nilpotent if the sequence {unx} converges to 0 uniformly in x. We*
* will
apply this definition to E*X and E_*X for dualisable X, using the In-adic topol*
*ogy.
In this case, both E*X and E_*X are finitely generated, so there is no differen*
*ce
between pointwise and uniform convergence.
Proposition 12.7.Let u: dX -! X be a self map of a dualisable spectrum. Then
the following are equivalent:
(a)K*u is nilpotent.
(b)K*u is nilpotent.
(c)E_*u is topologically nilpotent.
(d)E*u is topologically nilpotent.
(e)u is topologically nilpotent.
MORAVA K-THEORIES AND LOCALISATION 59
Proof.(a),(b): Duality.
(a),(e): Recall that u is topologically nilpotent if and only if 1Z ^ u is ni*
*lpotent
for all small Z. This is equivalent to (a) by the K"unneth theorem and Proposi-
tion 12.3.
(d))(b): We check by induction on k that u induces a topologically nilpotent
self map of (E=Ik)*X. Indeed, consider the short exact sequence
k+1
(E=Ik)*(X)=vk ae (E=Ik+1)*(X) i ann(vk; (E=Ik)*-2p (X)):
The subspace topology on ann(vk; (E=Ik)*(X)) is the same as the In-adic topology
by the Artin-Rees lemma. It is then easy to see that u induces a topologically
nilpotent map on (E=Ik+1)*X if it does on (E=Ik)*X. In the case k = n we have
(E=In)*X = K*X and this is finite and thus discrete so the self map is nilpoten*
*t.
(b))(d): It is easy to check that for any generalised Moore spectrum S=I of
height n, the spectrum E=I is K-finite so (E=I)*u is nilpotent. It follows eas*
*ily
N a
that if a 2 E*X then the composite NdX -u-!X -! E -! E=I vanishes for large
N. We know from Proposition 11.9 that the natural topology is the same as that
defined by the kernels of the maps E*X -! (E=I)*X. It follows that a O uN -! 0 *
*in
the In-adic topology. Thus the sequence (uN )* 2 EndE*(E*X) converges to zero
pointwise, and hence uniformly.
This shows that all our statements are equivalent except for (c). However, E_*
**u =
E*Du and by the above applied to Du this is topologically nilpotent if and only
if Du is, and Du is topologically nilpotent if and only if u is (by Corollary_1*
*1.8).
Thus (c) is equivalent to (e) also. |__|
Lemma 12.8. Let u: dX -! X be a self map of a small spectrum. Then there
is a unique idempotent e: X -! X such that u is the wedge of an isomorphism
deX -! eX with a nilpotent map d(1 - e)X -! (1 - e)X.
Proof.Let v :tX -! X be a good vn self map. We can choose a > 0 and b 2 Z
such that w = uavb has degree zero. By Corollary 8.12, we know that [X; X] =
ss0(DX ^X) is a finite ring. It follows that there exist r < s such that wr = w*
*s. We
then find that wk(s-r)= w(k+1)(s-r)as long as k(s - r) > r. Thus, if t is a mul*
*tiple
of s - r which is greater than r, we find that e = wt = uatvbtsatisfies e2 = e.
Thus e is an idempotent which commutes with u, which means that u respects the
splitting X = eX _ (1 - e)X. As uatvbt= 1 on eX we see that u: deX -! eX
is an isomorphism. On the other hand, we have uatvbt= 0 on (1 - e)X, so that
u: (1 - e)X -! (1 - e)X is nilpotent.
If e0is a different idempotent with the required properties then e = uatvbtis*
* the
wedge of an isomorphism on e0X with a nilpotent self map on (1 - e0)X. As e is_
idempotent, it is easy to deduce that e0= e. |__|
Proposition 12.9.Let u: dX -! X be a self map of a dualisable spectrum in
K. Then there is a unique idempotent e: X -! X such that u is the wedge of
an isomorphism deX -! eX with a topologically nilpotent map d(1 - e)X -!
(1 - e)X.
Proof.Choose a tower of generalised Moore spectra S=I(j) as in Proposition 4.22.
We will write X=I(j) for X ^ S=I(j). For each j Lemma 12.8 gives a unique
idempotent ej:X=I(j) -! X=I(j) such that u ^ 1 is a wedge of an isomorphism
60 M. HOVEY AND N. P. STRICKLAND
and a nilpotent in terms of the splitting given by ej. We claim that the follow*
*ing
diagram commutes:
ej
X=I(j) _________X=I(j)w
| |
| |
1^g|j |1^gj
| |
|u |u
X=I(j - 1)_____X=I(jw-e1):j-1
To see this, let v be a good vn self map of S=I(j). Using Remark 4.25 and
Proposition 4.4, we see that v induces compatible vn self maps of all spectra i*
*n the
diagram (all of which we call v) and that v O (1 ^ gj) = (1 ^ gj) O v. Note als*
*o that
(u ^ 1) O (1 ^ gj) = (1 ^ gj) O (u ^ 1). It follows from the proof of Lemma 12*
*.8
that there exist integers a; b such that ej = (u ^ 1)avb (interpreted as a self*
* map of
X=I(j)) and ej-1= (u ^ 1)avb (interpreted as a self map of X=I(j - 1)). It foll*
*ows
easily that the diagram commutes.
We next recall from Proposition 7.10 that X = holim-X ^ S=I(k). Moreover,
k
Corollary 8.12 shows that [X; X ^ S=I(k)] = ss0(DX ^ X ^ S=I(k)) is finite, so *
*there
is no lim1term and [X; X] = lim-[X; X ^ S=I(k)]. It follows that there is a uni*
*que
k
map e: X -! X such that the following diagram commutes for each j:
X __________Xwe
| |
1^jj| |1^jj
| |
|u |u
X=I(j) _____X=I(j):wej
The map e2 also has this property, so e2 = e. We also know that
(1 ^ jj) O (ue - eu) = ((u ^ 1)ej- ej(u ^ 1)) O (1 ^ jj) = 0
for all j, and thus that ue = eu, so u is compatible with the splitting X = eX _
(1 - e)X. Similar considerations show that u is an isomorphism on eX and that
u ^ 1 is nilpotent on (1 - e)X=I(j) for all j. It follows by Proposition 12.7_t*
*hat u
is topologically nilpotent on (1 - e)X. |__|
It follows from the previous proposition that the Krull-Schmidt theorem holds*
* in
D. (It was shown by Freyd [Fre66] that the Krull-Schmidt theorem holds for fini*
*te
torsion spectra, and our argument is much the same.)
Definition 12.10.We say that a spectrum X 2 D is indecomposable if X 6= 0 and
there do not exist nontrivial spectra Y; Z 2 D with X ' Y _ Z.
We will see in Proposition 12.17 that there are only a set of isomorphism cla*
*sses
of dualisable spectra, and hence of indecomposable spectra.
Proposition 12.11.Let X 2 D be an indecomposable spectrum, and write R =
[X; X]. Let I R be the set of topologically nilpotent self maps of X. Then I i*
*s a
two-sided ideal in R and R=I is a finite field.
Remark 12.12.Adams and Kuhn have shown [AK89 ] that in a slightly different
context, all possible finite fields can occur. Presumably this is true for us t*
*oo.
MORAVA K-THEORIES AND LOCALISATION 61
Proof.First, it follows immediately from Proposition 12.9 that every u 2 R is
either invertible or topologically nilpotent, so R = Rx q I. If a 2 I and b 2 R
then a 62 Rx so ab; ba 62 Rx so ab; ba 2 I. Now suppose that a; b 2 I. We claim
that c = a + b 2 I. If not, then c is invertible. By ourPprevious observation*
* we
have c-1b 2 I so c-1b is topologically nilpotent. Hence k(c-1b)k converges to*
* an
inverse for 1 - c-1b = c-1a, so a is invertible. This contradicts the assumptio*
*n that
a 2 I. Thus I is a two-sided ideal. It is clear that every nonzero element of R*
*=I is
invertible, so R=I is a division ring. Next, let J be the kernel of the evident*
* map
R -! End(K*X). This is clearly a two-sided ideal, and Proposition 12.7 tells us
that J I. Because K*X has finite dimension over K*, we see that R=J is finite
and thus that R=I is finite. A well-known theorem of Wedderburn says that a_fin*
*ite
division ring is a field. |__|
Proposition 12.13.For any spectrum X 2 D there are unique integers nY =
nY (X) 0, as Y runs through isomorphism classes of indecomposable spectra,W
such that nY = 0 for all but finitely many Y , and X is isomorphic to Y 2InY *
*Y .
(Here nY Y means the wedge of nY copies of the spectrum Y ).
Proof.We may assume that this holds for all X0such that dim(K*X0) < dim(K*X).
It is clear that it also holds if X is indecomposable. If X is decomposable we *
*can
write X = V _ W with V 6= 0 6= W and thus dim(K*V ) > 0 and dim(K*W ) > 0
and thus dim(K*V ) < dim(K*X)Wand dim(K*W ) < dim(K*X).WBy the induction
hypothesisWwe have V ' YnY (V )Y and W ' YnY (W )Y . It follows that
X ' Y (nY (V ) + nY (W ))Y , so we may take nY (X) = nY (V ) + nY (W ).
We still need to prove uniqueness. For this, we change notation and assume
that we have indecomposables U1; : :;:Ur and V1; : :;:Vs such that there exists*
* an
isomorphism f :U1 _ : :_:Ur ' V1 _ : :_:Vs. We need to show that r = s and
that after reordering the V 's we have Ui ' Vi for all i. Write g = f-1, and
fijfor the component of f mappingPUi to Vj, and gjifor the component of g
mappingPVj to Ui. We then have jgjifkj = ffiik:Uk -! Ui. In particular, we
have jgj1f1j = 1: U1 -! U1. Recall that the maps U1 -! U1 that are not
isomorphisms form a two-sided ideal (and thus an additive subgroup) in [U1; U1];
it follows that gj1f1j is invertible for some j. After renumbering the V 's, we*
* may
assume j = 1. This means that f11:U1 -!V1 is a split monomorphism but V1 is
indecomposable so f11is an isomorphism. Similarly g11is an isomorphism. We now
write U0 = U2 _ : :_:Ur and V 0= V2 _ : :_:Vs and redefine our notation slightly
so that f12is the component of f :U1_ U0 -!V1_ V 0mapping U1 to V 0and so on.
We thus have a matrix equation
f11 f21 g11 g21 1V1 0
f12 f22 g12 g22 = 0 1V 0 :
One can deduce directly that f22:U0 -!V 0is an isomorphism, with inverse given
by g22-g12g-111g21. By induction on the number of indecomposables, we can assume
that r - 1 = s - 1 and that after reordering we have Ui' Vi for i > 1. Thus_r =*
* s
and Ui' Vi for all i. |__|
We next exhibit some ideals in D; we conjecture that they are all the ideals.
Definition 12.14.Given k n we write Dk for the category of spectra X 2 D
such that X is a retract of Y ^ Z for some Y 2 D and some finite spectrum Z of
type at least k. It is easy to see that Dn = F.
62 M. HOVEY AND N. P. STRICKLAND
Proposition 12.15.Dk is an ideal in D and is closed under D. Moreover, given
X 2 D, the following are equivalent.
(a)X 2 Dk.
(b)X is a module over some generalised Moore spectrum S=I of height k.
(c)E_*X is Ik-torsion.
(d)E*X is Ik-torsion.
Proof.(a),(b): This is Proposition 4.17.
(b))(c): It is easy to see using the cofibrations which define S=I that E_*(X*
* ^
S=J) is Ik-torsion. It follows that E_*X is Ik-torsion.
(c))(b): By induction we may assume that there is a generalised Moore spectrum
S=J of type k - 1 such that X is a module over S=J. Let v be a good vk-1
self map of S=J. Let C be the category of those spectra Y such that for each
a 2 ss*(S=J ^ X ^ Y ) we have vN a = 0 for N 0. One sees easily that C is thick
and E 2 C. Thus D(S=J ^ X) 2 C by Theorem 8.9, and v acts nilpotently on
1 2 ss0(S=J ^ X ^ D(S=J ^ X)) = End(S=J ^ X). This means that v is nilpotent
as a self map of S=J ^ X, so that the evident map S=J ^ X -! S=(J; vpN) ^ X is a
split monomorphism for large N. As X -! S=J ^ X is also a split monomorphism,
we see that X is a module over S=(J; vpN).
We also know from Proposition 4.17 that Dk is an ideal. If X is a retract of
Y ^ Z for dualisable Y and Z, then DX is a retract of D(Y ^ Z) = DY ^ DZ. Thus
Dk is closed under D. It is then immediate from the isomorphism E*X =_E_*DX_
that (c),(d). |__|
Corollary 12.16.If X 2 K then the following are equivalent:
(a)X is small.
(b)E*X is finite.
(c)E_*X is finite.
(d)X is dualisable and K-nilpotent.
Proof.Using Theorem 8.6 we see that any of (a) : :(:d) implies that X is dualis*
*able,
so we may assume this throughout. We saw in Theorem 8.5 that (a) implies (b),(c)
and (d). By applying Proposition 12.15, we see that (b) implies (a) and (c) imp*
*lies
(a).
All that is left is to prove that (d) implies (a). Suppose that X is dualisab*
*le and
K-nilpotent. Let I be the category of those spectra Z 2 K such that Z is a modu*
*le
over S=I for some I of height n. We know from Proposition 4.17 that I is an ide*
*al,
and thus from Proposition 7.9 that it contains all K-nilpotent spectra. In part*
*icular,
we see that X is a retract of some S=I ^ X, and S=I ^ X is the smash product of*
* a __
small spectrum with a dualisable one so it is small, so X itself is small. *
* |__|
12.1. The semiring ss0D. Write ss0D for the set of isomorphism classes of spect*
*ra
in D. We start by verifying that this is really a set.
Proposition 12.17.The category D has 2@0 isomorphism classes.
Proof.There are only countable many finite spectra Y of type at least n, and for
each one we know that [LY; LY ] is finite, so LY has only finitely many retract*
*s. It
follows that ss0(F) is countable. Moreover, if U; V 2 F then [U; V ] is finite.*
* It follows
that there are at most @@00= 2@0 different towers of K-small spectra. If X 2 D *
*then
MORAVA K-THEORIES AND LOCALISATION 63
X ^ S=I is small and X = holim-X ^ S=I, so X is the inverse limit of one of the*
*se
I
towers. Thus |ss0D| 2@0.
Conversely, it is shown in [HMS94 , Proposition 9.3] that the p-adic integers*
*_embed
in the Picard group, which in turn embeds in ss0D. Thus |ss0D| 2@0. |_*
*_|
The set ss0D is a commutative semiring under _ and ^. The group of units is t*
*he
aforementioned Picard group of K. See [HMS94 , HS95, Str92] for discussion and
calculation of Picard groups. See also Section 14 and Section 15.1.
The subsets ss0Dk are ideals. The maps nY :ss0D -!N give an additive isomor-
phism
M
ss0D = N
Y 2I
where I denotes the set of isomorphism classes of indecomposable spectra. There
is also an interesting semiring homomorphism
d: ss0D -!N[t]=(t|vn|- 1)
defined by
|vn|-1X
d(X) = tidimFp Ki(X):
i=0
We can of course set t = 1 to get a semiring map ss0D -! N, or set t = -1 to get
a map ss0D -!Z. This factors through the ring K0D, which is the quotient of ss0D
in which we set Y = X + Z whenever there is a cofibre sequence X -! Y -! Z.
When X 2 F is small, we have two other measures of the "size" of X.
Definition 12.18.Given a finite Abelian p-group A we define the length to be
len(A) = logp|A|. Given a graded Abelian p-group A* which is periodic of period
|vpKn| for some K, we define
X
len(A*) = len(Ak)=pK 2 Z[1_p]:
0k<|vpKn|
We have normalised this so that len(K*) = 1, and thus len(A*) 2 N when A* is a
finite module over E*.
The length len(K*X) is defined whenever X is dualisable. It is not a very sen*
*si-
tive measure of the size of X, because len(K*(S=I)) = 2n for all I. If X is sma*
*ll we
can also define len(E*X) and len(ss*X) (we have the required periodicity becaus*
*e of
the existence of vn self maps). The following result shows that len(E*X) is a g*
*ood
measure of size.
Theorem 12.19. For any integer k 0, the set of isomorphism classes of K-small
spectra X with len(E*X) k is finite.
Proof.We will work with E(n) instead of E: since we are only considering finite
modules and small spectra, this makes no difference. Note that there are only f*
*initely
many E(n)*-modules M with lenM k. Now suppose we have a finite E(n)*-
module M. If M ~=E(n)*X for some K-local spectrum X, then Y is necessarily
small. The filtration theorem proved in [HS95 , Section 2] then tells us that M*
* has
a finite comodule filtration 0 = M0 M1 : : :Mt where t k and Mi=Mi-1~=
j(i)K* for some 0 j(i) < |vn|. Given Mi-1, the number of possibilities for Miis
64 M. HOVEY AND N. P. STRICKLAND
then determined by Ext1;j(i)E(n)*E(n)(K*; Mi-1). It follows that, as long as th*
*e groups
Exts;tE(n)*E(n)(K*; K*) are finite for all s and t, the number of realisable co*
*module
structures on M is finite.
Now fix a realisable E(n)*E(n)-comodule structure on M. Consider the category
C(M) of pairs (X; x), where X is a small spectrum and x: M -! E(n)*X is an
isomorphism of comodules. It is enough to show that C(M) has only finitely many
isomorphism classes. Recall from Proposition 6.5 that there is an Adams spectral
sequence
Est2(X; Y ) = Exts;tE(n)*E(n)(E(n)*X; E(n)*Y ) =) [X; Y ]t-s;
with EN = E1 for some constant N independent of X and Y . Moreover, given
maps E(n)*X -f! E(n)*Y g-!E(n)*Z which survive to Er, we have dr(gf) =
g*dr(f) + f*dr(g) (see [Mos68 , Theorem 2.1]. We say that objects (X; x) and (Y*
*; y)
in C(M) are Er-equivalent if the element yx-1 2 Hom E(n)*E(n)(E(n)*X; E(n)*Y ) =
E0;02(X; Y ) survives to Er. Our formula for dr(gf) shows that this is an equiv*
*alence
relation. Clearly any two objects in C(M) are E2-equivalent. If they are Er+1-
equivalent then they are Er-equivalent, and if they are EN -equivalent then the*
*y are
isomorphic. It is thus enough to show that each Er-equivalence class splits in*
*to
finitely many Er+1-equivalence classes.
If (X; x) and (Y; y) are Er-equivalent, we define
ffi(X; x; Y; y) = y*(x-1)*dr(y-1x) 2 Extr;r-1E(n)*E(n)(M; M):
As x and y are isomorphisms, we have ffi(X; x; Y; y) = 0 if and only if (X; x) *
*and
(Y; y) are Er+1-equivalent. Moreover, the formula for dr(gf) implies that
ffi(X; x; Z; z) = ffi(X; x; Y; y) + ffi(Y; y; Z; z)
whenever this makes sense. We conclude that the set of Er+1-equivalence classes
in a given Er-equivalence class bijects with a subgroup of Extr;r-1E(n)*E(n)(M;*
* M), so
it will be enough to show that this group is finite. Using our comodule filtrat*
*ion of
M, we find once again that it suffices to show that Exts;tE(n)*E(n)(K*; K*) is *
*finite
for all s and t.
It is easy to check using cobar resolutions that
Exts;tE(n)*E(n)(E(n)*; K*) = Exts;t*(K*; K*):
This is finite in each bidegree by the proof of of [Rav86, Theorem 6.2.10(a)]. *
* It
follows inductively that Exts;tE(n)*E(n)(E(n)*=Ik; K*) is finite and in particu*
*lar that
Exts;tE(n)*E(n)(K*; K*) is finite, as required. |*
*___|
13.K-nilpotent spectra
In this section, we give a number of characterisations of K-nilpotent spectra.
Theorem 13.1. Suppose X is a spectrum in K. Then the following are equivalent.
(a)X is a retract of X ^ Z for some finite spectrum Z of type at least n.
(b)X is K-nilpotent.
(c)X lies in the thick subcategory generated by the K-injectives.
(d)X ^S Y is K-local for all Y 2 S.
(e)The functor on K which takes Y to ss0(X ^K Y ), is a homology functor.
MORAVA K-THEORIES AND LOCALISATION 65
Before proving this, we need to address a different problem. Let X be a finite
spectrum of type m < n, with a good vm self map v :dX -! X. We need to
understand what happens to X and v under K-localisation.
As usual, if Z is a spectrum with a self-map f, we denote the cofibre of f by
Z=f.
Proposition 13.2.If Y 2 K and X is a finite spectrum of type m < n with
v :dX -! X a good vm self map of X, then the natural map
X ^ Y -! holim-(X ^ Y )=vk
is an equivalence. Furthermore we have a short exact sequence
0 -!lim-1ker(vk; ss*-dk-1(X ^ Y )) -!ss*(X ^ Y ) -!ss*(X ^ Y )^v-!0
Proof.First note that it does not matter whether we work in K or in S. Indeed, X
is finite, so X ^ Y is already K-local, so we have X ^ Y = bL(bLX ^ Y ). Also, *
*the
products in K and in S are the same. The homotopy inverse limit of a tower can *
*be
constructed as the fibre of a standard self map of the product of its terms, so*
* it is
also the same in K and S. g
We claim that the map X ^Y -! holim-(X ^Y )=vk is an X=v-equivalence. Indeed,
k
as v is strongly central, the map X=v ^ X -1^v--!X=v ^ X is trivial for k 2. T*
*hus
the map X=v ^ X ^ Y -! X=v ^ X=vk ^ Y is a split monomorphism for k 2, with
cofibre a suspension of X=v ^ X ^ Y . Thus we have a short exact sequence
ssi(X=v ^ X ^ Y ) ae ssi(X=v ^ X=vk ^ Y ) i ssi-dk-1(X=v ^ X ^ Y )
and a corresponding six-term long exact sequence of lim-and lim-1terms. However,
the maps of the tower induce multiplication by 1 ^ v on the cokernel terms, and
using strong centrality again, we find that the tower of cokernels is pro-isomo*
*rphic
to the zero tower. The tower of kernels is constant. Hence the map g is an X=v-
equivalence.
The map g is therefore an F (m + 1)-equivalence between K(n)-local, and hence
F (n)-local, spectra. Since m + 1 n, we see that g is an equivalence.
We now consider the Milnor exact sequence
0 -!lim-1ss*+1(X ^ Y=vk) -!ss*(X ^ Y ) -!lim-ss*(X ^ Y=vk) -!0:
To identify the terms in this sequence, note that we have short exact sequences
0 -!(ss*(X ^ Y ))=vk -!ss*(X ^ Y=vk) -!ker(vk; ss*-dk-1(X ^ Y )) -!0:
Taking inverse limits gives us the usual 6-term exact sequence
0 -!ss*(X ^ Y )^v-!lim-ss*(X ^ Y=vk) -!lim-ker(vk; ss*-dk-1(X ^ Y ))
-!lim-1ss*(X ^ Y )=vk -!lim-1ss*(X ^ Y=vk) -!lim-1ker(vk; ss*-dk-1(X ^ Y )) -!0:
Note that the sequence ss*(X ^ Y )=vk is Mittag-Leffler, so its lim-1term vanis*
*hes.
Thus lim-1ss*(X ^ Y=vk) = lim-1ker(vk; ss*-dk-1(X ^ Y )), so the first term in *
*the
Milnor sequence is as claimed.
It remains to prove that ss*(X ^ Y )^v= lim-ss*(X ^ Y=vk), or in other words
that the first map in the six term sequence is surjective. This is true because*
*_the_
surjective map ss*(X ^ Y ) -!lim-ss*(X ^ Y=vk) factors through it. |*
*__|
66 M. HOVEY AND N. P. STRICKLAND
Proof of Theorem 13.1.(a))(b): We know by Proposition 6.15 that LZ is K-
nilpotent. It follows that any retract of X ^ Z = X ^ LZ is also K-nilpotent.
(b),(c): this is Lemma 7.4.
(b))(d): If X is K-nilpotent, so is X ^S Y for any Y , so in particular X ^S Y
is K-local.
(d))(e): Suppose that X ^S Y is K-local for all Y , so that X ^K Y = X ^S Y .
It also follows that for any Z 2 S, the spectrum X ^S bCZ is both K-local and
K-acyclic, hence is zero. This implies that X ^K bLZ = X ^S bLZ = X ^S Z. In
particular, we find that
K_ S_
X ^K Yi= X ^S Yi:
i i
Given this, it is trivial to verify that Y 7! ss0(X ^K Y ) is a homology functo*
*r on K.
(e))(a): This is the most difficult part of the argument. For the rest of the
proof, all smash products, limits and so on are taken in K. The plan is to star*
*t with
the obvious fact that X is a retract of X ^ bLS. We then show that if Z is a fi*
*nite
type m spectrum with a vm self-map v, then X ^ bLZ is a retract of (X ^ bLZ)=vk
for some sufficiently large k. It will then follow by induction that X is a ret*
*ract of
X ^ bLZ for some finite type n spectrum Z.
So consider the functor H on K defined by H(Y ) = ss0(bLZ ^ X ^ Y ). By (e),
H is a homology functor on K. Our first goal is to show that there is a k such
that vkH*(Y ) is v-divisible for all Y . Indeed, suppose not. For each k, choos*
*e a
spectrum Yk and an element ak 2 H*(Yk) suchWthat bk = vkak is not v-divisible.L*
*We
can assume that bk is in degree 0. Let Y = Yk, so we have H*(Y ) = H*(Yk).
We know from Proposition 13.2 that H(Y ) maps onto its v-completion. However,
(bk) defines an element of the v-completion of H(Y ), since only finitely many *
*are
not in the image of any vl. Furthermore, since no bk is v-divisible, each maps *
*non-
trivially to the v-completion, so that (bk) can't be in the image of H(Y ), whi*
*ch is a
contradiction.
Hence there is a k such that vkH(Y ) is v-divisible for all Y . We will now s*
*how
that vkH(Y ) is in fact 0 for all Y . Indeed, by Proposition 13.2, the kernel o*
*f the
surjective map from H(Y ) to its v-completion is lim-1ker(vl; H(Y )), where we *
*have
left out the dimension shift. But we claim this tower is in fact Mittag-Leffler*
*, so
that it has no lim-1term. Indeed, suppose we have a class x in ker(vl; H(Y )) w*
*hich
is in the image of ker(vl+k; H(Y )). Then in particular, x is in vkH(Y ) so is*
* v-
divisible. Then for all i k, there is a w such that viw = x. In particular, w *
*is
in ker(vi+l; H(Y )), so x is in the image of ker(vl+i; H(Y )) for all i k. Thu*
*s the
tower is Mittag-Leffler, and we find that H(Y ) is v-complete, for all Y . In p*
*articular,
vkH(Y ) is a v-divisible subgroup of a v-complete group, so is trivial.
Now, take Y = DW for a finite type n spectrum W . Using Spanier-Whitehead
duality in K, we find that for any map W -! bLZ ^ X, the composite
k^1
W -! bLZ ^ X -v--!bLZ ^ X
is null. (Remember the smash product is taken in K). Thus,
vk ^ 1 : bLZ ^ X -! bLZ ^ X
MORAVA K-THEORIES AND LOCALISATION 67
is phantom, so by the last part of Theorem 9.5, v2k^ 1 is trivial. Thus, bLZ ^ *
*X is
a retract of bL(Z=v2k) ^ X. Since Z=v2k is a finite spectrum of type_m_+ 1, we *
*are
done. |__|
Proposition 13.3.If X is K-nilpotent and Y is arbitrary then X ^ Y , F (X; Y )
and F (Y; X) are K-nilpotent.
Proof.Let C be the category of those X such that X ^ Y , F (X; Y ) and F (Y; X)
are K-nilpotent; this is clearly thick. If X is a K-module, then X ^ Y , F (X; *
*Y )
and F (Y; X) are K-modules and therefore K-nilpotent. This shows that C contains
all K-injective spectra, and hence (by part (c) of Theorem 13.1) all K-nilpoten*
*t_
spectra. |__|
It follows from Proposition 13.3 that the K-nilpotent spectra form the coideal
generated by K as well as the ideal generated by K. One might then guess that if
X is in both the localising subcategory generated by K (equivalently, MX = X)
and the colocalising subcategory generated by K (equivalently bLX = X), then X
is K-nilpotent. This is false, however: the spectrum Y considered in Section 15*
*.1
is a counterexample.
14. Grading over the Picard group
Recall, from for example [HMS94 , Str92], that the Picard group Pic= Pic(K)
consists of isomorphism classes of spectra in K which are invertible under the *
*smash
product. In this section we will need to be very careful about the distinction *
*between
objects and isomorphism classes, so we make formal definitions as follows.
Definition 14.1.We say that a spectrum P 2 K is invertible if there is a spectr*
*um
Q 2 K such that P ^ Q ' S. We write P for the category of invertible spectra.
Given P 2 P we write [P ] for the isomorphism class of P . We write Picfor the
collection of these isomorphism classes, which is a set (rather than a proper c*
*lass)
by [HMS94 , Proposition 7.6] or by Proposition 12.17.
The following result is proved as [HMS94 , Theorem 1.3] but we give a differe*
*nt
argument here for the sake of completeness.
Proposition 14.2.Given P 2 K, the following are equivalent:
(a)P 2 Pic.
(b)K*P ' K* (up to suspension).
(c)E*P ' E* (up to suspension).
(d)E_*P ' E* (up to suspension).
Proof.(a))(b): If P ^ Q = S then K*(P ) K* K*(Q) = K*, so K*(P ) has rank
one and is isomorphic to K* up to suspension.
(b),(c): We may assume that K*P ' K* so that K*P is in even degrees. Then
E*P is pro-free by Proposition 2.5, and (E*P )=In = K*. It follows easily that
E*P = E*. The converse also follows from Proposition 2.5.
(b),(d): This is similar to (b),(c), using Proposition 8.4.
(b))(a): Since K*P is finite dimensional, we know that P is dualisable. Fur-
thermore, the group K*DP = Hom K*(K*P; K*) is also one-dimensional. Hence the
unit S -! DP ^ P of the ring spectrum DP ^ P is a K-equivalence, and thus_an
isomorphism. |__|
68 M. HOVEY AND N. P. STRICKLAND
There is an evident surjective homomorphism deg: Pic-! Z=|vn| which sends
[P ] to the degree in which K*P is concentrated.
The general properties of the Picard group are given in the following proposi*
*tion.
Proposition 14.3.
(a)The homomorphism Z -!Pic that takes m to bLSm|vn|extends to an injective
homomorphism Zp -!Pic.
(b)| Pic| = 2@0.
(c)If X is small, the orbit of X under the action of Picis finite.
(d)Pic is a profinite Abelian group, and the kernel of deg: Pic-! Z=|vn| is a
pro-p-group of finite index in Pic.
Proof.Part (a) is proved in [HMS94 , Section 9], and part (b) is an immediate
corollary of part (a) and Proposition 12.17. Part (c) follows from Proposition *
*14.2
and Theorem 12.19. Part (d) can also be deduced from the results of [HMS94 ]. H*
*ere
we will give an independent proof that Picis profinite, but we rely on [HMS94 ]*
* to
tell us that ker(deg) is p-local.
To prove that Picis profinite, choose a tower {S=J(i)} as in Proposition 4.22.
Let G(i) be the stabiliser in Picof S=J(i). Part (c) says that Pic=G(i) is fin*
*ite.
We claim first that G(i + 1) G(i). Indeed, suppose P 2 G(i + 1), so that
P ^ S=J(i + 1) ' S=J(i + 1). Then we have the composite
P -! P ^ S=J(i + 1) '-!S=J(i + 1) -!S=J(i)
induced by the unit. Since S=J(i) is a -spectrum, there is an induced map P ^
S=J(i) -f!S=J(i). The map E*f is easily seen to be an isomorphism, so f is an
equivalence.
To complete the proof, we need only show that if P 2 G(i) for all i, then P '*
* S.
Let Mibe the set of maps P -! S=J(i) such that the induced map E_*P -! E*=J(i)
is surjective. Because P ^ S=J(i) ' S=J(i), one can check that Mi is nonempty. *
*It
is also finite, and the sets Mi form an inverse system. It follows that the inv*
*erse
limit is nonempty. The Milnor sequence gives us a map P -! holim-S=J(i) = S.
i
It is easy to see that this induces an isomorphism E_*P ' E_*S, and thus_that
P ' S. |__|
Note that Proposition 14.3 does not rule out the possibility that Pic contains
an infinite product of copies of Zp. The main unanswered question about Picis
whether it is finitely generated as a profinite group.
We next address two related problems. Firstly, we have seen that most of Picis
a module over Zp. However, given P 2 Picand a 2 Zp we would like to be able to
define P (a)as an object rather than just an isomorphism class, or at least to *
*define
it up to canonical isomorphism. Secondly, given an element a 2 Picit is natural*
* to
choose P 2 P with [P ] = a and define ssa(X) = [P; X]. One would like to choose*
* the
spectra P compatibly for all a and collect the groups ssa(X) into a graded obje*
*ct
with commutative and associative pairings ssa(X) ssb(Y ) -! ssa+b(X ^ Y ). This
is not automatically possible: there is an obstruction in H3(Pic; Aut(S)), and *
*if it
vanishes then the solutions form a principal homogeneous space for H2(Pic; Aut(*
*S))
(here Picis acting trivially on Aut(S)). We believe that this fact is in the li*
*terature,
perhaps in the theory of Tannaka categories, but we have not managed to find a
reference. Rather than using this general theory, we will proceed more directl*
*y.
MORAVA K-THEORIES AND LOCALISATION 69
We have not been able to construct a grading over Picitself, but Theorem 14.11
provides a tolerable substitute.
We start with a discussion of signs, most of which is taken from [Riv72, Chap*
*ter
I]. We write R = [S; S], which is a commutative ring under composition (which
is the same as the smash product). For any X 2 K, the smash product gives a
natural map R -![X; X]. If X = P 2 P then this map is an isomorphism (because
[P; P ] = [S; P -1^ P ] = [S; S]). We write t = tP :[P; P ] -! R for the invers*
*e. One
can check that tP^Q (u ^ v) = tP (u)tQ (v) and tP (v O w) = tP (v)tP (w). Moreo*
*ver, if
we have maps P -u!Q v-!P then tP (vu) = tQ (uv).
Given a spectrum P 2 P, we have a twist map oP :P ^ P -! P ^ P and thus an
element fflP = t(oP ) 2 R. Of course for n 2 Z we have fflSn = (-1)n.
Lemma 14.4. If p > 2 then fflP = (-1)deg[P]. Even if p = 2, we have ffl2P= 1 a*
*nd
the map [P ] 7! fflP is a homomorphism Pic=2 -!Rx .
Proof.As o2 = 1 we have ffl2P= 1. It is not hard to check that fflP^Q = fflP ff*
*lQ and
that fflP = fflP0 if P ' P 0. We thus get a homomorphism ffl: Pic=2 -! Rx . N*
*ow
suppose that p > 2. Then K is commutative so external products in K homology are
commutative up to the usual sign; it follows that K*fflP = (-1)deg[P]. For nota*
*tional
convenience we will assume that deg[P ] is even so that K*fflP = 1. It is not h*
*ard to
see that e = (1 - fflP )=2 is an idempotent in R and K*e = 0. Thus K*eS = 0,_so
e = 0, so fflP = 1 as required. |__|
We write P0 for the category of those P 2 P such that deg[P ] = 0 and fflP = 1
(the second condition being redundant when p > 2). Because the symmetric group
on k letters is generated by adjacent transpositions, we see that it acts trivi*
*ally on
P (k)if P 2 P0. We write P0for the category of pairs (P; u) such that P 2 P0 and
u generates K0P . The morphisms (P; u) -!(Q; v) are required to send u to v. We
write Pic0and Pic0for the groups of isomorphism classes in P0 and P0.
If (P; u) 2 P0and X 2 F, we define
( (k)
ssk(P; u)(X) = [P ; X] ifk 0
[DP (|k|); X]ifk < 0
These are easily seen to be finite groups. It is not hard to construct associat*
*ive and
commutative (without signs) pairings
ssk(P; u)(X) ssl(P; u)(Y ) -!ssk+l(P; u)(X ^ Y ):
We would like to interpolate these groups for p-adic values of k. The main prob*
*lem
is to find a natural formulation of this not depending on any choices. Our solu*
*tion
involves the following definition.
Definition 14.5.Given a compact Hausdorff space X, let B(X) be the category
of locally-trivial bundles over X whose fibres are finite Abelian p-groups with*
* the
discrete topology. The morphisms are continuous bundle maps that are homomor-
phisms on each fibre.
Given a morphism f :A -!B in B(X), it is easy to see that for each x 2 X there
is a neighbourhood U of x and groups A0; B0 and a homomorphism f0:A0-! B0
such that the restriction of f over U is isomorphic to f0x 1: A0x U -! B0x U. It
follows from this that the kernel, cokernel and image of f all lie in B(X) and *
*thus
that B(X) is an Abelian category.
70 M. HOVEY AND N. P. STRICKLAND
A map OE: X -! Y gives a pullback functor OE*: B(Y ) -! B(X) in an evident
way. In particular, if G is an compact Hausdorff Abelian topological group and
oe :G x G -!G is the addition map, we have a functor oe*: B(G) -!B(G x G). We
also have an external tensor product functor : B(G) x B(G) -! B(G x G). In
this context, a pairing from A and B to C (where A; B; C 2 B(G)) means a map
A B -!oe*C.
We write Fx: B(X) -! Ab for the functor which sends A to the fibre Ax of A
over x.
Proposition 14.6.Given (P; u) 2 P0 there is a homology theory ss(P; u): F -!
B(Zp), together with natural pairings
ss(P; u)(W ) ss(P; u)(Z) -!oe*ss(P; u)(W ^ Z)
that are commutative and associative in a suitable sense. Moreover, for m 2 Z
there are natural isomorphisms Fm ss(P; u)(Z) = ssm (P; u)(Z) which are compati*
*ble
with the above pairings.
This will be proved after Proposition 14.8. We will denote Fm ss(P; u)(Z) by
ssm (P; u)(Z) even for non-integer values of m. Note that, if Z is a specific s*
*mall spec-
trum and m 2 Zp, then the local triviality of ss(P; u)(Z) implies that ssm (P; *
*u)(Z) =
ssi(P; u)(Z) for some integer i sufficiently close to m in the p-adic topology.*
* However,
the i in question depends on Z.
Note also that ss(P; u) is a homology theory defined only on small spectra. T*
*he
category B(Zp) does not have colimits, so in order to interpret ss(P; u) as a h*
*omology
theory on K, we would need to consider sheaves over Zp rather than locally triv*
*ial
bundles.
Definition 14.7.Let X 2 F be a small spectrum, and (P; u) an object of P0. We
say that a map v :P (pk)^ X -! X is a good (P; u) self map of X if
(a)The map v*:K*(P )pk K*(X) -!K*(X) is given by v*(upk x) = x for
all x 2 K*X.
(b)v ^ 1 is central in the graded ring ss*(P; u)F (X ^ X; X ^ X).
Note that there are no suspensions involved in the definition of a good (P; u*
*) self
map. Note also that a good (P; u) self map is an equivalence.
Proposition 14.8.Every small spectrum X admits a good (P; u) self map. If v
and v0 are two such maps then vpi= (v0)pj for some i and j. If v is a good (P; *
*u)
self map of X and w is a good (P; u) self map of Y then there exist integers i *
*and
j such that wpjf = fvpifor all maps f :X -! Y .
Proof.This is proved in a way which is closely parallel to the proof for intege*
*r-
graded vn self maps [Rav92a, Chapter 6]. If v is a map satisfying part (a) of t*
*he
definition, we form the ring spectrum R = F (X; X). The map v defines a map
v : P (pk)-!R, so we can form the map ad(v): P (pk)^ R -!R which measures the
difference between left multiplicationkand right multiplication by v. By our as*
*sump-
tion, the map K*v : K*P (p )-!K*R = Hom K*(K*X; K*X) takes the generator
upk to the identity map, which is obviously in the centre. Hence K*ad(v) = 0, *
*so
the telescope of the sequence
k) ad(v) (2pk) ad(v)
R ad(v)---!DP (p ^ R ---! DP ^ R ---! : : :
MORAVA K-THEORIES AND LOCALISATION 71
is zero. It follows using the smallness of R that ad(v) is nilpotent, and in th*
*e usual
way we deduce that some power vpjis central in ss*(P; u)F (X; X). We can apply *
*the
same argument to v^1 and thus show that vpj^1 is central in ss*(P; u)F (X(2); X*
*(2))
for large j, and thus that vpj is a good (P; u) self map.
Next, suppose that we have two good (P; u) self maps, say v and v0. We then
have K*(v - v0) = 0 and thus v - v0is nilpotent, by an argument with telescopes*
* as
above. Just as in [Rav92a], we deduce that vpi= (v0)pjfor some i and j. Asympto*
*tic
naturality follows from asymptotic uniqueness, again just as in [Rav92a]. Given*
* the
asymptotic naturality, it is not hard to see that the category of small spectra*
* that
admit a good (P; u) self map is thick. It is thus enough so show that a general*
*ised
Moore spectrum S=I of type n admits a good (P; u) self map, or even just a map
satisfying condition (a).
It follows from Proposition 14.3 that P (pk)^ S=I ' S=I for some k. Choose an
isomorphism v. Choose an element "u2 E_*P (pk)lifting upk . After replacing v by
a p-adic unit multiple of v, we may assume that v*"u= 1 (mod In). It follows us*
*ing
the cofibrations |vk|E=Ik vk-!E=Ik -! E=Ik+1 that vpn satisfies condition_(a), *
*as
required. |__|
Proof of Proposition 14.6.Let (P; u) be an object of P0, and X 2 F a small spec-
trum. Let C be a coset of pkZp in Zp for some k. A good (P; u) self map
v : P (pk)^ X -! X induces an isomorphism ssj+pk(P;Qu)(X) -! ssj(P; u)(X).
Let ssC (P; u)(X) be the set of elements a 2 j2C\Zssj(P; u)(X) such that there
exists a good (P; u) self map v :P (pk)^ X -! X such that aj = vaj+pk for all
j 2 C \ Z. If C0 is another coset and C0 C then there is an evident restriction
map ssC (P; u)(X) -!ssC0(P; u)(X). For i 2 Zp we define
ssi(P; u)(X) = lim-!ssC (P; u)(X):
C3i
We shall see shortly that this is consistent with our earlier definition when i*
* 2 Z.
If a 2 ssC (P; u)(X) and a0 2 ssC0(P; u)(X) and i 2 C \ C0 then we see using the
asymptotic uniqueness of (P; u) self maps that there is a coset C00with i 2 C00
C \ C0 such that the restriction of a + a0lies in ssC00(P; u)(X). Using this, w*
*e make
ssi(P; u)(X) into an Abelian group. We define
a
ss(P; u)(X) = ssi(P; u)(X);
i2Zp
so there is an evident map q :ss(P; u)(X) -! Zp with fibres ssi(P; u)(X). Given*
* an
element a 2 ssC (P; u)(X), we get an element ai2 ssi(P; u)(X) for each i 2 C. We
define U(C; a) = {(i; ai) | i 2 C} ss(P; u)(X), and we give ss(P; u)(X) the sm*
*allest
possible topology for which all sets of this form are open. k
Suppose we have integers j; k with k 0 and a good (P; u) self map v :P (p )^
X -! X. There is an evident map
= j;k;v:ssj(P; u)(X) -!ssj+pkZp(P; u)(X);
sending a 2 ssj(P; u)(X) to the system of elements (vm a | m 2 Z). Using asympt*
*otic
uniqueness again, it is not hard to check that this is an isomorphism. One can *
*also
deduce that ss(P; u)(X) is a locally trivial bundle over Zp, or in other words *
*an
object of the category B(Zp).
72 M. HOVEY AND N. P. STRICKLAND
Given a triangle X -! Y -! Z of small spectra, we can choose good (P; u) self
maps of X, Y and Z that are compatible with the maps of the triangle. Having
done so, it is easy to check that ss(P; u)(X) is an exact functor of X.
Similar arguments show that when i 2 Z there is a canonical isomorphism be-
tween our old definition of ssi(P; u)(X) and our new one.
Suppose that i; j 2 Zp. We would like to construct a natural map
i;j:ssi(P; u)(X) ssj(P; u)(Y ) -!ssi+j(P; u)(X ^ Y ):
Suppose that v :P (pk)^ X -! X and w :P (pk)^ Y -! Y are good (P; u) self maps
with v ^ 1 = 1 ^ w on X ^ Y , and that i0; j0 are integers congruent to i; j mo*
*dulo
pk. The smash product gives a pairing
i0;j0:ssi0(P; u)(X) ssj0(P; u)(Y ) -!ssi0+j0(P; u)(X ^ Y ):
The maps i0;k;vand so on give isomorphisms
ssi0(P; u)(X)' ssi(P; u)(X)
ssj0(P; u)(Y')ssj(P; u)(Y )
ssi0+j0(P; u)(X ^'Ys)si+j(P; u)(X ^ Y ):
It is natural to require that i;jshould be compatible with i0;j0under these iso*
*mor-
phisms. It is not hard to check using the strong centrality of good (P; u) self*
* maps
that there is a unique map i;jwhich has this compatibility for all choices of k*
*, i0,
j0, u and v. Given this, one can deduce the expected naturality, commutativity_*
*and
associativity properties. |__|
Corollary 14.9.Given an object (P; u) 2 P0, there are objects (P (k); uk ) in P0
for all k 2 Zp, defined up to canonical isomorphism, such that there is a natur*
*al
isomorphism
ssk(P; u)(X) = [P (k); X]
for all small spectra X. Moreover, there are canonical and coherent isomorphisms
P (k)^ P (l)= P (k+l). The spectrum P (k)is independent of u up to unnatural
isomorphism.
Proof.The functor X 7! ss-k(P; u)(X) is a homology theory on small spectra, so
by [HPS95 , Corollary 2.3.11] it extends canonically to a homology theory on al*
*l of K.
By Theorem 9.5 there is a spectrum P (k)representing this functor, so in partic*
*ular
ss-k(P; u)(X) = ss0(P (k)^ X) whenever X is small. This spectrum is unique up
to isomorphism, and the isomorphism is unique up to phantoms. However, for any
small spectrum Z we have [Z; P (k)] = ss-k(P; u)DZ which is finite, so Theorem *
*10.7
tells us that there are no phantom maps into P (k). This shows that P (k)is uni*
*que
up to canonical isomorphism. j
Let X be a nontrivial small spectrum, and v :P (p )^ X -! X a good (P; u) self
map. Choose an integer i congruent to k mod pj. Using v we construct a natural
isomorphism
ss-k(P; u)(X ^ Y ) = ss-i(P; u)(X ^ Y ) = ss0(P (i)^ X ^ Y )
for all small Y , and thus (by unique representability of homology theories) an*
* iso-
morphism P (k)^ X ' P (i)^ X. We know that K*P has dimension one over K*,
and it follows that K*P (k)has dimension one, and thus that P (k)2 P.
MORAVA K-THEORIES AND LOCALISATION 73
We still need to produce a canonical generator of K0P (k), however. To do thi*
*s,
suppose that we have small spectra X and Y , a map f :X -! Y , and good self ma*
*ps
v; w of X and Y such that wf = fv. If we construct isomorphisms P (k)^ X '
P (i)^ X and P (k)^ Y ' P (i)^ Y by the procedure outlined above, then we find
that they commute with the maps 1 ^ f. In particular, we can take X = S=I and
Y = S=I ^ S=I, choose a good self map v of S=I and take w = v ^ 1 = 1 ^ v. We
can then take f = j ^ 1 or 1 ^ j, where j :S -!S=I is the unit map. This gives *
*us
a commutative diagram
j^1
P (k)^ S=I____-P_(k)^-S=I1^^S=Ij
| |
| |
|u j^1 |u
P (i)^ S=I____-P_(i)^-S=I1^^S=I:j
If we apply K*, then it is easy to see (just by calculating everything) that the
equaliser of the top line is K*P (k)and the equaliser of the bottom line is K*P*
* (i), so
we get an isomorphism K*P (i)' K*P (k). One can check that this is independent
of the choices involved. We define uk 2 K*P (k)to be the image of ui under th*
*is
isomorphism.
A similar argument shows that fflP(k)= 1. Indeed, one can see as above that
the image of fflP(k)in [S; S=I] is the same as the image of fflP(i)for some int*
*eger i
depending on I. Since this image is the unit map j of S=I, and [S; S] = lim-[S;*
* S=I],
we find that fflP(k)= 1.
We next produce an isomorphism P (k)^ P (l)' P (k+l). Suppose that we have
finite spectra X and Y of type at least n, and maps S -! X and S -! Y . This
gives a map P (k)-!P (k)^ X, or equivalently an element of ss-k(P; u)F (P (k); *
*X).
Similarly, we have an element of ss-l(P; u)F (P (l); Y ). Using the pairings o*
*n the
groups ss*(P; u)(-) and the dualisability of P (k)and P (l)we get an element of
ss-k-l(P; u)F (P (k)^ P (l); X ^ Y ), and thus a map S -!P (k+l)^ F (P (k)^ P (*
*l); X ^
Y ), or equivalently a map P (k)^ P (l)-!P (k+l)^ X ^ Y . One can show that
these maps are compatible as X and Y run over the category of finite spectra
of type n equipped with a map in of S. We can therefore pass to the inverse
limit, which is just P (k+l)by Remark 4.20 and Proposition 7.10. This gives a
map P (k)^ P (l)-!P (k+l). There are no lim1terms because all groups involved
are finite, so the map is unique. One can show that it induces our earlier pair*
*ing
ss-k(P; u)Z ss-l(P; u)W -! ss-k-l(P; u)(Z ^ W ). Using this, one can check that
the maps P (k)^ P (l)-!P (k+l)have the expected coherence properties.
Finally, we show that our definition of P (k)is independent of u up to unnatu*
*ral
isomorphism. If j is an integer then P (j)is just an iterated smash power of P *
*or
DP and clearly does not depend on u. For any generalised Moore spectrum S=I
we can choose a good self map P (pi)^ S=I -! S=I and an integer j congruent to k
modulo pi. We then find that P (k)^ S=I is isomorphic to P (j)^ S=I independent*
*ly
of u. We are thus reduced to showing that if P 2 Picand P ^ S=I ' S=I for all I_
then P ' S. We have shown this in the proof of Proposition 14.3. |_*
*_|
We also want to be able to grade things over subgroups of Pic0of rank greater
than one. The following lemma will help us to do this.
74 M. HOVEY AND N. P. STRICKLAND
Lemma 14.10. Let (P; a) and (Q; b) be objects of P0. Let X be a small spectrum,
with good self maps v :P (pk)^ X -! X and w :Q(pl)^ X -! X. Then the following
diagram commutes.
l) (pk) _____o^1(pk) (pl) ________1^w k
Q(p ^ P ^ X wP ^ Q ^ X wP (p )^ X
| |
| |
1^v| |v
| |
|u |u
l) ____________________________________
Q(p ^ X w Xw
Proof.We may assume that k = l = 0, to simplify the notation. It is easy to
generalise Proposition 4.4 to show that w induces a natural transformation w0Y:*
*Q^
Y -! Y on the category of spectra Y which can be written as a retract of a spec*
*trum
of the form X ^ Z. By applying naturality to the twist map o :P ^ X -! X ^ P ,
we see that w0P^Xis the composite
-1
Q ^ P ^ X -1^o-!Q ^ X ^ P -w^1-!X ^ P -o-!P ^ X:
General nonsense about symmetric monoidal categories tells us that this is the *
*same
as the top line of the diagram in the lemma (think about the analogous statement
with vector spaces and tensor products). By applying naturality of w0 to the_map
v :P ^ X -! X, we conclude that the diagram commutes. |__|
We finish with a theorem about grading over groups of rank greater than one.
Theorem 14.11. Given objects (P1; u1); : :;:(Pr; ur) of P0, there is a homology
theory F :F -!B(Zrp) whose fibre over a point a_= (a1; : :;:ar) is
Fa_(X) = [P1(a1)^ : :^:Pr(ar); X]:
There are also natural pairings from F (X) and F (Y ) to F (X ^ Y ).
Note that this gives a grading over Zrpand not over the image of Zrpin Pic0.
Moreover, our construction uses a choice of basis for Zrp; it is not clear whet*
*her this
can easily be avoided.
Proof.We simply define F (X) to be the disjoint union of the sets Fa_(X) as a_r*
*uns
over Zrp. If we choose good (Pi; ui) self maps for each i, these give us isomor*
*phisms
Fa_(X) ' Fb_(X) for all b_sufficiently close to a_. These isomorphism appear to
depend on the order in which we use the different self maps, but Lemma 14.10 te*
*lls
us that this is not the case. Moreover, if we use a different system of self ma*
*ps then
by asymptotic uniqueness we find that the isomorphisms Fb_(X) ' Fa_(X) do not
change provided that b_is sufficiently close to a_. Thus, we get a bundle struc*
*ture_on
F (X). We leave the rest of the details to the reader. |*
*__|
15. Examples
15.1. The case n = 1. The height one case is quite straightforward and amenable
to calculation. However, we warn the reader that it rather atypical and often m*
*is-
leading as a guide to the case n > 1. The material in this section is well-know*
*n.
See [Rav84] or [Bou79] for the closely related theory of localisation with resp*
*ect to
K theory.
MORAVA K-THEORIES AND LOCALISATION 75
We shall assume that p > 2 for simplicity. The spectrum E is the 2(p-1)-perio*
*dic
Adams summand of KU^p, and K = E=p. For each a 2 1 + pZp < Zxpwe have
an Adams operation a: E -! E, with a b = ab. These are ring maps, and
their action on E* = Zp[v11] is given by a(v1) = ap-1v1. We choose a topologic*
*al
generator a of the group 1 + pZp ' Zp and define T = a - 1 2 E0E. It turns out
that E0E = Zp[[T;]]in particular, this ring is commutative. More generally, the*
* ring
E*E is the non-commutative power series ring E*[[T ]]in which bv1 = bp-1v1 b
and thus T v1 = ap-1v1T + (ap-1- 1)v1.
For any d 2 Zp we define Xd to be the fibre of the map a-ad = T +1-ad:E -!
E. One checks that E*Xd = E*[[T=]](T +1-ad) = E* and that b acts on E0Xd with
eigenvalue bd for all b 2 1+pZp. We can also determine ss*Xd. Note that a-ad a*
*cts
on ss2(p-1)kE with eigenvalue a(p-1)k- ad, which is a unit multiple of 1 - ad-(*
*p-1)k
or of p(d - (p - 1)k). It follows that ss*Xd has a summand Zp=(p(d - (p - 1)k))*
* in
dimension 2k(p-1)-1, and a summand Zp in dimension 2d if d is an integer divisi*
*ble
by p - 1. One can check that in this case the generator of ss2dXd is a K-equiva*
*lence
S2d -!Xd, so that Xd = bLS2d. More generally, it is not hard to show that Xd is
just the p-adic smash power (S2p-2)(d=(p-1))considered in Section 14. Note that
a-ad
the fibration bLS2d -! E ----! E shows that bLS2d is E-finite, as predicted by
Theorem 8.9. We know from [HMS94 ] that the Picard group consists of the spectra
N
kXd for d 2 Zp and 0 k < 2p-2, and that Pic' lim-Z=|vp1| = Z=(2p-2)xZp.
N
We next define Y to be the fibre of T 2- p: E -! E. This spectrum is a coun-
terexample to a number of plausible conjectures, as pointed out to us by Mike
Hopkins. Note that E*Y = E*[[T=]](T 2- p) = E* E*, so Y is dualisable. Be-
cause T acts on each homotopy group ssdE (for d 2 Pic) with eigenvalue in pZp we
find that the eigenvalue of T 2- p is a unit multiple of p and thus that ssdY =*
* Fp
when d = -1 (mod 2p - 2) and ssdY = 0 otherwise. Thus p: Y -! Y is a map of
dualisable spectra which induces the zero map of all Pic-graded homotopy groups.
However, p 6= 0 because E*Y = E* E* is torsion-free. Moreover, using the fibra-
tion Y=p2 -! E=p2 -! E=p2 one finds that pssd(Y=p2) = 0 for all d and thus that
p: Y=p2 -!Y=p2 is a map of small spectra that is nontrivial but acts trivially *
*on all
Pic-graded homotopy groups. This shows that all reasonable analogues of Freyd's
generating hypothesis [Fre66] in K are false. See [Dev96a] for more discussion *
*of the
generating hypothesis from the chromatic point of view. Appendix 2 of that paper
contains a different kind of counterexample to a certain analogue of the hypoth*
*esis.
It appears that Devinatz does not consider this to be evidence against the orig*
*inal
conjecture, and he is the expert.
We next show that Y does not lie in the thick subcategory generated by the
Picard group. Indeed, for any Z in that category we see easily that [Z; Y ] is*
* a
torsion group, but the identity map has infinite order in [Y; Y ]. Another inst*
*ructive
proof is to observe that when Z lies in the thick subcategory generated by the *
*Picard
group, the group Q E0Z is a finite-dimensional vector space over Qp on which
Tpacts_with eigenvalues in Qp. However, the eigenvalues of T acting on E0Y are
p62 Qp.
One might expect that a dualisable spectrum with finite homotopy groups would
be small, but again Y is a counterexample.
15.2. The case n = 2. We next analyse some calculations of Shimomura [Shi86]
in the light of the theory developed above. See also [Sad93]. We take n = 2 and
76 M. HOVEY AND N. P. STRICKLAND
p 5. Amongst other things, we obtain the following thought-provoking result.
Recall from [HMS94 ] that the map from Z to Picsending n to Sn extends over the
N
completion bZ= lim-Z=|vp2| = Z=(2p2-2)xZp. Recall also that there is a unique
N
translation-invariant measure on bZwith (bZ) = 1, called the .
Theorem 15.1. For all a 2 Z and all finite torsion spectra X the group ssabLX is
finite. However,
(i)ssabLS=p is finite for all a 2 bZsuch that a = 0 (mod |v1|).
(ii)If b = -1; -2; -3 or -4 then {a 2 bZ| a = b (mod |v1|) and ssabLS=p is fin*
*ite}
has Haar measure zero.
(iii)If a 62 {0; -1; : :;:-4} (mod |v1|) then ssabLS=p = 0.
We have not yet understood the more recent calculation of ss*L2S due to Shimo-
mura and Yabe [SY95 ].
Note that E(2)* and E(2)*E(2) are concentrated in degrees divisible by 2p - 2.
Thus if E(2)*X is generated in degrees divisible by 2p-2 then the whole cobar c*
*om-
plex which calculates H**E(2)*X = Ext**E(2)*E(2)(E(2)*; E(2)*X) is concentrated
in those degrees, as is the Ext group itself. Moreover, we know from [HS95 ] th*
*at
Hs*E(2)* = 0 when s > n2 + n = 6. The argument used there shows a fortiori
that Hs*E(2)*X = 0 when s > 6 and X is any of the spectra S=(p; vN1) used below,
where N may be infinite. As 2p - 2 > 6 we see that the E(2)-based Adams spectral
sequence collapses and sskLX = Hs;k+sE(2)*X for the unique s with 0 s < 2p-2
and s = -k (mod 2p - 2).
Shimomura calculates H**M11, where
M11= E*=(p; v11) = E(2)*S=(p; v11) = E*M2S1;
and thus H**M11= ss*M2S1. We would prefer to work with bLS, so we start
with some remarks about the necessary translation. We know that ss*LS=(p; vN1) =
H**E*=(p; vN1) is finite in each total degree, as the homotopy of any small obj*
*ect
is finite, so there are no lim-1terms and ss*bLS=p = lim-H**E*=(p; vN1). We sha*
*ll
N
write H**E*=(p; vN1) as a direct sum of groups Ai;N, where for fixed i the grou*
*ps
Ai;Nform a tower as N varies. As H**E*=(p; vN1) is finite we see that Ai;N= 0
for almostQallQi so the direct sum is the same as the product and thus ss*bLS=p*
* =
lim- iAi;N= ilim Ai;N.
N - N
Let ** be the cobar resolution for E(2)*=p. We have a finite number of sum-
mands in H**M11that are isomorphic to Fp[v1]=v11. Each such summand is gener-
ated by elements xr=vr1where xr 2 s;tsay and d(xr) is divisible by vr1and xr+1=*
*vr1
is homologous to xr=vr1. We refer to these as summands of type (1; s; t). We al*
*so
have summands isomorphic to Fp[v1]=vff1, where ff > 0. Each of these is generat*
*ed
by an element y=vff1where y 2 s;tsay and d(y) is divisible by vff1. We refer to
these as summands of type (ff; s; t). We now use the short exact sequence
v-N1 1 1 vN1 -N|v |1
-N|v1|E(2)*=(p; vN1) ---! E(2)*=(p; v1 ) = M1 --! 1M1:
This gives a short exact sequence
UN = H**(M11)=vN1-! H**E(2)*=(p; vN1) -!ann(vN1; H**M11) = VN ;
MORAVA K-THEORIES AND LOCALISATION 77
and thus a short exact sequence
lim-UN ae ss*bLS=p i lim-VN :
N N
Each summand of type (1; s; t) in H**M11contributes a summand of VN gener-
ated by xN and isomorphic to Fp[v1]=vN1. As N varies these form a tower isomorp*
*hic
to the evident tower
Fp[v1]=v1 j Fp[v1]=v21j Fp[v1]=v31j Fp[v1]=v41j : :;:
so the (graded) inverse limit is Fp[v1]. It is generated by a class x1 2 Hs;t.
On the other hand, consider a summand of type (ff; s; t) generated by y=vff1.*
* Write
fi = min(N; ff). We then get a summand of UN generated by d(y)=vfi1and isomorph*
*ic
to Fp[v1]=vfi1. We also get an isomorphic summand in VN generated by vN-fi1y. As
N varies, the summands in UN form a pro-constant tower and the summands in
VN form a pro-trivial tower. In the inverse limit we are left with a summand in
lim-UN isomorphic to Fp[v1]=vff1, generated by d(y)=vff12 Hs+1;t-ff|v1|.
N
Feeding Shimomura's calculations into this analysis gives the following descr*
*ip-
tion of ss*bLS=p. Firstly, there is a class i 2 ss-1bLS=p and an isomorphism ss*
**bLS=p =
E[i]A* for some other graded group A*. This group contains two copies of Fp[v1],
generated by 1 2 ss0 and h0 2 ss2p-3. We write B* for the direct sum of the oth*
*er
summands in A*, which is a v1-torsion group.
We need some new language to give a manageable description of B*. Firstly,
we will write numbers_in_terms of their base p expansions,_for example [123] =
p2+ 2p + 3. We write k for the digit p - k so thatP[1 2] = (p - 1)p + 2 for exa*
*mple.
We indicate repetitions by exponents, so [1n] = n-1i=0pi= (pn-1)=(p-1). We wr*
*ite
* for an undetermined string of digits, so that [*11] means any integer congrue*
*nt to
p + 1 modulo p2._We handle negative numbers by allowing infinite expansions, for
example -pn = [11 0n]. We write x " n for xn where typographically convenient.
Proposition 15.2 (Shimomura).The generators of B* are as follows. The entries
will be explained and justified after the table.
___________________________________________________________________
| | | | | | |
| | generator | parameters | width |period | degree |
|______|____________|_______________|_____________|_______|________|
| | [* t] | | | | |
| (a) | v2 h1 | t 6= 0 | [1] | p |1 - 2p |
|______|____________|_________________|____________|_______|_______ |
| | [* t _10n-1]| __ | __n-1 | n+1 | |
| (b) |v2 h0 |t 6= 1; n > 0 |[1 0 1 ] |p |2p - 3 |
|______|_____________|________________|_____________|______|_______ |
| | [* _1] | | __ | | |
| (c) | v2 g0 | | [1] | p |2p - 4 |
|______|____________|_______________|______________|_______|_______ |
| | [* t _2n] | __ __ | n | n+1 | |
| (d) |v2 g1v2 |t 6= 1; 2; n 0 |[1 2 ] + 1 |p | -2p |
|______|____________|_________________|_____________|______|_______|
| | [*_2_1_2n] | | n n|+2 | |
| (e) |v2 g1v2 | n 0 | [1 0 0 2 ] + 1p | | -2p |
|______|____________|______________|__________________|___|________|
| | [* t _2n] | __ | __n-1 | n+1 | |
| (f) |v2 aev2 |t 6= 2; n 0 | [1 0 1 ] |p | -3 |
|______|_____________|_______________|______________|______|______|_
In part (f ) the width is 1 when n = 0.
78 M. HOVEY AND N. P. STRICKLAND
What all this means is as follows. The group B* is a direct sum of modules
isomorphic to Fp[v1]=vw1generated by various elements x. These generators x come
in v2-periodic families. In the first column we identify the generator x by giv*
*ing its
image in H**E*=I2. The entries in the first column depend on various parameters
which are listed in the second column (except for the parameter "*" which always
occurs). It is implicit that t is an integer with 0 t < p. The third column *
*is
the integer w such that the summand in question is isomorphic to Fp[v1]=vw1. If
k
the entry in the fourth column is pk, then the generators in question form a vp*
*2-
periodic family as * varies. The last column is the degree of the generators mo*
*dulo
|v2| = 2(p2 - 1).
We now deduce our table from Shimomura's calculation of H**M11in [Shi86].
Proof.We start by remarking that our degrees are the usual ones, which are 2(p-*
*1)
times larger than Shimomura's degrees (see his (3.5.8)).
(a): This comes from Shimomura's (4.1.5). He has summands Fp[v1] in
H**M11for s 6= 0 (mod p). Our part (a) is the case n = 0. In (3.6.4) he defines
a0 = 1. By our earlier discussion, we get a generator in ss*bLS=p of width a0 =*
* 1 by
taking d(xs0)=v1. In (4.1.5) he states that d(xs0)=v1 is a unit multiple of vs2*
*h1 mod I2.
In (3.5.6) he defines h1 = tp1=v2 which has cohomological degree s = 1 and inte*
*rnal
degree t = 2(p - 1)p - 2(p2- 1) and thus total degree t - s = 1 - 2p. Our part *
*(a)
follows easily.
(b): This comes from_the case n > 0 in Shimomura's (4.1.5). We then have
an = pn+pn-1-1 = [101n-1] by (3.6.4). By (4.1.5) we see that d(xsn=van1) is a u*
*nit
n-pn-1
multiple of vsp2 h0, so these elements are generators of ss*bLS=p of width a*
*n.
As s 6= 0 (mod p) we have t 2 {0;_: :;:p - 2} such that s = t + 1 (mod p) and t*
*hus
spn-pn-1 = [*t0n]+pn-pn-1 = [*t10n-1]. In (3.6.5) h0 is defined as t1 2 1;2p-2
so the degree is 2p - 3.
(c): This comes from the summands Fp[v1] in Shimomura's (4.1.6).
Here t 2 pZ and d(vt2V )=vp-11= -vt+p-12g0 mod I2. We thus_have generators
vt+p-12g0 of width p-1, and the general form of t+p-1 is [*1]. We learn from (3*
*.5.7)
that g0 2 2;2p-2so the total degree is 2p - 4.
(d): Shimomura has summands Fp[v1] for numbers m of the form
spn where either s 62 {0; -1} (mod p) or s = -1 (mod p2). Our part (d) is the
case s 62 {0; -1} (mod p), so in Shimomura's notation we have ffl(s) = 0 and th*
*us
A(m) = 2 + (p + 1)(pn - 1)=(p - 1). He shows that d(ym )=vA(m)1is a unit multip*
*le
of ve(m)2g1 mod I2, where in the case ffl(s) = 0 we have e(m) = m - (pn - 1)=(p*
* - 1).
The general form of m - pn is [*t0n]_with t = 0; : :;:p - 3, so the general_for*
*m of
e(m) is [*t0n] + [10n] - [1n] = [*t2n] + 1, so the generators are v2 " [*t2n](g*
*1v2).
(We have separated out one factor of v2 to avoid special behaviour when n = 0.)
We learn from (3.5.7) that g1 2 2;2-2pso the total degree is -2p.
(e): This comes from the summands Fp[v1] where m = spn with
s = -1 (mod p2) so ffl(s) = 1 and so A(m) = 2 + pn(p2 - 1) + (p + 1)(pn -
1)=(p - 1). Once again d(ym )=vA(m)1is a unit multiple of ve(m)2g1 mod_I2,_but *
*now
e(m) = m -_pn(p_- 1)_- (pn - 1)=(p_-_1)._ The general form of m is [*110n] so
e(m) = [*110n] - [10n] - [1n] = [*212n] + 1. The total degree is -2p again.
(f): This comes from the summands Fp[v1] in Shimomura's Theo-
rem 4.4. Here n 0 and s 6= -1 (mod p) and an = pn + pn-1 - 1 as before, except
MORAVA K-THEORIES AND LOCALISATION 79
that a0 = 1. His Proposition 4.3 shows that d(xsnGn)=van1is a unit multiple of
ve(n;s)2ae, where e(n; s) = spn-(pn-1)=(p-1)_(see his Lemma 4.2). The general f*
*orm_
of (s - 1)pn is [*t0n] with t 6=_2and then e(n; s) = [*t0n] + [10n] - [1n] = [**
*t2n] + 1,
so the generators are v2 " [*t2n](aev2). We know from (3.5.7) that ae 2_3;0so_t*
*he
total degree is -3. |__|
Remark 15.3.Our table only gives the Z-graded homotopy groups of bLS=p. How-
ever, if a 2 Z and b 2 Zp then we still have ssa+b|v2|bLS=p = lim-ssa+b|v2|LS=(*
*p; vN1),
N
and the terms in the inverse limit and the maps between them are locally consta*
*nt
as b varies. One can deduce that our table also describes ssa+b|v2|bLS=p if we *
*allow
the symbol * to denote an arbitrary infinite sequence of p-adic digits.
15.3. Finiteness questions. We next study the finiteness or otherwise of the
groups ssa+b|v2|bLS=p. (These are of course vector spaces over Fp so that "fini*
*te"
means the same as "finitely generated".) It is convenient to formulate the prob*
*lem
more generally. Let M be a group graded over Zp, for example Mb = ssa+b|v2|bLS=*
*p.
We assume that it is a graded module over Fp[u], where u has degree one; in our
applications u = vp+11.
Definition 15.4.Suppose that 0 and 0 OE; w < p . Let A(OE; ; w) be the Zp-
graded group which is a direct sum of copies of Fp[u]=uw with one copy generated
in each degree congruent to OE mod p . Thus A(OE; ; w) is a p-adically interpol*
*ated
version of OE|v2|Fp[u; vp2 ]=uw. Given a with 0 a < w and b 2 p Zp we write
uavb2x for the generator of A(OE; ; w)a+b+OE. The mnemonic is that OE; and w a*
*re
the phase, wavelength and width of the group. We also write supp(A(OE; ; w)) =
{b 2 Zp | A(OE; ; w)b 6= 0}.
Definition 15.5.GivenPintegers 0 r s and OE 2 Zp we let OEtbe the t-th p-adic
digit of OE (so OE = t0 OEtpt and 0 OEt< p) and define
B(r; s; OE) = { 2 Zp | t= OEt forr t < s}:
We say that a set of this form is an (r; s)-block.
Lemma 15.6. If 2p w then B(; ; OE + p ) supp(A(OE; ; w)).
P -1
Proof.Write = OE+p . If 2 B(; ; ) then we define k = p + j=0( t-OEt)pt.
It is not hard to check that 1 k 2p - 1 < w. Moreover, we have
-1X X X
- k - OE = - ( t- OEt) - = (OEt- t)pt+ ( t- t)pt:
t=0 t< t
It is clear that t = OEt when t < , and by assumption we have t = t when
t < . It follows that - k - OE is divisible by p so that ukv2-k-OEx define*
*s_a
nonzero element of A(OE; ; w) . |__|
L
Proposition 15.7.Suppose that M = iA(OEi; i; wi), where the numbers i are
unbounded and the numbers pi =wiare bounded above. Then {a 2 Zp | Ma is finite}
has Haar measure zero.
Proof.As the numbers pi =wi are bounded above, we can find ff 0 such that
2pi-ff wi for all i. As the numbers i are unbounded, we may assume (after
80 M. HOVEY AND N. P. STRICKLAND
passing to a subgroup and reindexing if necessary) that our direct sum is index*
*ed
by the natural numbers and that i> i-1+ff (where -1 means 0). We now write
Bi= B(i- ff; i; OEi+ pi-ff) supp(A(OEi; i; wi)):
It will be enough to show that C = {a | a lies in infinitely many of theBsetsi}*
* has
measure one. Note that the conditions a 2 Bi depend on disjoint subsets of the
digits ofTa (because i> i-1+ ff). It follows that given any finite set I of int*
*egers
we have ( j2IBcj) = (1-p-ff)|I|, and thus that the correspondingTmeasure is ze*
*ro
if I is infinite. In particular if J isSfiniteTthen ( j62JBcj) = 0. There ar*
*e only
countably many such sets J, so we have ( J j62JBj) = 0. This union is just the
complement of C, so (C) = 1. |___|
L
Proposition 15.8.Suppose that M = iA(OEi; i; wi) and define OE0i= pi -(wi-
1) - OEi. If OEi-! 1 and OE0i-!1 then Ma is finite for all a 2 Z.
Proof.If a 2 Z then for almost all i we have OEi > |a| and OE0i> |a|, and in pa*
*r-
ticular OE0i> 0. It is clear that OEi = |xi| is the lowest positive degree in *
*which
A(OEi; i; wi) is nonzero, and -OE0i= |v-12uwi-1xi| is the highest negative degr*
*ee in
which A(OEi; i; wi) is nonzero, so A(OEi; i; wi)a = 0 for all but finitely many*
* i. It_is_
easy to see that A(OEi; i; wi)a is finite for all i, so Ma is finite. *
* |__|
We now want to transfer these results to modules that are graded in the more
usual way. For each a 2 Z and b 2 Zp we can define ssa+b|v2|bLS=p = ssb|v2|bLS-*
*a=p
using the p-adic powers of S|v2|2 P0 that we considered earlier.NWe can patch t*
*hese
groups together to get a group graded over bZ= lim-Z=|vp2| = Z=2(p2 - 1) x Zp.
N
(We do not need to worry about making coherent choices of isomorphisms here, as
we are simply counting group orders.) We now define A0(OE; ; w) to be the obvio*
*us
interpolation of the Z-graded group OEFp[v1; vp2 ]=vw1. Note that when w is lar*
*ge
this is almost |v1|-periodic. In this context we write OE0= p |v2| - (w - 1)|v1*
*| - OE =
w-1
-|v-p2v1 x|, where x is the generator of A0(OE; ; w).
It is not hard to deduce the following from Propositions 15.7 and 15.8.
L
Proposition 15.9.Suppose that M = iA0(OEi; i; wi), where
(a)The integers OEi are all congruent to a fixed number OE modulo |v1| = 2p - *
*2.
(b)The sequences i, wi, OEi and OE0iall tend to infinity.
(c)The numbers pi =wi are bounded above.
Then Ma is finite for all a 2 Z, but {a 2 bZ| a = OE (mod |v1|) and Ma_is_finit*
*e}
has measure zero. |__|
We can now prove our result about ss*bLS=p.
Proof of Theorem 15.1.By a thick subcategory argument it suffices to verify the
finiteness statement for X = S=p, so we work with S=p throughout.
Let C* be the direct sum of the entries (b), (d), (e) and (f) in Proposition *
*15.2.
The omitted entries, together with the Fp[v1]-free summands generated by 1 and *
*h0,
are all concentrated in degrees 0; : :;:-3 modulo |v1| and are finite in all de*
*grees.
We also have ss*bLS = E[i]A* with |i| = -1. It will thus suffice to show that C*
** is
finite in all integer degrees, but the set of degrees a congruent to -1, -2 or *
*-3 mod
|v1| such that Ca is finite has measure zero. This will follow from Proposition*
* 15.9
once we have determined the parameters , OE, OE0and w for the families in entri*
*es (b),
MORAVA K-THEORIES AND LOCALISATION 81
(d), (e) and (f). This is a straightforward calculation. It is easy to see that*
* in each
case the period and the width w tend to infinity as n does, but that p =w p.
Moreover, if we set * = 0 we get a generator x whose degree satisfies 0 |x| < *
*|vp2|,
which implies that OE = |x|. Given this, it is immediate that OE tends to infin*
*ity as n
does. A little more calculation is required to find the numbers OE0. For exampl*
*e, in
entry (b) we have OE = 2(pt + p - 1)pn-1(p2 - 1) + 2p - 3 and w = pn + pn-1 - 1
and = n + 1 so
OE0= p |v2| - (w - 1)|v1| - OE
= 2pn+1(p2 - 1) - 2(pn + pn-1 - 2)(p - 1) - 2(pt + p - 1)pn-1(p2 - 1) - 2p + 3
= 2(p - 1 - t)pn-1(p2 - 1) + 2p - 1:
As 0 t < p - 1 we see that this tends to infinity as well. The other cases ar*
*e __
similar (although the case t = p - 1 in entry (f) needs to be done separately).*
* |__|
16.Questions and conjectures
In this section we assemble a long list of questions that we have not been ab*
*le to
answer. Some of them have a long history, but others seem not to have been asked
before.
Problem 16.1. When X is a small spectrum in K, is the number len(ss*X) defined
in Definition 12.18 always an integer? If not, is there a universal constant M *
*such
that pM len(ss*X) is always an integer?
Problem 16.2. Is sskbLS always a finitely generated module over Zp for all k 2 *
*Z?
Theorem 15.1 suggests that this is true but subtle.
Problem 16.3. It would be pleasant to have a natural topology on ss0D making it
a topological semiring. One idea would be to define UZ(X) = {Y | Z ^ Y ' Z ^ X}
and declare the sets UZ(X) (as Z runs over F) to be a basis of neighbourhoods of
X. This has some good properties and some bad ones. For example, take n = 1
and p > 3. Then we can define aWv1 self map v :S2p-2=p -! S=p, which becomes
an isomorphism in K. Write X = p-2i=0S2i=p, so that 2X ' X 6' X. It seems
that X cannot be separated from X, so the topology is not Hausdorff, and the
subspace ss0F is not discrete. We hope nonetheless that some minor modifications
will yield a satisfactory theory.
Problem 16.4. We can define a group K0(F) as the monoid of isomorphism classes
of small spectra under the wedge operation, modulo relations [Y ] = [X] + [Z] f*
*or
every cofibre sequence X -! Y -! Z. There is a homomorphism O: K0(F) -! Z
defined by
X
O[X] = (-1)klen(EkX):
0k<|vn|
There is also a homomorphism :K0(F) -!Z[1_p], defined by
[X] = len(sseven(X)) - len(ssodd(X)):
What is the relationship between O and ? Is O an isomorphism? If not, is Q O
an isomorphism? Can one say anything about the higher Waldhausen K-theory
of F? There is a paper by Waldhausen about this [Wal84] but he assumed that
82 M. HOVEY AND N. P. STRICKLAND
the Telescope Conjecture would turn out to be true. Ravenel discusses this brie*
*fly
in [Rav93] in the light of his disproof of the Telescope Conjecture [Rav92b].
Problem 16.5. Is Hopkins' chromatic splitting conjecture true? The simplest form
of this says that K(n - 1)*LK(n)S has rank two over K(n - 1)*. A much more
elaborate version is explained in [Hov95a], where some interesting consequences*
* are
deduced. Part of it can be rephrased as saying that
^n
Ln-1LK(n)S = Ln-1S^p^ (S _ Ln-jS1-2j):
j=1
One possible approach is to consider the functor v-1k((E=Ik)*X) for k < n. This*
* is
a cohomology theory on the category of E-finite spectra; it would be interestin*
*g to
know how it relates to K(k)*X.
Problem 16.6. A problem related to the previous one is as follows: in the langu*
*age
of Definition 6.7, what is the relationship between the support and cosupport o*
*f an
E-local spectrum X? In particular, X is K-local if and only if cosupp(X) = {n},
and the examples suggest that in this case we have supp(X) = {m; m + 1; : :;:n}
for some m. Is this always the case?
Problem 16.7. Is there a cofinal set of ideals I for which there is a generalis*
*ed
Moore spectrum of type S=I which is a commutative and associative ring spectrum?
Given a map BP*=I -! BP*=J of BP*BP -comodules, is there a map S=I -! S=J
inducing it? What about connecting maps for short exact sequences of comodules?
How large a diagram of such maps can be made to commute on the nose? The first
step here is clearly to find a convenient and conceptual formulation of the pro*
*blem.
A good answer would be helpful for the theory of unstable vn periodicity, among*
*st
other things.
Problem 16.8. What are the invariant prime ideals in E*? The obvious conjecture
is that they are just the ideals Ik = (vj | j < k) for k n. Many people believ*
*e,
as we once did, that this is an easy consequence of Landweber's classification *
*of
invariant prime ideals in BP*, but this does not seem to be the case. We also
conjecture that every invariant radical ideal is prime. This would also help to*
* prove
that the categories Dk are the only nontrivial ideals in D.
Appendix A. Completion
In this appendix, we study the theory of (usually infinitely generated) modul*
*es
over a regular local ring, that satisfy certain completeness properties. In par*
*ticular,
this applies to the modules E*X and E_*X over E*, for X 2 K. Much of the theory
presented here is obtained by specialising the results of [GM92 ] to our simpler
context. Presumably a lot of it is known in some form to commutative algebraist*
*s.
Let R be a Noetherian graded ring with a unique maximal homogeneous ideal m,
which is generated by a regular sequence of homogeneous elements (x0; : :;:xn-1*
*).
All modules, ideals and so on are required to be graded, and all maps and eleme*
*nts
are required to be homogeneous.
A.1. Ordinary completion. Recall that the completion of a module M at m is
M^m= lim-M=mkM:
k
MORAVA K-THEORIES AND LOCALISATION 83
Theorem A.1.
(a)(M^m)^m= M^m.
(b)If M is finitely generated then M^m= R^mR M, and this is an exact functor
of M.
(c)R^mis flat over R.
(d)If f :M -! N is epi, then so is the induced map M^m-! N^m.
Proof.(a): Because M^mis defined as the inverse limit of {M=mkM}, there is a
map ssk: M^m-! M=mkM. Because the maps in the inverse system are epi, the map
ssk is also epi. It is clear that ssk factors through an epimorphism M^m=mkM^m-!
M=mkM. On the other hand, the obvious map M -! M^minduces a map M=mkM -!
M^m=mkM^m. One can check that these maps are mutually inverse, so that
(M^m)^m= lim-M^m=mkM^m= lim-M=mkM = M^m
k
(b) and (c) are well-known consequences of the Artin-Rees lemma.
(d): It is easy to see that the induced maps M=mnM -! N=mnN and mnM -!
mnN are epi, and thus that mnM=mn+1M -! mnN=mn+1N is also epi. Let Kn be
the kernel of the map M=mnM -! N=mnN. We get a diagram as follows, in which
the columns and the last two rows are exact.
L __________wmnM=mn+1M _____mnN=mn+1Nww
v| v v
| | |
| | |
|u |u |u
Kn+1 v_________M=mn+1M_w ________N=mn+1Nww
| | |
| | |
| |u |u
|u |u |u
Kn v___________M=mnM_w __________N=mnNww
It follows from the snake lemma that the map Kn+1 -! Kn is epi, and thus that
lim-1Kn = 0. We therefore have a short exact sequence limKn -!limM=mnM -!
n - - __
lim-N=mnN, so M^m-! N^mis epi as claimed. |__|
Note that completion does not preserve monomorphisms and is not right exact.
For example, we can take R = Z(p)and m = (p). Then the completion of Z(p)-!
Q isLZp -! 0, so completionLdoes not preserve monomorphisms. Next, consider
M = kZ and f = kpk : M -! M. One can show that the element (p; p2; : :):
is nonzero in the cokernel of f : M^m-! M^mbut is zero in the completion of the
cokernel. Thus, completion is not right exact.
A.2. Local homology. We next recall the theory of local (co)homology and the
derived functors of the completion functor. Recall that one can define the lef*
*t-
derived functors of any additive functor F . One only needs F to be right exact
in order to prove that the zeroth derived functor of F is F (which is false in *
*the
84 M. HOVEY AND N. P. STRICKLAND
present context). Our response is to take the zeroth derived functor as the "co*
*rrect"
definition of completion.
For any x 2 R we let Ko(x) denote the complex R -!R[1=x], with R in degree
0 and R[1=x] in degree 1. We also write Ko(m) = Ko(x0) : : :Ko(xn-1); there
is then a natural map Ko(m) -!R. Note that Ko(m) is a complex of flat modules.
In fact, Ko(m) is the finite acyclisation of R determined by R=m in the derived
category of R, so it is determined by the ideal m up to canonical quasiisomorph*
*ism
k
over R. To see this, write Ko(x) as the colimit of the complexes R x-!R and ten*
*sor
these complexes together. We also write P Ko(x) for the complex of projectives
R R[t] (1;tx-1)-----!R[t], which is quasiisomorphic to Ko(x). Here the map tx *
*- 1 :
R[t] -! R[t] is the R[t]-module map which takes 1 to tx - 1. We write P Ko(m) =
P Ko(x0) : : :P Ko(xn-1), which is quasiisomorphic to Ko(m) by flatness. The
local homology and cohomology groups of a module M are
L*M = Hm*(M) = H*(Hom (P Ko(m); M))
H*m(M) = H*(P Ko(I) M) = H*(Ko(I) M):
The last equality again uses the flatness of Ko(m). Because R is a regular loc*
*al
ring, we have
(
Hkm(R) = k < n 0
k = n R=(x10; : :;:x1n-1)
There are important questions to address about the extent to which this descrip*
*tion
of Hnm(R) is natural; however, we shall not address them here.
It follows from the above that Ko(m) is actually quasiisomorphic to nHnm(R)
(in other words, the complex whose only nonzero term is Hnm(R), concentrated in
degree n). Note that this is a complex of injectives.
It turns out that the functors LkM are the left derived functors of completio*
*n,
in the following sense.
Theorem A.2.
(a)There are natural maps M -jM-!L0M -fflM-!M^m. Moreover, ffl is an epimor-
phism, and the composite M -! M^mis the obvious map.
(b)There is a short exact sequence
lim-1TorRs+1(R=mk; M) ae LsM i lim TorRs(R=mk; M):
k - k
In particular, there is a short exact sequence
lim-1TorR1(R=mk; M) ae L0M i M^m:
k
(c)For any short exact sequence M0 -!M -! M00there is a long exact sequence
Lk+1M00-! LkM0 -!LkM -! LkM00-! Lk-1M0:
(d)There is a natural isomorphism
LsM = Extn-sR(Hnm(R); M):
Moreover, both sides vanish if s < 0 or s > n. (Thus L0 is right exact and
Ln is left exact).
(e)If M is projective then fflM is an isomorphism.
MORAVA K-THEORIES AND LOCALISATION 85
Proof.In [GM92 ] Greenlees and May construct functors Lk and a natural map
ffl : L0M -! M, satisfying (c) and (e). In Theorem 2.5, they prove that their g*
*roups
LkM are the same as Hmk(M) as defined above. They prove (b) as Proposition 1.1;
it follows that fflM is an epimorphism, as stated in (a). The map Ko(m) -!R ind*
*uces
a map
j :M = Hom (R; M) -!H0(Hom (Ko(m); M)) = L0M:
One can check that the composite M -j!L0M -ffl!M^mis the obvious map.
In our case, we know that P Ko(m) is a cochain complex of projectives, whose
only cohomology is Hnm(R), concentrated in degree n. Moreover, P Ko(m) vanishes
above degree n. It follows that Pk = P Kn-k(m) defines a projective resolution *
*of
Hnm(R), and thus that LkM = Hk(Hom (P Ko(m); M)) = Extn-kR(Hnm(R); M)._This_
proves (d). |__|
Proposition A.3.There is a natural map : LsM LtN -! Ls+t(M N), which
is commutative, associative and unital in the obvious sense.
Proof.The universal property of Ko(m) as the finite acyclisation of R gives a
quasiisomorphism :P Ko(m) -! P Ko(m) R P Ko(m) compatible with the maps
P Ko(m) -! R- P Ko(m) R P Ko(m), which is unique up to homotopy. This
gives a map of chain complexes
Hom R(P Ko(m); M) R Hom R(P Ko(m); N) -!Hom R (P Ko(m); M N):
By taking homology, we get the advertised pairing. Using the uniqueness of (a*
*nd
the analogous map P Ko(m) -!P Ko(m)3 ), we see that the pairing is commutative,_
associative, and unital. |__|
Proposition A.4.If M is finitely generated then the map : M R L0N -!
L0M R L0N -! L0(M R N) is an isomorphism, and L0M = M^m. In particular,
R=mk L0N = N=mkN.
Proof.Because L0 is additive, it is clear that Rk R L0N = (L0N)k = L0(Nk) =
L0(Rk N). One can check that the map is just . For an arbitrary finitely
generated module, choose an exact sequence Rj -!Rk -!M -! 0, and use the use
the right exactness of L0 to conclude that M R L0N = L0(M R N). In particular,
M^m= M R R^m= M R L0R = L0(M R R) = L0M:
|___|
A.3. L-complete modules.
Definition A.5.An R-module M is L-complete if jM :M -! L0M is an isomor-
phism. We write M for the category of R-modules, and bM for the subcategory of
L-complete modules.
Theorem A.6.
(a)For any M 2 M, the modules M^mand LkM lie in bM. In particular, L20M =
L0M.
(b)If M 2 bMthen LkM = 0 for k > 0.
(c)L0M = 0 , M^m= 0 , M = mM.
(d)If M 2 bMand M = mM then M = 0.
(e)Mb is an Abelian subcategory of M, which is closed under extensions.
86 M. HOVEY AND N. P. STRICKLAND
(f)L0 : M -!Mb is left adjoint to the inclusion bM-! M.Q
(g)If {Mk} is a collection of L-complete modules, then kMk is L-complete. If
they form an inverse system then lim-Mk is L-complete. If they form a tower
k
then lim-1Mk is also L-complete.
k
Proof.It is proved as Theorem 4.1 in [GM92 ] that if N = M^mor N = LkM then
L0N = N and LkN = 0 for k > 0. It follows that M^mand LkM are L-complete.
It also follows that if M is L-complete and k > 0 then LkM = LkL0M = 0. This
proves (a) and (b).
(c): We have epimorphisms L0M -! M^m-! M=mM, so L0M = 0 ) M^m=
0 ) M = mM. Suppose that M = mM. We shall prove that L0M = 0, using
an argument supplied by John Greenlees. It follows that M = mkM, so that
R=mk M = 0 for all k. Using the short exact sequences
mk=mk+1 -!R=mk+1 -!R=mk
and the fact that mk=mk+1 is a free module over R=m, we get an exact sequence
TorR1(mk=mk+1; M) -!TorR1(R=mk+1; M) -!TorR1(R=mk; M) -!0:
From this we see that the maps in the tower
TorR1(R=mk; M)- TorR1(R=mk+1; M)- : : :
are surjective, so that lim-1TorR1(R=mk; M) = 0. Using part (b) of Theorem A.2,
k
we see that L0M = 0 as claimed.
(d): If M 2 bMand M = mM then M = L0M = 0 by (c).
(e): First, we claim that the image of any map f :M0 -! M00of L-complete
modules is L-complete. To see this, factor f as a composite M0 q-!M -j!M00, with
q epi and j mono, so that M is the image of f. We have a diagram as follows:
q j
M0 ________Mwwv_______M00w
| | |
| | |
j0|' j| ' |j00
| | |
| | |
|u |u |u
L0M0 _____wL0MwL0q___wL0M00L0j
Note that L0q is epi because L0 is right exact. Because the left square commute*
*s,
we see that j is epi; because the right square commutes, we see that it is mono.
Thus j is an isomorphism, and M is L-complete.
Next, suppose that N0 -!N -! N00is a short exact sequence, and that any two
of the terms are L-complete. Part (c) of Theorem A.2 gives a long exact sequence
relating the Lk-groups of N0, N and N00, from which it is easy to see that the *
*third
of these is also L-complete. Thus bMis closed under extensions, and under kerne*
*ls
and cokernels of epimorphisms and monomorphisms. For any map f :M0 -!M00as
in the previous paragraph, we have ker(f) = ker(q) and cok(f) = cok(j); it foll*
*ows
that bMis closed under kernels and cokernels, and thus that it is Abelian.
(f): Suppose that N is L-complete and M is arbitrary. We need to show that
j : M -! L0M induces an isomorphism j*M: Hom (L0M; N) -!Hom (M; N). There
is a map : Hom (M; N) -! Hom (L0M; N), defined by the commutativity of the
MORAVA K-THEORIES AND LOCALISATION 87
following diagram:
f
M _______wN
| aeae|o
j | (f) ae |j
M | ' | N
| ae |
|uae |u
L0M _____wL0NL0f
It is clear that j*MO = 1, so that j*M is epi. Suppose that f 2 ker(j*M), so
that f can be factored as L0M -q!M0 -g!N, where M0 is the cokernel of jM .
By the argument given above, we have an epimorphism j*M0: Hom (L0M0; N) -!
Hom(M0; N). However, because L0 is right exact and idempotent, we see that
L0M0 = 0. It follows that g = 0 and thus f = 0. Thus j*M: Hom (L0M; N) -!
Hom(M; N) is an isomorphism, as required. Q Q
(g): It is easy to see from the definitions that Hms( kMk) = kHms(Mk). It
follows that a product of L-complete modules is L-complete. If the modules {Mk}
form an inverse system involving various maps u: Mk -! Ml then lim-Mk is the
Q Q k
kernel of a map kMk -! uMl, so it is L-completeQby (d). IfQthe inverse system
is a tower, then lim-1Mk is the cokernel of a map kMk -! kMk, and thus is
k __
L-complete. |__|
__
Corollary A.7.bMis a symmetric monoidal category with tensor product M N =
L0(M N).
Proof.It suffices to show that the map L0: L0(L0M L0N) -! L0(M N)
induced by the pairing of Proposition A.3 is an isomorphism. Consider two modul*
*es
M; N 2 M. Let M0 be the cokernel of M -! L0M. Because L0 is idempotent and
right exact, we have L0M0 = 0. It follows that M0 = mM0, or in other words that
the natural map m R M0 -!M0 is surjective. This implies that m R M0R N -!
M0R N is also surjective, so that L0(M0R N) = 0. Using the right exactness of
L0, we see that the map L0(j 1): L0(M R N) -!L0(L0M R N) is surjective.
Similarly, the map L0(L0M R N) -! L0(L0M R L0N) is surjective, and thus
L0(j j): L0(M R N) -! L0(L0M R L0N) is surjective. On the other hand,
one sees from the definitions that L0 O L0(j j) = 1. It follows that L0_is_an
isomorphism, with inverse L0(j j). |__|
We next prove a useful criterion for modules to be L-complete.
Proposition A.8.If M is L-complete and a 2 m then lim-0M = lim-1M = 0,
where the limits refer to the tower
M -a M -a M -a : :::
Conversely, if (x0; : :;:xn-1) is a regular system of parameters and the above *
*holds
with a = xi for each i, then M is L-complete.
Proof.Let Co be the complex (of projectives)
R[t] at-1---!R[t]:
88 M. HOVEY AND N. P. STRICKLAND
One checks directly that Hs Hom(Co; M) = lim-sM. Moreover, there is an obvious
short exact sequence Co -! P Ko(a) -! R. It follows that lim-0M = lim-1M = 0 if
and only if Hom (P Ko(a); M) is quasiisomorphic to M = Hom (R; M).
Suppose that M is L-complete, so that M ' Hom (P Ko(m); M). The complex
Ko(m) is quasiisomorphic to Ko(a) Ko(m). Indeed, Ko(m) R[1=a] is acyclic,
since Ko(m) is finitely acyclic and R[1=a] is finitely local. Thus P Ko(m) is h*
*omotopy
equivalent to P Ko(a) P Ko(m). Therefore
Hom (P Ko(a); M)' Hom (P Ko(a); Hom(P Ko(m); M))
' Hom (P Ko(a) P Ko(m); M)
' M:
It follows that lim-0M = lim-1M = 0.
Conversely, let (x0; : :;:xn-1) be a regular system of parameters, and suppose
that M ' Hom (P Ko(xi); M) for each i. As P Ko(m) = P Ko(x0): :P:Ko(xn-1), __
we see easily that M ' Hom (P Ko(m); M), so M is L-complete. |__|
A.4. Pro-free modules.
Theorem A.9. Let M be an L-complete R-module. The following are equivalent:
1.Every regular sequence of parameters x_is regular on M.
2.Some regular sequence of parameters x_is regular on M.
3.TorRs(M; R=m) = 0 for all s > 0.
4.TorR1(M; R=m) = 0.
5.M = L0F = Fm^for some free module F .
6.M is a projective object of bM.
Definition A.10.If M 2 bMsatisfies the conditions of Theorem A.9, we say that
M is pro-free.
Proof of Theorem A.9.(1))(2): clear.
(2))(3): Suppose that x_= (x0; : :;:xn-1) is a regular system of parameters f*
*or
R, and that x_is also regular on M. Write Im = (x0; : :;:xm-1 ), so that In = m.
We therefore have short exact sequences
R=Im -xm-!R=Im -! R=Im+1
and
M=Im M -xm-!M=Im M -! M=Im+1 M:
The first of these also gives a long exact sequence
TorRs+1(R=Im+1 ; M) -!TorRs(R=Im ; M) xm--!TorRs(R=Im ; M) -!
TorRs(R=Im+1 ; M) -!TorRs-1(R=Im ; M):
By comparing this with the second short exact sequence, we see inductively that
TorRs(R=Im ; M) = 0 for all s > 0 and all m, in particular TorRs(R=m; M) = 0 for
s > 0.
(3))(4): clear.
(4))(5): Suppose that TorR1(M; R=m) = 0. Choose a basis {_ei| i 2 I} for
M=mMLover R=m, and choose elements ei 2 M lifting _ei. These give a map
F = IR -! M, and thus a map f :L0F -! M. By proposition A.4, we see that
MORAVA K-THEORIES AND LOCALISATION 89
R=m L0F = R=m F ' R=m M, so that R=m cok(f) = 0. By part (d)
of Theorem A.6, we see that cok(f) = 0, so f is epi. We therefore get an exact
sequence
0 = TorR1(R=m; M) -!R=m R ker(f) -!R=m R L0F -'!R=m R M -! 0:
It follows that R=m R ker(f) = 0, so that ker(f) = 0. Thus f is an isomorphism,
as required.
(5))(1): Suppose that x_= (x0; : :;:xn-1) is a regular system of parameters f*
*or
R.Q Write Im = (x0; : :;:xm-1 ). Let S be a set, and a = (as)s2S an element of
s2SR=Im . We shall say that a converges to zero (or write a -!0) if for all k *
*we
have as 2 mkR=Im for all but finitely many s. One can check directly that
!^
M Y
R=Im = {a 2 R=Im | a -!0}:
k2S m s2S
We write Fm for this module. Using the above description, one can check directly
that the sequence
Fm -xm-!Fm -! Fm+1
L
is short exact. It follows that x_is regular on F0 (which is the same as L0( S*
* R)
by part (e) of Theorem A.2).
(5))(6): Suppose that F is free and that M = L0F . Then for N 2 bMwe have
bM(M; N) = M(F; N) by part (f) of Theorem A.6. This is clearly an exact functor
of N, as required.
(6))(4): Suppose that P is a projective object of bM. Choose a free module F
and an epimorphism F -! P . This gives an epimorphism L0F -! L0P = P , which
must split because P is projective. It follows that TorR1(R=m; P ) is a summand*
*_of
TorR1(R=m; L0F ), which vanishes because (5))(4). |__|
Corollary A.11.The product of any family of pro-free modules is pro-free.
Proof.This follows easily from condition (1). |___|
Corollary A.12.Mb has enough projectives.
Proof.Given M 2 bM, choose a free module F and an epimorphism F -! M._This_
gives an epimorphism L0F -! L0M = M. |__|
Proposition A.13.If f :P -! Q is a map of pro-free modules such that the
induced map P=mP -! Q=mQ is a monomorphism, then f is a split monomor-
phism.LIn particular,Qif {Pk} is a family of pro-free modules, then the natural*
* map
L0 k Pk -! kPk is a split monomorphism.
Proof.For the first claim, choose elements es 2 P for s 2 S giving a basis for
P=mP as an R=m-vector space, and then choose elements e0t2 Q for t 2 T such
that {fes | s 2 S} q {et | t 2 T } gives a basis for Q=mQ. This gives an obvious
90 M. HOVEY AND N. P. STRICKLAND
diagram
M M
R _____w R
S SqT
| |
| |
| |
| |
|u |u
P _________Qw
As in the proof of (4))(5) above, we see that the vertical maps are isomorphisms
after applying L0. The top horizontal map is a split monomorphism, so the bottom
one is also.
Now consider a family {Pk} of pro-free modules. By proposition A.4, we know
that M M M
R=m R L0 Pk = R=m R Pk = R=m R Pk:
k Q k Q k
It is easy to see that R=m R kPk = kR=m R Pk. Thus, the first part of_this
proposition applies. |__|
Appendix B. Small objects in other categories
In this appendix we investigate when various other categories of local spectr*
*a have
no small objects. We assume that all spectra and Abelian groups are p-local. We
write I for the Brown-Comenetz dual of the sphere, so that [X; I] = Hom (ss0X; *
*Q=Z)
and IX = F (X; I). Here Q=Z denotes Q=Z(p)in keeping with our conventions. If
L: S -! S is a localisation functor, we say that a spectrum X is L-small if it *
*is a
small object in the category of L-local spectra. Localisation functors split na*
*turally
into two categories: we say that a localisation functor L has a finite local if*
* there
is a nonzero finite spectrum X such that LX = X. Similarly, we say that L has a
finite acyclic if there is a nonzero finite X such that LX = 0. It is an old co*
*njecture
of the first author that every localisation functor either has a finite local o*
*r a finite
acyclic: it certainly cannot have both.
Definition B.1.We say that a spectrum X is finite-dimensional if it admits a
finite filtration 0 = Xa Xa+1 : :X:b= X for some integers a b, such that
Xk=Xk-1 is a wedge of copies of Sk. Clearly if Y is a finite-dimensional CW
complex then a1 Y is a finite-dimensional spectrum. We define the width of X
to be the minimum possible value of b - a.
Remark B.2.It is not hard to show that X has a filtration as above if and only *
*if
sskX = 0 for k a and HbX is free and HkX = 0 for k > b. It follows that any
retract of X admits a filtration of the same type.
Lemma B.3. Let L be a localisation functor on the category of spectra, and SL
the category of L-local spectra. If X is an L-small spectrum then it is a retra*
*ct of
-N LY for some integer N and some finite-dimensional suspension spectrum Y .
Proof.Let Xk denote the k'th space of X. We assume some foundational setting in
which this is an infinite loop space with ssjXk = ssj-kX for j 0. It is well-k*
*nown
that X is the telescope of the spectra -k1 Xk. It is not hard to conclude that X
is also the telescope of the spectra -k1 Yk, where Yk is the 2k-skeleton of a CW
MORAVA K-THEORIES AND LOCALISATION 91
complex weakly equivalent to Xk. As localisation functors preserve telescopes, *
*we
see that X = LX is the telescope of the spectra L-k1 Yk. By smallness, it must_
be a retract of one of these terms. |__|
W Q
Lemma B.4. For any set A, the spectrum ( i2AS)^pis a retract of i2AS^p.
Proof.ByQPropositionLA.13 (with R = Zp) we see that there is a set B A such
that i2AZp = (Q j2B Zp)^p.L It follows that for any finitely generated Abelian
groupWM we haveQ i2AM^p= ( j2B M)^p. We can use the set B to define a map
( j2BS^p)^p-! i2AS^p, and by applying the previous remark to MW= sskS we
conclude thatQthis map is an isomorphism. It follows that Wp^= ( i2AS^p)^pis_a
retract of i2AS^pas claimed. |__|
Lemma B.5. If X is a nontrivial finite spectrum then coloc = coloc~~, and
this contains Z^pfor all finite-dimensional spectra Z. Moreover, coloc =
coloc~~~~, and this contains all finite-dimensional spectra.
Proof.Consider a generalised Moore spectrum S=I = S=(J; v). It is easy to
see that S=(J; vk) lies in the thick subcategory generated by S=I, and thus that
(S=J)^v= holim-S=(J; vk) 2 coloc. If J 6= 0 then |v| > 0 and S=J is (-1)-
k
connected and simple connectivity arguments show that (S=J)^v= S=J. When
J = 0 we know that v is just a power of p and (S=J)^v= S^p. By induction on
the height we conclude easily that coloc~~~~ = coloc~~~~. If X is an arbitra*
*ry
nontrivial finite spectrum then X generates the same thick subcategory as some
generalised Moore spectrum, so coloc = coloc~~~~. This part of the argument
is essentially [Hov95a, Lemma 3.7].
Next, let X be a finite-dimensional spectrum, with filtration as in Definitio*
*n B.1.
Lemma B.4 tells us that (Xk=Xk-1)^p2 coloc~~~~, and it follows easily that X^p2
coloc~~~~.
Next, recall that Z^p= F (S-1=p1 ; Z), so we have a fibration F (SQ; Z) -!Z -!
Z^p, in which the fibre is a rational spectrum. It is easy to see that whenever*
* V is a
nontrivial rational spectrum, we have coloc = coloc = {all rational spe*
*ctra}.
By taking Z = S, we conclude that S 2 coloc~~~~. We also conclude that
HQ 2 coloc~~~~ = coloc~~~~. This shows that coloc~~~~ = coloc~~~~ as
claimed. Finally, we can take Z to be a finite-dimensional spectrum. We have se*
*en
that Z^p2 coloc~~~~ and that F (SQ; Z) is rational and thus in coloc, so_it
follows that Z 2 coloc~~~~ = coloc~~~~. |__|
Theorem B.6. Suppose L is a localisation functor with a finite local.
(a)If LHQ 6= 0, then L is the identity on finite-dimensional spectra. If LHQ =
0, then L is p-completion on finite-dimensional spectra.
(b)Every L-small spectrum is finite, and is torsion if LHQ = 0.
Proof.(a): Let Z be a finite-dimensional spectrum. The category SL of L-local
spectra is a colocalising category which contains a nontrivial finite spectrum,*
* so by
Lemma B.5 it contains Z^p. If LHQ 6= 0 then SL contains a nontrivial rational
spectrum so it contains all rational spectra, and in particular it contains F (*
*SQ; Z).
We have a fibration F (SQ; Z) -! Z -! Z^pso we conclude that Z 2 SL and thus
LZ = Z. On the other hand, suppose that LHQ = 0. If V is a rational spectrum we
must have LV = 0, for otherwise SL would contain a nontrivial rational spectrum
92 M. HOVEY AND N. P. STRICKLAND
and thus would contain HQ. Thus LF (SQ; Z) = 0 so our fibration shows that
LZ = LZ^p= Z^p.
(b): We begin by showing that any L-small spectrum is finite-dimensional. In
case LHQ 6= 0, this follows from part (a) and Lemma B.3. If LHQ = 0, then the
telescope holim-!(p; X) of an L-small spectrum X computed in SL is zero. Applyi*
*ng
smallness to the canonical map X -! holim-!(p; X), we find that 0 = pm :X -! X
when m 0. Thus X is a retract of X=pm , which is a retract of -nY=pm (because
LY=pm = Y=pm by part (a)). It followsWthatLX is finite-dimensional and torsion.
We next claim that we have [X; iS] = i[X; S] for any set of indices i. If
LHQ 6= 0 then this is clear, because X is small in SL and the wedge in S is
finite-dimensional and thus L-local and thusWtheWsame as the coproduct in SL. In
the case LHQW= 0 weLknow that D(S=pm ) ^ iS = iD(S=pm ) is L-local so
[X ^ S=pm ; iS] = i[X ^ S=pm ; S] and we have seen that X is a retract of
X ^ S=pm for some m so the claim follows.
Let C be the full subcategory of finite-dimensional spectra Y such that
1_ M
[Y; S]* = [Y; S]*:
k=0 k
This is clearly thick, and contains both X and all finite spectra. We shall pro*
*ve by
induction on the width of Y that if Y 2 C then Y is finite. We may assume that Y
has width greater than zero, so there is a cofibration Z -! Y -! W in which W is
a wedge of spheres of the same dimension and Z has width strictly less than that
of Y . By the defining property of C, there is a finite subwedge V of W such th*
*at
the map Y -! W factors through V . Let T be the cofibre of the map Y -! V , so
that T 2 C. By applying the octahedral axiom to the maps Y -! V ae W we get a
cofibration T ae Z i U in which the first map is a split monomorphism, so T is
a retract of Z and thus has width less than that of Y . It follows that T is fi*
*nite,_and
thus that Y is finite. We conclude that our L-small spectrum X must be finite. *
* |__|
We now investigate when LF (n) is L-small. It is convenient to use the notati*
*on
for the collection of L-local spectra, and to think of as a generalised B*
*ousfield
class. If L = LE , then = . We can then use notation such as ,
which is the local objects corresponding to the acyclics for the functor X 7! L*
*X ^
F (n) = L(X ^ F (n)).
Proposition B.7.Suppose L is a localisation functor on S. Then the following
are equivalent.
(a)LF (n) is L-small. W
(b)If {Xff} is a set of L-local spectra, then ffXff^ F (n) is L-local.
(c)The natural map LS ^ X ^ F (n) -! L(X ^ F (n)) is an isomorphism for all
X.
(d)= :
Furthermore, LF (n) is a small generator of SL if and only if = .
MORAVA K-THEORIES AND LOCALISATION 93
Proof.(a))(b): Suppose LF (n) is L-small, and {Xff} is a set of L-local spectra.
Then we have _ _
[F (n); L Xff]*=[LF (n); L Xff]*
M
= [LF (n); Xff]*
M
= [F (n); Xff]*
_
= [F (n); Xff]*:
W W
It follows using duality that the map Xff^ F (n) -!L( Xff^ F (n)) is an equi*
*v-
alence.
(b))(c): Consider the collection of all X such that the map LS ^ X ^ F (n) -!
L(X ^ F (n)) is an equivalence. This category is clearly thick and contains S.
Part (b) implies that it is localising, so it is all of S.
(c))(d): This is clear. W
(d))(a): We first show that if Xffis L-local for all ff, then Xff^ F (n) is*
* also
L-local. To see this, choose FW(n) to be a ring spectrum (take F (n) = DY ^ Y
for a finite type n Y ). Then Xff^ F (n) is an LS ^ F (n)-module spectrum,
soWin particular is LS ^ F (n)-local. Since we are assuming part (d), we see t*
*hat
Xff^ F (n) is L ^ F (n)-local, and in particular is L-local.
We therefore have _ _
[LF (n); L Xff]= [F (n); L Xff]
_
= [S; L( Xff^ DF (n))]
_
= [S; Xff^ DF (n)]
_
= [F (n); Xff]
M
= [F (n); Xff]
M
= [LF (n); Xff]
as required.
To complete the proof, note that LF (n) is a generator of SL if and only if s*
*s*L(X^
F (n)) = [DF (n); LX] = 0 implies that LX = 0. This is true if and only_if =
. |__|
Corollary B.8.Suppose L is a localisation functor with a finite local. Then SL
has a nonzero small object if and only if for some n, and in this
case the small objects in SL are Cn for the smallest such n. The category SL is*
* an
algebraic stable homotopy category if and only if L = LF(n)for some n, and in t*
*his
case F (n) is a small generator.
Proof.The small objects in SL are all finite by Theorem B.6. Furthermore, LS is
either S or the p-completion of S, and in either case we have = ~~~~. The_
corollary then follows from Proposition B.7. |__|
Proposition B.7 also applies to localisations with finite acyclics. For examp*
*le, we
get the following corollary.
Corollary B.9.Let E = K(m) _ K(m + 1) _ : :K:(n) for some 0 m n < 1.
Then SE is an algebraic stable homotopy category with small generator LE F (m).
94 M. HOVEY AND N. P. STRICKLAND
It would be interesting to investigate these categories, as well as SF(n).
In order to apply Corollary B.8, we need a criterion to determine when
for some n.
Lemma B.10. Suppose for some n. Then LI 6= 0.
Proof.It suffices to show that DF (n) ^ I 6= 0 for all n. But DF (n) ^ I =_IF (*
*n)
and IX is never zero unless X is. |__|
Lemma B.10 is more useful than it appears: in fact, we conjecture that its co*
*n-
verse holds as well. There are very few spectra that detect I.
Lemma B.11. BP ^ I = 0.
Proof.This follows from Ravenel's results in [Rav84]. He proves in Lemma 3.2 th*
*at
[P (1); S]* = 0, where P (1) = BP=p = BP ^ S=p. As D(S=p) = S-1=p we see that
[BP; S=p]* = 0. As S=p has finite homotopy groups, we see that S=p = I2(S=p)
so [BP; S=p]* = Hom (ss*(BP ^ I(S=p)); Q=Z) = 0. Thus BP ^ S=p ^ I = 0. We
also have S[1_p] ^ I = 0 because ss*I is a torsion group. As ~~~~= ~~~~_ ~~~~, we
conclude that BP ^ I = 0. |___|
Corollary B.12.We have X ^ I = 0 if X is a BP -module. In particular K(n),
HZ, HFp and HQ = SQ are I-acyclic. Furthermore, I itself is I-acyclic.
Proof.The only thing to prove is that I is I-acyclic. But the homotopy groups o*
*f I
are concentrated in non-negative dimensions and are all finite except for ss0I *
*= Q=Z.
It follows that I is in the localising subcategory generated by HFp. Since_HFp *
*is
I-acyclic, so is I. |__|
W Now recall that a spectrum is said to be harmonic if it is W -local, where W =
0n<1 K(n).
Corollary B.13.There are no nonzero small objects in the following categories.
(a)The BP -local category.
(b)The harmonic category.
(c)The HZ-local category.
(d)The HFp-local category.
(e)The I-local category.
Proof.All of these localisation functors kill I, so it suffices to check that t*
*hey do
have a finite local. It is well-known that all finite spectra are BP -local and*
* HZ-
local, and that all finite torsion spectra are HFp-local ( [Bou79, Theorem 3.1]*
*).
Similarly, all finite spectra are harmonic by [Rav84, Corollary 4.5]. To see th*
*at S=p
is I-local, note that I2(S=p) = S=p since the homotopy groups of S=p are finite.
When Z is I-acyclic, we have
[Z; S=p] = [Z; F (I(S=p); I)] = [Z ^ I(S=p); I] = [Z ^ D(S=p) ^ I; I] = 0
as required. |___|
MORAVA K-THEORIES AND LOCALISATION 95
References
[Ada74] J. F. Adams, Stable homotopy and generalised homology, University of Ch*
*icago Press,
Chicago, 1974.
[AK89] J. F. Adams and N. J. Kuhn, Atomic spaces and spectra, Proc. Edinburgh *
*Math. Soc.
(2) 32 (1989), no. 3, 473-481.
[And95] M. Ando, Isogenies of formal group laws and power operations in the coh*
*omology
theories En, Duke Math. J. 79 (1995), no. 2.
[Bak91] A. Baker, A1 structures on some spectra related to Morava K-theory, Qu*
*art. J.
Math. Oxf 42 (1991), 403-419.
[Bor53] A. Borel, Sur la cohomologie des espaces fibres principaux et des espac*
*es homogenes
de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207.
[Bou79] A. K. Bousfield, The localization of spectra with respect to homology, *
*Topology 18
(1979), 257-281.
[CS96] J. D. Christensen and N. P. Strickland, Phantom maps and homology theor*
*ies, 25
pp., 1996, to appear in Topology.
[Dev92] E. S. Devinatz, Small ring spectra, J. Pure Appl. Alg. 81 (1992), 11-16.
[Dev96a]E. S. Devinatz, The generating hypothesis revisited, Preprint, 1996.
[Dev96b]E. S. Devinatz, Morava modules and Brown-Comenetz duality, Preprint, 19*
*96.
[DHS88] E. S. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable h*
*omotopy
theory, Ann. of Math. (2) 128 (1988), 207-241.
[EKMM96]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules *
*and alge-
bras in stable homotopy theory, Amer. Math. Soc. Surveys and Monographs*
*, vol. 47,
American Mathematical Society, 1996.
[Fra92] J. Franke, On the construction of elliptic cohomology, Math. Nachr. 158*
* (1992), 43-
65.
[Fra96] J. Franke, Uniqueness theorems for certain triangulated categories poss*
*essing an
Adams spectral sequence, Preprint, 1996.
[Fre66] P. Freyd, Stable homotopy, Proceedings of the Conference on Categorical*
* Algebra (La
Jolla, 1965) (S. Eilenberg, D. K. Harrison, S. Mac Lane, and H. R"ohrl,*
* eds.), 1966,
pp. 121-172.
[GM92] J. P. C. Greenlees and J. P. May, Derived functors of I-adic completion*
* and local
homology, J. Alg. 149 (1992), no. 2, 438-453.
[GM95a] J. P. C. Greenlees and J. P. May, Completions in algebra and topology, *
*Handbook of
Algebraic Topology (Amsterdam) (Ioan M. James, ed.), Elsevier, Amsterda*
*m, 1995.
[GM95b] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, vol. 113*
*, Mem. Amer.
Math. Soc., no. 543, American Mathematical Society, 1995.
[GS96a] J. P. C. Greenlees and H. Sadofsky, The Tate spectrum of vn-periodic co*
*mplex oriented
theories, Math. Z. 222 (1996), 391-405.
[GS96b] J. P. C. Greenlees and N. P. Strickland, Varieties and local cohomology*
* for chromatic
group cohomology rings, 48 pp., Submitted to Topology, 1996.
[HG94] M. J. Hopkins and B. H. Gross, The rigid analytic period mapping, Lubin*
*-Tate space,
and stable homotopy theory, Bull. Amer. Math. Soc. 30 (1994), 76-86.
[HMS94] M. J. Hopkins, M. E. Mahowald, and H. Sadofsky, Constructions of elemen*
*ts in
Picard groups, Topology and Representation Theory (E. M. Friedlander an*
*d M. E.
Mahowald, eds.), Contemp. Math., vol. 158, Amer. Math. Soc., 1994, pp. *
*89-126.
[Hov95a]M. Hovey, Bousfield localization functors and Hopkins' chromatic splitt*
*ing conjecture,
The Cech Centennial (Mila Cenkl and Haynes Miller, eds.), Contemp. Math*
*., vol. 181,
Amer. Math. Soc.,1995, pp. 225-250.
[Hov95b]M. Hovey, Cohomological Bousfield classes, J. Pure Appl. Algebra 103 (1*
*995), 45-59.
[HPS95] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homoto*
*py theory,
To appear in Mem. Amer. Math. Soc., 1995.
[HR92] M. J. Hopkins and D. C. Ravenel, Suspension spectra are harmonic, Bol. *
*Soc. Mat.
Mexicana 37 (1992), 271-280, This is a special volume in memory of Jose*
* Adem, and
is really a book. The editor is Enrique Ramirez de Arellano.
[HRW96] M. J. Hopkins, D. C. Ravenel, and W. S. Wilson, Morava Hopf algebras an*
*d spaces
K(n) equivalent to finite Postnikov systems, Preprint, 1996.
[HS] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II*
*, To appear
in Ann. of Math.
96 M. HOVEY AND N. P. STRICKLAND
[HS95] M. Hovey and H. Sadofsky, Invertible spectra in the E(n)-local stable h*
*omotopy cat-
egory, To appear in J. London Math. Soc., 1995.
[HS96] M. Hovey and H. Sadofsky, Tate cohomology lowers chromatic Bousfield cl*
*asses, Proc.
Amer. Math. Soc., 124 (1996), 3579-3585.
[Lan76] P. S. Landweber, Homological properties of comodules over MU*(MU) and B*
*P*(BP),
Amer. J. Math. 98 (1976), 591-610.
[LS69] R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear for*
*m for Hopf
algebras, Amer. J. Math. 91 (1969), 75-94.
[Mar83] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland, 1983.
[Mil92] H. R. Miller, Finite localizations, Bol. Soc. Mat. Mexicana 37 (1992), *
*383-390, This is
a special volume in memory of Jose Adem, and is really a book. The edit*
*or is Enrique
Ramirez de Arellano.
[Mor85] J. Morava, Noetherian localizations of categories of cobordism comodule*
*s, Ann. of
Math. (2) 121 (1985), 1-39.
[Mos68] R.M.F. Moss, On the composition pairing of Adams spectral sequences, Pr*
*oc. London
Math. Soc. 18 (1968), 179-192.
[Nas96] C. Nassau, On the structure of P(n)*(P(n)) for p = 2, preprint, 1996.
[Rav84] D. C. Ravenel, Localization with respect to certain periodic homology t*
*heories, Amer.
J. Math. 106 (1) (1984), 351-414.
[Rav86] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres,*
* Academic
Press, 1986.
[Rav92a]D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, An*
*nals of Math-
ematics Studies, vol. 128, Princeton University Press, 1992.
[Rav92b]D. C. Ravenel, Progress report on the telescope conjecture, Adams Memor*
*ial Sym-
posium on Algebraic Topology Volume 2 (Nigel Ray and Grant Walker, eds.*
*), 1992,
London Mathematical Society Lecture Notes 176, pp. 1-21.
[Rav93] D. C. Ravenel, Life after the telescope conjecture, NATO Adv. Sci. Inst*
*. Ser. C Math.
Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 205-222.
[Riv72] N. Saavedra Rivano, Categories Tannakiennes, Springer-Verlag, Berlin, 1*
*972, Lecture
Notes in Mathematics, Vol. 265.
[Rud86] Y. B. Rudyak, Exactness theorems for the cohomology theories MU ,BP and*
* P(n),
Mat. Zametki 40 (1986), no. 1, 115-126,141. English translation: Math. *
*Notes 40
(1986) 562-569.
[RW80] D. C. Ravenel and W. S. Wilson, The Morava K-theories of Eilenberg-MacL*
*ane spaces
and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691-*
*748.
[Sad93] H. Sadofsky, Hopkins' and Mahowald's picture of Shimomura's v1-Bockstei*
*n spectral
sequence calculation, Algebraic topology (Oaxtepec, 1991) (Martin Tango*
*ra, ed.),
1993, Contemp. Math. 146, Amer. Math. Soc., pp. 407-418.
[Shi86] K. Shimomura, On the Adams-Novikov spectral sequence and products of fi*
*-elements,
Hiroshima Math. J. 16 (1986), 209-224.
[Spa66] E. H. Spanier, Algebraic topology, McGraw-Hill, 1966.
[Str92] N. P. Strickland, On the p-adic interpolation of stable homotopy groups*
*, Adams
Memorial Symposium on Algebraic Topology Volume 2 (Nigel Ray and Grant *
*Walker,
eds.), 1992, London Mathematical Society Lecture Notes 176, pp. 45-54.
[Str96] N. P. Strickland, Products on MU-modules, 34 pp., Submitted to Trans. A*
*mer. Math.
Soc., 1996.
[SY95] K. Shimomura and A. Yabe, The homotopy groups ss*(L2S0), Topology 34 (1*
*995),
no. 2, 261-289.
[Wal84] F. Waldhausen, Algebraic K-theory of spaces, localization, and the chro*
*matic fil-
tration of stable homotopy theory, Algebraic Topology Aarhus 1982 (Ib M*
*adsen and
Robert Oliver, eds.), Lecture Notes in Mathematics, vol. 1051, Springer*
*-Verlag, 1984,
pp. 173-195.
Index
atomic, 24 K0(F), 81
Krull-Schmidt theorem, 60
BG, 48
Bousfield class, 32-33, 43 L, 6
Brown-Comenetz duality, 52-55, 90 L-complete, 8, 45, 85-90
B(X), 69 Landweber exactness, 5, 9-14
len, 63, 81
Cf, 44 Lf, 44
bC, 6 bL, 6, 44-45
chromatic splitting, 82 local homology, 83-85
cohomology functor, 50 localising subcategory, 6, 38, 39, 43
coideal, 6
colocalising subcategory, 6, 38, 39, 43M, 6, 40
completion, 82-90 MASS, 15
cophantom, 51, 55 minimal weak colimit, 7, 10, 50
cosupport, 36, 82 MMU , 4
D, 46 monochromatic,m6,o40-41nogenic, 7
DE, 5 Moore spectrum, 21, 26, 28, 53, 82
deg, 68 MU, 4, 10
DMU , 4, 22 -spectrum, 24
dualisable, 7, 34, 46-48, 57-64 dMU, 4
E, 4, 5, 7-9, 15-18, 45 mwlim-!, 7, 50
E(n), 4
even finite spectrum, 9 nilpotent
evenly generated, 9-12 map, 36, 57
E_*X, 45 spectrum, 6, 34, 39, 42, 64-67
topologically, 57-60
F, 46 Noetherian, 15-18
F(m), 6
F-small, 7, 34, 47 P, 67
finite, 7, 34, 46, 47, 62 P0, 69
finite-dimensional, 90 phantom, 5, 9, 11, 30, 51, 55
Fn, 30 ss(P; u), 70
Pic, 67
generalised Moore spectrum, 91 Pic0, 69
good (P; u) self map, 70 Picard group, 52, 63, 67-74
good vn element, 24 Pic0, 69
good vn self map, 22, 27, 38, 57 P0, 69
Pro(F), 29-31
Haar measure, 76, 79 pro-free, 88-90
harmonic, 94
holim-!, 7 representable, 51, 52
holim,-7
homology functor, 50-52 S,s4,e5miring, 62-64
ideal, 6, 42, 43, 61, 62 Seq(F), 29
bI, 52, 54 *, 15
In, 4 (k)*,*16
injective b , 15
comodule, 14 S=I, 26, 28, 31
spectrum, 14, 19-21, 34 small, 7, 34, 46, 50, 62, 90-94
invariant prime, 82 smashing, 32, 37
spectral sequence
K, 4 Adams, 14, 15, 34, 35
K, 6 universal coefficient, 14
K(n), 4 stable homotopy category, 6-7
97
98 M. HOVEY AND N. P. STRICKLAND
algebraic, 7, 41
monogenic, 7, 33
strongly central, 21, 22, 24, 25
element, 24
support, 36, 82
T(m), 6
telescope conjecture, 37
thick, 6, 36, 37
Tn, 30
Tow(F), 29
tower, 29, 30, 32
width, 90
wk, 4
MORAVA K-THEORIES AND LOCALISATION 99
MIT, Cambridge, MA
E-mail address: hovey@math.mit.edu
Trinity College, Cambridge CB2 1TQ, UK
E-mail address: neil@dpmms.cam.ac.uk
~~