= : i2I
Consequently, The composition
:K (n) ^ _S|ei|! K(n) ^ K(n) ^ X:*
* ! K(n) ^ X
i2I
Proof: The proposition follows from the next two results of _
Ravenel [15]. _|_| is then a weak equivalence. |_|
PropositionL1.2.et v : kX ! X be a self mapPofraospectrumpositionF1.5.or any tw*
*o spectra X and Y , the natural map
X. Let X=vX and v-1X denote the cofiber of v and the colimit of
the sequence (1.5.1)K(n)*X K(n)*K(n)*Y ! K(n)*X ^*
* Y
:::-k|v|X v-!-(k+1)|v|X ! ::: is an equivalence.
12 Hopkins and SmithNilpotence II *
* 13
Proof: Consider the map (1.5.1) as a transformationPofrfunc-oof: Since 1 2 ss*
**E is non-nilpotent, for some prime p and
tors of Y . The left side satisfies the Eilenberg-Steenrodfaxiomsor some n 1,
since K(n)*Y is a flat (in fact free) K(n)*-module. The rightKside(n)*E 6= 0:
satisfies the Eilenberg-Steenrod axioms by definition. The trans-
formation is an isomorphism when Y is the sphere,Shenceifornallce K(n) and E ar*
*e both fields, it follows from Lemma 1.7 that
Y . _|_| K(n)o^fEsisubothsapwedgeeofnsuspensi*
*onssofiK(n)oandnaswedgeof E. In particular, E is a retract of a wedge of
suspensions of K(n). The result_ther*
*efore follows from the next
Proof of Corollary 5: If for some n, K(n)*Ep=r0,othenpEodoessition. |_|
not detect the nonnilpotent map
: S0! K(n): PropositionL1.9.et M have the homoto*
*py type of a wedge of
suspensions of K(n) (fixed n). If E *
*is a retract of M, then E itself
If K(n)*E 6= 0, then by Proposition 1.4 has the homotopy type of a wedge of *
*suspensions of K(n).
E*ff = 0 ) K(n)*ff = 0;
so the result reduces to Theorem 3. _|_| Lemmae1.10.Thewhomotopyihomomorphism*
*cinducedzbymtheaHur-p
Propositions 1.4 and 1.5 portray the Morava K-theories as^being1M : M S0^ M *
*! K(n) ^ M
a lot like fields. One formulation of Theoremi3sisathaththeyoaremtheomorphism o*
*f K(n) -modules.
prime fields of the category of spectra. *
* *
A (skew) field is a ring, all of whose modules are free.
Proof: The map in question is a we*
*dge of suspensions of the
map
DefinitionA1.6.ring spectrum E is a field if E*X isjaRfree:E*-K(n) S0^ K(n) ! *
*K(n) ^ K(n);
module for all spectra X.
so it suffices to prove the claim wh*
*en M is K(n). In this case the
This property also admits a geometric expression.result is a consequence of t*
*he formula [16],
jR(vn) = vn: _|_|
Lemma 1.7.If E is a field, then E ^X has the homotopy type of
a wedge of suspensions of E
Proof: This is very similar to the proof ofL1.7.e_|_|mmaL1.11.et f : M ! N be*
* a map of wedges of suspensions
of K(n). The homotopy homomorphism
ss*f : ss*M ! ss*N
PropositionL1.8.et E be a field. Then E has the homotopy type
of a wedge of suspensions of K(n) for some n.is a map of K(n)*-modules.
14 Hopkins and SmithNilpotence II *
* 15
Proof: Consider the following commutative1diagram:.4.Morava K-theories and du*
*ality
M -f---!N We will often use the device of repl*
*acing a self map of a finite
?? ? spectrum n
y ?y f : X ! X
1^f with its Spanier-Whitehead dual
K (n) ^-M---!K(n) ^ N:
n! X ^ D*
*X;
The right vertical arrow is the inclusion of a wedge summandDsincef : S
N admits the structure of a K(n)-module spectrum.aItmthereforeap from the n-sph*
*ere to the ring spectrum X ^ DX. If V is
suffices to prove that the composition inducestahmapeoffK(n)*-inite dimensional*
* K(n) vector space K(n) X, then the ring
modules. The left vertical arrow does by LemmaK1.10(andnthe) (X ^ DX) is natura*
*lly i*somorphic to the *ring
bottom horizontal arrow is a map of K(n)-module spectra. _|_|*
V V * End(V ):
Proof of Proposition 1.9: Since M has the homotopy type of a
wedge of suspensions of K(n)'s, it can be givenTthehstructureeofeaffect in Mora*
*va K-theory of the duality map
K(n)-module spectrum. Let i : E ! M, and p : M ! E be the flip dual*
*ity0
inclusion and retraction mappings respectively. By LemmaX1.11,^ DX !DX ^ X ! S
the composite i O p induces a homomorphism ofiK(n)*-moduless to send an endomor*
*phism to its trace (in the graded sense). Let
* *
* *
ss*M ! ss*M: {ei} V be a basis of V , and {ei} *
*V the corresponding dual
basis. The effect of the other duali*
*ty map
Choose a basis {ei} of the image of this map, and represent0it by
_ S ! X ^ DX
S|ei|! M: is to send 1 2 K(n) P * *
*to ei ei2 V V *
*. In particular,
The map
i_ j Lemma 1.12.The duality map S0! X^DX *
*induces a non-zero
N = K(n) ^ S|ei|! K(n) ^ M ! M homomorphism in K(n)-homology if and*
* only if K(n)*X 6= 0.
then gives rise to an isomorphism
ss*N imagesofs*(i O p); 2.Proofs of Theorems 3 and 7
since it sends the obvious basis of ss*N toStheobasism{ei}.eTheof the condition*
*s in Theorem 3 require the case n = 1, and
composite some of them don't. When the target *
*spectrum is finite, the case
N ! M p-!E n = 1 is superfluous.
is the desired homotopy equivalence. _|_|
Lemma 2.1.Let X and Y be finite spec*
*tra. For m 0
16 Hopkins and SmithNilpotence II *
* 17
i)K(m)*X HFp*X K(m)* The map
(1): S0! T
ii)K(m)*Y HFp*Y K(m)* f
iii)K(m)*f = HFp*f 1K(m)*for every f:: X !cYan be thought of as the infinite s*
*mash product of f.
*
* 0
Proof: This follows from the Atiyah-HirzebruchLspectralese-mmaL2.3.et E be a *
*ring spectrum with unit : S ! E. The
quence, using the fact that |vm| ! 1 as m !f1.o_|_|llowing are equivalent:
i)E ^ T is contractible;
(1): S0! E ^*
* T is null;
CorollaryI2.2.f f is either a self-map of a finiteispectrumior) ^ f
an element in the homotopy of a finite ring spectrum,itheifollowingi) ^ f(n): S*
*0! E ^ X(n)is null for n 0;
are equivalent: (n) 0 (n)
iv)1E^ f : E E ^ S ! E ^ X is nu*
*ll for n 0.
i)K(m)*f is nilpotent for m 0
ii)HFp*f is nilpotent: Proof: i))ii) and iv))i) are immed*
*iate. Since
If |f| 6= 0 then both of these conditions hold. (n)
lim-!E ^ X E ^ T;
Proof: If |f| 6= 0 then, from dimensionalaconsiderations,nd since homotopy gr*
*oups commute with direct limits, a null ho-
motopy of S0! E ^ T must occur at so*
*me S0! E ^ X(n)for
HFp*fi= 0 for i 0. n 0. This gives ii)) iii). The impl*
*ication iii))iv) is the only
*
* 0! E ^ X(n)is null
It then follows from 2.1 that K(m)*fi= 0 foroi;mne0.rWhenequiringtEhtoebenasrin*
*gospectrum.iIfsSthe first map in the following factorization of 1 ^ f(n)@
|f| = 0, part (3) of 2.1 applies to every power of f. The result *
* E
follows easily from this. _|_| E ^ S0! E ^ E ^ X(n)! E ^ X(*
*n)
Let f : S0! X be a map of spectra. Consider the homotopy _
direct limit T of the sequence This completes the proof. |_|
(2.2.1)S0! X ! X ^ X ! X ^ X ^ X ! :::; Proof of Theorem 3: Part i) follow*
*s from part iii), since the
in which the map X(n)! X(n+1)is given by iteratedfmultiplicationofactorslthro*
*ughliteratedosmashing.wPartsii)from part i) since multiplication in the @
f ^ 1X(n): X(n) S0^ X(n)! X(n+1): ss
*X ^ DX andK(n)*X ^ DX
The n-fold composition
corresponds, under Spanier-Whitehead*
* duality, to composition in
S0! ...! X(n)
[X;X]* andEndK(n)*(K(n)*X):
is the iterated smash product
Replacing
f(n)= f ^ ...^ f: f : F ! X
18 Hopkins and SmithNilpotence II *
* 19
with ii)A map
Df : S0! DF ^ X f : F ! X
in part iii) if necessary, we may assume thathFa=sS0.tThehresulte property that
reduces to Theorem 2 once it is shown that
f(m)^ 1Z: F(m)^ Z ! X(m)^ Z
1BP^ f(m)
is null for m 0. From Lemma 2.3 (with the obviousinotation)s null for m 0 if *
*and only if
this is equivalent to showing that BP ^ T is contractible.KIn(viewn)*(f ^ 1Z) =*
* 0
of the Bousfield equivalence
for all 0 n 1.
:
it is enough to show that P(n)^T is contractiblePforrnoo0.fAgain: In part i), t*
*he map f(m)is already null for m 0 by
from 2.3 this equivalent to showing that part iii) of Theorem 3. The only if *
*part of ii) is clear. Letting E
S0! P(n) ^ T be the ring spectrum Z ^ DZ and repl*
*acing
is null for n 0. Now let n grow to infinity. Since f ^ 1Z: F ^ Z ! X ^ Z
lim-!P(n) HFp; with its Spanier-Whitehead dual
the map F ! X ^ Z ^ DZ
S0! lim-!P(n) ^ T
*
* _
is null by assumption. Since homotopy commutesrwithedirectdlim-uces the if part*
* to i). |_|
its, the nullhomotopy arises at some
S0! P(n) ^ T: Lemma 2.5.Let X f-!Y g-!Z be a seque*
*nce of maps. The map
This completes the proof of Theorem 3. _|_|Cf! CgOfinduced by g gives rise to a*
* cofibration
The proof of Theorem 7 requires a slight modification ofCthef! CgOf! Cg:
third assertion of Theorem 3, and a useful cofibration (2.6).
Proof: Consider the following diag*
*ram in which the rows and
CorollaryL2.4.et F and Z be finite spectra,cEoalringuspectrum,mns are cofibrati*
*ons:
and X an arbitrary spectrum.
f
i)If a map f : F ! X ^ E satisfies X----!Yf----!Cfl??
K(n)*(f) = 0 for0all n 1; flfl g?y ?y
then for m 0, the composite X-gOf---!Z----!CgOf
(m) 1^ ?? ??
F(m)f---!(X ^ E)(m) X(m)^ E(m)--!X(m)^ E y y
is null. Cg----!?
20 Hopkins and SmithNilpotence II *
* 21
The upper right square is a pushout. It followssthatithenbottomce Cl(Y ) Cl(X)*
*. Part ii) of Corollary 2.4 then gives that
arrow is a homotopy equivalence. This completes1theYproof.^_|_|f(m)is null for *
*m 0. This means that
Y ^ Cf(m) Y _ (Y ^ F(m)) for*
* m;0
CorollaryL2.6.et f : X ! Y and g : Z ! W be two maps. *
* _
There is a cofibration so Y 2 C. This completes the proof o*
*f Theorem 7. |_|
X ^ Cg! Cf^g! Cf^ G:
Proof: Apply the lemma to the factorization3.vnself-maps
f ^ g = f ^ 1YO 1X^ g: _|_| The purpose of the next two sections*
* is to establish Theorem 9.
The "only if" part, that X 62 Cnimpl*
*ies that X does not admit a
vnself-map is easy: if for some j < *
*n K(j)*X 6= 0, and if v is a vn
self-map, then the cofiber Y of v is*
* a finite spectrum satisfying
Proof of Theorem 7: It suffices to establish
K(n)*Y= 0
(2.6.1) if X 2 C and X 2 Cnthen Cn C, K(j)*Y6= 0;
for it then follows that C = Cm, where
contradicting the fact that Cn Cj. T*
*he proof of the "if" part
m = min{n | Cn:C} falls into two steps. In this sectio*
*n it is shown that the property
of admitting a vnself-map is generic*
*. It then remains to construct
Since everything has been localized at p, setfor each n, a spectrum Xnwith a vn*
*self-map. This is done in
Cl(X) = {n 2 N | K(n)*(X) 6= 0}: sectionF4.or any spectrum X, the ele*
*ment p 2 [X;X]
*
* *is a v0self-map
With this notation, (2.6.1) becomes: satisfying condition (*) of Theorem *
*9. We therefore need only
consider vnself-maps when n 1. Beca*
*use of this, unless otherwise
(2.6.2if)X 2 C and Cl(Y ) Cl(X), then Y 2 C.mentioned, in this and the next se*
*ction, we will work entirely in the
category C1.
Suppose, then, that X 2 C. Then so is Z ^ XAforsanymZe2nC0.tioned in section *
*1.4 a self-map
Let f : F ! S0be the fiber of the duality map S0! X ^ DX.
Then Y ^Cf2 C. Setting g = f(m-1)in Corollary 2.6 and smashingkF ! F
with the identity map of Y gives a cofibration
Y ^ F ^ Cf(m-1)! Y ^ Cf(m)! Y ^ Cf^ F(m-1):oftaofiniteaspectrummcorresponds*
*,aunderpSpanier-Whiteheadfduality,rom the k-sphere to the ring spectrum
It follows that Y ^ Cf(m)2 C for all m.
By 1.12, K(n)*f 6= 0 if and only if n =2Cl(X), so that R = F ^ DF:
K(n)*(1Y^ f) = 0 for all n,
22 Hopkins and SmithNilpotence II *
* 23
DefinitionL3.1.et R be a finite ring spectrum,Pnr>o0:oAnfele-: Since we are wor*
*king over a Z(p)-algebra it follows that
ment
ff 2 ss*R pk(x - y) = 0
is a vn-element if for some k. The result now follows b*
*y expanding
(
N pN
K(m)*ff is a unitif m = n xp = (y + (x - y))
nilpotentotherwise:
using the binomial theorem. _|_|
Lemma 3.2.Let R be a finite ring spectrum, and ff 2 ss*R a vn-
element. There exist integers i and j such that
( Lemmae3.5.LetlRebemaefinitenringtspe*
*ctrum,.andFffo2rsskRsaovn-me i > 0, ffiis in the center of ss
*
* *R.
K(m)*ffi = 0 if m 6= n
vjnif m = n
Proof: Raising ff to a power, if n*
*ecessary, we may assume that
K(m)*ff is in the center of K(m)*R f*
*or all m. Let l(ff) and r(ff)
Proof: It follows from lemma 2.2 that HFp*ffbisenilpotent.tRais-he elements o*
*f End(ss R) given by left and right multiplication
ing ff to a power, if necessary, we may supposebthatyHFp*fff=f0..ItSince R 2 C *
*the diff*erence l(ff)-r(ff) has finite order. Since
then follows from lemma 2.2 that K(m)*ff = 0 for all but finitely 1
many m. Raising ff to a further power, if necessary,Kit(canmbe)*(l(ff) -=r(ff))*
*0 for all;m
arranged that K(m)*ff = 0 for m 6= n.
The assertion K(n)*ffi= vjnis equivalent tolthe(assertionfthatf)-r(ff) is nil*
*potent by Theorem 3. The result now follows from
ffi= 1 2 K(n)*R=(vn- 1). The ring K(n)*R=(vn-31).has4a. _|_|
finite group of units, and so i can be taken to be the order of this
group. _|_|
Lemma 3.6.Let ff;fi 2 ss*R be vn-ele*
*ments. There exist integers
i and j with ffi= fij.
CorollaryI3.3.f f : kF ! F is a vnself-map, then there exist
integers i;j with the property that
( Proof: Raising ff and fi to powers*
* if necessary, we may assume
K(m)*fi = 0 if m 6=_n|_| that K(m)*(ff_-=fi)0 for all m. The *
*result follows, as above,
multiplicationibyfvjnm:= n from 3.4. |_|
Lemma 3.4.Suppose that x and y are commutingCelementsoofraollaryI3.7.f f and g *
*are two v self-maps of F, then fiis
Z(p)-algebra. If x-y is both torsion and nilpotent,hthenoformNot0,opic to gjfor*
* some i annd j. _| |
*
* _
N pN
xp = y :
24 Hopkins and SmithNilpotence II *
* 25
CorollaryS3.8.uppose X and Y have vnself-mapsNvXoandwvY.suppose that Y is a ret*
*ract of X, and let i : Y ! X and
There are integers i and j so that for everypZ:andXevery! Y be the inclusion an*
*d retraction mappings respectively.
Choose a vnself-map v of X which com*
*mutes with iOp. The map
f : Z ^ X ! Y
p O v O i
the following diagram commutes: is easily checked to be a v *
* _
nself-map*
* of Y . |_|
Z ^ Xf----!Y
? ?
1^vXi?y ?yvYj Corollary 3.10.The full subcategory *
*of C1consisting of spectra
admitting a vnself-map satisfying co*
*ndition (*) of Theorem 9 is
Z ^ Xf----!Y: thick.
Proof: This is similar to 3.9, and*
* involves checking that the
Proof: The spectrum DX ^Y has two vnself-maps:iDvX^1Yntegers which arise in 3*
*.6-3.8 are powers of p. In fact, the only
and 1DX^vY. By Corollary 3.7 there are integerspilandajcforewhichwhere an integ*
*er which is not a power of p comes up is in using
DvXi^1Yis homotopic to 1DX^vYj. The result now3follows.from7 to arrange that K(*
*m) v is in the center of End (K(m) X).
Spanier-Whitehead duality. _|_| But this is guaranteed at the outset*
* b*y condition (*). _|_| K(m)* *
CorollaryT3.9.he full subcategory of C1consisting4of.spectraCad-onstruction of *
*vnself-maps
mitting a vnself-map is thick.
4.1.Preliminaries
Proof: Call the subcategory in question C.TNotehthateXe2xCaifmples of self-ma*
*ps needed for the proof of Theorem 9 are
and only if X 2 C. To check that C is closedcunderocofibrationsnstructed using *
*the Adams spectral sequence
it therefore suffices to show that if
Exts;tA[H*Y;H*X] ) [X;Y ]t*
*-s
(3.9.1) X ! Y ! Z
which relates the mod p cohomology o*
*f X and Y as modules over
is a cofibration with X and Y in C, then Z istinhC.eUsingSCorol-teenrod algebra*
* to [X;Y ]*. The spectral sequence is usually
lary 3.8 choose the vnself-maps vX and vY ofdXiandsYpsolthatayed in the (t-s;s)*
*-plane, so that the groups lying in a given
vertical line assemble to a single h*
*omotopy group. With this con-
kX ----!kY----!kZ vention the "filtration jumps" are v*
*ertical in the sense that the
? ? difference between two maps represen*
*ting the same class in
vX?y vY?y
Exts;tA[H*Y;H*X]
X ----!Y ----!Z:
represents a class in
commutes. The induced map vZ: kZ ! Z is easily seen to be a 0 0
vnself-map. ExtsA;t[H*Y;H*X];
26 Hopkins and SmithNilpotence II *
* 27
with s0> s, and t - s = t0- s0. using the long exact sequence coming*
* from
There are many criteria for convergence of the Adams_spectral
sequence. A simple one, which is enough for the presentCpurpose==B M ae C==B M*
* i M:
is [1] Recall that for p = 2, the dual St*
*eenrod algebra is
Lemma 4.1.If X a finite spectrum and Y is a connective spec-A*= F2[1;2;:::]
trum with the property that each sskY is a finite abelian p-group,i
then the Adams spectral sequence converges strongly to |i| = 2 - 1
and for p odd
[X;Y ]*:
A*= [o0;o1;:::] Fp[1;2;:::]
If B C are Hopf-algebras over a field k, the forgetful functor|o i
i| = 2p - 1
C-modules! B-modules |i| = 2(pi- 1):
has both a left and a right adjoint. The leftTadjointhe subalgebra of the Steen*
*rod algebra generated by
1;:::;Sq2nwhen p = 2
M 7! C M Sq
B fi;P1;:::;Pn-1when p is odd, a*
*nd n 6= 1
carries projectives to projectives, and so prolongs tofaichangewofhen p is odd *
*and n = 0
rings isomorphism is denoted An. It is the finite sub *
*Hopf-algebra which is annihilated
by the ideal
(4.1.1) Ext*C[C M;N] Ext*B[M;N]:
B (2n+11;2n2;:::;n+1;n+2;:::)p *
*= 2
When M is a C-module this can be combined with the "shearing(pn1;:::;n;n+1;on+1*
*;:::)p 6= 2:
isomorphism"
The augmentation ideal of A==Anis 2p*
*n(p-1)-connected. The fact
C M! C==B M C==B = C k that the connectivity goes to infini*
*ty with n plays an important role
B B in the Approximation Lemma 4.5.
X It is customary to give the dual S*
*teenrod algebra the basis of
c m7! c0i c00im
X monomials in the 's and o's. With th*
*is convention, the Adams-
(c)= c0i c00i; Margolis elements are
sdual tops(s < t)
to give another "change of rings" isomorphism Pt t
ae
Ext*C[C==B M;N] Ext*B[M;N]: Qndual toonpnodd+1p = 2:
The difference between ExtCand ExtBcan thereforeEbeameasuredch Qnis primitive, *
*and together they generate an exterior
by the augmentation ideal sub Hopf-algebra of the Steenrod alg*
*ebra. The Pstall satisfy
___
C==B= ker{ffl : C==B;! k} (Pst)p= 0;
28 Hopkins and SmithNilpotence II *
* 29
but are primitive only when s = 0. The Adams-MargolisTelementsheoremI4.3.f M is*
* a connective A-module with
are naturally ordered by degree
H(M;d) = 0 for|d| n;
|Pst|= 2ps(pt- 1) then
|Qn|= 2pn- 1: Ext*;*A[M;Fp]
has a vanishing line of slope 1=n. *
*_|_|
In general, there is no way to pre*
*dict the intercept of the van-
4.2.Vanishing lines ishing line, but there is the follow*
*ing:
Given an A-module M, and an Adams-Margolis element d, the
Margolis homology of M, H(M;d), is the homologyPofrtheocomplexpositionS4.4.uppo*
*se that M is a connective A-module, and
(M*;d*), with that y = mx + b
Mn= M n 2 Z is a vanishing line for Ext*;*A[M;Fp*
*]. If N is a (c - 1)-connected
d2n= d A-module, then
( p-1 s y = m(x - c) + b
d2n+1= d ifd = Pt is a vanishing line for
d ifd = Qn:
Ext*;*A[M N;Fp]:
When X is a spectrum the symbol H(X;d) will be used to denote
H(H*X;d). The Margolis homology groups are periodic of period k
1 if p even, or if d = Qn, and are periodic ofPperiodr2ootherwise.of: Let N be *
*the quotient of N by the elements of degree
greater than k, and Nkj Nkthe submod*
*ule consisting of elements
of degree > j. There is an exact seq*
*uence
DefinitionL4.2.et M be an A-module. A line Nk k j
j! N ! N :
y = mx + b Since M is connective,
is a vanishing line of M N = lim-!M Nk
Ext*;*A[M;Fp] k
and
if Exts;tA[M N;Fp] = limExts;t[M*
* Nk;F ];
-! A *
* p
Exts;tA[M;Fp] = 0 fors > m(t - s) + b: k
so it suffices to prove the result f*
*or each Nk. This is trivial for
The following result, due to Anderson-Davisk[2]