=

:
Consequently,

:
Proof: The proposition follows from the next two results of
Ravenel [15]. __|_ |
Proposition 1.2. Let v : kX ! X be a self map of a spectrum
X. Let X=vX and v-1 X denote the cofiber of v and the colimit of
the sequence
: :-:k|v|X -v!-(k+1)|v|X ! : : :
Nilpotence II 11
respectively. Then there is an equality of Bousfield classes

:
it is enough to show that P (n) ^ T is contractible for n 0. Again
from 2.3 this equivalent to showing that
S0 ! P (n) ^ T
is null for n 0. Now let n grow to infinity. Since
lim-!P (n) HFp ;
the map
S0 ! lim-!P (n) ^ T
is null by assumption. Since homotopy commutes with direct lim-
its, the nullhomotopy arises at some
S0 ! P (n) ^ T:
This completes the proof of Theorem 3. __|_ |
The proof of Theorem 7 requires a slight modification of the
third assertion of Theorem 3, and a useful cofibration (2.6).
Corollary 2.4. Let F and Z be finite spectra, E a ring spectrum,
and X an arbitrary spectrum.
i) If a map f : F ! X ^ E satisfies
K(n)*(f) = 0 for all0 n 1;
then for m 0, the composite
(m) 1^
F (m)-f--! (X ^ E)(m) X(m) ^ E(m) --! X(m) ^ E
is null.
Nilpotence II 19
ii)A map
f : F ! X
has the property that
f(m) ^ 1Z : F (m)^ Z ! X(m) ^ Z
is null for m 0 if and only if
K(n)*(f ^ 1Z ) = 0
for all 0 n 1.
Proof: In part i), the map f(m) is already null for m 0 by
part iii) of Theorem 3. The only if part of ii) is clear. Letting E
be the ring spectrum Z ^ DZ and replacing
f ^ 1Z : F ^ Z ! X ^ Z
with its Spanier-Whitehead dual
F ! X ^ Z ^ DZ
reduces the if part to i). __|_ |
Lemma 2.5. Let X -f!Y -g!Z be a sequence of maps. The map
Cf ! CgOf induced by g gives rise to a cofibration
Cf ! CgOf ! Cg:
Proof: Consider the following diagram in which the rows and
columns are cofibrations:
X ---f-! Y ----! Cf
flfl ? ?
fl g?y ?y
X --gOf--!Z ----! CgOf
?? ?
y ?y
Cg ----! ?
Nilpotence II 20
The upper right square is a pushout. It follows that the bottom
arrow is a homotopy equivalence. This completes the proof. __|_ |
Corollary 2.6. Let f : X ! Y and g : Z ! W be two maps.
There is a cofibration
X ^ Cg ! Cf^g ! Cf ^ G:
Proof: Apply the lemma to the factorization
f ^ g = f ^ 1Y O 1X ^ g: __|_ |
Proof of Theorem 7: It suffices to establish
(2.6.1) if X 2 C and X 2 Cn then Cn C,
for it then follows that C = Cm , where
m = min {n | Cn C }:
Since everything has been localized at p, set
Cl(X) = {n 2 N | K(n)*(X) 6= 0}:
With this notation, (2.6.1) becomes:
(2.6.2) if X 2 C and Cl(Y ) Cl(X), then Y 2 C.
Suppose, then, that X 2 C. Then so is Z ^ X for any Z 2 C0.
Let f : F ! S0 be the fiber of the duality map S0 ! X ^ DX.
Then Y ^Cf 2 C. Setting g = f(m-1) in Corollary 2.6 and smashing
with the identity map of Y gives a cofibration
Y ^ F ^ Cf(m-1)! Y ^ Cf(m)! Y ^ Cf ^ F (m-1):
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
K(n)*(1Y ^ f) = 0 for all n,
Nilpotence II 21
since Cl(Y ) Cl(X). Part ii) of Corollary 2.4 then gives that
1Y ^ f(m) is null for m 0. This means that
Y ^ Cf(m) Y _ (Y ^ F (m)) for m 0 ;
so Y 2 C. This completes the proof of Theorem 7. __|_ |
3. vn self-maps
The purpose of the next two sections is to establish Theorem 9.
The "only if" part, that X 62 Cn implies that X does not admit a
vn self-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
K(n)*Y = 0
K(j)*Y 6= 0;
contradicting the fact that Cn Cj. The proof of the "if" part
falls into two steps. In this section it is shown that the property
of admitting a vn self-map is generic. It then remains to construct
for each n, a spectrum Xn with a vn self-map. This is done in
section 4.
For any spectrum X, the element p 2 [X; X]* is a v0 self-map
satisfying condition (*) of Theorem 9. We therefore need only
consider vn self-maps when n 1. Because of this, unless otherwise
mentioned, in this and the next section, we will work entirely in the
category C1.
As mentioned in section 1.4 a self-map
kF ! F
of a finite spectrum corresponds, under Spanier-Whitehead duality,
to a map from the k-sphere to the ring spectrum
R = F ^ DF:
Nilpotence II 22
Definition 3.1. Let R be a finite ring spectrum, n > 0: An ele-
ment
ff 2 ss*R
is a vn-element if
(
K(m)*ff is a unit if m = n
nilpotent otherwise:
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
(
K(m)*ffi = 0 if m 6= n
vjn if m = n
Proof: It follows from lemma 2.2 that HFp *ff is nilpotent. Rais-
ing ff to a power, if necessary, we may suppose that HFp *ff = 0. It
then follows from lemma 2.2 that K(m)*ff = 0 for all but finitely
many m. Raising ff to a further power, if necessary, it can be
arranged that K(m)*ff = 0 for m 6= n.
The assertion K(n)*ffi = vjnis equivalent to the assertion that
ffi = 1 2 K(n)*R=(vn - 1). The ring K(n)*R=(vn - 1) has a
finite group of units, and so i can be taken to be the order of this
group. __|_ |
Corollary 3.3. If f : kF ! F is a vn self-map, then there exist
integers i; j with the property that
(
K(m)*fi = 0 if m 6= n __|_ |
multiplication by vjnif m = n:
Lemma 3.4. Suppose that x and y are commuting elements of a
Z(p)-algebra. If x-y is both torsion and nilpotent, then for N 0,
N pN
xp = y :
Nilpotence II 23
Proof: Since we are working over a Z(p)-algebra it follows that
pk(x - y) = 0
for some k. The result now follows by expanding
N pN
xp = (y + (x - y))
using the binomial theorem. __|_ |
Lemma 3.5. Let R be a finite ring spectrum, and ff 2 sskR a vn-
element. For some i > 0, ffi is in the center of ss*R.
Proof: Raising ff to a power, if necessary, we may assume that
K(m)*ff is in the center of K(m)*R for all m. Let l(ff) and r(ff)
be the elements of End (ss*R) given by left and right multiplication
by ff. Since R 2 C1 the difference l(ff)-r(ff) has finite order. Since
K(m)* (l(ff) - r(ff))= 0 for all m;
l(ff) - r(ff) is nilpotent by Theorem 3. The result now follows from
3.4. __|_ |
Lemma 3.6. Let ff; fi 2 ss*R be vn-elements. There exist integers
i and j with ffi = fij.
Proof: Raising ff and fi to powers if necessary, we may assume
that K(m)* (ff - fi)= 0 for all m. The result follows, as above,
from 3.4. __|_ |
Corollary 3.7. If f and g are two vn self-maps of F , then fi is
homotopic to gj for some i and j. __|_ |
Nilpotence II 24
Corollary 3.8. Suppose X and Y have vn self-maps vX and vY .
There are integers i and j so that for every Z and every
f : Z ^ X ! Y
the following diagram commutes:
Z ^ X --f--! Y
? ?
1^vXi?y ?yvYj
Z ^ X --f--! Y:
Proof: The spectrum DX ^ Y has two vn self-maps: DvX ^ 1Y
and 1DX ^vY . By Corollary 3.7 there are integers i and j for which
DvX i^1Y is homotopic to 1DX ^vY j. The result now follows from
Spanier-Whitehead duality. __|_ |
Corollary 3.9. The full subcategory of C1 consisting of spectra ad-
mitting a vn self-map is thick.
Proof: Call the subcategory in question C. Note that X 2 C if
and only if X 2 C. To check that C is closed under cofibrations
it therefore suffices to show that if
(3.9.1) X ! Y ! Z
is a cofibration with X and Y in C, then Z is in C. Using Corol-
lary 3.8 choose the vn self-maps vX and vY of X and Y so that
kX - ---! kY ----! kZ
? ?
vX?y vY?y
X - ---! Y ----! Z:
commutes. The induced map vZ : kZ ! Z is easily seen to be a
vn self-map.
Nilpotence II 25
Now suppose that Y is a retract of X, and let i : Y ! X and
p : X ! Y be the inclusion and retraction mappings respectively.
Choose a vn self-map v of X which commutes with i O p. The map
p O v O i
is easily checked to be a vn self-map of Y . __|_ |
Corollary 3.10. The full subcategory of C1 consisting of spectra
admitting a vn self-map satisfying condition (*) of Theorem 9 is
thick.
Proof: This is similar to 3.9, and involves checking that the
integers which arise in 3.6-3.8 are powers of p. In fact, the only
place where an integer which is not a power of p comes up is in using
3.7 to arrange that K(m)*v is in the center of End K(m)*(K(m)*X).
But this is guaranteed at the outset by condition (*). __|_ |
4. Construction of vn self-maps
4.1. Preliminaries
The examples of self-maps needed for the proof of Theorem 9 are
constructed using the Adams spectral sequence
Exts;tA[H*Y; H*X] ) [X; Y ]t-s
which relates the mod p cohomology of X and Y as modules over
the Steenrod algebra to [X; Y ]*. The spectral sequence is usually
displayed in the (t - s; s)-plane, so that the groups lying in a given
vertical line assemble to a single homotopy group. With this con-
vention the "filtration jumps" are vertical in the sense that the
difference between two maps representing the same class in
Ext s;tA[H*Y; H*X]
represents a class in
0;t0* *
ExtsA [H Y; H X];
Nilpotence II 26
with s0> s, and t - s = t0- s0.
There are many criteria for convergence of the Adams spectral
sequence. A simple one, which is enough for the present purpose
is [1]
Lemma 4.1. If X a finite spectrum and Y is a connective spec-
trum with the property that each sskY is a finite abelian p-group,
then the Adams spectral sequence converges strongly to
[X; Y ]*:
If B C are Hopf-algebras over a field k, the forgetful functor
C-modules ! B-modules
has both a left and a right adjoint. The left adjoint
M 7! C M
B
carries projectives to projectives, and so prolongs to a change of
rings isomorphism
(4.1.1) Ext*C[C M; N] Ext*B[M; N]:
B
When M is a C-module this can be combined with the "shearing
isomorphism"
C M ! C==B M C==B = C k
B B
X
c m 7! c0i c00im
X
(c)= c0i c00i;
to give another "change of rings" isomorphism
Ext *C[C==B M; N] Ext*B[M; N]:
The difference between ExtC and ExtB can therefore be measured
by the augmentation ideal
______
C==B = ker{ffl : C==B ! k};
Nilpotence II 27
using the long exact sequence coming from
______
C==B M ae C==B M i M:
Recall that for p = 2, the dual Steenrod algebra is
A* = F2[1; 2; : :]:
|i| = 2i- 1
and for p odd
A* = [o0; o1; : :]: Fp[1; 2; : :]:
|oi| = 2pi- 1
|i| = 2(pi- 1):
The subalgebra of the Steenrod algebra generated by
Sq1; : :;:Sq2n when p = 2
fi; P1; : :;:Pn-1when p is odd, and n 6= 1
fi when p is odd and n = 0
is denoted An. It is the finite sub Hopf-algebra which is annihilated
by the ideal
n+1 2n
(21 ; 2 ; : :;:n+1 ; n+2 ; : :):p = 2
n
(p1; : :;:n; n+1 ; on+1 ; : :):p 6= 2:
The augmentation ideal of A==An is 2pn(p-1)-connected. The fact
that the connectivity goes to infinity with n plays an important role
in the Approximation Lemma 4.5.
It is customary to give the dual Steenrod algebra the basis of
monomials in the 's and o's. With this convention, the Adams-
Margolis elements are
s
Ptsdual to pt (s < t)
ae
Qn dual to on p odd
n+1p = 2:
Each Qn is primitive, and together they generate an exterior
sub Hopf-algebra of the Steenrod algebra. The Ptsall satisfy
(Pts)p = 0;
Nilpotence II 28
but are primitive only when s = 0. The Adams-Margolis elements
are naturally ordered by degree
|Pts|= 2ps(pt- 1)
|Qn| = 2pn - 1:
4.2. Vanishing lines
Given an A-module M, and an Adams-Margolis element d, the
Margolis homology of M, H(M; d), is the homology of the complex
(M*; d*), with
Mn = M n 2 Z
d2n = d
( p-1 s
d2n+1 = d ifd = Pt
d ifd = Qn:
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
1 if p even, or if d = Qn, and are periodic of period 2 otherwise.
Definition 4.2. Let M be an A-module. A line
y = mx + b
is a vanishing line of
Ext*;*A[M; Fp]
if
Exts;tA[M; Fp] = 0 for s > m(t - s) + b:
The following result, due to Anderson-Davis [2] and to Miller-
Wilkerson [10] relates vanishing lines to Margolis homology groups.
It has not been stated in its strongest form.
Nilpotence II 29
Theorem 4.3. If M is a connective A-module with
H(M; d) = 0 for |d| n;
then
Ext*;*A[M; Fp]
has a vanishing line of slope 1=n. __|_ |
In general, there is no way to predict the intercept of the van-
ishing line, but there is the following:
Proposition 4.4. Suppose that M is a connective A-module, and
that
y = mx + b
is a vanishing line for Ext*;*A[M; Fp]. If N is a (c - 1)-connected
A-module, then
y = m(x - c) + b
is a vanishing line for
Ext*;*A[M N; Fp]:
Proof: Let Nk be the quotient of N by the elements of degree
greater than k, and Nkj Nk the submodule consisting of elements
of degree > j. There is an exact sequence
Nkj! Nk ! Nj:
Since M is connective,
M N = lim-!M Nk
k
and
Exts;tA[M N; Fp] = lim-!Exts;tA[M Nk ; Fp];
k
so it suffices to prove the result for each Nk . This is trivial for
k < c, so suppose k c, and by induction, that the result is true
for k0< k. Suppose that (s; t) satisfies
s > m(t - s - c) + b
Nilpotence II 30
and consider the exact sequence
M Nkk-1! M Nk ! M Nk-1 :
By induction,
Exts;tA[M Nk-1 ; Fp] = 0:
The module Nkk-1is just a sum of copies of kF p_the A-module
which consists of F p in degree k, and zero elsewhere. It follows
that
Exts;tA[M Nkk-1; Fp]
is a product of copies of
Exts;tA[M kF p; Fp] Exts;t-kA[M; Fp];
which is zero since
s > m(t - s - c) + b
> m((t - k) - s) + b: __|_ |
Lemma 4.5 (Approximation lemma). Let M be a connective
A-module, and suppose that Ext*;*A[M ; Fp] has a vanishing line of
slope m. Given b, for n 0 the restriction map
Exts;tA[M; Fp] ! Exts;tAn[M; Fp]
is an isomorphism when
s m(t - s) + b:
Proof: The result follows from the exact sequence
______
A==An M ae A==An M i M;
______
Proposition 4.4, and the fact that the connectivity of A==An can
be made arbitrarily large by taking n to be large. __|_ |
Nilpotence II 31
4.3. Morava K-theories and the Adams spectral sequence
We need to be able to examine the K(n)-Hurewicz homomorphism
from the point of view of the Adams spectral sequence. This can
be done, but it is a little easier to work with the connected cover
k(n) of K(n). The spectrum k(n) is a ring spectrum, with
k(n)* = Fp[vn] K(n)* = Fp[vn; v-1n]:
Lemma 4.6. The transformation k(n)*X ! K(n)*X extends to
a natural isomorphism
v-1nk(n)*X K(n)*X:
Proof: Since localization is exact, both sides satisfy the exact-
ness properties of a homology theory. They agree when X is the
sphere, hence for all X. __|_ |
Corollary 4.7. If k(n)*X is finite then K(n)*X = 0.
Proof: If k(n)*X is finite, then for j 0, k(n)jX = 0. This
means that for each x 2 k(n)*X, vmnx = 0 for m 0. The result
then follows from lemma 4.6. __|_ |
Since k(n) is a ring spectrum, the mod p cohomology H*k(n)
is a coalgebra over the Steenrod algebra. It has been calculated by
Baas and Madsen [5]
Proposition 4.8. As a coalgebra over the Steenrod algebra,
H*k(n) A==E[Qn]:
It follows that the E2-term of the Adams spectral sequence for
ss*k(n) ^ X is isomorphic to
Exts;tE[Qn][H*X; Fp];
and that the map of E2-terms induced by the Hurewicz homomor-
phism is the natural restriction. __|_ |
Nilpotence II 32
Corollary 4.9. If X is a finite spectrum and H(X; Qn) = 0, then
K(n)*X = 0.
Proof: The group
Ext *;*E[Qn][H*(X); Fp]
is the cohomology of the complex
H*X Qn!H*X Qn!H*X Qn!: : ::
This means that for s > 0, the graded abelian group
Exts;*E[Qn][H*X; Fp]
is isomorphic to the Margolis homology group H(X; Qn). The van-
ishing of these groups implies that
Ext *;*E[Qn][H*X; Fp] Ext*;0E[Qn][H*X; Fp] H*X
is finite, and hence that k(n)*X is finite. The result then follows
from Corollary 4.7. __|_ |
4.4. Examples of self maps
The key to constructing self-maps is the following result of the
second author [19]. An account appears in [14].
Theorem 4.10 (Smith). For each n = 1; 2; : :t:here is a finite
spectrum Xn satisfying
i) The Adams spectral sequence
Exts;tE[Qn][H*Xn ^ DXn; Fp] ) k(n)*Xn ^ DXn
collapses;
Nilpotence II 33
ii)The Margolis homology groups H(Xn ^ DXn; d) are zero if
|d| < |Qn|:
Theorem 4.11. The spectrum Xn is in Cn \ Cn-1 and has a vn
self-map satisfying (*) of Theorem 9.
The proof of Theorem 4.11 uses the Adams spectral sequence
and the following consequence of the results of Wilkerson [20]. The
proof is in the appendix to this section.
Theorem 4.12. Suppose that B C are finite, connected, graded,
cocommutative Hopf-algebras over a field k of characteristic p > 0.
If
b 2 Ext*;*B[k; k];
then for N 0, bpN is in the image of the restriction map
Ext*;*C[k; k] ! Ext*;*B[k; k]: __|_ |
Proof of Theorem 4.11: That Xn is in Cn \ Cn-1 follows from
Corollary 4.9.
For the construction of the self-map it is slightly cleaner to
work from the point of view of finite ring spectra. Thus let R be
the finite ring spectrum Xn ^ DXn. The ring ss*R is an algebra
over ss*S0, and the image of ss*S0 in ss*R is in the center (in the
graded sense). Similarly, if B A is a sub Hopf-algebra, the ring
Ext *;*B[H*R; Fp]
is a central algebra over Ext*;*B[F p; Fp].
To show that Xn admits a vn self-map satisfying condition (*)
of Theorem 9 it suffices to exhibit an element
v 2 ss*R
satisfying
M pN
(4.12.1) k(n)*vp = vn . 1; for some M; N > 0;
(4.12.2) the map k(m)*v is nilpotent when m 6= n.
Nilpotence II 34
Step 1: First to find an approximation to a vn self-map in the
E2-term of the Adams spectral sequence. Let
n-1
vn 2 Ext1;2pE[Qn][F p; Fp]
be the element represented by vn 2 k(n)*. We need to find a
N ;pN (2pn-1)*
w 2 ExtpA [H R; Fp]
restricting to vpNn. 1, for N 0. By 4.3, the bigraded group
Ext *;*A[H*R; Fp]
has a vanishing line of slope 1=2(pn - 1). Using the approximation
lemma, an integer n can be chosen for which the restriction map
(4.12.3) Ext s;tA[H*R; Fp] ! Exts;tAn[H*R; Fp]
is an isomorphism if
s > ____1____2(pn(-t1)- s):
By Theorem 4.12 there is an element
w"2 Ext*;*An[F p; Fp]
restricting to vpNn2 Ext*;*E[Qn][F p; Fp]. The class w can be taken to
be the image of "w. 1 under the isomorphism (4.12.3).
Step 2: This construction of the class w actually gives some-
thing more. Since
Ext *;*An[F p; Fp]
is in the center (in the graded sense) of
Ext*;*An[H*R; Fp];
the class w commutes with every
ff 2 Exts;tA[H*R; Fp]
with
(4.12.4) s ____1____2(pn(-t1)- s):
Nilpotence II 35
Step 3: Now to choose a power of w which survives the Adams
spectral sequence. The differentials in the Adams spectral sequence
are derivations, and the values of drw lie in the region (4.12.4). This
means that
dr-1w = 0 ) drwp = 0:
Sincebd1w = 0 it follows that dbwpb = 0. The possible values of
drwp for r > b lie in the region
s > ____1____2(pn(-t1)- s);
which is above the vanishing line. This means that the class wpb is
a permanent cycle.
Step 4: For simplicity, replaceNw with wpb, and adjust the in-
teger N so that w restricts to vpn . 1. Let
v 2 ss*R
be a representative of w. We willNsee that this is the desired class.
The difference k(n)*(v - vpn ) is represented by a class in
Ext s;tE[Qn][H*R; Fp]
with s > 1=2(pn - 1) (t - s). Some power of k(n)*(v - vpNnb) is
therefore represented by a class above the vanishing line of
Ext *;*E[Qn][H*R; Fp]
(which has slope 1=2(pn - 1)), and hence is zero. Lemma 3.4 then
gives that that
M pMN
k(n)*vp = vn M 0:
This proves property (4.12.1)
Property (4.12.2) is trivial when m < n, since R 2 Cn. When
m > n, it is a consequence of the fact that the Adams spectral
sequence k(m)*R has a vanishing line of slope 1=2(pm - 1), and
that the powers of v are represented by classes lying on the line
s = ____1____2(pn(-t1)- s)
which has a larger slope. This completes the proof. __|_ |
Nilpotence II 36
4.5. Proof that Theorem 7 implies the Nilpotence Theo-
rem
This subsection is included to satiate any curiosity aroused by the
claim made after the statement of Theorem 7. Since the argument
is not necessary for establishing any of the results of this paper, it
is included only as a sketch.
In [7, Section 1] the Nilpotence Theorem is reduced to showing
that if R is a connective, associative ring spectrum, and
ff 2 ss*R
is in the kernel of the MU-Hurewicz homomorphism then ff is
nilpotent. This in turn is easily reduced to the case when R is
localized at p and MU is replaced with BP . The case |ff| 0 is
easy, so it may be assumed that |ff| > 0.
Let __ffbe the map
(4.12.5) |ff|R ff^1!R ^ R ! R:
The map (4.12.5) induces multiplication by BP*ff = 0 in BP ho-
mology. This means that from the point of view of the Adams-
Novikov spectral sequence, composition withe __ffmoves the homo-
topy to the right along a line of positive slope.
Step 1: The construction used to produce the spectra Xn of
this section can be used to construct finite torsion free spectra Yn
with the property that H*Yn, as a module over An is free over
An==E, where E is the sub-Hopf-algebra
[Q0; : :;:Qn]:
See [19].
Step 2: Use the spectral sequence of [16, Theorem 4.4.3] to
show that
Exts;tBP*BP[BP*; BP*R ^ Yn]
has a vanishing line with slope tending to zero as n ! 1.
Nilpotence II 37
Step 3: It follows from the vanishing line that for n 0 the
spectrum
Yn ^ ff-1 R
is contractible.
Step 4: Now use Theorem 13 to conclude that Yn is Bousfield
equivalent to the sphere, hence that ff-1 R is contractible, hence
that ff is nilpotent.
5. Endomorphisms, up to nilpotents
5.1. N-endomorphisms
The vn self maps form an endomorphism (up to nilpotent elements)
of the category Cn. It turns out that these are the only endomor-
phisms of this kind that can occur in the category of finite spectra.
Definition 5.1. Let C be a full subcategory of C0 which is closed
under suspensions. A collection v, of self-maps
vX : kX X ! X X 2 C
satisfying is an N-endomorphism of C if
i) The map vX is the composite
kX --flip--!kX ^ S1
??
y vX ^1S1
X ^ S1 --flip--!X
ii) for each f : X ! Y in C there are integers i and j with
ikX = jkY , such that the following diagram commutes:
N X --f--! N Y
? ?
vXi?y ?yvYj
X ----! Y:
f
Nilpotence II 38
An N-endomorphism is an F -endomorphism if the integers i and
j can be taken to be powers of p.
Two N-endomorphisms v and v0 will be identified if for each
X 2 C there are integers i and j with vX i = v0Xj. Two F -
endomorphisms v and v0 will be identifiedjif for each X 2 C there
are integers i and j with vX pi= v0Xp.
Remark 5.2.
(1) If v is an N-endomorphism of a category C, and f : X ! Y
an isomorphism with X 2 C, then defining vY to be
kf
kY - ---! kX
??
y vX
X - ---! Y
f-1
extends v to the full subcategory obtained from C by adjoining
the suspensions of Y . Because of property ii), the resulting N-
endomorphism is independent of the choice of isomorphism f. In
this way an N-endomorphism can always be extended uniquely to
a full subcategory which is closed under suspensions and isomor-
phisms. This procedure will be used without comment, so once
an N-endomorphism has been defined on a subcategory C of fi-
nite spectra, it will be taken to be extended to the smallest full
subcategory containing C, which is closed under suspensions and
isomorphisms. Among other things, this means that if vX is de-
fined, so is vX^S1 and
vX^S1 = vX ^ 1S1:
(2) An N-endomorphism is of degree zero if all of the integers kX
are zero. If an N-endomorphism is not of degree zero, then none of
the integers kX is zero, and the maps vX can all be chosen to have
finite order. Given two spectra X; Y 2 C, the maps vX and vY
can be chosen in such a way that the integers kX and kY coincide.
With this arrangement, given a map
f 2 [X; Y ]*;
Nilpotence II 39
if there are integers i and j for which
viYO f = f O vjX;
then it must be the case that i = j. This same discussion applies
to any finite collection of elements of C.
Example 5.3.
(1) Taking vX to be nilpotent defines an F -endomorphism.
(2) Taking each vX to be a multiple of the identity defines an
actual endomorphism .
(3) Suppose C Cn. Taking vX to be a vn self-map defines an
N-endomorphism . Taking vX to be a vn self-map satisfying con-
dition (*) of Theorem 9 defines an F -endomorphism .
5.2. Classification of N-endomorphisms
The above list of examples turns out to be complete.
Theorem 5.4. Suppose that v is an N-endomorphism of a full
subcategory C C0 which is closed under suspensions. Then vX
is nilpotent for every X, some power of vX is a multiple of the
identity, or C Cn for some n and vX is a vn self-map.
Of course, these possibilities aren't exclusive. If X 2 Cn+1 Cn
any vn self-map of X is nilpotent.
Corollary 5.5. Suppose the X 2 C0, and that v 2 [X; X]* is in
the center. Then v is nilpotent, a power of v is a multiple of the
identity, or v is a vn self-map.
Proof: Let C be the full subcategory of C0 consisting of the
suspensions of X. The map v determines an N-endomorphism of
C, so the result follows from Theorem 5.4. __|_ |
Nilpotence II 40
Theorem 11 is an immediate consequence of Corollary 5.5 and
Theorem 9.
The proof of Theorem 5.4 falls into two parts. First it is shown
that an N-endomorphism extends uniquely to a thick subcategory.
It then suffices to construct, for each n, a spectrum Xn 2 Cn \ Cn+1
whose only non-nilpotent self-map is a vn self-map.
First to dispense with the N-endomorphisms of degree zero.
Proposition 5.6. If X is in C0, and v : X ! X is in the center
of [X; X]* = ss*X ^ DX then there are integers m and n for which
vn = multiplication by m.
Proof: Since
ss*X ^ DX Q HQ*X ^ DX End HQ*X;
The map HQ*v must be in the center of End HQ*X which consists
of the endomorphism "multiplication by a constant". Since the
Hurewicz map HQ* factors through HZ*, this constant must be
an integer k. The map w = v - k then has finite order.
Since all of the eigenvalues of HFp *wp-1 are equal to 0 or 1, the
map HFp *w(p-1)pN is an idempotent for N 0. Replace w with
w(p-1)pN. The map w still has finite order, and is in the center of
[X; X]*. Define connective spectra A1 and A2 by
A1 = w-1 X
A2 = (1 - w)-1 X
The map
X ! A1 _ A2
induces an isomorphism on both mod p and rational homology,
hence on homology with coefficients in Z(p). It is therefore a ho-
motopy equivalence, and in particular A1 and A2 are finite.
The ring of self-maps [X; X]* can be written as a ring of 2 x
2 matrices, in which the ij-entry is in [Aj; Ai]. The map w is
represented by the matrix
w|A1 0
0 0 ;
Nilpotence II 41
whose (1; 1) entry is an equivalence. Given a map f : kA2 ! A1,
let "fbe the map
0 f
0 0 :
Then
ad (w)f"= w|A1f0 00 :
Since w is central, and w|A1 is an equivalence this means that f
is null. By Lemma 5.7 below, it follows that one of A1 and A2
is contractible. If A1 is contractible, then w is nilpotent, and the
result follows from Lemma 3.4. If A2 is contractible, then 1 - w
is nilpotent, HQ*X = 0, andNwe may assume that the integer k
is 0, soMthat w = v(p-1)p . It then follows from Lemma 3.4 that
v(p-1)p = 1 for M 0. This completes the proof. __|_ |
We have used
Lemma 5.7. If A and B are non-contractible p-local finite spec-
tra, then [A; B]* 6= 0.
Proof: Since
HFp *DA ^ B = hom [HFp *A; HFp *B] 6= 0;
the spectrum DA ^ B is not-contractible. It therefore has a non-
zero homotopy group. Now use the isomorphism
ss*DA ^ B [A; B]*: __|_ |
5.3. Some technical tools
The next few results are a bit technical, but they come up several
times.
Nilpotence II 42
Lemma 5.8. Suppose that M is a bimodule over the ring Z(p)[v],
and for m 2 M let
ad (v)m = vm - mv:
If there are integers i, j and k, for which
i) k ad(vi)m = 0, and
ii)ad(vj) ad(vi)m = 0,
then
ad vijk m = 0:
Clearly, k can be taken to be a power of p, so that if i and j
are powers of p, then so is ijk.
Lemma 5.9. Suppose M is a bimodule over the ring Z[v]. Let
ad (v) : M ! M be the operator ad(v)m = vm - mv. Then there
is a formula
X n
(5.9.1) ad (vn)m = adi(v)m . vn-i:
i