Errata for "Nilpotence and periodicity in stable homotopy theory" by Dou-
glas C. Ravenel, February 1, 1994, edition. Most of these were found by Peter
Landweber.
Major revisions:
In the index (pages 205-209), reduce all page numbers by 2.
Page 9, insert between 1.5.4 and last paragraph: Some comments are in
order about the definition of a vn-map given in the theorem. First, X is not
logically required to have type n, but that is the only case of interest. If X
has type > n, then the trivial map satisfies the definition, and if X has type
< n, it is not difficult to show that no map satisfies it (3.3.11). Second, it *
*does
not matter if we require K(m) *(f) to be trivial or merely nilpotent for m > n.
If it is nilpotent for each m > n, then some iterate of it will be trivial for *
*all
m > n. For d > 0 this follows because some iterate of H*(f) must be trivial
for dimensional reasons, and K(m)*(f) = K(m)* H*(f) for m 0. The case
d = 0 occurs only when n = 0, for which the theorem is trivially true since the
degree p map satisfies the definition.
Page 35-36, omit 3.4.6 and the preceding two paragraphs, and replace the
"sketch of proof" by the following:
Proof of Theorem 3.4.2. Note that Cp;0= C (p)by convention and we have a
decreasing filtration
C (p)= Cp;0 Cp;1 . .C.p;n . . .
with \n0 Cp;n= {0} by Corollary 3.3.9(ii).
Now suppose C C (p)is thick. If C 6= {0}, choose the largest n so that
Cp;n C. Then C 6 Cp;n+1, and we want to show that C = Cp;n, so we need
to verify that C Cp;n.
Let M be a comodule in C but not in Cp;n+1. Thus v-1mM = 0 for m < n
but v-1nM 6= 0. Choosing a Landweber filtration of M in C ,
0 = F0M F1M . . .FkM = M;
all FsM are in C, hence so are all the subquotients
FsM=Fs-1M = dsMU*=Ip;ms:
Since v-1nM 6= 0, we must have
v-1n(MU*=Ip;ms) 6= 0
for some s, so some ms is no more than n. This ms must be n, since a smaller
value would contradict the assumption that C Cp;n. Hence we conclude
that
MU*=Ip;n2 C: (3:4:6)
1
Now let N be in Cp;n; we want to show that it is also in C. Then v-1n-1M = 0,
so each subquotient of a Landweber filtration of N is a suspension of MU*=Ip;m
for some m n. Since MU*=Ip;n2 C by (3.4.6), it follows that MU*=Ip;n2 C
for all m n. Hence the Landweber subquotients of N are all in C, so N itself
is in C. ||
Page 50, replace Proof of 5.1.5 (the rest of this section) with:
Proof of Corollary 5.1.5. Let R = DW ^ W and let e : S0 ! R be the adjoint
of the identity map. R is a ring spectrum (A.2.8) whose unit is e and whose
multiplication is the composite
DW^De^W 0
R ^ R = DW ^ W ^ DW ^ W ---------! DW ^ S ^ W = R:
The map f : X ! Y is adjoint to ^f: S0 ! DX ^ Y , and W ^ f is adjoint
to the composite
^f e^DX^Y
S0 -! DX ^ Y ---------! R ^ DX ^ Y = F;
which we denote by g. The map W ^ f(i)is adjoint to the composite
g(i)
S0 ---------! F (i)= R(i)^ DX(i)^ Y (i)-! R ^ DX(i)^ Y (i);
the latter map being induced by the multiplication in R.
By 5.1.4 it suffices to show that MU ^ g(i)is null for large i. Let Ti =
R ^ DX(i)^ Y (i)and let T be the direct limit of
f^ T2^f^
S0 -g!T1 T1^-!T2 -! T3 -! . .:.
The desired conclusion will follow from showing that MU ^ T is contractible,
and our hypothesis implies that K(n) ^ T is contractible for each n.
Now we need to use the methods of Chapter 7. Since we are in a p-local
situation, it suffices to show that BP ^T is contractible. Using 7.3.2 and the *
*fact
that K(n) ^ T is contractible, it suffices to show that P (m) ^ T is contractib*
*le
for large m.
Now for large enough m,
K(m)*(W ^ f) = K(m)* H*(W ^ f) and
P (m)*(W ^ f) = P (m)* H*(W ^ f):
Our hypothesis implies that both of these homomorphisms are trivial, so P (m)^
T is contractible as required. ||
2
Page 99, replace last sentence of paragraph after 9.0.1 with:
We claim that h(f^) is nilpotent if MU*(f) is.
To see this, observe that if MU*(f) = 0, then MU ^ f-1 X is contractible,
where f-1 X denotes the homotopy direct limit of
X -f! -dX -f! -2dX -f! . .:.
Since X is finite, this means that for large enough m, the composite
m
md X f-! X -! MU ^ X
is null. Then h(fcm) = h(f^)m = 0, so h(f^) is nilpotent.
3
Misprints and minor corrections:
Page xi, line 2 of second paragraph: algebraic patterns
Page xi, line 4 of second paragraph: Northwestern
Page xiv, line 6 of second paragraph: realizability
Page xiv, line 8 of second paragraph: chosen
Chapter 1
Page 1, line 3: three sections
Page 1, last line: [(X; x0); (Y; y0)].
Page 4, line -9: point_out the properties
Page 5, (1.3.2): E i+1(X):_
Page 5, line 3 of 1.3.3: E *(f) = 0
Page 6, paragraph after 1.4.2: Actually this is the weakest of the three for*
*ms
of the nilpotence theorem; the other two (5.1.4 and 9.0.1) are equivalent and
imply this one. __
Page 6, line 1 after 1.4.1: E *to
Page 6, line 3 after 1.4.2: bordism theory,
Page 6, line 3 above 1.5: Chapter 3
Page 8, line 5: If X is as above then
Page 8, line 7: if X is simply connected and not contractible.
Page 8, line 5 of 1.5.4: as a vn-map; see page 53.
Page 9, lines -5 and -2: asymptotically
Chapter 2
Page 11, line 1: In this chapter
Page 12, line 3 after 2.1.1: It is easy to construct a finite CW-complex of
dimension 2
Page 13, last line before 2.2.6: was proved in [CMN79].
Page 15, line 2 above 2.3.4: three of the fundamental
Page 17, line 1 of 2.4.2(ii): following composite:
Page 17, line 5 of 2.4.2(ii): of its suspensions.)
Page 18, line 6 after (2.4.3): so we can use 2.3.4(iii)
Page 18, line above (2.4.4): and consider the diagram
Page 18, line 1 of (2.4.4):
fe _____-j
tdX _________Xk+ek=-X Xk+ek+e= Sk+e
Page 18, line 2 after (2.4.4): ife-1 = fe.
Page 19, line 2 of (2.4.5):
tdXk+1k+1_____-tdXk+ek+1i
4
Page 19, line 3 of (2.4.5):
tdXk+2k+2_____-tdXk+ek+2i
__ Page 20, line 5 of second paragraph in x2.5: known to be isomorphic to
H*(Y ; Q).
Page 20, line 3 above (2.5.1): maps from the inverse
Page 22, line 4: For sufficiently large
Page 22, line 5 after diagram: the form g1fi11
Page 22, line 6 after diagram: the cofibre of fi11
Page 23, line 3 of diagram: W (2) = Cfi11
Chapter 3
Page 25, line 3 of 3.1.1: compact smooth manifold
Page 25, line 3 above 3.1.2: e.g. a nonsingular complex algebraic variety
Page 31, line 5 after 3.3.6: finitely generated module M
Page 31, lines 5 and 4 above 3.3.7: Now L is not Noetherian, but it is a
direct limit of Noetherian rings, so finitely presented modules over it admit
similar filtrations.
Page 32, line 2 above 3.3.11: Another consequence
Page 34, line 2 of 3.4.2: p-local modules in C
Page 34, line after 3.4.2: We will give the proof_of_this result below.
___Page_34, insert after 3.4.3: The condition v-1n-1MU*(X) = 0 is equivalent to
K(n - 1)*(X) = 0, in view of 1.5.2(v) and 7.3.2(d) below.
Chapter 4
Page 42, line 3 of 4.3.4: elements of finite order
Chapter 5
Page 45, first sentence: In this chapter we will derive the thick subcategory
theorem (3.4.3) from a variant of the nilpotence theorem (5.1.4 below) with the
use of some standard tools from homotopy theory, which we must introduce
before we can give the proof.
Page 48, 5.2.1 (i): For any spectrum Y , the graded group [X; Y ]* is isomor-
phic to ss*(DX ^ Y ), and this isomorphism is natural in both X and Y . In
particular, DS0 = S0.
Page 48, 5.2.1(ii): This isomorphism is reflected in Morava K-theory, namely
(since K(n)*(X) is free over K(n)*)
Hom K(n)*(K(n)*(X); K(n)*(Y )) ~=K(n)*(DX ^ Y ):
In particular for Y = X, K(n)*(e) 6= 0 when K(n)*(X) 6= 0. Similar statements
hold for ordinary mod p homology. For X = S0, this isomorphism is the identity.
5
Page 48, line 3: which we will prove at the end of Section 5.2, using some
methods from Chapter 7.
Page 48, 5.2.1(iii): DDX ' X and [X; Y ]* ~=[DY; DX]*.
Page 48, insert at end of 5.2.1: (vi) The functor X 7! DX is contravariant.
Page 51, bottom of page, insert: Equivalently (by 7.3.2(d) and 1.5.2(v)), n
is the smallest integer such that C contains an X with v-1nBP*(X) 6= 0.
Chapter 6
Page 56, sentence beginning on line -8: Hence 5.1.5 tells us that ad(f^) is
nilpotent and the argument above applies to give the desired result.
Chapter 7
Page 71, diagram:
W ___________-Ef^ X
| | Q
| | Q Q E^X
j^W | j^E^X | Q
| | Q
|? |? Qs
E ^ W _______-EE^^Ef^ X _______-Em^^X:X
Page 74, 7.2.9, line 4: Then FBA(p) is the free
Page 80, bottom line: X ' limLnX:
Chapter 8
Page 87, line 9: LnY
Page 95, (8.6.2): lim1ssi(CnX)
Page 95, Def 8.6.3: A spectrum Y is E-convergent if the E-based Adams
spectral sequence for Y converges to ss*(Y ) and there is a nondecreasing funct*
*ion
s(i) such that,
Page 95, line -4, put a space after "theorem"
Page 95, line 1 after (8.6.2) and page 96, line 5: lim1
Chapter 9
Page 99, line 3 after 9.0.1: let R = DX ^ X
Page 100, replace first sentence with: Proof that 9.0.1 implies 5.1.4: (In t*
*he
following argument, MU could be replaced by any ring spectrum for which 9.0.1
holds.)
Page 100, third displayed formula:
j^X
S0 -f!X ---------! MU ^ X
Page 100, fourth displayed formula:
j^Xff
S0 -f!Xff---------! MU ^ Xff
6
Appendix A
Page 121, omit A.1.6 and preceding sentence.
Page 124, line 6 of A.2.8: are each homotopic to the identity on E
Page 124, bottom of page: such that the following diagram commutes up to
homotopy
E ^ E ^ M _______-Em^^ME
| |
| |
E^ | |
| |
|? |?
E ^ M ___________-M
and the composite
j^M
M = S0 ^ M ---------! M ^ M -! M
is homotopic to the identity.
Page 126, line 4 above A.3: There is a Hurewicz theorem for connective
spectra
Page 130, first sentence of bottom paragraph: This theorem can used to
construct a spectrum by constructing the cohomology theory it represents.
Page 141, line 8:
E*(E ^ X) = E*(E) ss*(E)E*(X):
Appendix B
Page 148, line 1 after B.1.7: For a paracompact base space X
Page 155, line 1: E*(E) and the map
Page 162, last line of B.4.7: with coefficients in R
Page 163, line 2 of Proof of B.4.10: w(x; y) = logF(F (x; y)) - logF(x) -
logF(y).
Page 171, line 2 of part (c) of Proof: at most one formal summand
Page 171, line 1 of B.5.16: an ideal J S
Page 173, third displayed formula:
0 -! TorBP*1(M; BP*=(p)) -! M -p! M -! M=pM -! 0:
Page 173, three lines above B.6.2:
0 -! TorBP*1(M; BP*=InM) -! M=In-1M -p! M=In-1M -! M=InM -! 0:
Thus we see that TorBP*1(M; BP*=In) = 0 provided that M=In-1M is vn-torsion
free, i.e., that multiplication by vn in M=In-1M is monic.
Page 174, line 2: for each positive integer n
Page 174, line 9 after B.6.2: In all three cases
Page 175, bottom line: conjugate of oi
7