10 M. HOVEY AND N. P. STRICKLAND
Corollary 2.14. A spectrum X is evenly generated if and only if there is a cofibre
sequence
P - Q - X -?* EP,
where P and Q are retracts of wedges of even finite spectra and 8 is a phantom
map.
Proof If there is such a cofibre sequence, then there is a short exact sequence
[Z,P]-[Z,Q]-»[Z,X]
for all finite Z, since 8 is phantom. It follows easily that X is evenly generated. The
converse follows from the Proposition and the construction in [CS98, Proposition
4.6].

Corollary 2.15. Suppose X is evenly generated and Y is a spectrum such that Y*
25 concentrated in even degrees. Let ?*(X, Y) be the graded group of phantom maps
from X to Y. Then
?2fc(X,Y) = 0
72k-1(X,Y) = [X,Y]2k-1.
In particular, this holds if X,Y 6 £'.
Proof The cofibre sequence of Corollary 2.14 gives an exact sequence
[EQ,Y]* - [EP, Y]* [X,Y]* - [P,Y]" - [Q,Y]*.
If W is an even finite, then we see by induction on the number of cells that [TV, Y]* is
concentrated in even degrees. This implies that [P, Y]* and [Q, Y]* are concentrated
in even degrees. As 8 is phantom, we see that the image of 8* consists of phantoms.
If / :
HkX
Y is phantom then the composite
EfcP

EfcX
Y is also phantom,
but any phantom map out of a wedge of finite spectra is zero, and it follows that
/ factors through 8. Thus, 7*(X, Y) is precisely the image of 8*. The corollary
follows easily.
We point out that tS2k~1{X^ Y) can be nonzero in the situation of Corollary 2.15,
even when X and Y are Landweber exact. For example, let F denote the fiber of
the map K from the complex iiT-theory spectrum to its p-completion. Then
F —* K is a phantom map. Also K2kF = 0 and iT2k-iF is the rational vector space
Zp/Z(p). Thus F is a wedge of odd suspensions of HQ, and so
(J2k"1(HQ^K)
is
nonzero for all k.
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, M is an R-module spectrum,
and X is evenly generated. Suppose as well that i?* and M* are concentrated
in even degrees. Then R*X is flat, has projective dimension at most 1, and is
concentrated in even degrees. Furthermore, we have
M*X = M* ®Rm R*X
M2kX
= Hom|£ (#*X, M*)
M
2 f c - i
x =
Ext^f(P*X, Af*).
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