Corollary 2.14. A spectrum X is evenly generated if and only if there is a cofibre
P - Q - X -?* EP,
where P and Q are retracts of wedges of even finite spectra and 8 is a phantom
Proof If there is such a cofibre sequence, then there is a short exact sequence
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

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 / :
Y is phantom then the composite

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
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
= Hom|£ (#*X, M*)
2 f c - i
x =
Ext^f(P*X, Af*).
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