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 — 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*).