MORAVA K-THEORIES AND LOCALISATION 11

Proof. Choose a cofibre sequence P — Q — X — SP as in Proposition 2.14. If

W is an even finite, it is easy to see by induction on the (even) cells that R*W is

free over i?* and concentrated in even degrees. Since R*X = lim i?*W, we

—• Aes-(X)

see that R*X is also concentrated in even degrees and is a filtered colimit of free

modules, so is Sat. One can also check by induction on the cells, using the fact that

M* is evenly graded, that the natural map M*®^ i?* W — * M* W is an isomorphism

for W an even finite. Taking colimits, we find that M* ®Rm R*X — M*X is also

an isomorphism.

Similarly, one can check by induction on the cells that when W is an even finite,

the natural map [W, M]* — Homj^ (U* W, M*) is an isomorphism. As P and Q are

retracts of wedges of such W, we see that R*P and P*Q are projective over i?* and

that [Q,M]* = Homfl-l(jR*Q,M*) and [P,M]* = Hom^ (i?*P,M*). In particular,

these groups are concentrated in even degrees. Because 6 is phantom, we have a

short exact sequence

R*P—-+ R*Q — » R*X,

which is a projective resolution of i?*X. We now apply the functor [—,M]* to the

cofibration P — Q — X to get an exact sequence

HomgHA.Q.M*) - Hom^

1

(i?*P,M*) - [X,M]* -

Horn^ (R*Q, M*) - Hom^ (R*P, M*),

and thus a short exact sequence

Ext^*+1(ii*X,M*)-

[X,M]* -»Hom^(i?*X,M*).

The first term is concentrated in odd degrees and the last one in even degrees, so

the sequence splits uniquely. •

Note that this proposition implies that spectra such as P(n) and K(n) are not

evenly generated for n 0. Indeed, P(ri)*P(n) and K(ri)*K(n) both contain

a Bockstein element in degree 1. Similarly, HZ and HFP are not evenly gener-

ated. In fact, the only M£7-module spectra that are evenly generated are the even

Landweber exact M[/"-module spectra. One can prove this by using the fact that

X AF(n) is a retract of (MUAF(n))AX and is therefore evenly graded. Applying

this to the spectra S/I of Section 4 shows that vn acts injectively on X*/I.

Corollary 2.17. Let M and N be MU-module spectra in £'. Then

[M,N]2k

= llom%^MU(MU*M,MU*N).

= Hom^(MZ7*M,iV*)

[M,N)2k~l

=Ext)£^MU(MU*M,MU*N)

= Ext^(MC/*M,AT*).

Ifsl then

Exts^MU(MU*M,MU*N)

= 0.

Proof. By Landweber exactness we have X*N =

X*MU®MU*

AT * and in particular

MU*N = MU*MU

®MU*

AT*. This is an extended comodule, so for any comodule

C* we have

H.omMu*Mu{C*,MU*N) = HomMC/*^*,

A7*).