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*).
Previous Page Next Page