MORAVA K-THEORIES AND LOCALISATION
13
Now let N be another spectrum in £', and consider the following diagram.
[M,N]MU HomMt/,(M*,iV*)
/
[M,N] - ^ HomMu.Mu(MU.M,MU*N).
Here
[M,N]MU
denotes the group of M£/-module maps, and all the groups are
groups of degree-zero maps. The bottom map is an isomorphism by Proposi-
tion 2.17. The map / sends a map u: M* iV* to
1 0 u: MU+MU
®MU*
M* = MU*M - MU*N = MU*MU ®Mu. N*.
It is easy to check that this is injective and that the diagram commutes. It follows
that 7r* is injective. Now suppose we have a map v: M* » iV* of MC/*-modules.
We then have a unique map u: M N such that MU*u = f(v). A diagram chase
shows that u is a map of MU-module spectra (up to a phantom term which is zero
as usual), and /(7r*(u)) = f(v) so 7r*(u) = v. Thus 7r* is an isomorphism.
Proposition 2.21. If A G £ and 7r*(A) is a commutative MU*-algebra then there
is a unique product on A making it into a commutative MU-algebra spectrum.
Proof For any X and Y we have a pairing MU*X
0MC/*
MU*Y MU*(X A Y)
and thus a pairing
-A*(X)
®A .
A*{Y) = A*
®MU*
MU*X
®A*
A*
®MU*
MU*Y -
A* 0MC/, MC/*(X A Y) = A*(X A Y).
This is easily seen to be commutative, associative and unital. Now write A as a
minimal weak colimit of finite spectra Aa- Then A A A = mwlim A
a
A AQ by
Lemma 2.18. We have
A°(Aa A Ap) = A0(DAa A DAp) = (A*(DAa) ®A* A*(DAp))0.
Therefore the maps ia: Aa -A, when thought of as elements of -Ao(£M.c*)) give rise
to a compatible collection of maps Aa AAp A, and hence a map A A A A. This
map is unique up to phantoms, and it is commutative, associative, and unital up
to phantoms. However, all the relevant phantom groups vanish by Proposition 2.19
and Corollary 2.15.
Proposition 2.22. If A is a Landweber exact ring spectrum and M is an A-module
spectrum then there are universal coefficient spectral sequences of A*-modules
Ext^(A*X,M*) =
Mt+sX
Toif;t(M*,A*X)=Mt+sX.
Proof The first spectral sequence follows from Proposition 2.12, Proposition 2.13
and [Ada74, Theorem 13.6]. The second is constructed by the same methods.
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