12
M. HOVEY AND N. P. STRICKLAND
More generally, if we resolve iV* by injective MU*-modules and apply the functor
MU*MU 0Mt/* (—) we get a resolution of MU*N by injective comodules. Using
this it is not hard to check that
Exts^MU(C*,MU*N)
= E x t
5
^ (C*, JV„)
for all s. The rest of the corollary is proved in Proposition 2.16.
Lemma 2.18. Let {Ma} and {Np} be filtered diagrams of finite spectra. Then
mwlim Ma A Np = mwlim Ma A mwlim Mp.
a,/? a p
Proof, We know from [HPS97, Theorem 4.2.3] that all the relevant diagrams have
minimal weak colimits. Write M = mwlim Ma and N = mwlim Np and L =
—*•
a
—»•
p
mwlim Ma A Np, Let ia: Ma M and jp: Np N andfca/?:Ma ANp L
be the obvious maps. The maps ia A jp: Ma ANp L are compatible as a and /?
vary, so the weak colimit property gives a map / : L MAN with fokap = iaAjp.
We claim that this is an isomorphism; it suffices to check that 7T*L = 7r*(M A iV).
As 7T * is a homology theory, we have 7r*L = lim 7r*(Ma A Np). As 7r*(— A N)
is a homology theory, we have 7r*(M A N) = lim 7r*(Ma A iV). By the same
•—*
a
logic, we have irJMa A N) = lim nJMa A Np), It follows that 7r*(M A N) =
lim 7r*(MQ ANp), as required. D
Proposition 2.19. Suppose X and Y are evenly generated. Then X AY is evenly
generated,
Proof, It is clear that if U and W are even finites, so is U A W, Given this, the
proposition follows immediately from Proposition 2.13 and Lemma 2.18.
In fact, it is also true that £' is closed under the smash product, though we do
not need this.
Proposition 2.20. If M E
£;
then M admits a canonical structure as an MU-
module spectrum {in the traditional homotopical sense). If N is another spectrum
in
£7
then the degree-zero MU-module maps M N biject with the MX)^-module
maps M* iV*.
Proof For any spectrum X we have a natural map
e: MU*(MU AX) = MU*MU ®Mu* MU*X - MU+X
of left MU*-modules, and thus a natural map
(Af A MU)*X = M*{MU AX) = M* ®MUm MU*(MU A X) -
By Brown representability, we get a map v: M A MU M which is unique mod
phantoms. It is not hard to check that this is associative and unital mod phantoms.
However, Proposition 2.19 shows that M A MU and M A MU A MU are evenly
generated, so Corollary 2.15 tells us that there are no degree-zero phantom maps
M A MU M or M A MU A MU M, Thus v is unique, associative and unital,
and M £.
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