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 € £.