4

M. HOVEY AND N. P. STRICKLAND

2)i\/ir- When p 2 there is a unique commutative product on E(n) G DMC/, but

we avoid using this here so that we can handle all primes uniformly.

We now define

E = E(n) = E(n) AMu MU-

This is clearly a module over MU with a given map MU — » E, and one can check

that this gives an isomorphism

E* = (E(n)*)$n = Zp[vuv2,... , v „ - i , t ^ f t .

It is again well-defined up to non-canonical isomorphism under MU, and it ad-

mits a non-canonical associative ring structure. If p 2 then there is a unique

commutative product on E as an object of 2 5 ^ .

Because E(n) is an MU-module under MU, there is a canonical map

£(n)*

®MU*

MU*X - E(n)*X.

This map is an isomorphism, by the Landweber Exact Functor Theorem [Lan76].

Similarly, we have an isomorphism

E*

8MU*

MU*X - E*X.

It follows that the homology theory represented by E(n) is independent of the

choice of object E(n) G

*DMU

up to canonical isomorphism, and thus the underly-

ing spectrum of E(n) is well-defined up to an isomorphism that is canonical mod

phantoms (see [HPS97, Section 4] for a discussion of phantoms and representabil-

ity). We shall show later that the relevant group of phantoms is zero, so as a

spectrum E(n) is well-defined up to canonical isomorphism. We shall also show

that there is a canonical commutative ring structure on this underlying spectrum.

Similar remarks apply to E.

We can also define M£/-modules MU/Ik €

TMU

for 0 & n in the evident

way, and then define

E{n)/Ik = MU/h AMU E(n)

E/Ik = MU/Ik AMU E

K = K{n) = E(n)/In = E/In.

It is clear that 7r*(E(ri)/Ik) = E(ri)*/Ik and so on. In particular we have K(ri)* =

E(n)*/In = E*/In = F p ^

1

] . These MC/-modules admit (non-unique) associative

products, so (E(ri)/Ik)*X is canonically a module over E(ri)*/Ik. Similar remarks

apply to E/Ik.

There are evident cofibrations

X^k-VE/Ik^E/Ik-E/Ik+1,

and similarly for E(ri)/Ik*

We also know from [Bak91] that there is an essentially unique Aoo structure on

the spectrum E. It is widely believed that this can be improved to an

JE?OO

structure,

and that the maps MU — E characterised by Matthew Ando [And95] (which do

not include the map MU — E considered above) can be improved to Eoo maps.

Unfortunately, proofs of these things have not yet appeared. Nonetheless, just using

the AQO structure we can still use the methods of [EKMM96] to define a derived

category *DE of left i?-modules. This is a complete and cocomplete triangulated

category, with a smash product functor A: § x

TE

—

*DE*