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
MU*X - E(n)*X.
This map is an isomorphism, by the Landweber Exact Functor Theorem [Lan76].
Similarly, we have an isomorphism
MU*X - E*X.
It follows that the homology theory represented by E(n) is independent of the
choice of object E(n) G
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
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 ^
] . 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
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
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

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