By an evident induction we conclude that E*X is concentrated in even de-
grees, that the sequence {^o--- ^n-i} is regular on E*X and that K*{X) =
J S * ( X ) / ( V
, . . .
-i) = K*
E*X. It follows from Theorem A.9 that E*X is
pro-free. D
2.1. Landweber exactness. We next recall the theory of Landweber exact ho-
mology theories, and prove some convenient extensions. Many of the theorems
we prove were proved by Pranke [Pra92] in the case where 7r*M is countable; our
methods are a generalisation of his.
Definition 2.6. An MU*-module M* is said to be Landweber exact if the sequence
(tuo,'itfi,...) is regular on M*. We write £* for the category of Landweber exact
modules that are concentrated in even degrees. We also write £ for the category
of MZ7-module spectra M such that 7r*(M) G £*. Maps in £ are M?7-module
maps. Finally, we write £ ? for the category of finite spectra X such that H*X
is free and concentrated in even degrees. We refer to such an X as an even finite
spectrum. Note that any even finite spectrum has a finite filtration where the
filtration quotients are finite wedges of even spheres.
The basic result is as follows.
Theorem 2.7 (Landweber). / / M* G £* then the functor M*
MU*X is a
homology theory. Thus (by Brown representability), there is a spectrum M equipped
with a natural isomorphism M*X M*
MU*X. This M is unique up to
isomorphism, and the isomorphism is canonical modulo phantoms.
Proof. See [Lan76, Theorem 2.6].
The following result summarises Proposition 2.21 and Proposition 2.20, which
are proved below. It justifies the statements made in Section 1 about the uniqueness
of E and E(n) and their ring structures.
Theorem 2.8. The functor 7r*: £ £* is an equivalence of categories. The in-
verse functor sends commutative Ml)^-algebras to commutative MU-algebra spec-
It is convenient to introduce a new category £' at this point; it will follow from
the theorem that £' = £.
Definition 2.9. £' is the category of spectra M such that M* is concentrated in
even degrees, with a given MC/*-module structure on M* and a stable natural iso-
MU*X M*X which is the identity when X = S. Morphisms
of £' must preserve the module structure.
The converse of the Landweber exact functor theorem [Rud86] shows that if
M G £;, then M* G £*. Theorem 2.7 says that 7r*: £' £* is essentially surjective
on objects.
In order to show that 7r* is an equivalence of categories, we introduce the follow-
ing definition.
Definition 2.10. A spectrum X is evenly generated if and only if, for every finite
spectrum Z and every map Z X\ there is an even finite spectrum W and a
factorisation Z -^ W X of / .
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