MORAVA K-THEORIES AND LOCALISATION
3
Our debt to Mike Hopkins will be very obvious to anyone familiar with the sub-
ject. We have been heavily influenced by his point of view and a large number of our
results were previously known to him. We also thank Matthew Ando, Dan Chris-
tensen, Chun-Nip Lee, John Palmieri and Hal Sadofsky for helpful conversations
about the subject matter of this paper.
1. BASIC DEFINITIONS
Fix a prime p and an integer n 0. We shall localise all spectra at p\ in
particular, we shall write MU for what would normally be called MU(P). We write
S for the category of p-local spectra.
1.1. Th e spectra E(ri), E(ri) and K(n). We next want to define the spectra
E = E(n) and K = K(n). It is traditional to do this using the Landweber
exact functor theorem and Baas-Sullivan theory. Here we will use the more recent
techniques of [EKMM96] instead of the Baas-Sullivan construction.
It is well-known that the integral version of MU has a natural structure as an Eoo
ring spectrum or (essentially equivalently) a commutative 5-algebra in the sense
of [EKMM96]. It follows from [EKMM96, Theorem VIII.2.2] that the same applies
to our p-local version. We can thus use the framework of [EKMM96, Chapters
II-III] to define a topological closed model category MMU of MC/-modules. The
associated homotopy category (obtained by inverting weak equivalences) is called
the derived category of MCZ-modules and written
TMU-
It is a stable homotopy
category in the sense of [HPS97]. There is a "forgetful" functor TMU § and a
left adjoint MU A (-) : S - TMu.
k
Let Wk £ 7r2(pfc_i)MC/ be the coefficient of
xp
in the p-series of the universal for-
mal group law over MU* (so wo = p) and write In = (wo,... , w
n
_i). We can con-
struct an object w^MU/In of *DMu by the methods of [EKMM96, Chapter V] (see
also [Str96]). Using [EKMM96, Theorem VIII.2.2] again, we can Bousfield-localise
MU with respect to w^MU/In to get a strictly commutative M£/-algebra which
we call MU, As explained in [GM95a], the homotopy ring of MU is (w^MU*)^.
Next, consider the graded ring
E{n)* = Z(p)[vu...
,vn\Kl\
\vk\ = 2 ( / - 1).
There is a unique p-typical formal group law over this ring with the property that
F
\P)F{X)
= expF(px) +F ^2 VkxP '
0kn
(Thus we take V& to be a Hazewinkel generator rather than an Araki generator.)
This gives a map MU* E(n)* (using Quillen's theorem that the formal group law
over MU* is universal), and one can check that this makes Z(
p
)[vi,... , vn] into the
quotient of MU* by a regular sequence (see [Str96, Proposition 8.15] for details).
Moreover, the image of Wk is Vk modulo Ik = (wo*---
J ^ - I ) -
It follows from
the results of [EKMM96, Chapter V] that there is an M/7-module E{n) e 1MU
with a given map MU E(n) inducing an isomorphism 7r*E(n) = E(n)* of MU*-
modules. It is shown in [Str96] that this is unique up to non-canonical isomorphism
under MU, and that it admits a non-canonical associative product. The resulting
i£(n)*-module structure on E(n)*X (for any spectrum X) is nonetheless canonical,
because it is derived from the M?7-module structure of the spectrum E{ri) A l G
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