We will denote by K = E∧
K(n) the version of Morava K-theory associated
to E, it is known that E is K-local in the category of E-modules and we can consider
the localisation LK (E ∧ E) for which
E = π∗(LK (E ∧ E)).
By [6, proposition 2.2], this localisation can be taken either with respect to K in
the category of S-modules, or with respect to E ∧ K in the category of E-modules.
By [2, lemma 7.6], the homotopy π∗(LK M) viewed as a module over the local ring
(E∗, m) is L-complete.
We will write Map(X, Y ) for the set of all functions X −→ Y and
for the set of all continuous functions if X, Y are topologised.
A detailed discussion of the relevant K(n)-local Galois theory of Lubin-Tate
spectra can be found in section 5.4 and chapter 8 of , and we adopt its viewpoint
and notation. In particular, En nr is a K(n)-local Galois extension of LK(n) with
proﬁnite Galois group
= Z Sn,
where Sn is the usual Morava stabiliser group which can be viewed as the full
automorphism group of a height n Lubin-Tate formal group law Fn deﬁned over
, and also as the group of units in the maximal order of a central division
algebra over Qp of Hasse invariant 1/n. The p-Sylow subgroup Sn
Sn has index
− 1) and Sn is the semi-direct product
Sn = Fpn
The proﬁnite group Z acts as the Galois group
In particular, the closed subgroup nZ Z is the stabiliser of Fpn and En
(En nr)h(nZ); similarly, E(n) (En nr)hZ.
Our ﬁrst result is a generalisation of a well known result, see  for example.
Theorem 5.2. For E as above, there are natural isomorphisms of E0-algebras
is a pro-free L-complete E∗-module.
Theorem 5.3. Let E be a Lubin-Tate spectrum as above.
is an L-complete Hopf algebroid.
(b) the maximal ideal m E∗ is invariant.
is a pro-free E∗-module.
(d) There are isomorphisms of K∗ = E∗/m-algebras
E∗/m[θk : k
: 1) ⊗Fp[u,u−1] E∗/m.
Now let us consider the reduction K∗E in greater detail. First note that the
pair (K∗,K∗E) is a Z-graded Hopf algebroid. Now
K∗ = F[u,
where F ⊆
and |u| = 2. Since
= vn under the map BP −→ K classifying a
is invariant. This suggests that we might usefully change
to a Z/2(pn − 1)-grading on K∗-modules by setting
= 1. To emphasise this
regrading we write (−)• rather than (−)∗. In particular, K• = F(u).