16 ANDREW BAKER

We will denote by K = E∧

E(n)

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∗

∨

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

Mapc(X,

Y )

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 [14], 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

Gn

nr

= 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

Fpn ⊆

¯

F

p

, 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

0

Sn has index

(pn

− 1) and Sn is the semi-direct product

Sn = Fpn

× Sn.0

The proﬁnite group Z acts as the Galois group

Gal(W

¯

F

p

/W Fp)

∼

=

Gal(¯p/Fp)

F

∼

= Z.

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 [1] for example.

Theorem 5.2. For E as above, there are natural isomorphisms of E0-algebras

E∗

∨E(n)

∼

=

Mapc(Sn,E∗).

Furthermore,

Mapc(Sn,E∗)

is a pro-free L-complete E∗-module.

Theorem 5.3. Let E be a Lubin-Tate spectrum as above.

(a) (E∗,E∗

∨E)

is an L-complete Hopf algebroid.

(b) the maximal ideal m E∗ is invariant.

(c) E∗

∨E

is a pro-free E∗-module.

(d) There are isomorphisms of K∗ = E∗/m-algebras

K∗E

∼

=

E∗

∨E/E∗ ∨Em

∼

=

E∗/m[θk : k

1]/(θpn

−

up −1θ

: 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,

u−1],

where F ⊆

¯

F

p

and |u| = 2. Since

upn−1

= vn under the map BP −→ K classifying a

complex orientation,

upn−1

is invariant. This suggests that we might usefully change

to a Z/2(pn − 1)-grading on K∗-modules by setting

upn−1

= 1. To emphasise this

regrading we write (−)• rather than (−)∗. In particular, K• = F(u).

16

We will denote by K = E∧

E(n)

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∗

∨

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

Mapc(X,

Y )

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 [14], 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

Gn

nr

= 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

Fpn ⊆

¯

F

p

, 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

0

Sn has index

(pn

− 1) and Sn is the semi-direct product

Sn = Fpn

× Sn.0

The proﬁnite group Z acts as the Galois group

Gal(W

¯

F

p

/W Fp)

∼

=

Gal(¯p/Fp)

F

∼

= Z.

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 [1] for example.

Theorem 5.2. For E as above, there are natural isomorphisms of E0-algebras

E∗

∨E(n)

∼

=

Mapc(Sn,E∗).

Furthermore,

Mapc(Sn,E∗)

is a pro-free L-complete E∗-module.

Theorem 5.3. Let E be a Lubin-Tate spectrum as above.

(a) (E∗,E∗

∨E)

is an L-complete Hopf algebroid.

(b) the maximal ideal m E∗ is invariant.

(c) E∗

∨E

is a pro-free E∗-module.

(d) There are isomorphisms of K∗ = E∗/m-algebras

K∗E

∼

=

E∗

∨E/E∗ ∨Em

∼

=

E∗/m[θk : k

1]/(θpn

−

up −1θ

: 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,

u−1],

where F ⊆

¯

F

p

and |u| = 2. Since

upn−1

= vn under the map BP −→ K classifying a

complex orientation,

upn−1

is invariant. This suggests that we might usefully change

to a Z/2(pn − 1)-grading on K∗-modules by setting

upn−1

= 1. To emphasise this

regrading we write (−)• rather than (−)∗. In particular, K• = F(u).

16