L-COMPLETE HOPF ALGEBROIDS 15

Now we recall some facts from [13, lemma A1.1.13] about the extension of Hopf

algebroids

(D, Φ) −→ (A, Γ) −→ (A, Γ ),

where Γ is the Hopf algebra associated to Γ and Φ is unicursal. We have the

following identiﬁcations:

Γ = A ⊗Φ Γ, D = A ΓA, Φ = A ⊗D A.

The map of Hopf algebroids Γ −→ Γ is normal and

Φ = Γ

Γ

A = A

Γ

Γ ⊆ Γ.

Furthermore, for any left Γ-comodule M, A

Γ

M is naturally a left Φ-comodule

and there is an isomorphism of A-modules

(4.2) A

Γ

M

∼

=

A

Φ

(A

Γ

M).

Proposition 4.4. Let M be a Γ-comodule. If when viewed as a Γ -comodule, M

has non-trivial primitive Γ -subcomodule A

Γ

M, then the primitive Γ-subcomodule

A ΓM is non-trivial.

Proof. Combine Lemma 4.1 and (4.2).

Our next result is immediate.

Theorem 4.5. Let (k, Γ) is a Hopf algebroid over a ﬁeld. If the associated

Hopf algebra (k, Γ ) is unipotent, then (k, Γ) is unipotent.

5. Lubin-Tate spectra and their Hopf algebroids

In this section we will discuss the case of a Lubin-Tate spectrum E and its

associated Hopf algebroid (E∗,E∗ ∨E), where E denotes any of the 2-periodic spectra

lying between E(n) (by which we mean the 2-periodic version of the completed

2(pn

− 1)-periodic Johnson-Wilson spectrum E(n)) and En

nr

discussed in [2], see

especially section 7. The most important case is the ‘usual’ Lubin-Tate spectrum

En for which

π∗(En) = W Fpn

[[u1,...,un−1]][u±1],

but other examples are provided by the K(n)-local Galois subextension of En

nr

over

E(n) in the sense of Rognes [14]. In all cases, E∗ = π∗(E) is a local ring with

maximal ideal induced from that of E(n)∗, and we will write m for this. The

residue ﬁeld E∗/m is always a graded subﬁeld of the algebraic closure

¯

F p[u,

u−1]

of

Fp[u,

u−1].

Remark 5.1. Since all of the spectra considered here are 2-periodic we will

sometimes treat their homotopy as Z/2-graded and as it is usually trivial in odd

degrees, we will often focus on even degree terms. However, when discussing reduc-

tions modulo a maximal ideal, it is sometimes more useful to regard the natural

periodicity as having degree 2(pn − 1) with associated Z/2(pn − 1)-grading; more

precisely, we will follow the ideas of [4] and consider gradings by

Z/2(pn

− 1) to-

gether with the non-trivial bilinear pairing

ν :

Z/2(pn

− 1) ×

Z/2(pn

− 1) −→ {1, −1};

ν(¯

i,

¯)

j =

(−1)ij,

where

¯

i denotes the residue class i (mod 2(pn − 1)).

15

Now we recall some facts from [13, lemma A1.1.13] about the extension of Hopf

algebroids

(D, Φ) −→ (A, Γ) −→ (A, Γ ),

where Γ is the Hopf algebra associated to Γ and Φ is unicursal. We have the

following identiﬁcations:

Γ = A ⊗Φ Γ, D = A ΓA, Φ = A ⊗D A.

The map of Hopf algebroids Γ −→ Γ is normal and

Φ = Γ

Γ

A = A

Γ

Γ ⊆ Γ.

Furthermore, for any left Γ-comodule M, A

Γ

M is naturally a left Φ-comodule

and there is an isomorphism of A-modules

(4.2) A

Γ

M

∼

=

A

Φ

(A

Γ

M).

Proposition 4.4. Let M be a Γ-comodule. If when viewed as a Γ -comodule, M

has non-trivial primitive Γ -subcomodule A

Γ

M, then the primitive Γ-subcomodule

A ΓM is non-trivial.

Proof. Combine Lemma 4.1 and (4.2).

Our next result is immediate.

Theorem 4.5. Let (k, Γ) is a Hopf algebroid over a ﬁeld. If the associated

Hopf algebra (k, Γ ) is unipotent, then (k, Γ) is unipotent.

5. Lubin-Tate spectra and their Hopf algebroids

In this section we will discuss the case of a Lubin-Tate spectrum E and its

associated Hopf algebroid (E∗,E∗ ∨E), where E denotes any of the 2-periodic spectra

lying between E(n) (by which we mean the 2-periodic version of the completed

2(pn

− 1)-periodic Johnson-Wilson spectrum E(n)) and En

nr

discussed in [2], see

especially section 7. The most important case is the ‘usual’ Lubin-Tate spectrum

En for which

π∗(En) = W Fpn

[[u1,...,un−1]][u±1],

but other examples are provided by the K(n)-local Galois subextension of En

nr

over

E(n) in the sense of Rognes [14]. In all cases, E∗ = π∗(E) is a local ring with

maximal ideal induced from that of E(n)∗, and we will write m for this. The

residue ﬁeld E∗/m is always a graded subﬁeld of the algebraic closure

¯

F p[u,

u−1]

of

Fp[u,

u−1].

Remark 5.1. Since all of the spectra considered here are 2-periodic we will

sometimes treat their homotopy as Z/2-graded and as it is usually trivial in odd

degrees, we will often focus on even degree terms. However, when discussing reduc-

tions modulo a maximal ideal, it is sometimes more useful to regard the natural

periodicity as having degree 2(pn − 1) with associated Z/2(pn − 1)-grading; more

precisely, we will follow the ideas of [4] and consider gradings by

Z/2(pn

− 1) to-

gether with the non-trivial bilinear pairing

ν :

Z/2(pn

− 1) ×

Z/2(pn

− 1) −→ {1, −1};

ν(¯

i,

¯)

j =

(−1)ij,

where

¯

i denotes the residue class i (mod 2(pn − 1)).

15