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
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