L-COMPLETE HOPF ALGEBROIDS 17
The right unit generates a second copy of K• in K•E and there is an element
θ0 =
ηL(u)−1ηR(u)
which satisfies the relation
θ0n−1
p
= 1.
The coproduct makes θ0 group-like,
ψ(θ0) = θ0 θ0.
Now it is easy to see that K•E contains the unicursal Hopf algebroid
(5.1) K• ⊗Fp K• = F ⊗Fp F(u, θ0) = F ⊗Fp F[u,
θ0]/(upn−1
1,θ0
pn−1
1),
where θ0,u have degrees
¯,
0
¯
2 Z/2(pn 1) respectively.
Ignoring the generator u and the grading, we also have the ungraded Hopf
algebroid
(F, F ⊗Fp F(θ0)) = (F, F ⊗Fp F[θ0]/(θ0
pn−1
1))
which is a subHopf algebroid of (F,K¯E).
0
Since F is a Galois extension of Fp with Galois group a quotient of Z, we obtain
a ring isomorphism
F ⊗Fp F

=
F
Gal(F/Fp)∗.
If Gal(F/Fp) is finite this has its usual meaning, while if Gal(F/Fp) is infinite we
have
F
Gal(F/Fp)∗
=
Mapc(Gal(F/Fp),
F).
Of course, if Gal(F/Fp) is finite this interpretation is still valid but then all maps
Gal(F/Fp) −→ F are continuous. In each case, we obtain an isomorphism of Hopf
algebroids
(5.2) F ⊗Fp F(θ0)

= Map(Gal(F/Fp) Fpn
×
, F).
Now we consider the associated Hopf algebra over the graded field K•,
K• ⊗F⊗Fp
F(u,θ0)
K•E = K•[θk : k 1]/(θk
pn
θk : k 1)
whose zero degree part is
(5.3) F[θk : k 1]/(θk
pn
θk : k 1)

=
Mapc(Sn, 0
F).
The right hand side fits into the framework of Example 3.6, so this Hopf algebra
over F is unipotent. Tensoring up with K• we have the following graded version.
Theorem 5.4. The Hopf algebra (K•,K• ⊗F⊗Fp
F(u,θ0)
K•E) is unipotent.
Remark 5.5. The identification of (5.3) can be extended to all degrees of
K• ⊗F⊗Fp
F(u,θ0)
K•E. To make this explicit, we consider
Mapc(Sn,
Fur) with the
action of Gal(F/Fp) Fpn
×
induced from the action on Sn used in defining Gn
nr
and
the F-semilinear action of Gal(F/Fp) Fpn
×
on
Fur
obtained by inducing up the
r-th power of the natural 1-dimensional representation of Fpn
×
. Then
[K• ⊗F⊗Fp
F(u,θ0)
K•E]2r

=
Mapc(Sn,
Fur)Gal(F/Fp)Fpn
×
,
where the right hand side is the set of continuous Gal(F/Fp) Fpn
×
-equivariant
maps. This is essentially a standard identification appearing in work of Morava
and others in the 1970’s.
17
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