The right unit generates a second copy of K• in K•E and there is an element
which satisﬁes the relation
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,
where θ0,u have degrees
2 ∈ Z/2(pn − 1) respectively.
Ignoring the generator u and the grading, we also have the ungraded Hopf
(F, F ⊗Fp F(θ0)) = (F, F ⊗Fp F[θ0]/(θ0
which is a subHopf algebroid of (F,K¯E).
Since F is a Galois extension of Fp with Galois group a quotient of Z, we obtain
a ring isomorphism
F ⊗Fp F
If Gal(F/Fp) is ﬁnite this has its usual meaning, while if Gal(F/Fp) is inﬁnite we
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
(5.2) F ⊗Fp F(θ0)
= Map(Gal(F/Fp) Fpn
Now we consider the associated Hopf algebra over the graded ﬁeld K•,
K•E = K•[θk : k 1]/(θk
− θk : k 1)
whose zero degree part is
(5.3) F[θk : k 1]/(θk
− θk : k 1)
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
K•E) is unipotent.
Remark 5.5. The identiﬁcation of (5.3) can be extended to all degrees of
K•E. To make this explicit, we consider
Fur) with the
action of Gal(F/Fp) Fpn
induced from the action on Sn used in deﬁning Gn
the F-semilinear action of Gal(F/Fp) Fpn
obtained by inducing up the
r-th power of the natural 1-dimensional representation of Fpn
where the right hand side is the set of continuous Gal(F/Fp) Fpn
maps. This is essentially a standard identiﬁcation appearing in work of Morava
and others in the 1970’s.