INTRODUCTION TO POLYNOMIAL FORMAL GROUPS
AND HOPF ALGEBRAS
L. CHILDS
ABSTRACT.
The purpose of this paper is to explain why studying polynomial formal groups
is of interest in the study of finite, local, commutative Hopf algebras over valuation
rings of local fields.
1991 Mathematics subject classification numbers: 14L15, 14L05
Let p be a prime number and let R be the valuation ring of a local field K
containing Qp. The unifying theme of this monograph is to construct classes of fi-
nite, p-elementary jR-group schemes, or equivalently, commutative, cocommutative,
jpower rank i?-Hopf algebras of exponent p.
Finite cocommutative Hopf algebras over K are, in principle, well-understood:
they are all if-forms of group rings of finite groups and are obtainable by Galois
descent. (A if-form of KG is a if-Hopf algebra so that L ®K H = LG for some
finite extension L of if.) In fact, the well-known correspondence between Galois
extensions of K and finite Gal(if/if)-sets [Wa79, §6.3] extends to the classifica-
tion of commutative, cocommutative, finite if- Hopf algebras and their principal
homogeneous spaces [By95, section 1].
However, over R, finite, commutative, cocommutative Hopf algebras are much
less well understood. We begin with a brief survey of known results about these
algebras.
The first systematic classification result was that of Tate and Oort [TO70], who
showed that i2-Hopf algebras of rank p are in bijective correspondence with
{b e R : b divides p}
R*p-X '
with b in R corresponding to Hb = R[x]/(xp bx). H^ is a Hopf order over R in the
dual (KG)*, where G is Cp, the cyclic group of order p, if and only if b G if
*p_1.
This classification implies that the number of jR-Hopf algebras of rank p increases
with [if : Qp].
The Tate-Oort classification was extended in the 1970's in two directions.
M. Raynaud [Ra74] generalized the methodology, as well as the classification,
of Tate-Oort, to construct a class of group schemes of exponent p and order pn
over JR, n 1, or equivalently, Hopf algebra orders over R in forms of KG for G
elementary abelian of order q pn. He assumed that the residue field k of R
contains ¥q. Raynaud algebras are jR-Hopf algebras H of rank q which admit an
action of ¥q. Then
R¥qx
acts on the kernel of the counit map e : H R, and,
Complete Memoir received by the editor March 11, 1996.
1
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