Galois objects with normal bases for free Hopf algebras of prime degree

Abstract

Listen

Translate article

Suppose R is a commutative ring and H a commutative/cocommutative Hopf algebra which is a free R module of prime rank. The set of isomorphism classes of R algebras which are Galois H objects forms an abelian group (see Chase and Sweedler [a]). In this paper we show that when H is an algebra over a special subring of the p-adic integers containing a full set of (p 1)st roots of unity, then the subring consisting of those Galois objects which are isomorphic to H as H* module (H* = Hom,(H, R)) is isomorphic to UE~(R)/[U~(R)]~, where the notation U,(R) means units of R congruent to 1 modx and 5 is a (p 1)st root of an element b of R which will be seen to characterize the Hopf algebra H. The paper is organised as follows. We briefly review notation related to Hopf algebras, integrals, and Galois objects in Section 0. In Section 1 we present the description of Hopf algebras of order p given by Tate and Oort in their paper “Group Schemes of Prime Order” [S]. Such a Hopf algebra H is associated with a pair of constants a, b. In fact Hz R[x]/<x” ax) and H* x R[x]/( xp bx), where ab = wI, (some unit of R). In Section 2 we describe the H* module structure of H and the H* module structure of a Galois H object S, and we state what it means for S to have a normal basis (S z H as H* module). In Section 3 we assume R contains a (p l)st root of 6. We show RZ/pZ c H from which it follows that S contains a “Kummer order” C, modp S, with S, = R and SiS, = Si+i.