Monogenic Hopf orders and associated orders of valuation rings

AbstractLet R be a complete discrete valuation ring of mixed characteristic with perfect residue field, and let H be a finite local commutative R-Hopf algebra. We consider when there exists a finite extension of the field of fractions of R, whose valuation ring is a Galois H-object. If this occurs then H is monogenic. Conversely, if H is also cocommutative and H is monogenic, then there exists a valuation ring which is a Galois H-object. To prove this result, we represent H as the kernel of an isogeny of a special type between formal groups over R. We deduce that if A is a finite abelian R-Hopf algebra, such that both A and its dual are local, then A is the associated order of a valuation ring if and only if the dual of A is monogenic