 
Summary: ON PRIMITIVE AND REALISABLE CLASSES
A. Agboola
Version of January 11, 2000
Abstract. Let S be a scheme, and let G be a finite, flat, commutative group scheme over S. In
this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is
realisable. This gives an affirmative answer to a question of Waterhouse. We also discuss applications
to locally free classgroups and to Selmer groups of abelian varieties.
Introduction
Let F be a number field with ring of integers OF , and suppose that G=OF is a finite, flat,
commutative group scheme of exponent N . Then G = Spec(B), where B is an OF Hopf
algebra. Let H 1 (OF ; G) denote flat cohomology of Spec(OF ) with coefficients in G, and write
G D = Spec(A) for the Cartier dual of G. The group H 1 (OF ; G) parametrises isomorphism
classes of twisted forms of B. It may be shown that each twisted form C of B is a locally free
Amodule, and so determines a class (C) in the Picard group Pic(G D ) of G D .
In recent years, a large amount of work has been done concerning the Amodule structure of
twisted forms of B. The initial motivation for this work was the study of the Galois module
structure of rings of integers : in many cases, the twisted form C may be viewed as an order in
the ring of integers of some (in general wildly ramified) extension of F . An important aspect of
this theory is the class invariant homomorphism / which is defined by
/ : H 1 (OF ; G) \Gamma! Pic(G D )
