Summary: On Primitive and Realisable Classes
Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.
(Received: 1 July 1999; accepted: 15 February 2000)
Abstract. Let S be a scheme, and let Gbe a ˘nite, Łat, 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 af˘rmative answer to a question of Waterhouse. We also discuss
applications to locally free classgroups and to Selmer groups of Abelian varieties.
Mathematics Subject Classi˘cations (2000). 11Gxx, 11Rxx, 14Kxx.
Key words. class invariants, classgroups, realisable classes, primitive classes, group schemes,
Let F be a number ˘eld with ring of integers OF , and suppose that GaOF is a ˘nite,
Łat, commutative group scheme of exponent N. Then G peB, where B is
an OF -Hopf algebra. Let H1
OF Y G denote Łat cohomology of peOF with
coef˘cients in G, and write GD
peA for the Cartier dual of G. The group