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ON PRIMITIVE AND REALISABLE CLASSES Version of January 11, 2000
 

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 )

  

Source: Agboola, Adebisi - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics