On primitive and realisable classes (Q2721347)

Let \(F\) be a number field with ring of integers \({\mathcal O}_F\). Let \(G = \text{ Spec} ({\mathcal B})\) where \({\mathcal B}\) is an \({\mathcal O}_F\)-Hopf algebra. Then \(G/{\mathcal O}_F\) is a finite, flat commutative group scheme. The isomorphism classes of twisted forms \({\mathcal C}\) of \({\mathcal B}\) are parametrised by \(H^1({\mathcal O} _F, G)\), the flat cohomology of \(\text{ Spec} ({\mathcal O}_F)\) with coefficients in \(G\). In many cases the twisted form \({\mathcal C}\) can be viewed as an order in the ring of integers of some extension of \(F\). The original motivation for this work was the study of the Galois module structure of the ring of integers. An important aspect of the theory of twisted forms is the class invariant homomorphism introduced by \textit{W. C. Waterhouse} [Trans. Am. Math. Soc. 153, 181-189 (1971; Zbl 0208.48401)]: NEWLINE\[NEWLINE \psi \colon H^1({\mathcal O}_F,G) \to \text{ Pic} (G ^D)NEWLINE\]NEWLINE given by \(\psi({\mathcal C}) = ({\mathcal C})( {\mathcal B})^{-1}\), where \(G^D\) is the Cartier dual of \(G\). NEWLINENEWLINENEWLINEThe paper is concerned with describing the image of \(\psi\). An element in \(\text{Pic}(G^D)\) is realisable if it is in the image of \(\psi\). \textit{L. N. Childs} and \textit{A. R. Magid} [J. Pure Appl. Algebra 4, 273-286 (1974; Zbl 0282.14015)] showed that every realisable class is primitive. Waterhouse raised the question if the converse is true. This work settles, in almost all cases, that Waterhouse's question has an affirmative answer. NEWLINENEWLINENEWLINEThe proof used the work of Childs and Magid and the work of \textit{L. Breen} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 339-352 (1975; Zbl 0313.14001)]. \textit{N. Byott} [J. Algebra 201, 284-316 (1998; Zbl 0911.16010)] has given a precise description of the realisable classes, but it does not seem to be easy to use this description to determine whether or not every primitive class is realisable. In the last section, the author describes some consequences of his result on torsion points of abelian varieties.