Monogenic Hopf algebras and local Galois module theory
Let \(K\) be an unramified extension of the \(p\)-adic rational numbers with valuation ring \(R\) and residue field \(k\). Let \(H\) be a monogenic, Abelian, local-local \(R\)-Hopf algebra. The author shows that \(H\) represents the kernel of an isogeny of one-dimensional formal groups over \(R\). Kummer theory of formal groups (c.f. the reviewer and \textit{D. J. Moss} [Advances in Hopf algebras, Lect. Notes Pure Appl. Math. 158, 1-24 (1994; Zbl 0826.16035)]) then yields a description of all field extensions \(L\) of \(K\) for which the valuation ring is a free module over the linear dual (over \(R\)) of \(H\). The author's method is to use Dieudonné module theory to show that any monogenic Abelian local-local Hopf algebra over \(k\) has a smooth resolution by one-dimensional \(k\)-formal groups, then, using Honda systems (c.f. \textit{J.-M. Fontaine} [C. R. Acad. Sci., Paris, Sér. A 280, 1273-1276, 1423-1425 (1975; Zbl 0331.14022, Zbl 0331.14023)]), shows that for such a Hopf algebra \(H\) over \(R\), a smooth resolution for \(H/pH\) over \(k\) can be lifted to a resolution of \(H\) by one-dimensional formal groups over \(R\).
scientific article; zbMATH DE number 1956742
scientific article