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Lubin-Tate formal groups and module structure over Hopf orders. (English) Zbl 0979.11053

This article begins with a systematic account of the theory of Hopf order actions on rings of integers, following M. J. Taylor’s ideas [Ill. J. Math. 32, 428-452 (1988; Zbl 0631.14033)]. More precisely one has a Dedekind ring \(O_K\) of characteristic 0 with quotient field \(K\); \(\mathcal C\) is an integrally closed finite \(O_K\)-algebra, which is Hopf Galois under a finite commutative \(O_K\)-Hopf algebra \(\mathcal A\). Let \(A=K\mathcal A\), and \(C=K\mathcal C\). Then \(A\) becomes a group ring \(\overline K[G]\) on base extension with the algebraic closure \(\overline K\); at level \(K\), the whole situation can be described simply by \(\Omega= \text{Gal}(\overline K/K)\)-sets. (This is probably known to experts, and mentioned in a note of the reviewer [Commun. Algebra 24, 737-747 (1996; Zbl 0890.16021)]). If we now switch the point of view and begin with an \(A\)-Hopf Galois extension \(C\) over \(K\), we find that going back to level \(O_K\) cannot be done so canonically. There are three competing choices for the integral version of \(A\): \(\mathcal A\), the maximal order in \(A\); \({\mathcal A}^\circ = O_{\overline K}[G]^\Omega\); and finally \({\mathcal A}^{\text{ass}}\), the so-called associated order. For descending \(C\) one always takes \(\mathcal C\), the maximal order in \(C\). It is not clear when any of the three integral versions of \(A\) is a Hopf algebra. The authors prove the following remarkable result (Prop. 4.6): \({\mathcal A}^\circ\) is a Hopf algebra iff the maximal order in the dual Hopf algebra \(B\) of \(A\) has discriminant 1 (i.e. is etale). In this case, this maximal order and \({\mathcal A}^\circ\) are Cartier duals of each other. For \(G\) abelian, Prop. 4.8 shows that \(\mathcal A\) is Hopf iff it is etale; loosely speaking this means that the maximal order in a \(K\)-Hopf algebra is very rarely Hopf over \(O_K\). The authors also give information on the index of \({\mathcal A}^{\text{ass}}\) in \(\mathcal A\).
In Section 6 this general apparatus is applied to Hopf extensions afforded by Lubin-Tate formal groups. Theorem 6.11 establishes freeness of \(\mathcal C\) over \({\mathcal A}^{\text{ass}}\). The results of this section, which are of a local nature, have been applied to a global setting (elliptic curves) by W. Bley in his habilitation thesis (Augsburg 1998).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11S23 Integral representations
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References:

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