## 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

### Keywords:

Hopf orders; formal groups; descent; associated orders

### Citations:

Zbl 0631.14033; Zbl 0890.16021
Full Text:

### References:

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