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Nontrivial tame extensions over Hopf orders. (English) Zbl 1018.11056

Let \(l\) be a prime \(>3\) and \(K = Q(\zeta_{l^m})\) be the cyclotomic field generated by a primitive \(l^m\)-th root of unity, with ring of integers \(S\). Let \(\lambda = \zeta_{l^m} -1\), and \(C_l = \langle \sigma \rangle \) be the cyclic group of order \(l\). Suppose \(l\) satisfies Vandiver’s Conjecture. Let \(\Lambda_j\) be the Larson order of rank \(l\) in \(K[C_l]\), where \(\Lambda_j = S[\frac {\sigma -1}{\lambda^j}]\), where \(0 \leq j \leq l^m -1\). N. Byott [J. Algebra 177, 409-433 (1995; Zbl 0838.16009)] defined the notion of semilocal principal homogeneous space for a Hopf order over the ring of integers of a number field. The authors show that there is a semilocal principal homogeneous space \(M\) over the linear dual \(B_j\) of \(\Lambda_j\) that is not a free \(\Lambda_j\)-module. In particular, one can choose \(M\) to be the full ring of integers of a degree \(l\) Kummer extension \(L\) of \(K\). Semilocal principal homogeneous spaces for \(B_j\) are tame \(\Lambda_j\)-extensions, hence locally free, so this result is an extension to Larson orders over suitable cyclotomic fields of the converse of the Hilbert-Speiser Theorem of C. Greither, D. R. Replogle, K. Rubin and A. Srivastav [J. Number Theory 79, 164-173 (1999; Zbl 0941.11044)]. The authors also find non-free semilocal principal homogeneous spaces over certain Larson orders in \(K[C_p^2]\) that are also Raynaud orders. Their approach to these results is to consider the extension \(R(\Lambda)\) to Hopf orders by Byott [op. cit.] of L. McCulloh’s [J. Reine Angew. Math. 375/376, 259-306 (1987; Zbl 0619.12008)] set of realizable classes in the group \(Cl(\Lambda)\) of classes of locally free \(S[\Lambda]\)-modules, and extend a criterion of Greither et. al. [op. cit.] using Swan modules for the intersection of \(R(\Lambda)\) and the kernel group \(D(\Lambda)\) to be nontrivial.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
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