Hopf orders and a generalization of a theorem of L. R. McCulloh
Let \(K\) be an algebraic number field with ring of integers \(R\) and \(N\) a tame finite normal extension of \(K\) with Galois group \(G\). Assume that \(G\) is the additive group of a finite field \(F\). The multiplicative group \(C=F\setminus\{0\}\) of \(F\) operates on \(G\), and let \(I\) be the Stickelberger ideal in \(\mathbb{Z} C\). The integral closure \(S\) of \(R\) in \(N\) represents an element \([S]\) of the locally free class group \(\text{Cl}(RG)\), and a theorem of McCulloh says that these elements form a subgroup of \(\text{Cl}(RG)\) which is explicitly given in terms of \(I\). The author generalizes this result to orders \(X\) which he calls ``semilocal principal homogeneous spaces''. He observes that \(N\) is tame over \(K\) if and only if at all primes \(P\ni\text{char }F\) of \(K\), the ring \(S_P\) is a principal homogeneous space over the dual Hopf order \(\text{Map}(G,R)\). Therefore, he starts with any Hopf order \(A\) in \(KG\) and considers \(R\)-orders \(X\) such that the \(X_P\) are principal homogeneous spaces over the dual Hopf order \(B\) of \(A\). The principal homogeneous spaces over \(B\) form an abelian group which coincides with the flat cohomology group \(H^1(\text{Spec }R,\text{Spec }B)\). In analogy with \textit{M. J. Taylor}'s class invariant homomorphism [Ill. J. Math. 32, No. 3, 428-452 (1988; Zbl 0631.14033)], the author defines a class \(\psi(X)\in\text{Cl}(A)\). Assuming that \(A\subset KG\) is invariant under \(C\subset\text{Aut }KG\), he proves that \(\text{Im }\psi=I\cdot\text{Cl}_0 (A)\), where \(\text{Cl}_0(A)\) denotes the kernel of the natural map \(\text{Cl }A \to \text{Cl }R\) induced by the augmentation of \(A\).
scientific article; zbMATH DE number 823511