Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois structures
This paper offers an interesting contribution to Galois module theory in the setting of Hopf Galois theory. The governing question is the following: Given a finite extension \(L/K\) of local fields and a \(K\)-algebra \(A\) acting on \(L\), under which conditions will the ring \(O_L\) of integers be free over the so-called associated order \(\mathfrak A=\{\alpha\in A \mid \alpha O_L\subset O_L\}\)? The classical setting is a \(G\)-Galois extension \(L/K\), and \(A=K[G]\). If \(L/K\) is tame, then the associated order \(\mathfrak A\) is exactly \(O_K[G]\), and Noether's theorem says that \(O_K\) is free over it. Childs' monograph [\textit{L. N. Childs}, Taming wild extensions: Hopf algebras and local Galois module theory. Mathematical Surveys and Monographs. 80. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0944.11038)] proved much more generally that we have freeness if \(\mathfrak A\) is a Hopf order in \(A=K[G]\). Without any hypothesis one should not expect an affirmative answer, there are counterexamples.NEWLINENEWLINEThe present paper generalises this result to Hopf Galois extensions \(L/K\) in the sense of Greither-Pareigis (see loc. cit. or the original paper in [\textit{C. Greither} and \textit{B. Pareigis}, J. Algebra 106, 239--258 (1987; Zbl 0615.12026)]). The Hopf algebra \(H\) which plays the role of \(A\) now is a form of a group ring. The extension \(L/K\) need not even be Galois in the classical sense. There are two cases that are captured: either \(L/K\) is unramified, or the degree of \(L/K\) is coprime to the residue characteristic and \(H\) is commutative. In both cases the author shows that the associated order is Hopf. Basically this is descent theory. In the latter case the associated order turns out to be the \textit{maximal} order. In the former case, the situation is just the opposite: if \(E/K\) is a Galois extension splitting \(H\) (so \(E\otimes_K H\) is a group ring \(E[N]\)), then \(\mathfrak A\) is gotten by taking the \(\text{Gal}(E/K)\)-fixed points of the \textit{minimal} order \(O_E[N]\).NEWLINENEWLINEIn the final section, the author uses standard techniques to combine the local results into the following global result: Let \(L/K\) be a domestic extension of number fields, which is \(H\)-Galois for a finite commutative \(K\)-Hopf algebra \(H\). Then \(O_L\) is locally free over the associated order. (An extension \(L/K\) is called \textit{domestic} if no prime of \(K\) lying over a divisor \(p\) of the degree \([L:K]\) is ramified in \(L\). Note that this is considerably more restrictive than tameness.)
scientific article; zbMATH DE number 6032696