Hopf algebras and local Galois module theory

by Lindsay N. Childs

This is a research area in algebra and algebraic number theory which has led to a half-dozen doctoral theses at Albany in the past 6 years.

Galois module theory centers around the idea of understanding the ring of integers S of an extension L of an algebraic number field K with ring of integers R as an H-module, where H is some appropriate order over R in a K-Hopf algebra A which measures L. In the classical case, L is a tamely ramified Galois extension of K with Galois group G, A is the group ring KG, and H = RG: then S is locally free of rank one as an RG-module: this is a theorem of Emmy Noether from the 1930's.

For Galois extensions of local fields which are wildly ramified (i.e. not tamely ramified), S is not a free RG-module, and the structure of S as an RG-module is difficult to describe. Thus any sort of reasonable Galois module theory for wild(ly ramified) extensions has been problematical for many years. The first good result in the area was by Leopoldt (l959). He replaced the group ring RG by the associated order of S in KG, namely, the set A of elements of KG which map S into itself. A is an order over R in KG, that is, an R-algebra which contains RG and is finitely generated as R-module, and for wild extensions A is larger than RG. Leopoldt showed that if K = Q, the rational numbers, and L is any abelian extension of Q (wild or tame), then S is free as an A-module, thereby generalizing the 19th century Hilbert-Speiser theorem for tame abelian extensions of Q.

Two basic results created an interesting new approach to wild extensions.

One is the discovery that if the associated order of S in L is a Hopf order in A, that is, is closed under the comultiplication on KG, then S is free of rank one as an H-module. This natural generalization of Noether's theorem first appeared for H an order in A = KG, G the Galois group of L/K, in a paper of Childs, "Taming wild extensions with Hopf algebras", Trans. Amer. Math. Soc., vol. 304 (1987), pp. 111-140, which built upon the 1984 Albany Ph. D. thesis of Susan Hurley (c.f. L. Childs and S. Hurley, Trans. Amer. Math. Soc., vol. 298 (1986), pp. 763-778). The most general version of Noether's theorem is in a paper of Childs and D. Moss, in "Advances in Hopf Algebras", ed. by J. Bergen and S. Montgomery, Marcel Dekker, 1994, pp. 1-24.

The other is the discovery by Greither and Pareigis (J. Algebra, vol. 106 (1987), pp. 239-258, that field extensions in characteristic zero may have many Hopf Galois structures: thus the A referred to three paragraphs above is in general far from unique. In fact, N. Byott has shown (in Comm. Algebra, 1996) that a Galois extension L/K with Galois group G has a unique Hopf Galois structure iff G is cyclic of order n where n and (Euler's phi function)(n) are coprime.

Research of Childs and his students and collaborators in recent years has been devoted to questions related to trying to understand some of the consequences of these discoveries. This research has largely centered on four problems:

Classification of Hopf algebras (of finite rank, usually abelian) over valuation rings of local number fields (c.f. doctoral theses of Underwood (1992), Koch (1995) and Smith (1997), as well as subsequent papers of Underwood, and papers of Childs and Zimmermann and the forthcoming Memoir by Childs, Greither, Moss, Sauerberg and Zimmermann).

Understanding the principal homogeneous spaces, or equivalently, the Hopf Galois extensions, for classes of Hopf algebras (c.f. the doctoral thesis of D. Moss (1994)).

Better understanding the classification of Hopf Galois structures on various kinds of extensions of local fields (c.f. doctoral theses of Kohl (1996) and Tse (1997); and

Seeking criteria for deciding if a given extension of local fields has an associated Hopf order in some Hopf algebra over which the extension is Hopf Galois (here the best recent work is that of N. Byott, at Exeter).

The last question is perhaps the ultimate goal, but in dealing with this goal it seems useful (and of intrinsic interest) to better understand the other three problems. The paper of Childs in volume 2 (1996) of the New York Journal of Mathematics (pp. 86-102), which deals with this last problem, illustrates the interplay of these questions.

December 8, 1997