Nigel P. Byott

Classes réalisables d'extensions non abéliennes

Abstract Let k be a number field and O k its ring of integers. Let Γ be a finite group, N/k a Galois extension with Galois group isomorphic to Γ, and O N the ring of integers of N. Let ℳ be a maximal O k -order in the semi-simple algebra k[Γ] containing O k [Γ], and Cl(ℳ) its locally free classgroup. When N/k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O N the class of , denoted , in Cl(ℳ). We define the set ℛ(ℳ) of realizable classes to be the set of classes c ∈ Cl(ℳ) such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to Γ, and for which . In the present article, we prove, by means of a fairly explicit description, that ℛ(ℳ) is a subgroup of Cl(ℳ) when , where V is an 𝔽2-vector space of dimension r ≧ 2, C a cyclic group of order 2 r − 1, and p a faithful representation of C in V; an example is the alternating group A 4. In the proof, we use some properties of the binary Hamming code and solve an embedding problem connected with Steinitz classes. In addition, we determine the set of Steinitz classes of tame Galois extensions of k, with the above group as Galois group, and prove that it is a subgroup of the classgroup of k.

Integral Galois module structure for elementary abelian extensions with a Galois scaffold

This paper justifies an assertion in [Eld09] that Galois scaffolds make the questions of Galois module structure tractable. Let k be a perfect field of characteristic p and let K = k((T )). For the class of characteristic p elementary abelian p-extensions L/K with Galois scaffolds described in [Eld09], we give a necessary and sufficient condition for the valuation ring OL to be free over its associated order AL/K in K[Gal(L/K)]. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

Some self-dual local rings of integers not free over their associated orders

Let p be a prime number, and let K be a finite extension of the rational p -adic field ℚ p . Let L/K be a finite abelian extension with Galois group G , and let L , K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG . Defining the associated order of the extension L/K by L can be viewed as a module over the ring , and a fortiori over the group ring K G .

Sufficient conditions for large Galois scaffolds

Abstract Let L / K be a finite, Galois, totally ramified p-extension of complete local fields with perfect residue fields of characteristic p > 0 . In this paper, we give conditions, valid for any Galois p-group G = Gal ( L / K ) (abelian or not) and for K of either possible characteristic (0 or p), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper [BCE] . But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] from characteristic p to characteristic 0. This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring O K that lie in K [ G ] for G an elementary abelian p-group.

ON THE MIXING PROPERTIES OF PIECEWISE EXPANDING MAPS UNDER COMPOSITION WITH PERMUTATIONS, II: MAPS OF NON-CONSTANT ORIENTATION

For an integer m ≥ 2, let 𝒫m be the partition of the unit interval I into m equal subintervals, and let ℱm be the class of piecewise linear maps on I with constant slope ±m on each element of 𝒫m. We investigate the effect on mixing properties when f ∈ℱm is composed with the interval exchange map given by a permutation σ ∈ SN interchanging the N subintervals of 𝒫N. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map fsf(x) = mx mod 1.

Two constructions with galois objects

Let G, H be Hopf algebras which are finitely generated projective modules over the ground ring R. We use the ideal of integrals of the dual Hopf algebra H* of H to give a new description of Gal(l)(A) where A ∈ Gal(H) and i: G → H is a Hopf algebra morphism which is both monk and epic in the category of finitely generated projective R-modules. We also give a new description of the group operation on Gal(H) for H cocommutative