A valuation criterion for normal basis generators in equal positive characteristic

Thomas, Lara

doi:10.1016/j.jalgebra.2008.05.024

Cited by 11 publications

(17 citation statements)

References 6 publications

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“…The second author was supported by the European Union (European Social Fund -ESF) and Greek national funds through the Operational Program"Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) -Research Funding Program: THALIS. another active research area concerning local and finite fields, see the work in [6,50] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers

Karanikolopoulos

Kontogeorgis

2019

Trans. Amer. Math. Soc.

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We study p-group Galois covers X → P 1 with only one fully ramified point in characteristic p > 0. These covers are important because of the Harbater-Katz-Gabber compactification theorem of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero p-rank: we focus on curves that admit a big action. Moreover we initiate the study of the Galois module structure of polydifferentials.

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Section: Introductionmentioning

confidence: 99%

Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers

Karanikolopoulos

Kontogeorgis

2019

Trans. Amer. Math. Soc.

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“…In any case, our method is necessarily restricted to totally ramified extensions of ppower degree, since if L/K admits an A-scaffold, then it possesses a "valuation criterion": there is an integer b such that any element of L of valuation b is a free generator of L over A (see Proposition 2.12). This property, which can be viewed as a strong version of the Normal Basis Theorem, has been studied in a number of papers [BE07, Tho08,Eld10,Byo11,dSFT12], and can only hold when L/K is totally ramified and of p-power degree (see [dSFT12, Proposition 1.2] for the Galois case).…”

Section: Introductionmentioning

confidence: 99%

Scaffolds and generalized integral Galois module structure

Byott¹,

Childs²,

Elder³

2018

Ann. inst. Fourier

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Let L/K be a finite, totally ramified p-extension of complete local fields with residue fields of characteristic p > 0, and let A be a K-algebra acting on L. We define the concept of an A-scaffold on L, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where L/K was Galois and A = K[G] for G = Gal(L/K). When a suitable A-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in A. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with L/K. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois p-extensions in characteristic p. We also apply our results to the non-classical situation where L/K is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power K-Hopf algebra.

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“…In §2, we review the facts we shall need from Hopf-Galois theory, and prove several preliminary results in the case of p-extensions. These show, in effect, that the relevant Hopf algebras behave similarly to the group algebras considered in [9]. In §3 we develop some machinery to handle extensions whose degrees are not powers of p. In [4], such extensions were treated by reducing to a totally and tamely ramified extension.…”

Section: Introductionmentioning

confidence: 99%

“…We complete the proof of Theorem 2 in §4. The ramification groups, which play an essential role in the arguments of [4] and [9], are not available in the Hopf-Galois setting, but their use can be avoided by working directly with the inverse different. Finally, in §5, we give an example of a family of extensions which are not Galois, but to which Theorem 2 applies.…”

Section: Introductionmentioning

confidence: 99%

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Byott¹

2011

J. Théor. Nombres Bordeaux

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Let S/R be a finite extension of discrete valuation rings of characteristic p > 0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let D S/R be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element ρ of L with vL(ρ) ≡ −vL(D S/R ) − 1 (mod n) generates L as an Hmodule. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

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A valuation criterion for normal basis generators in equal positive characteristic

Cited by 11 publications

References 6 publications

Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers

Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers

Scaffolds and generalized integral Galois module structure

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

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