Parametrizing finite order automorphisms of power series rings

Basson, Dirk (Dirk Johannes) (2010-12)

Thesis (MSc (Mathematics))--University of Stellenboswch, 2010.

Thesis

ENGLISH ABSTRACT: In the work of Green and Matignon it was shown that the Oort-Sekiguchi conjecture is equivalent to a local question of lifting automorphisms of power series rings. The Oort-Sekiguchi conjecture asks when an algebraic curve in characteristic p can be lifted to a relative curve in characteristic 0, while keeping the same automorphism group. The local formulation asks when an automorphism of a power series ring over a field k of characteristic p can be lifted to an automorphism of a power series ring over a discrete valuation ring with residue field k of the same order as the original automorphism. This thesis looks at the local formulation and surveys many of the results for this case. At the end it presents a new theorem giving a Hensel's Lemma type sufficient condition under which lifting is possible.

AFRIKAANSE OPSOMMING: Green en Matignon het bewys dat die Oort-Sekiguchi vermoede ekwivalent is aan `n lokale vraag oor of outomorfismes van magsreeksringe gelig kan word. Die Oort-Sekiguchi vermoede vra of `n algebra ese kromme in karakteristiek p gelig kan word na `n relatiewe kromme in karakteristiek 0, terwyl dit dieselfde outomorfisme groep behou. Die lokale vraag vra wanneer `n outomorfisme van `n magsreeksring oor `n liggaam k van karakteristiek p gelig kan word na `n outomorfisme van `n magsreeksring oor `n diskrete waarderingsring met residuliggaam k, terwyl dit dieselfde orde behou as die aanvanklike outomorfisme. Hierdie tesis fokus op die lokale vraag en bied `n opsomming van baie bekende resultate vir hierdie geval. Aan die einde word `n nuwe stelling aangebied wat voorwaardes stel waaronder hierdie vraag positief beantwoord kan word.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/5243
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