9 research outputs found
Degree Extensions of Arbitrary Valuation Rings and "Best "
No full textWe prove the explicit characterization of the so-called "best f" for degree
Artin-Schreier and degree Kummer extensions of Henselian valuation
rings in residue characteristic . This characterization is mentioned briefly
in [Th16, Th18]. Existence of best is closely related to the defect of such
extensions and this characterization plays a crucial role in understanding
their intricate structure. We also treat degree Artin-Schreier defect
extensions of higher rank valuation rings, extending the results in [Th16], and
thus completing the study of degree extensions that are the building blocks
of the general theory.Comment: 13 page
Ramification theory for degree p extensions of arbitrary valuation rings in mixed characteristic (0, p )
No full text
Upper Ramification Groups for Arbitrary Valuation Rings
No full textT. Saito established a ramification theory for ring extensions locally of
complete intersection. We show that for a Henselian valuation ring with
field of fractions and for a finite Galois extension of , the
integral closure of in is a filtered union of subrings of which
are of complete intersection over . By this, we can obtain a ramification
theory of Henselian valuation rings as the limit of the ramification theory of
Saito. Our theory generalizes the ramification theory of complete discrete
valuation rings of Abbes-Saito. We study "defect extensions" which are not
treated in these previous works.Comment: 44 page
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Ramification Theory for Arbitrary Valuation Rings in Positive Characteristic
No full textOur goal is to develop ramification theory for arbitrary valuation fields, that is compatible with the classical theory of complete discrete valuation fields with perfect residue fields. We consider fields with more general (possibly non-discrete)
valuations and arbitrary (possibly imperfect) residue fields. The “defect” case, i.e., the case where there is no extension of either the residue field or the value group, gives rise to many interesting complications. We present some new results for Artin-Schreier extensions of valuation fields in positive characteristic (\cite{V1}). These results relate the ``higher
ramification ideal" of the extension with the ideal generated by the inverses of Artin-Schreier
generators via the norm map. These are further related to K\" ahler differentials, which has been
shown in previous work of Kato and others to offer refined information about wild ramification
in the imperfect residue field case. We also introduce a generalization and further refinement of Kato's refined Swan conductor in this case. Similar results are true in the mixed characteristic case (\cite{V2})
Local Oort groups and the isolated differential data criterion
No full textIt is conjectured that if k is an algebraically closed field of
characteristic p > 0, then any branched G-cover of smooth projective k-curves
where the "KGB" obstruction vanishes and where a p-Sylow subgroup of G is
cyclic lifts to characteristic 0. Obus has shown that this conjecture holds
given the existence of certain meromorphic differential forms on P_1^k with
behavior determined by the ramification data of the cover. We give a more
efficient computational procedure to compute these forms than was previously
known. As a consequence, we show that all D_25- and D_27-covers lift to
characteristic zero.Comment: Minor edits, still 16 page
