Topics in higher ramification theory
Franz-Viktor Kuhlmann, Anna Rzepka
TL;DR
This paper advances higher ramification theory in positive residue characteristic by developing ramification ideals $I_E$ and their relation to defect, and then computing differents and traces in prime-degree Artin–Schreier and Kummer extensions. It develops methods to compute ramification ideals, traces, and differents, and links these to Kähler differentials via annihilators, showing that equality between differents and annihilators holds in classical cases but can fail in general. The authors classify defect scenarios, distinguishing defectless from defective extensions, and provide explicit generators for valuation rings in unibranched prime-degree extensions, together with detailed analyses of the role of valuation bases. They also examine the Kähler differentials across defectless and defect cases, connect them to ramification data, and discuss deeply ramified fields through the vanishing of differentials. Overall, the work yields concrete, computable invariants (ramification ideals, differents, traces, norms, and Kähler differentials) that illuminate the structure of extensions with and without defect and their ramification behavior, with implications for local uniformization and related valuation-theoretic problems.
Abstract
We introduce and study several notions in the setting of higher ramification theory, in particular ramification ideals and differents. After general results on the computation of ramification ideals, we discuss their connection with defect and compute them for Artin-Schreier extensions and Kummer extensions of prime degree equal to the residue degree, with or without defect. We present an example that shows that nontrivial defect in an extension of degree not a pime may not imply the existence of a nonprincipal ramification ideal. We compute differents for the mentioned extensions of prime degree, after computing the necessary traces, and discuss the question when they are equal to the annihilator of the Kähler differentials of the extension. Further, we introduce and study the ideal generated by the differents of the elements of the upper valuation rings in such extensions.