Artin-Schreier-Witt lifts of purely inseparable extensions
S. Srimathy
TL;DR
This work addresses the problem of lifting purely inseparable residue extensions in equicharacteristic discrete valued fields by introducing a novel framework of genomes ($\mathcal{G}$) and $\mathcal{G}$-weaves. It proves the existence and explicit construction of $\mathcal{G}$-cyclic extensions for any $\mathcal{G}$ and, for any purely inseparable modular $l/k$, of $\mathcal{G}$-cyclic lifts for every $l/k$-admissible $\mathcal{G}$, yielding Artin–Schreier–Witt lifts of arbitrary degree with residue $l/k$. The approach shows there is no cap on the wild ramification index in this equicharacteristic setting, in contrast to the mixed characteristic case, and provides a structure theorem for ASW extensions alongside tower and tensor-product lifting results. Practically, the results enable flexible and explicit construction of cyclic lifts over tensor products of ASW and purely inseparable modular extensions and answer questions raised in ramification theory surveys about possible ramification patterns.
Abstract
Given a discrete valued field $K$ of positive characteristic, we study the cyclic lifting problem of purely inseparable extensions of the residue field. We prove that unlike the mixed characteristic case, cyclic lifts of any finite purely inseparable modular extension exist and show how to explicitly construct them. Moreover, given such a residual extension, we prove the existence of Artin-Schreier-Witt lifts of any finite degree. This follows from a more general construction based on the notion of $\mathcal{G}$-weaves and $\mathcal{G}$-cyclic extensions where $\mathcal{G}$ is an arbitrary gene over $K$. In particular, this gives an affirmative answer to a question in \cite{ramification_survey} as well as implies that there is no cap on the wild ramification index unlike the mixed characteristic case. We also show some interesting applications such as constructing cyclic lifts of fields that are isomorphic to the tensor product of a purely inseparable modular extension and an Artin-Schreier-Witt extension. Finally, we prove a structure theorem of Artin-Schreier-Witt extensions over a discrete valued fields which restricts the ramification type of intermediate field extensions complementing the above results.