Kummer-Artin-Schreier-Witt Theory

We study the problem of lifting the Artin–Schreier–Witt isogeny from characteristic $p>0$ to characteristic $0$, which is central to the lifting problem for Galois covers of algebraic schemes in positive characteristic. We introduce a new technique that associates a Kummer class, representing a tamely ramified cyclic extension, to a Witt vector via Matsuda’s Kummer–Artin–Schreier–Witt theory. This viewpoint leads to an explicit construction of a lift of the isogeny over a concrete base ring. Our results lay the groundwork for further applications, including the study of inseparable extensions and Kato’s refined Swan conductor.

💡 Research Summary

The paper addresses the long‑standing problem of lifting the Artin–Schreier–Witt isogeny, which classifies wildly ramified cyclic extensions in characteristic p, to characteristic 0 (mixed characteristic (0, p)). This problem is central to the broader question of lifting Galois covers of curves, especially for abelian groups, because every abelian cover can be obtained as a pull‑back of a suitable isogeny of algebraic groups. The authors focus on the cyclic case of order p^s and the exact sequence of group schemes
0 → ℤ/p^s → W_s → V_s → 0
over an algebraically closed field k of characteristic p. While Sekiguchi and Suwa proved the existence of a flat group scheme W_s over R = ℤ_{(p)}