Ramified Galois Covers via Monoidal Functors

Abstract

Abstract. We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame G-covers between normal varieties for finite and étale group schemes and we prove that, if G is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of G-covers is reducible.

Moduli of formal torsors

“…Notice that points of ∆ G over a eld L, namely G-torsors over L((t)), can also be seen as (not necessarily connected) Galois extensions of L((t)) and, taking integers, as special covers of L[[t]] with an action of G. It is therefore natural to ask and indeed this has been our initial approach to the problem, if one can dene a moduli space of special G-covers of B[[t]] for varying B or, more precisely, give a dierent moduli interpretation of G-torsors of B((t)) in terms of covers of B[[t]], in the spirit of [Ton17] and [Ton14]. We don't have a precise answer to this question, but in [TY19,Yas] we give partial answers.…”