Conductors and the moduli of residual perfection

Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A which agrees with the classical Artin conductor when the residue field is perfect. The definition rests on two results of perhaps wider interest: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space. Finally, I show that this conductor extends the non-logarithmic variant of Kato's conductor to representations of rank greater than one.