The Drinfeld-Grinberg-Kazhdan theorem and embedding codimension of the arc space

We prove an extension of the theorem of Drinfeld, Grinberg and Kazhdan to arcs with arbitrary residue field. As an application we show that the embedding codimension is generically constant on each irreducible subset of the arc space which is not contained in the singular locus. In the case of maximal divisorial sets, this relates the corresponding finite formal models with invariants of singularities of the underlying variety.