The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory

Abstract

Let k be a field. In this article, we provide an extended version of the Drinfeld-Grinberg-Kazhdan Theorem in the context of formal geometry. We prove that, for every formal scheme V topologically of finite type over Spf(k[[T ]]), for every non-singular arc γ ∈ L∞(V )(k), there exists an affine noetherian adic formal k-scheme S and an isomorphism of formal k-schemes L∞(V )γ ∼= S ×k Spf(k[[(Ti)i∈N]]). We emphasize the fact that the proof is constructive and, when V is the completion of an affine algebraic k-variety, effectively implementable. Besides, we derive some properties of such an isomorphism in the direction of singularity theory.