On the Grinberg - Kazhdan formal arc theorem

TL;DR

This paper provides a simplified proof of the Grinberg-Kazhdan theorem on the structure of formal neighborhoods of arcs in algebraic varieties, removing the characteristic zero restriction.

Contribution

It offers a concise proof of the theorem applicable over arbitrary fields, extending its validity beyond characteristic zero.

Findings

The formal neighborhood decomposes into a smooth infinite-dimensional part and a finite type scheme.

The proof does not rely on the characteristic zero assumption.

The result broadens the applicability of the original theorem.

Abstract

Let X be an algebraic variety over a field k, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. Grinberg and Kazhdan proved that if k has characteristic 0 then the formal neighborhood of f in L(X) admits a decomposition into a product of an infinite-dimensional smooth piece and a piece isomorphic to the formal neighborhood of a closed point of a scheme of finite type. We give a short proof of this theorem without the characteristic 0 assumption.