Formal neighborhoods in arc spaces

TL;DR

This paper explores the structure of formal neighborhoods in arc spaces, providing proofs and discussing variants of a key theorem applicable to certain morphisms like smooth morphisms and embeddings.

Contribution

It offers a complete proof of the Greenberg-Kazhdan-Drinfeld theorem and discusses two versions for the relative case, expanding its applicability.

Findings

Proves the Greenberg-Kazhdan-Drinfeld theorem with examples.

Introduces two versions of the relative theorem.

Shows applicability to smooth morphisms and closed embeddings.

Abstract

The theorem of Greenberg-Kazhdan-Drinfeld describes the formal neighborhood of a closed arc. After giving a complete proof with examples, two possible versions for the relative case of the theorem are discussed. Each one is shown to hold for some classes of morphisms, including smooth morphisms and closed embeddings in smooth variety.