# Jet closures and the local isomorphism problem

## TL;DR

This paper investigates whether isomorphisms of all local jet schemes imply the original morphism is an isomorphism, introducing jet-based closures to analyze the problem and providing partial solutions with both negative and positive results.

## Contribution

It introduces jet scheme-based closure operations and relates them to the local isomorphism problem, offering new insights and partial resolutions.

## Key findings

- Negative answer in general cases
- Positive results in specific geometric situations
- New closure operations among ideals

## Abstract

If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.