Arc Stability and Schemic Motivic Integration

TL;DR

This paper introduces a functorial approach to motivic integration for schemes, establishing a change of variables formula and stability properties, and defining motivic volume for schemes with smooth reduction.

Contribution

It provides a new, functorial framework for motivic integration, extending previous work and linking stability properties to integrability and motivic volume definitions.

Findings

Established a change of variables formula for motivic integration.

Proved schemes with smooth reduction have a natural motivic volume.

Connected stability properties to the integrability of motivic functions.

Abstract

We present a general, functorial approach to Motivic Integration for separated schemes of finite type in lieu of recent work by Hans Schoutens on the subject. Presented is a change of variables formula and a hierarchy of stability properties which are closely connected to the integrability of Motivic functions for schemes. Most notably, we prove that if a scheme has smooth reduction then it has a natural geometric definition of motivic volume in the Grothendick ring of the formal motivic site.