A Generalized Theorem of Katz and Motivic Integration

TL;DR

This paper extends Katz's theorem to motivic integration over generalized arc spaces, facilitating new approaches to cohomology studies of these spaces and broadening the scope of motivic volume calculations.

Contribution

It introduces a generalized theorem of Katz applicable to motivic integration on limit points beyond traditional settings, advancing the study of cohomology in generalized arc spaces.

Findings

Extended Katz's theorem to new motivic integration contexts

Developed methods for computing motivic volume in generalized arc spaces

Provided tools for cohomology analysis of these spaces

Abstract

In what follows, we are interested in an extension of a theorem of Nicholas Katz, which will be useful in studying the cohomology of generalized arc spaces develop by Hans Schoutens. As is well known, one is typically interested in the motivic volume of a definable subset of $\mathcal{X} \times X \times \mathbb{Z}^n$ where $\mathcal{X}$ is a scheme over $k((t))$ and $X$ the special fiber of $\mathcal{X}$ . Schoutens has introduced the possibility of developing a motivic integration for limit points other than $k[[t]]$.