Cohomologie \'etale des espaces d'arcs

TL;DR

This paper proves a structure theorem for the arc space of a scheme, generalizing previous results and confirming a conjecture, then constructs a derived category of sheaves with finiteness properties.

Contribution

It establishes a global structure theorem for arc spaces, extending Drinfeld-Grinberg-Kazhdan's theorem and confirming a conjecture of Kollar and Nemethi.

Findings

Arc space locally modeled as product of infinite dimensional affine space and a stratified scheme.

Constructed a stable derived category of constructible sheaves on the arc space.

Proved a finiteness result for the cohomology of these sheaves.

Abstract

We establish a structure theorem on the arc space of a $k$-scheme of finite type. More precisely, we show that the arc space is locally for the pro-smooth toplogy a product of an infinite dimensional affine space and of a non-noetherian scheme, stratified by $k$-schemes of finite type, that glues the different formal completions of the arc space. This statement globalizes Drinfeld-Grinberg-Kazhdan's theorem and proves a conjecture of Kollar and Nemethi. Then, we use this theorem to construct a derived category of constructible sheaves, stable by six operations and we show a finiteness result for the cohomology.