Cohomologie \'etale des espaces d'arcs
Alexis Bouthier

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 -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 -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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology
