Motivic nearby cycles functors, local monodromy and universal local acyclicity

TL;DR

This thesis explores applications of Ayoub's motivic nearby cycles functor, including a generalization of Grothendieck's local monodromy theorem and the study of universal local acyclicity for motives, revealing new properties of the functor.

Contribution

It extends classical monodromy results to motives and demonstrates that universal local acyclicity can be characterized via motivic nearby cycles, with new structural insights into the functor.

Findings

Inertia acts quasi-unipotently on motives' étale cohomology.

Universal local acyclicity over 1D regular bases is detected by motivic nearby cycles.

Unipotent motivic nearby cycles form a direct summand with rational coefficients.

Abstract

In this thesis we give two applications of Ayoub's motivic nearby cycles functor: First we give a generalization of Grothendieck's classical local monodromy theorem. In the classical setup we show that the inertia group acts quasi-unipotently on the \'etale cohomology of sheaves 'coming from motives'. Second we study the notion of universal local acyclicity for motives and show that for \'etale motives universal local acyclicity over an excellent 1- dimensional regular base scheme is detected by the motivic nearby cycles functor. Along the way we prove properties of the motivic nearby cycles functor which might be of independent interest. In particular we show that with rational coefficients the unipotent motivic nearby cycles functor is a direct summand of the total motivic nearby cycles functor.