Motivic nearby cycles functors, local monodromy and universal local acyclicity

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.