The quotient map on the equivariant Grothendieck ring of varieties

TL;DR

This paper establishes that the quotient map on the G-equivariant Grothendieck ring of varieties is well defined under certain conditions, extending previous results to wild actions and applying this to motivic nearby fibers.

Contribution

It proves the well-definedness of the quotient map on the equivariant Grothendieck ring for both tame and wild actions, generalizing existing results and applying to motivic invariants.

Findings

The quotient map is well defined in the tame case.

The quotient map is well defined in the wild case with a modified ring.

The quotient of the motivic nearby fiber is a well-defined invariant.

Abstract

For a scheme S with a good action of a finite abelian group G having enough roots of unity we show that the quotient map on the G-equivariant Grothendieck ring of varieties over S is well defined with image in the Grothendieck ring of varieties over S/G in the tame case, and in the modified Grothendieck ring in the wild case. To prove this we use a result on the class of the quotient of a vector space by a quasi-linear action in the Grothendieck ring of varieties due to Esnault and Viehweg, which we also generalize to the case of wild actions. As an application we deduce that the quotient of the motivic nearby fiber is a well defined invariant.