The abelian fundamental group with modulus in mixed characteristic

TL;DR

This paper introduces a new definition of the abelian fundamental group with modulus for schemes over mixed characteristic bases, extending previous work in equal characteristic, and proves a Lefschetz-type theorem in this setting.

Contribution

It generalizes the abelian fundamental group with modulus to mixed characteristic schemes and establishes a Lefschetz theorem for semi-stable schemes.

Findings

Defined abelian fundamental group with modulus in mixed characteristic

Proved Lefschetz-type theorem for semi-stable schemes

Extended Kerz--Saito's framework to mixed characteristic

Abstract

We define the abelian fundamental group with modulus of a regular flat scheme over a discrete valuation ring, taking into account wild ramification along a divisor. Our definition provides a mixed-characteristic analogue of the abelian fundamental group with modulus introduced by Kerz--Saito for smooth schemes over a perfect field. In this setting, we prove a Lefschetz-type theorem for strictly semi-stable schemes: restriction to a hypersurface of sufficiently large degree relative to the ramification induces an isomorphism of the abelian fundamental groups.