Compatible systems and ramification

TL;DR

This paper extends Deligne's theorem on compatible systems of $ ext{l}$-adic sheaves to higher-dimensional schemes, demonstrating compatibility along boundaries and ramification, and confirming $ ext{l}$-independence conjectures in equicharacteristic cases.

Contribution

It generalizes compatibility results of $ ext{l}$-adic sheaves to higher dimensions and boundary stratifications, and establishes ramification compatibility and $ ext{l}$-independence in new settings.

Findings

Compatibility along the boundary up to stratification.

Compatibility of ramification in compatible systems.

Validation of $ ext{l}$-independence conjectures in equicharacteristic cases.

Abstract

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.