Compatible systems of $\ell$-adic sheaves

TL;DR

This paper introduces a new notion of compatibility for families of $ ext{ell}$-adic sheaves on schemes, showing it is preserved under key functors and establishing independence results for characteristic cycles.

Contribution

It defines a compatibility concept for $ ext{ell}$-adic sheaves and proves its stability under six functors, nearby cycles, and $ ext{ell}$-adic $ ext{epsilon}$-factors, with independence results for characteristic cycles.

Findings

Compatibility is preserved by six functors.

Compatibility is preserved by nearby cycles and epsilon-factors.

Characteristic cycles are independent of $ ext{ell}$ for compatible families.

Abstract

We introduce a notion of compatibility for families $(\mathcal{F}_{\ell})_{\ell}$ of bounded constructible $\ell$-adic complexes of \'etale sheaves on schemes. For schemes of finite type over a field, this notion is preserved by the usual six functors. We prove that the compatibility of a family is preserved by the nearby cycles functor and by the linearized $\varepsilon$-factors introduced recently by the author. We establish independence of $\ell$ for the characteristic cycles and characteristic $\varepsilon$-cycles of compatible families.