GEOMETRIC LOCAL -FACTORS IN HIGHER DIMENSIONS

Quentin Guignard

doi:10.1017/s1474748021000037

Abstract

AbstractWe prove a product formula for the determinant of the cohomology of an étale sheaf with $\ell $ -adic coefficients over an arbitrary proper scheme over a perfect field of positive characteristic p distinct from $\ell $ . The local contributions are constructed by iterating vanishing cycle functors as well as certain exact additive functors that can be considered as linearised versions of Artin conductors and local $\varepsilon $ -factors. We provide several applications of our higher dimensional product formula, such as twist formulas for global $\varepsilon $ -factors.

Highlights

Let k be a perfect field of positive characteristic p and let k be an algebraic closure of k
We prove a product formula for the determinant of the cohomology of anetale sheaf with l-adic coefficients over an arbitrary proper scheme over a perfect field of positive characteristic p distinct from l
The local contributions are constructed by iterating vanishing cycle functors as well as certain exact additive functors that can be considered as linearised versions of Artin conductors and local ε-factors

Summary

Introduction

Let k be a perfect field of positive characteristic p and let k be an algebraic closure of k. It is natural to ask whether it is possible to give a similar product formula determining the line det(RΓ(X,F ))−1 up to unique isomorphism, functorially in F , when the base field k is algebraically closed It will be clear from the proof of Theorem 1.2 that a positive answer to the latter question in the case X = P1k implies a positive answer for any proper k-scheme X. Let us, notice that in the twist formula 1.9, we can allow G to be a twisted local system, and this yields a more precise conclusion in the discussion following Theorem 1.9 regarding the commutation of the formation of Saito’s characteristic class with proper pushforward. The latter case is handled in 5.4 by choosing an arbitrary pencil and by applying to the latter the product formula from 5.1, as well as the functoriality properties from Section 3

1.13. Conventions and notation

Preliminaries

Nearby and vanishing cycles

Gabber-Katz extensions and examples of contractions