Milne's correcting factor and derived de Rham cohomology II

TL;DR

This paper extends Milne's correcting factor, linking it to the Euler characteristic of derived de Rham cohomology, to a broader class of schemes over finite fields under certain resolution of singularities assumptions.

Contribution

It generalizes Milne's result from smooth projective varieties to arbitrary separated schemes of finite type over finite fields, assuming resolution of singularities.

Findings

Milne's correcting factor equals the Euler characteristic of derived de Rham cohomology modulo Hodge filtration.

Generalizes the interpretation of Milne's factor to schemes with singularities.

Connects Geisser's generalization of Milne's factor to $eh$-cohomology with compact support.

Abstract

Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field $\mathbb{F}_q$, is the Euler characteristic of the derived de Rham cohomology of $X/\mathbb{Z}$ modulo the Hodge filtration $F^n$. In this note, we extend this result to arbitrary separated schemes of finite type over $\mathbb{F}_q$ of dimension at most $d$, provided resolution of singularities for schemes of dimension at most $d$ holds. More precisely, we show that Geisser's generalization of Milne's factor, whenever it is well defined, is the Euler characteristic of the $eh$-cohomology with compact support of the derived de Rham complex relative to $\mathbb{Z}$ modulo $F^n$.