Milne's Correcting Factor and Derived De Rham Cohomology. II

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 $F^n$.