Milne's Correcting Factor and Derived De Rham Cohomology II

Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field F_q, is the Euler characteristic of the derived de Rham cohomology of $X/\{Z}$ modulo the Hodge filtration $Fⁿ$. In this note, we extend this result to arbitrary separated schemes of finite type over 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 Z modulo $Fⁿ$.

2010 Mathematics Subject Classification: 14G10, 14F40, 11S40, 11G25

Keywords and Phrases: Zeta functions, Special values, Derived de Rham cohomology, eh-cohomology

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