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 $F^{n}$. 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 $F^{n}$.

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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