Milne's correcting factor and derived de Rham cohomology

TL;DR

This paper reveals that Milne's correcting factor, an important invariant in number theory, equals the Euler characteristic of the derived de Rham complex modulo the Hodge filtration, providing a new geometric interpretation.

Contribution

It establishes a direct link between Milne's correcting factor and the derived de Rham complex, offering a novel geometric perspective on this number-theoretic invariant.

Findings

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

Provides a geometric interpretation of Milne's correcting factor in terms of derived de Rham cohomology.

Connects number-theoretic invariants with algebraic geometric structures.

Abstract

Milne's correcting factor is a numerical invariant playing an important role in formulas for special values of zeta functions of varieties over finite fields. We show that Milne's factor is simply the Euler characteristic of the derived de Rham complex (relative to $\mathbb{Z}$) modulo the Hodge filtration.