The quadratic Artin conductor of a motivic spectrum

TL;DR

This paper introduces a quadratic Artin conductor for motivic spectra over smooth proper schemes, providing a quadratic refinement of the Grothendieck-Ogg-Shafarevich formula through a new formula relating quadratic Euler characteristic, rank, and conductor.

Contribution

It defines the quadratic Artin conductor for motivic spectra and establishes a formula linking it with quadratic Euler characteristic and rank, refining classical results.

Findings

Established the quadratic Artin conductor for motivic spectra.

Proved a formula relating quadratic Euler characteristic, rank, and the quadratic Artin conductor.

Provided a quadratic refinement of the Grothendieck-Ogg-Shafarevich formula.

Abstract

Given a motivic spectrum $K$ over a smooth proper scheme which is dualizable over an open subscheme, we define its quadratic Artin conductor under some assumptions, and prove a formula relating the quadratic Euler characteristic of $K$, the rank of $K$ and the quadratic Artin conductor. As a consequence, we obtain a quadratic refinement of the classical Grothendieck-Ogg-Shafarevich formula.