On the splitting principle for cohomological invariants of reflection groups

TL;DR

This paper proves Serre's splitting principle for cohomological invariants of finite orthogonal reflection groups over fields with characteristic coprime to the group order, extending to Witt- and Milnor-Witt K-theory invariants.

Contribution

It establishes the splitting principle for these invariants in new contexts, including Rost's cycle modules and Witt- and Milnor-Witt K-theory, under specific characteristic conditions.

Findings

Proves Serre's splitting principle for cohomological invariants with Rost's cycle modules.

Extends the principle to Witt- and Milnor-Witt K-theory invariants.

Validates the principle when the field characteristic is coprime to the group order.

Abstract

Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ with values in Rost's cycle modules (over $\mathrm{k}_{0}$) if the characteristic of $\mathrm{k}_{0}$ is coprime to $|W|$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants.