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Totally decomposable algebras with involution and R-triviality

M. Archita, Karim Johannes Becher

TL;DR

The paper proves that for a totally decomposable $K$-algebra with involution of the first kind with $ ext{ind}(A) ≤ 2$, the projective group of proper similitudes $PSim^+(A,σ)$ is $R$-trivial. It develops the framework of similarity factors $G(A,σ)$ and connects $R$-equivalence to norm data from extensions where the involution becomes hyperbolic, using quadratic-form and Pfister-form techniques. The main contribution handles non-split, orthogonal-involution cases (degree divisible by 8) by decomposing $(A,σ)$ into Pfister-twisted quaternion components and applying a Pfister-subform analysis to constrain similarity factors. These results extend prior rationality and $R$-triviality conclusions to a broader class of algebras with involution and provide a practical route to verify $R$-triviality via quadratic-form invariants.

Abstract

We show that the group of proper projective similitudes of a totally decomposable algebra with involution of the first kind over a field of characteristic different from 2 is R-trivial.

Totally decomposable algebras with involution and R-triviality

TL;DR

The paper proves that for a totally decomposable -algebra with involution of the first kind with , the projective group of proper similitudes is -trivial. It develops the framework of similarity factors and connects -equivalence to norm data from extensions where the involution becomes hyperbolic, using quadratic-form and Pfister-form techniques. The main contribution handles non-split, orthogonal-involution cases (degree divisible by 8) by decomposing into Pfister-twisted quaternion components and applying a Pfister-subform analysis to constrain similarity factors. These results extend prior rationality and -triviality conclusions to a broader class of algebras with involution and provide a practical route to verify -triviality via quadratic-form invariants.

Abstract

We show that the group of proper projective similitudes of a totally decomposable algebra with involution of the first kind over a field of characteristic different from 2 is R-trivial.
Paper Structure (4 sections, 13 theorems, 11 equations)

This paper contains 4 sections, 13 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Similarity factors
  3. Pfister forms
  4. The main result

Key Result

Theorem 1

If $(A,\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\operatorname{\mathsf{ind}} A\leqslant 2$, then ${\bf PSim}^+(A,\sigma)$ is $R$-trivial.

Theorems & Definitions (26)

  • Theorem : \ref{['main']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 16 more