Local-global principle for triangularizability and diagonalizability of matrices

TL;DR

The paper develops a local-global principle for triangularizability and diagonalizability of matrices in $ ext{M}_n( ext{O}_k)$ over number fields, proving the principle when $ ext{O}_k$ is a PID and identifying obstructions when it is not. It recasts the problem in terms of two arithmetic varieties, $X_M$ (triangularization) and $Y_M$ (diagonalization), and shows the Brauer--Manin obstruction governs diagonalizability, while a stratified Brauer--Manin obstruction is conjectured to govern triangularizability. The authors prove the diagonalization result at the level of connected components of $Y_{M, ext{red}}$, establishing weak approximation and showing that $M$ is diagonalizable over $k$ whenever these components admit $k$-points. In the triangularization case, they introduce and develop the stratified obstruction, verifying it in special Jordan-form-like cases and outlining a program for its general validity via fibrations and spreading-out techniques. Overall, the work provides a geometric framework linking linear algebra over rings of integers to modern obstruction theory in arithmetic geometry, with concrete results and conjectures guiding future study.

Abstract

Given a number field $k$ with the ring of integers $\mathcal{O}_k$ and a matrix $M\in \mathrm{M}_{n}(\mathcal{O}_k)$. We prove that if $\mathcal{O}_k$ is a principal ideal domain, the local-global principle for triangularizability and diagonalizability of $M$ holds. To explain the possible failures of the local-global principle, we prove that the stratified Brauer--Manin obstruction is the only obstruction to the local-global principle for triangularizability and diagonalizability of $M$ in some special cases.

Theorems & Definitions (52)

• Theorem 1.1: Theorem \ref{['bm']}
• Theorem 1.2: Theorem \ref{['th5.3']}
• Lemma 2.1: SZ14 and Lv20
• Lemma 2.2: Dem
• Lemma 2.3: HW16
• Remark 2.4
• Lemma 2.5: bgg
• Remark 2.6
• Lemma 3.1
• proof