Local-global principle for triangularizability and diagonalizability of matrices

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.