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Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized? I would be specially interested in the case where $L$ is an automorphism and $\mathcal{R}$ is the valuation ring associated to a complete non-archimedean field.

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  • $\begingroup$ Do you have any good criterion when the valuation ring is $\mathbb Z_p$? $\endgroup$
    – Z. M
    Commented Mar 24, 2022 at 10:38
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    $\begingroup$ It is necessary for the eigenvalues to be in $R$ and for the reduction $\overline{L} : (R/\mathfrak{m})^n \to (R/\mathfrak{m})^n$ to be diagonalisable. Also the projections onto the eigenspaces projections would need to lift. Therefore I would guess that at some point Hensel's Lemma will need to be invoked. But I'm not sure if a Henselian ring is really all that's needed to turn the necessary condition into a sufficient one. It's certainly sufficient to separate the eigenvalues that are not $\mathfrak{m}$-close to each other. But what happens to close eigenvalues, I do not know. $\endgroup$
    – Johannes Hahn
    Commented Mar 24, 2022 at 11:15
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    $\begingroup$ Aside to my earlier comment: Choosing the $U_\alpha$ in advance such that $U_\alpha={\cal R}^n\cap KU_\alpha$ and $\bigoplus_{\alpha} U_\alpha\neq {\cal R}^n$, and then taking the $\alpha$-s to be in $1+{\frak m}^n$ for $n$ sufficiently large (to guarantee that $L({\cal R}^n)\subseteq {\cal R}^n$) gives rise to non-diagonal $L$ such tha $L_{R/\frak m}$ is diagonalizable. This works for any discrete valuation ring $\cal R$. $\endgroup$
    – Uriya First
    Commented Mar 24, 2022 at 12:08
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    $\begingroup$ If $L$ has distinct eigenvalues modulo $\mathfrak{m}$, then it is diagonalisable over $\mathcal{R}$. By the way, people who work with modular forms care a lot about this kind of diagonalization problem (with $L$ being a Hecke operator) -- a very considerable chunk of Wiles' proof of Fermat's last theorem is about understanding the sizes of congruence ideals, which are precisely there to measure the failure of diagonalisability of operators on $\mathbf{Z}_p$-modules. $\endgroup$
    – David Loeffler
    Commented Mar 24, 2022 at 21:35
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    $\begingroup$ The intro of this paper by Ribet link.springer.com/content/pdf/10.1007/BF01393341.pdf is one reference for the modular forms angle. $\endgroup$
    – David Loeffler
    Commented Mar 27, 2022 at 17:25

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