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Suppose $F$ is a finite extension of $\mathbb Q_p$, and $X$ is a rigid variety over $F$. I saw in proposition 3.7 of Oswal, Shankar, Zhu, and Patel - A $p$-adic analogue of Borel's theorem: "Let $\mathbb L$ be a local system on $X$ with prime to $p$ monodromy".

What is the definition of "prime to $p$" monodromy?

I guess this is the definition: Let $\pi_1$ be the étale fundamental group of $X$, and let $$\pi_1'=\varprojlim_{\substack{\text{open normal $H$ such that} \\ \text{$\pi_1/H$ has prime to $p$ order}}}\pi_1/H,$$ and we say $\mathbb L$ has prime to $p$ monodromy if the action of $\pi_1$ on geometric fibers of $\mathbb L$ induces an action of $\pi_1'$ (since $\pi_1'$ is a quotient of $\pi_1$).

But I don't know whether this definition is true.

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  • $\begingroup$ What was the paper? $\endgroup$
    – LSpice
    Commented Mar 9 at 1:44
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    $\begingroup$ What does it mean for a definition to be untrue? Your definition is also what I would guess if I saw that statement (with no further helpful context). $\endgroup$
    – Will Sawin
    Commented Mar 9 at 1:56

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