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Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and $p\mathbb{Z}_p [ X , X^{-1}]$ respectively. These rings can be realized as subrings of the Tate rings $\mathbb{Z}_p \langle X \rangle$ and $\mathbb{Z}_p \langle X , X^{-1} \rangle$ respectively.

If $f = \sum_{n \in \mathbb{Z}} a_n X^n$ is an element of $\mathbb{Z}_p \{ X , X^{-1}\}$, is it always true that the element $f_+ = \sum_{n \in \mathbb{N}} a_n X^n$ of $\mathbb{Z}_p \langle X \rangle$ is in $\mathbb{Z}_p \{ X\}$ ?

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  • $\begingroup$ Please could you give a reference for the definitions of $\mathbb{Z}_p\{X\}$ and $\mathbb{Z}_p\{X,X^{-1}\}$, or include them in the question? (I am not an expert, so am guessing slightly what Henselization means in this context.) Specifically, is is true that $(1-p(X+X^{-1})^{-1}$ is in $\mathbb{Z}_p\{X,X^{-1}\}$? $\endgroup$
    – Mark Wildon
    Commented Apr 6, 2016 at 19:23
  • $\begingroup$ You can look into Raynaud's "Anneaux henseliens". If $B$ is the integral closure of $\mathbb{Z}_p[X]$ in $\mathbb{Z}_p \langle X \rangle$, then $\mathbb{Z}_p \{ X\}$ is the localization of $B$ w.r.t $1 + B \cap p\mathbb{Z}_p \langle X \rangle$. The same construction applies to $\mathbb{Z}_p [X,X^{-1} ]$. In particular $(1-p(X+X^{-1})^{-1}$ is in $\mathbb{Z}_p\{X,X^{-1}\}$. $\endgroup$
    – js21
    Commented Apr 8, 2016 at 6:57
  • $\begingroup$ Should I understand that if $f = (1-p(X+X^{-1})^{-1}$, then $f_+$ is not in $\mathbb{Z}_p\{X\}$ ? $\endgroup$
    – js21
    Commented Apr 8, 2016 at 7:05
  • $\begingroup$ That was my guess, and it still looks plausible to me now. But I don't have a proof. $\endgroup$
    – Mark Wildon
    Commented Apr 8, 2016 at 10:41

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