Pullback to the closed point of Z_l(1) on a henselian local ring

Question

Let $X=\mathrm{Spec}(A)$ be the spectrum of a henselian local ring and denote $\mathbb{Z}_l(1)$ the $l$-adic Tate twist on $X$, where $l$ is prime to the residual characteristic (let's say I work on the proétale site and define $\mathbb{Z}_l(1):=R\lim \mu_{l^n}$, if that makes sense ?). Denote also $i:x\to X$ the closed point of $X$. How do I go about determining $i^\ast \mathbb{Z}_l(1)$ ?

More generally, I would appreciate some references or explanations on how $\mathbb{Z}_l(1)$ on a scheme $X$ behaves under $f^\ast$ for a morphism $f:Y\to X$ (exluding the case where $f$ is étale; in that case $f^\ast \mathbb{Z}_l(1)=\mathbb{Z}_l(1)$).

Answer 1

The functor $i_{proet}^{\ast}$ commutes with limits by [1, cor. 6.1.5], so $$i_{proet}^\ast \mathbb{Z}_l(1)=i_{proet}^\ast\lim \nu^\ast \mu_{l^n}=\lim i^\ast_{proet}\nu^\ast \mu_{l^n} = \lim \nu^\ast i^\ast \mu_{l^n}= \lim \nu^\ast \mu_{l^n}=\mathbb{Z}_l(1)$$ where $\nu$ is the geometric morphism from the proétale to the étale topos, and the second-to-last equality holds because $\mu_{l^n}$ is étale as $l$ is invertible.

[1] Bhatt, Bhargav; Scholze, Peter, The pro-étale topology for schemes, Bost, Jean-Benoît (ed.) et al. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-805-3/pbk). Astérisque 369, 99-201 (2015). ZBL1351.19001.