My question is about a statement in Lang's Cyclotomic Fields, Ch. 8, $\S$2, although I've modified the notation a little.
Let $R$ be a complete discrete valuation ring with fraction field $K$, uniformizing parameter $\pi$, and residue field of finite order. Let $A$ be the Lubin-Tate formal group for $\pi$. Then we have the injective homomorphism $a\mapsto [a]$ of $R$ into $\text{End}(A)$. If $x\in\overline{K}$ is a primitive $\pi^n$-torsion point of $A$ then the map $a\mapsto [a]x$ induces an isomorphism $R/\pi^n\cong A[\pi^n]$. This isomorphism in turn induces an isomorphism $(R/\pi^n)^\times\cong\text{Gal}(K(A[\pi^n])/K)$. Letting $n$ go to infinity, we get an isomorphism $\phi:R^\times\xrightarrow{\sim}\text{Gal}(K(A_\text{tors})/K)$. What is this isomorphism explicitly?
I think that, for $y\in A[\pi^n]$, it should be $\phi(u)(y)=[u]y$ because $u\in R^\times$ corresponds to the $R$-module automorphism $a\mapsto ua$ of $R$ (or $R/\pi^n$) and this will induce $[a]x\mapsto [ua]x=[u]([a]x)$ on $A[\pi^n]$. However, Lang says that $u\in R^\times$ acts as $[u^{-1}]$. Why is this?
For background, the reason I'm interested is that Cassou-Nogues and Taylor, in Elliptic Functions and Rings of Integers cite this section of Lang to claim that the Artin symbol $(u,K)$ acts as $[u^{-1}]$ on torsion points of $A$. Don't know if that's relevant. Thanks!