I'm trying to understand a certain function on a specific adic space I'm stuck on something silly and is probably due to my lack of understanding of the points in this case. This is Proposition 3.3.5 of Scholze's MSRI lecture notes.


Let $A=\mathbb{Z}\_p [[T]]$, and let $Y=Spa(A,A)^{an}$ . Then one can define a unique continuous surjection $\kappa: Y \to [0,\infty]$ given by $\kappa(x)=\frac{\log |T(\tilde x)|}{\log |p(\tilde x)|}$ where $\tilde x$ is the maximal generalization of $x$. Now the fact that confuses me is that $\kappa(x)=0$ if and only if $|p(x)|=0$. I can see what $|p(X)|=0$ implies $\kappa(x)=0$, but why is the converse true? In other words why is $\log |T(\tilde x)| \in [-\infty,0)$ and not in $[-\infty,0]$? Why can I not have $|T(\tilde x)|=1$?.


Thanks



Relative position of elements in adic space
Presumably you are taking the $(p,T)$-adic topology on $A$. In this topology the element $T$ is topologically nilpotent, so for any $x \in \mathrm{Spa}(A,A)$ you have $|T(x)|<1$. 


