Induces a Weierstraß domain an injection of affinoid K-algebras?

Question

Let $X=$Sp$A$ be an affinoid $K$-space and $X(f_1,\ldots,f_r)=\{x \in X ; |f_i(x)| \leq 1\}$ for $f_i \in A$ be a Weierstraß domain in $X$.

The inclusion induces a canonical morphism of affinoid K-algebras

\begin{equation} A \rightarrow A\langle f_1,\ldots, f_r\rangle=A\langle T_1,\ldots,T_r\rangle/(T_i-f_i ; i=1, \ldots, r). \end{equation}

First I thought it is obviously an injection but after finding no literature mention this fact I got sceptical.

Now I am wondering if this morphism is an injection or rather can failed to be an injection?

Answer 1

this morphism is injective. by induction, you just have to check the map $A\to A<f>$ is injective. this map is the composition of two maps: $A\to A[f]\to A<f>$ the first map is injective because if $g\in A$ is in the kernel then we must have $g\in (T-f)$ but by checking the degrees you can see this is impossible.

the second map is injective because it's just the completion with respect to the topology induced from $A$ and this topology is Hausdorf.