Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now consider the map
$\phi\otimes_{k}\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right] \otimes_{k} K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right] \to A\otimes_{k}A$.
Is the sequence $T_{1}\otimes 1 - 1\otimes T_{1},\dots,T_{n}\otimes 1 - 1\otimes T_{n} \in A\otimes_{k}A$ a regular sequence?
I am already struggling with the special case $n=1$ and $k=K$. Here is question is as follows. Write $T:=T_{1}$. Does the map $\phi\otimes_{k}\phi\colon K\left[T^{\pm}\right] \otimes_{K} K\left[T^{\pm}\right] \to A\otimes_{K}A$ send $T\otimes 1 - 1\otimes T$ to a non-zero divisor in $A$?
See stackexchange for the same question: https://math.stackexchange.com/questions/4930093/do-%c3%a9tale-coordinates-give-rise-to-a-regular-sequence-of-diagonal-elements
