# Do étale coordinates give rise to a regular sequence of diagonal elements?

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

• You meant to write: "... in $A \otimes A$", right? Also the subscripts seem to be wrong (they should probably run from $1$ to $n$).
– R.P.
Commented Jun 10 at 11:37
• @R.P. Yes, thank you! Commented Jun 12 at 14:16