Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal O_{Y,Q}=\mathcal O_{X,P}[T_1,\ldots,T_k]/(T_1^n-x_1,\ldots,T_k^n-x_k)$$
for certain $x_i\in\mathcal O_{X,P}$. In other words, we have adjoined certain $n$-th roots.
Now, let $R\subseteq Y$ be the ramification divisor and $H=\pi(R)$ the branch locus (both now with the reduced subscheme structure). Let $\mathcal I(H)$ be the ideal sheaf of $H$ and $\mathcal I(H)_P$ the stalk at $P$. Can I conclude that this ideal $\mathcal I(H)_P\subseteq \mathcal O_{X,P}$ is generated by the product $x_1\cdots x_k$? Or, correspondingly, can I conclude that $\mathcal I(R)_Q$ is generated by the product $y_1\cdots y_k$?
Edit. As pointed out in the answer by Dmitry Vaintrob, I want to assume $n$ not divisible by $\mathrm{char}(\Bbbk)$. Furthermore, we assume that the $x_i$ are reduced and coprime.
