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Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (of the same dimension) which makes $B$ integral over $A$.

Let $\mathfrak{m}=(x_1,\ldots,x_d)$ be a (fixed) regular system of parameters of $A$ and assume that there exist principal prime ideals $\mathfrak{q}_i\subset B$ such that $\mathfrak{q}_i\cap A=(x_i)$. By going up, $\sum_i\mathfrak{q}_i=\mathfrak{n}$. Is it possible to choose a regular system of parameters $\mathfrak{n}=(y_1,\ldots,y_d)$ for $B$ such that $\varphi(x_i)=y_i^{n_i}$ for certain integers $n_i$ and such that $y_i$ generats $\mathfrak{q}_i$? If not, can I add any conditions that will give me such a result?

2 fixed a typo

Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras which makes $B$ integral over $A$. If

Let $\mathfrak{m}=(x_1,\ldots,x_d)$ is be a (fixed) regular system of parameters of $A$, is A$and assume that there exist principal prime ideals$\mathfrak{q}_i\subset B$such that$\mathfrak{q}_i\cap A=(x_i)$. By going up,$\sum_i\mathfrak{q}_i=\mathfrak{n}$. Is it possible to choose a regular system of parameters$\mathfrak{n}=(y_1,\ldots,y_d)$for$B$such that$\varphi(x_i)=y_i^{n_i}$for certain integers$n_i$? n_i$ and such that $y_i$ generats $\mathfrak{q}_i$? If not, can I add any conditions that will give me such a result?

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# Local coordinate system under finite integral extension

Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras which makes $B$ integral over $A$. If $\mathfrak{m}=(x_1,\ldots,x_d)$ is a regular system of parameters of $A$, is it possible to choose a regular system of parameters $\mathfrak{n}=(y_1,\ldots,y_d)$ for $B$ such that $\varphi(x_i)=y_i^{n_i}$ for certain integers $n_i$? If not, can I add any conditions that will give me such a result?