# Primes of the power series rings

Let $$A_n \colon= K[[X_1,\ldots,X_n]]$$ be a $$n$$-variable formal power series ring. By setting $$X_n \mapsto 0$$, we obtain a natural surjection $$\begin{equation*} \psi_{n,n-1} \colon A_n \twoheadrightarrow A_{n-1}. \end{equation*}$$

Choose a monic polynomial equation $$\begin{equation*} f(T) := T^e + a_{1}T^{e-1} + \ldots + a_{e-1}T + a_e, \end{equation*}$$ where $$a_1,\ldots,a_e \in (X_1,\ldots,X_n)A_n$$. We shall consider the finite extension $$B_n$$ over $$A_n$$ defined as follows$$\colon$$

$$B_n \colon= A_n[T]/(f(T))$$.

By setting $$B_{n-1} \colon= A_{n-1} \otimes_{A_n} B_n$$, we have the following surjection induced by $$\psi_{n,n-1}\colon$$ $$\begin{equation*} \phi_{n,n-1} \colon B_n \twoheadrightarrow B_{n-1}. \end{equation*}$$

Suppose that a prime ideal $${\frak P}_n$$ of $$A_n$$ satisfies the following surjection onto a prime ideal $${\frak P}_{n-1}$$ of $$A_{n-1}\colon$$
$$\begin{equation*} (\sharp) \quad \psi_{n,n-1} \colon {\frak P}_n \twoheadrightarrow {\frak P}_{n-1}. \end{equation*}$$

## $$\begin{equation*} \phi_{n,n-1} \colon {\frak Q}_n \twoheadrightarrow {\frak Q}_{n-1}~ ? \end{equation*}$$

• What do you mean by deleting $X_n$? Setting it to $0$? – Wojowu Mar 6 at 16:30
• Yes, I suppose that $X_n$ goes to $0$. – Rinmyaku Mar 6 at 18:49