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Let $k$ a field and $R= k[[X_1,X_2, ..., X_n]]$ the regular local ring of formal power series with maximal ideal $(X_1,X_2, ..., X_n)R$. How to prove that $R$ is a regular ring?

My attempts: $k$ is clearly regular, so firstly I tried to do it inductively, therefore to show following statement: If $A$ is local regular with max ideal $m_A$, then $A[[X]]$ is also local regular with obvious maximal ideal $(X)$.

But I don't know how to make here the induction step.

user267839
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    $R/\mathfrak{m}$ and $\mathfrak{m}/\mathfrak{m}^2$ are easily computed, so you just need to show that $R$ has the right Krull dimension. –  Mar 22 '18 at 15:42
  • Here I don't know how to prove that $dimA[[X]] = dim(A) +1$. By induction hypotheses $m_A$ is generated by $dim(A)$ elements as $k$-algebra, so $(X)$ is indeed generated by $dim(A)+1$ ements (as $k$ algebra) – user267839 Mar 23 '18 at 18:24

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