Localization in power series

Question

Let $A$ be a comm. ring with unity.

Say $\mathfrak{M}$ is a maximal ideal of $A[[X]]$. Is following statement generally true? \begin{align*} (A[[X]])_\mathfrak{M} \cong (A_{\mathfrak{M} \cap A}[[X]])_\mathfrak{M} \end{align*}

I don't see why it is wrong but can't show it either. Can someone give me a hint to construct a counter example/proof of the statement?

Answer 1

No, try with $A = \Bbb{Z}, \mathfrak{m} = (2,X)$.

Any element of $A[[X]]_{\mathfrak{m}}=(A[[X]]-\mathfrak{m})^{-1}A[[X]]$ is of the form $$\frac{g}{1+2n+Xf} = \frac{g}{2n+1}\sum_{k\ge 0} (-\frac{Xf}{2n+1})^k\tag{1}$$ with $f,g\in A[[X]],n\in \Bbb{Z}$.

When expanding the RHS of $(1)$ as a power series $\in \Bbb{Q}[[X]]$ you'll get an element of $\Bbb{Z}[\frac1{2n+1}][[X]]$.

On the other hand $A_{\mathfrak{m}\cap A}[[X]] $ contains $\sum_{k\ge 0} \frac{X^k}{2k+1}$ for which no such $n$ exists.