Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $\hat R$ be the $\mathfrak m$-adic completion of $R$. $J$ be an ideal of $R$ with $\sqrt J=\mathfrak m$. Let $\hat J =J\hat R$ be the $\mathfrak m$-adic completion of $J$ regarded as an ideal of $\hat R$. I know that $\hat {J^n}={\hat J}^n, \forall n\ge 1$. My question is, are the following true:
(1) the completion map $R \to \hat R$ induces an isomorphism $R/J^n \to \hat R/\hat J^n, \forall n\ge 1$ ?
(2) the completion map $R \to \hat R$ induces an isomorphism $J^n/J^{n+1} \to \hat J^n/\hat J^{n+1}, \forall n\ge 0$ ?
[Note that question (2) just amounts to asking whether the associated graded rings $\oplus_{n\ge 0} J^n/J^{n+1}$ and $\oplus_{n\ge 0} \hat J^n/\hat J^{n+1}$ are canonically isomorphic or not ]
Note that any module of finite length is complete. In particular, $R/J^n \cong \widehat{R/J^n}$ and $J/J^{n+1} \cong \widehat{J/J^{n+1}}$. But completion is exact, so $\widehat{R/J^n} \cong \hat{R}/\hat{J}^n$ and $\widehat{J/J^{n+1}} \cong \hat{J}/\hat{J^{n+1}}$.