Localization at a maximal ideal and quotients. 
If we have a commutative ring $R$ and a maximal ideal $m$, then is 
  $m/m^2$ isomorphic to $m_m/m^2_m$? 

Thx.

It is enough to show that $R/\mathfrak{m} \cong R_\mathfrak{m}/\mathfrak{m}_\mathfrak{m}$, since $\mathfrak{m}/\mathfrak{m}^2$ and $\mathfrak{m}_\mathfrak{m}/\mathfrak{m}_\mathfrak{m}^2$ are just the base changes of the $R$-module $\mathfrak{m}$ to these respective rings.
This is straightforward with universal properties.  $R\to R/\mathfrak{m}$ and $R\to R_\mathfrak{m} \to R_\mathfrak{m}/\mathfrak{m}_\mathfrak{m}$ are both universal with respect to maps that send $\mathfrak{m}$ to $0$ and $R\setminus \mathfrak{m}$ to invertible elements.
