Is the ring of formal power series in $n$ variables the colimit of the powers of its maximal ideal

Question

Let $\mathbb C[[t_1, ..., t_n]]$ be the noetherian local ring of formal power series in $n$ variables with a unique maximal ideal $\mathfrak{m}=(t_, ..., t_n)$. Then, there is a descending chain of ideals of $\mathbb C[[t_1, ..., t_n]]$ given by $(1)\supset\mathfrak{m}\supset\mathfrak{m}^2\supset\cdots\mathfrak{m}^p\supset\cdots$. It is a well-known fact that every module is the direct limit (colimit) of its finitely generated submodules. Is it true that $\mathbb C[[t_1, ..., t_n]]\cong colim_p\mathfrak{m}^p$ as $\mathbb C[[t_1, ..., t_n]]$-modules?

Answer 1

Yes, but for trivial reasons; I don't think this is the question you meant to ask. The colimit over any diagram $F : J \to C$ of shape $J$ such that $J$ has a terminal object $1$ is just the evaluation $F(1)$, which here is the unit ideal. So the same would be true of any descending chain of ideals starting at the unit ideal, in any ring.

There is a nontrivial statement which looks a bit like this, which is that $\mathbb{C}[[t_1, \dots t_n]]$ is the limit of the quotients $\mathbb{C}[[t_1, \dots t_n]/m^p$.