# Approximating modules over complete local rings

Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Is the canonical functor $$\mathsf{Mod}(A) \to \varprojlim_n ~ \mathsf{Mod}(A/\mathfrak{m}^n),~ M \mapsto (M/\mathfrak{m}^n M)_n$$ an equivalence of categories? If not, does it hold when we restrict to finitely generated modules?

Edit: Meanwhile I've read this many times for finitely generated modules. This seems to be a well-known statement which is related to Grothendieck's existence theorem. Any elementary reference?

• I think the RHS can be identified with those $A$-modules with a "continuous" action. – Zhen Lin May 3 '14 at 23:33
• @Zhen Lin: Ok, this sounds reasonable ... – Martin Brandenburg May 4 '14 at 9:48

The answer is no in general. Let $$A=\mathbb Z_p$$ (which is complete local Noetherian with maximal ideal $$(p)$$), and let $$M = \mathbb Q_p$$. Then because $$\mathbb Q_p$$ is $$p$$-divisible, we have $$M / p^k = 0$$ for each $$k \in \mathbb N$$.
More generally, I suppose we're seeing that if $$A$$ is local with nonzero maximal ideal $$\mathfrak m$$, and if $$A$$ is a domain, then the field of fractions of $$A$$ is a nonzero module which is killed by this functor.