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?

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

It seems plausible with the finitely-generated restriction.


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