3
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.