I am having trouble seeing why the following two isomorphisms should hold for a Noetherian ring $A$ and ideals $I$,$J$ of $A$:

  1. $$\varprojlim A/(I+J)^n \cong \varprojlim A/I^n+J^n$$
  2. $$\varprojlim_m (\varprojlim_n A/(I^n + I^m)) \cong \varprojlim A/(I^n + J^n)$$

Any help or comments will be greatly appreciated.

  • $\begingroup$ Note that $I^n+J^n\subseteq (I+J)^n$ for every $n$; therefore, there is a natural map from $A/(I^n+J^n)$ to $A/(I+J)^n$ for every $n$, and these maps commute with the relevant structure maps; therefore, you have a natural map from $\varprojlim A/(I^n+J^n)$ to $\varprojlim A/(I+J)^n$; perhaps you can show that this is an isomorphism? Or that $\varprojlim A/(I^n+J^n)$ with the corresponding mappings into the $A/(I+J)^n$ has the appropriate universal property? $\endgroup$ Dec 5 '11 at 6:12

The isomorphism (1) comes from the fact that the systems $((I+J)^n)_n$ and $(I^n+J^n)_n$ define the same topology on $A$: $$ I^n+J^n \subseteq (I+J)^n, \quad (I+J)^{2n} \subseteq I^n+J^n.$$

Isomorphism (2) is more complicated (at least I don't have a simple proof). Fix an $m$. For any $n\ge m$, there is a canonical exact sequence $$ 0\to J^m/(J^m\cap (I^n+J^n)) \to A/(I^n+J^n)\to A/(I^n+J^m)\to 0.$$ Passing to the limit, we get a canonical exact sequence $$ 0 \to \varprojlim_n J^m/(J^m\cap (I^n+J^n)) \to \hat{A}:=\varprojlim_n A/(I^n+J^n)\to A_m:=\varprojlim_n A/(I^n+J^m)\to 0.$$ The exactness on the right comes from the surjectivity of $$J^m/(J^m\cap (I^{n+1}+J^{n+1}))\to J^m/(J^m\cap (I^n+J^n)).$$ As above, the $J^m\cap (I^n+J^n)$ define the same topology as the $J^m\cap (I+J)^n$. By Artin-Rees lemma, the latter define the same topology as the $J^m(I+J)^n$. So the lefthand side term in the above exact sequence is $(J^m)^{\hat{}}=J^m \hat{A}$. Therefore $A_m\simeq \hat{A}/J^m \hat{A}$. Passing to the limit we get $$\varprojlim_m A_m\simeq \hat{A}.$$


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