6
$\begingroup$

Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology coming from the $I$-adic topology on $M$. I wonder how much this can be generalized to the case where $A$ is not necessarily commutative. For example, let $\mathcal{R}$ be the valuation ring of a complete, discretely valued non-archimedean field $K$, and let $\pi$ be a uniformizer of $\mathcal{R}$. Assume you have an almost-commutative filtered algebra $A$ over $\mathcal{R}$ which is two-sided noetherian, and complete with respect to the $\pi$-adic topology. Are there any conditions on $A$ which assure that the Artin-Rees lemma holds for the $\pi$-adic topology? Notice that the previous statement covers, for example, the case where $A$ is the universal enveloping algebra of a finitely generated Lie algebra over $\mathcal{R}$, and many other objects which have a PBW-type theorem.

$\endgroup$

1 Answer 1

3
$\begingroup$

There is some discussion of this in Rowen's "Ring Theory", volume I, Section 3.5, with additional references therein.

Exercise 19 on p. 462 in op. cit. states that a polycentral ideal $I$ of a noetherian ring $A$ has the Artin-Rees property, i.e., for every f.g. left module $M$ and a f.g. submodule $N\subseteq M$, there is $i\geq 1$ with $N\cap I^i M\subseteq IN$. (Applying this to $I^nN$ in place of $N$ shows that the $I$-adic topology on $N$ conicides with the topology induced from the $I$-adic topology of $M$.)

Here, an ideal $I$ is called polycentral if there is a chain of ideals $I=I_0\supseteq I_1\supseteq\dots\supseteq I_{t+1}=0$ such that $I_i = I_{i+1} +\sum_{r=1}^{s(i)} a_r A$ with $a_1,\dots a_{s(i)}\in A$ central modulo $I_{i+1}$. In particular, any ideal generated by central elements is polycentral.

In the context of your question, $I=\pi A$ is polycentral since it is generated by a central element, so it has the Artin-Rees property.

Actually, it is noted in op. cit. that the usual proof of the Artin-Rees lemma in the commutative case carries over in the non-commutative case for ideals generated by central elements.

$\endgroup$

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.