On the first Local Cohomology module of a complete local ring of depth $1$

Question

Let $(R,\mathfrak m)$ be $\mathfrak m$-adically complete Noetherian local ring of depth $1$. Thus the local cohomology module $H^1_{\mathfrak m}(R)$ is a non-zero Artinian module.

My question is:

How to show that the module $H^1_{\mathfrak m}(R)$ is a finitely generated $R$-module if and only if the set $ \{P\in \mathrm{Ass}(R): \dim (R/P)=1\}$ is empty ?

I can show this if $R$ is Cohen-Macaulay, by using Grothendieck local duality, but otherwise, I have no idea in general.

Please help.

Answer 1

This doesn't really answer the main question, but it does give a related condition on when local cohomology is finitely generated.

Consider the following result:

Lemma. If $(R,\mathfrak{m})$ is a local ring, then an artinian $R$-module $N$ is finitely generated if and only if $\text{Att}(N)\subseteq\{\mathfrak{m}\}$.

We can combine this with the non-vanishing result of Macdonald and Sharp:

Theorem Let $(R,\mathfrak{m})$ be local and $M$ a $d$-dimensional non-zero finitely generated $R$-module. Then $H_{\mathfrak{m}}^{d}(M)\neq 0$ and $$\text{Att}(H_{\mathfrak{m}}^{d}(M))=\{ \mathfrak{p}\in\text{Ass}(M):\text{dim}\,R/\mathfrak{p}=d\}.$$

You can see that if $R$ is a (not necessarily complete) equidimensional local ring of dimension $d$, that the claim in the original question holds since $\{ \mathfrak{p}\in\text{Ass}(R):\text{dim}\,R/\mathfrak{p}=d\}$ will be all the minimal primes.

Information on attached primes can be found in Matsumura or in Brodmann and Sharp's book on Local Cohomology.

Do you have a reason to impose a depth, rather than dimension, condition on your ring, and do you actually know if the statement in the original question is true? I am not able to find a reference for it.