Relationship between the depth of a local Cohen-Macaulay ring and its associated graded ring

Question

For a Noetherian local ring $(R, \mathfrak m)$ , let $\mathrm{gr}_{\mathfrak m} (R):= \oplus_{n \ge 0} \mathfrak m^n/\mathfrak m^{n+1}$ be the associated graded ring.

It is known that $\dim R=\dim gr_{\mathfrak m}(R)$ .

My question is: What is known about the relationship between $\operatorname {depth} R$ and $\operatorname {depth} \mathrm {gr}_{\mathfrak m} (R)$ ? I am especially interested in the case when either $R$ or $gr_{\mathfrak m}(R)$ is a Cohen-Macaulay ring.

It is easy to show that $\operatorname {depth} R>0$ if $\operatorname {depth} gr_{\mathfrak m} (R)>0$

Answer 1

In general, imposing the standard homological conditions on $\operatorname{gr}_{\mathfrak{m}}(R)$ is stronger than imposing those same conditions on $R$. For example:

1. If $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay, then so is $R$.
2. If $\operatorname{gr}_{\mathfrak{m}}(R)$ is Gorenstein, so is $R$.
3. If $\operatorname{gr}_{\mathfrak{m}}(R)$ is a complete intersection, so is $R$.

but all the converses are very far from true in general. There are many nice references for these; I'll point to "Connections between a Local Ring and Its Associated Graded Ring" by Fröberg since it has them all conveniently located in one place. https://core.ac.uk/download/pdf/82296230.pdf

An exception to the rule is when $R$ is regular, in which case $\operatorname{gr}_{\mathfrak{m}}(R)$ is also regular (this is easy to see/standard). As for your specific question, $\operatorname{depth}\operatorname{gr}_{\mathfrak{m}}(R)$ is very hard to control and, in general, requires strong hypotheses on $R$. For instance, Sally proved the following:

Suppose $R$ is Cohen-Macaulay and has minimal multiplicity, i.e., $e(R)=\mu_R(\mathfrak{m})-\dim R+1$. Then $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay.

But this fails to be true even taking one step away to Cohen-Macaulay rings of almost minimal multiplicity, i.e., assuming $e(R)=\mu_R(\mathfrak{m})-\dim R+2$. Sally considered these rings as well; she showed if such a ring $R$ is Gorenstein, then so is $\operatorname{gr}_{\mathfrak{m}}(R)$, and she showed $\operatorname{gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay as long as $R$ does not have the maximal possible type equal to $e(R)-2$. From this, she conjectured the following:

Suppose $R$ is Cohen-Macaulay and has almost minimal multiplicity. Then $\operatorname{gr}_{\mathfrak{m}}(R)$ is almost Cohen-Macaulay in the sense that $\operatorname{depth} \operatorname{gr}_{\mathfrak{m}}(R) \ge \dim R-1$.

It took 15 years, but this conjecture was eventually solved in the affirmative, independently, by Rossi-Valla and Wang, using very different techniques.

Beyond these the general case becomes intractable quickly, though there is a wealth of interesting research on understanding homological properties of associated graded rings/modules.

The book "Syzygies and Hilbert Functions" by Irena Peeva is a good reference for much of this discussion, especially the historical context of work on Sally's conjecture, and has references to Sally's papers.