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 $\operatorname {depth} gr_{\mathfrak m} (R)>0$

  • $\begingroup$ It is not true that depth $R >0$ implies depth $gr_\mathfrak{m}(R)>0$. For instance, for a field $k$, $k[[t^4,t^5,t^{11}]]\cong k[[x_1,x_2,x_3]]/(x_1^4-x_2 x_3, x_2^3-x_1x_3, x_3^2-x_1^3 x_2^2)$ is a Cohen-Macaulay one-dimensional ring but its associated graded ring has depth zero. $\endgroup$
    – Francesco
    Oct 29, 2019 at 16:38
  • $\begingroup$ @Franceso: yeah I meant to say only one if ... good example $\endgroup$
    – user521337
    Oct 29, 2019 at 18:46

1 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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .