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

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

• 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. Oct 29, 2019 at 16:38
• @Franceso: yeah I meant to say only one if ... good example Oct 29, 2019 at 18:46

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.