Auslander’s $\delta$-invariants of Gorenstein local rings

Abstract:

Let (R, $\mathfrak {m}$, k) be a Gorenstein local ring with associated graded ring $G(R)$. It is conjectured that for any integer $n > 0$, Auslander’s $\delta$-invariant $\delta (R/{\mathfrak {m}^n})$ of $R/{\mathfrak {m}^n}$ equals 1 if and only if ${\mathfrak {m}^n}$ is contained in a parameter ideal of R. In an earlier paper we showed that the conjecture holds if $G(R)$ is Cohen-Macaulay. In this paper we prove that the conjecture has an affirmative answer if depth $G(R) = \dim R - 1$ and R is gradable. We also prove that if R is not regular and depth $G(R) \geq \dim R - 1$, then $\delta (R/{\mathfrak {m}^2}) = 1$ if and only if R has minimal multiplicity.