Generalized-type lower bound in the second extremal Hilbert-coefficient case

Determine whether a lower bound for the generalized type \(\operatorname{type}_I(A)\) can be obtained, analogous to Theorem \ref{bd}, for a Cohen–Macaulay local ring \((A,\mathfrak m)\) and an \(\mathfrak m\)-primary integrally closed ideal \(I\) satisfying \(e_2(I)=e_1(I)-e_0(I)+\lambda(A/I)+1\), and determine whether equality in such a bound forces the associated graded ring \(G(I)\) to be Cohen–Macaulay.

Background

The paper establishes, in Theorem \ref{bd}, a lower bound for the generalized type $\operatorname{type}_I(A)$ when an integrally closed $\mathfrak m$-primary ideal satisfies the equality $e_2(I)=e_1(I)-e_0(I)+\lambda(A/I)$. Theorem \ref{typeq} further proves that equality in this lower bound implies that $G(I)$ is Cohen–Macaulay.

The unresolved question concerns the next extremal numerical case, $e_2(I)=e_1(I)-e_0(I)+\lambda(A/I)+1$. In this setting, the paper derives structural results involving the Hilbert coefficients, the Ratliff–Rush filtration, and Cohen–Macaulayness of associated graded rings, but does not establish an analogue of the generalized-type bound or determine whether equality in such a bound would imply Cohen–Macaulayness of $G(I)$.