The equality of Elias-Valla and the associated graded ring of maximal ideals (Q456818)

Let \((A,\mathfrak{m})\) be a Noetherian local ring of dimension \(d > 0\), and let \(\{\mathrm{e}_i(I)\}\) denote the Hilbert coefficients of an \(\mathfrak{m}\)-primary ideal \(I\). Corso proved that, given a parameter ideal \(Q\) which forms a reduction for the maximal ideal, the following inequality holds: NEWLINE\[NEWLINE 2\mathrm{e}_0(\mathfrak{m})-\mathrm{e}_1(\mathfrak{m})+\mathrm{e}_1(Q) \leq v(A) -d+2, NEWLINE\]NEWLINE where \(v(A) = \ell_A(\mathfrak{m}/\mathfrak{m}^2)\) is the embedding dimension of \(A\). He also conjectured that, when equality holds and the ring \(A\) is Buchsbaum, then the associated graded ring \(G\) of \(\mathfrak{m}\) is Buchsbaum as well. The ring \(A\) is said to be Buchsbaum if \(\ell_A(A/\mathfrak{q})-\mathrm{e}_0(\mathfrak{q})\) is independent of the choice of the parameter ideal \(\mathfrak{q}\) in \(A\). The conjecture was known for \(1\)-dimensional rings (Corso) and \(1\)-dimensional Buchsbaum modules (Rossi and Valla).NEWLINENEWLINE\noindent In the main theorem, the author gives an affirmative answer to the conjecture for all positive dimensional rings. This also generalizes a previous result of \textit{J. Elias} and \textit{G. Valla} [J. Pure Appl. Algebra 71, No. 1, 19-41 (1991; Zbl 0733.13007)] on the Cohen-Macaulayness of \(G\).