The structure of Sally modules and Buchsbaumness of associated graded rings (Q2873893)

Let \(A\) be a Noetherian local ring with maximal ideal \({m}\) and dimension \(d >0\). Let \(I\) be an \({m}\)- primary ideal and \(Q\) a parameter ideal contained in \(I\) as a reduction; that is \(Q \subset I\) and \(I^{n+1}=QI^n\) for some integer \(n\gg 0\). There exist integers \(\left\{ e_i(I)\right\}\) for \(0 \leq i \leq d\) such that NEWLINE\[NEWLINEl(A/I^{n+1})= e_0(I) {n+d \choose d} -e_1(I) {n+d-1 \choose d-1}+ \ldots + (-1)^de_d(I)NEWLINE\]NEWLINE where \(l_A(M)\) denotes the length of an \(A\)-module \(M\) and \(n\gg 0\). The \(\left\{ e_i(I)\right\}\) are called the Hilbert coefficients of \(A\) with respect to \(I\). If \(A\) is Cohen-Macaulay, then the following inequality NEWLINE\[NEWLINE2e_0(I)-e_1(I) \leq 2l_A(A/I)+ l_A(I/I^2+Q)NEWLINE\]NEWLINE was proven to be true, first by \textit{J. Elias} and \textit{G. Valla} [J. Pure Appl. Algebra 71, No. 1, 19--41 (1991; Zbl 0733.13007)] then by \textit{A. Guerrieri} and \textit{M. E. Rossi} [J. Algebra 199, No. 1, 40--61, Art. No. JA977194 (1998; Zbl 0899.13017)]. They also showed that having the equality is equivalent to having \(I^3=QI^2\) and \(Q \cap I^2= QI\).NEWLINENEWLINE When \(A\) is an arbitrary Noetherian local ring then the inequality NEWLINE\[NEWLINE2e_0(I)-e_1(I)+e_1(Q) \leq 2l_A(A/I)+ l_A(I/I^2+Q)NEWLINE\]NEWLINE also holds [\textit{A. Corso}, Commun. Algebra 37, No. 12, 4503--4515 (2009; Zbl 1188.13002)] and [\textit{M. E. Rossi} and \textit{G. Valla}, Hilbert functions of filtered modules. Lecture Notes of the Unione Matematica Italiana 9. Berlin: Springer (2010; Zbl 1201.13003)].NEWLINENEWLINE The author of the present paper investigates when the equality holds if \(A\) is not Cohen-Macaulay. He extends the results of Elias-Valla [loc. cit.] and Guerrieri-Rossi [loc. cit.], to an arbitrary Noetherian local ring. The equality is associated with the Buchsbaumness of the associated graded ring of \(I\). The proof relies on the Sally module of \(I\) with respect to \(Q\).