Generalized stretched ideals and Sally's Conjecture

Given a finite module $M$ over a Noetherian local ring $(R, \m)$, we introduce the concept of $j$-stretched ideals on $M$. Thanks to a crucial specialization lemma, we show that this notion greatly generalizes (to arbitrary ideals, and with respect to modules) the classical definition of stretched $\m$-primary ideals of Sally and Rossi-Valla, as well as the notion of minimal and almost minimal $j$-multiplicity given recently by Polini-Xie. For $j$-stretched ideals $I$ on a Cohen-Macaulay module $M$, we show that ${\rm gr}_I(M)$ is Cohen-Macaulay if and only if two classical invariants of $I$, the reduction number and the index of nilpotency, are equal. Moreover, for the same class of ideals, we provide a generalized version of Sally's conjecture (proving the almost Cohen-Macaulayness of associated graded rings). Our work unifies the approaches of Rossi-Valla and Polini-Xie and generalizes simultaneously results on the (almost) Cohen-Macaulayness %and almost Cohen-Macaulayness of associated graded modules by several authors, including Sally, Rossi-Valla, Wang, Elias, Rossi, Corso-Polini-Vaz Pinto, Huckaba and Polini-Xie.