2011/11/30 by Paolo Mantero, Yu Xie, Mantero, Paolo +1
Mathematics · Medicine · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Intracranial Aneurysms: Treatment and Complications #math.AC
paper · pdf · doi:10.48550/arxiv.1112.0055
arxiv created 2011/11/30 · openalex publication_date 2011/11/30 · arxiv updated 2011/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 grI(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.