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ON THE ASYMPTOTIC BEHAVIOR OF THE LINEARITY DEFECT

  • HOP D. NGUYEN (a1) (a2) and THANH VU (a3)

Abstract

This work concerns the linearity defect of a module $M$ over a Noetherian local ring $R$ , introduced by Herzog and Iyengar in 2005, and denoted $\text{ld}_{R}M$ . Roughly speaking, $\text{ld}_{R}M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. It is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$ -module $M$ , each of the sequences $(\text{ld}_{R}(I^{n}M))_{n}$ and $(\text{ld}_{R}(M/I^{n}M))_{n}$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence $(\text{ld}_{R}C_{n})_{n}$ where $C$ is a finitely generated graded module over a standard graded algebra over $R$ .

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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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