Koszul property and finite linearity defect over $g$-stretched local rings
Do Van Kien, Hop D. Nguyen
TL;DR
The paper tackles Herzog–Iyengar’s question on whether finite linearity defect of the residue field forces Koszulness, proving a positive answer for Cohen–Macaulay local rings of almost minimal multiplicity with char $k=0$. It introduces and analyzes $g$-stretched rings (where $ rak m^2$ is principal) and develops a reduction to $ ext{dim }R\\le1$, a complete description of one-dimensional $g$-stretched rings via Matsuoka’s structure theorem, and a robust weak Koszul filtration framework to control linearity defects. A key outcome is a numerical characterization of $g$-stretched Koszul rings and partial progress toward Herzog–Iyengar’s question, with several consequences for absolute Koszulness and stability under completion. The results connect almost minimal multiplicity, dimension reduction, and explicit ring structures to derive Koszulness from finite linearity defect, offering new tools and open questions in the local Koszul theory.
Abstract
The linearity defect is a measure for the non-linearity of minimal free resolutions of modules over noetherian local rings. A tantalizing open question due to Herzog and Iyengar asks whether a noetherian local ring $(R,\mathfrak{m})$ is Koszul if its residue field $R/\mathfrak{m}$ has a finite linearity defect. We provide a positive answer to this question when $R$ is a Cohen-Macaulay local ring of almost minimal multiplicity with the residue field of characteristic zero. The proof depends on the study of noetherian local rings $(R,\mathfrak{m})$ such that $\mathfrak{m}^2$ is a principal ideal, which we call $g$-$stretched$ local rings. The class of $g$-stretched local rings subsumes stretched artinian local rings studied by Sally, and generic artinian reductions of Cohen-Macaulay local rings of almost minimal multiplicity. An essential part in the proof of our main result is a complete characterization of one-dimensional complete $g$-stretched local rings. Beside partial progress on Herzog-Iyengar's question, another consequence of our study is a numerical characterization of all $g$-stretched Koszul rings, strengthening previous work of Avramov, Iyengar, and Şega.