The Koszul Property for Truncations of Nonstandard Graded Polynomial Rings
Caitlin M. Davis, Boyana Martinova
TL;DR
The paper studies truncations of nonstandard graded polynomial rings and proves that for every integer $e$, the associated graded module $\operatorname{gr_{\mathfrak m}}(S_{\ge e})$ has a linear $\operatorname{gr_{\mathfrak m}}(S)$-resolution, i.e. $S_{\ge e}$ is a nonstandard Koszul module. The authors develop an inductive, filtration-based approach that transfers linear resolutions from smaller (in the number of variables) pieces to the truncations, using the Horseshoe Lemma and a Koszul-type double complex to glue pieces together. This generalizes the standard-graded phenomenon that truncations have linear resolutions and connects to the behavior of Betti tables for nonstandard truncations. The results motivate conjectures about nonstandard Koszulness of nonstandard Veronese subrings and highlight open questions about extending these techniques to arbitrary modules and multigraded settings.
Abstract
We prove that truncations of nonstandard graded polynomial rings are (nonstandard) Koszul modules in the sense of Herzog and Iyengar. This provides an analogue of the fact that such truncations have linear resolutions in the standard graded case.