Short Koszul modules

TL;DR

This paper investigates graded modules with linear resolutions over standard graded algebras, establishing conditions under which the algebra is Koszul and analyzing properties of modules with specific Hilbert series.

Contribution

It proves that modules with certain Hilbert series imply the algebra is Koszul and characterizes when modules with constant Betti numbers are linear.

Findings

If a module has Hilbert series of form p s^d + q s^{d+1}, then the algebra is Koszul.

Modules with constant Betti numbers lead to specific Hilbert series for the algebra.

Under certain conditions, generic modules are linear.

Abstract

This article is concerned with graded modules M with linear resolutions over a standard graded algebra R. It is proved that if such an M has Hilbert series $H_M(s)$ of the form $ps^d+qs^{d+1}$, then the algebra R is Koszul; if, in addition, M has constant Betti numbers, then $H_R(s)=1+es+(e-1)s^{2}$. When $H_R(s)=1+es+rs^{2}$ with $r\leq e-1$, and R is Gorenstein or $e=r+1\le 3$, it is proved that generic R-modules with $q\leq(e-1)p$ are linear.