Short Koszul modules

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 $p{s}^d+q{s}^{d+1}$, 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 $H_R(s)=1+es+r{s}^2$ with $r\le e-1$, and R is Gorenstein or $e=r+1\le 3$, it is proved that generic R-modules with $q\le (e-1)p$ are linear.