On Rees algebras of ideals and modules with weak residual conditions

Let $E$ be a module of projective dimension one over $R=k[x_1,\dots ,x_d]$. If $E$ is presented by a matrix $\phi$ with linear entries and the number of generators of $E$ is bounded locally up to codimension $d-1$, the Rees ring $\mathcal{R}(E)$ is well understood. In this paper, we study $\mathcal{R}(E)$ when this generation condition holds only up to codimension $s-1$, for some $s<d$. Moreover, we provide a generating set for the ideal defining this algebra by employing a method of successive approximations of the Rees ring. Although we employ techniques regarding Rees rings of modules, our findings recover and extend known results for Rees algebras of perfect ideals with grade two in the case that $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}E=1$.