On Rees algebras of linearly presented ideals and modules

TL;DR

This paper investigates the algebraic structure of Rees algebras for linearly presented ideals and modules, extending previous results by relaxing certain conditions and employing generic Bourbaki ideals.

Contribution

It generalizes the description of Rees algebra defining ideals for linearly presented ideals and modules under weaker assumptions than prior work.

Findings

Determined the defining ideal of Rees algebra under relaxed codimension conditions.

Extended results to Rees algebras of linearly presented modules of projective dimension one.

Generalized previous work by P. H. L. Nguyen.

Abstract

Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let $\varphi$ denote its Hilbert-Burch matrix. When $\varphi$ has linear entries, the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood under the additional assumption that the minimal number of generators of $I$ is bounded locally up to codimension $d-1$. In the first part of this article, we determine the defining ideal of $\mathcal{R}(I)$ under the weaker assumption that such condition holds only up to codimension $d-2$, generalizing previous work of P.~H.~L.~Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one.