Rate and syzigies of modules over Veronese subrings

TL;DR

This paper investigates the growth of minimal free resolutions of modules over Veronese subrings, establishing bounds on their rate and regularity, and extends known results with new inequalities and proofs.

Contribution

It provides new bounds on the rate of modules over Veronese subrings and extends previous results, including a simplified proof of Backelin's theorem.

Findings

Bound on rate of modules over Veronese subrings.

If generated in degree zero, modules have zero regularity over Veronese subrings.

Simplified proof of Backelin's theorem.

Abstract

Let $K$ be a field, $R$ be a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, $\rate_R(M)$, is a measure of the growth of the shifts in the minimal graded free resolution of $M$. In this paper, we study the rate of Veronese modules of $M$. More precisely, it is shown that $\rate_{R^{(c)}}(M)\leq \lceil \max\{\rate_{R}(M),\rat(R)\}/c\rceil+\max\{0,\lceil t^{R}_{0}(M)/c\rceil\},$ for all $c\geq 1$. This extends a result of Herzog et al. As a consequence of this, if $M$ is generated in degree zero, then $\reg_{R^{(c)}}(M)=0$, for all $c\geq \max\{\rate_{R}(M), \rat(R)\}$. Also, for powers of the homogeneous maximal ideal $\m$ of $R$, it is shown that $\rate_{R^{(c)}}(\m^{s}(s))\leq \lceil \rat(R)/c\rceil$, for all $c\geq 1$. In particular case, we give a simple proof to a theorem of Backelin.