Rate and syzigies of modules over Veronese subrings

Let $K$ be a field, $R$ be a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, ${\text{rate}}_R(M)$, 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 $\rate_{R^{©}}(M)\leq \lceil \max{\rate_{R}(M),\rat®}/c\rceil+\max{0,\lceil t^{R}_{0}(M)/c\rceil},$forall$c\geq 1$. This extends a result of Herzog et al. As a consequence of this, if $M$ is generated in degree zero, then ${\text{reg}}_{R^{(c)}}(M)=0$, for all $c\ge \max \{{\text{rate}}_R(M),\text{rat}(R)\}$. Also, for powers of the homogeneous maximal ideal $\text{m}$ of $R$, it is shown that ${\text{rate}}_{R^{(c)}}({\text{m}}^s(s))\le \lceil \text{rat}(R)\mathrm{/}c\rceil$, for all $c\geq 1$. In particular case, we give a simple proof to a theorem of Backelin.