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 $M$, $\rate_R(M)$, 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.