On the rate of graded modules

Let $K$ be a field, $R$ 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 find upper bounds for this invariant. More precisely, let $(A,\mathfrak{n})$ be a regular local ring and $I\subseteq \mathfrak{n} ^t$ be an ideal of $A$, where $t\geq 2$. We prove that if $(B=A/I, \mathfrak{m} =\mathfrak{n} /I)$ is a Cohen-Macaulay local ring with multiplicity $e(B)= \binom{h+t-1}{h}$, where $h=embdim(B)-dim B$, then $rat(gr_{\mathfrak{m}}(B))=t-1$ and for every $B$-module $N$, which annihilated by a minimal reduction of $\mathfrak{m}$, $rate_{gr_{\mathfrak{m}}(B)}(gr_{\mathfrak{m}}(N))\leq t-1$.