I would like to know if the following fact is true.
Let $K$ be a field, $S=K[x_1,\dots,x_n]$ be the standard polynomial ring, and $I$ be a graded ideal of $S$.
It's known that, up to isomorphism, there is only one minimal graded free $S$-resolution of $I$, $$ {\bf F}:0\rightarrow F_s\rightarrow F_{s-1}\rightarrow \dots \rightarrow F_0\rightarrow I\rightarrow 0, $$ with $F_i=\bigoplus_{j}S(-j)^{\beta_{i,j}}$, where $\beta_{i,j}$ are the graded Betti numbers of $I$.
My question is, given any other $S$-resolution of $I$, non-minimal, $$ {\bf G}:0\rightarrow G_r\rightarrow G_{r-1}\rightarrow \dots \rightarrow G_0\rightarrow I\rightarrow 0, $$ with $G_i=\bigoplus_{j}S(-j)^{\gamma_{i,j}}$, is it true that $\gamma_{i,j}\ge\beta_{i,j}$ for all $i,j\ge 0$?
If it is not true in general, is it true for some special kind of non-minimal resolution?
Does some references exist in the literature?