I am a bit new to commutative algebra. It is an algorithmic question rather than mathematical.
Suppose that I have a graded polynomial ring $R = \oplus_{i \geq 0} R_i$ and a graded module $M=\oplus_{i \geq 0} M_i$ over $R$. According to Hilbert's syzygy theorem, I should stop resolving $M$ when the syzygy $\Omega_kM$ is graded free for $k \geq 1$: $$ 0 \rightarrow \Omega_1M \rightarrow F_0 \rightarrow M \\ 0 \rightarrow \Omega_2M \rightarrow F_1 \rightarrow \Omega_1M \\ \vdots \\ 0 \rightarrow \Omega_kM \rightarrow F_{k-1} \rightarrow \Omega_{k-1}M, $$ where $F_i$ is an appropriate free module, $\Omega_kM = \oplus_{i \geq 0} R(-d_i)$ (i.e. graded free over $R$), $d_i = \deg(\eta_i)$, and $\{\eta_i\}_i$ are basis.
My question is "How can we find that $\Omega_kM$ is graded free" in an algorithmic sense to stop calculating syzygies?
Edit: Link to the solution: syzygy_freeRes.pdf