What does the Hilbert Syzygy theorem state?

Question

Let $G$ be a finite group with $n$ elements acting on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation. And let $\mathbb{Q}[x_1,\cdots,x_n]^G = \mathbb{Q}[g_1,\cdots,g_m]$ . Then the $g_i$ might be the roots of some polynomials $s_j(y_1,\cdots,y_m) \in \mathbb{Q}[y_1,\cdots,y_m]$ , $j=1,\cdots,r$ called "syzygies". What does the Hilbert Syzygy theorem state in this situation?