Suppose $S_j$ is a homogeneous syzygy of multidegree $\gamma_j$ in $S(G)$, where $G=\{g_1,\dots,g_t\}$. Show that $S_j G=\Sigma_{i=1}^{t} c_ix^{\alpha(i)}g_i$ has multidegree $< \gamma_j$.
Now, I express $S_j$ as $S_j=(c_1 x^{\alpha(1)},\dots,c_t x^{\alpha(t)})$. The condition for $S_j$ to have a certain multidegree $\gamma$ is that $\alpha(i)+multideg(g_i)=\gamma$ whenever $c_i \ne 0$; therefore, doing a simple observation of the terms, I obtain that $S_j G$ has multidegree exactly equal to $\gamma_j$. Where am I wrong?