4.7 Annihilator of a module

Lemma 4..9   Let $\;R=Loc_<K[x_1,\ldots , x_n]/(h_1,\ldots , h_p)\;$, $M\ \subseteq R^m$.
$Ann_R(R^m/M):=\left\{ \,g\in R\,\vert\:g R^m\subset M\,\right\}$ is generated by first entries of syzygies of the module

$$\left( \begin{array}{c\vert c\vert c\vert c\vert c} e_1 & M &... ... \ddots & 0\\ e_m & 0 & \cdots & 0 & M \\ \end{array}\right)$$

where e_i is the i-th unit vector in R^m.
(We identify a matrix with the module generated by it columns.)