Minimal graded free resolution ($F_{\bullet }$)

Over $S=k[X_1,\dots ,X_n]$ with its standard grading and $\mathfrak{m}=(X_1,\dots ,X_n)$, a graded free resolution is minimal when $d_i(F_i)\subseteq \mathfrak{m}{F}_{i-1}$ for every $i\ge 1$. It is constructed by choosing minimal homogeneous generators for each successive syzygy module, using the graded Nakayama lemma. After tensoring with $k=S/\mathfrak{m}$, all its differentials vanish, so $F_i/\mathfrak{m}{F}_i\cong {\mathrm{Tor}}_i^S(M,k)$; these dimensions and degrees are intrinsic to $M$.