= Minimal graded free resolution
{title2=$F_\bullet$}
Over $S=k[X_1,\ldots,X_n]$ with its standard grading and $\mathfrak m=(X_1,\ldots,X_n)$, a graded <free resolution> is minimal when $d_i(F_i)\subseteq\mathfrak mF_{i-1}$ for every $i\ge1$. 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 $F_i/\mathfrak mF_i\cong\operatorname{Tor}_i^S(M,k)$; these dimensions and degrees are intrinsic to $M$.
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