Hilbert syzygy theorem
= Hilbert syzygy theorem
{c}
{wiki=Hilbert's_syzygy_theorem}
Every <finitely generated module> with a grading over the <polynomial ring> $S=k[X_1,\ldots,X_n]$ has a finite graded <free resolution> of length at most $n$. The <Koszul resolution> of $k$ has length $n$, so $\operatorname{Tor}_i^S(M,k)=0$ for $i>n$. In a <minimal graded free resolution>, this <Tor functor> is $F_i/(X_1,\ldots,X_n)F_i$; the <graded Nakayama lemma> forces $F_i=0$ for $i>n$.