Hilbert syzygy theorem

Name
This result is termed the Hilbert syzygy theorem.

Verbal statement
Minimal resolutions over a multivariate polynomial ring over a field of finitely generated graded modules have length bounded by the number of variables.

Symbolic statement
Let $$A = k[x_1, \dots, x_n]$$ be a polynomial ring over a field $$k$$ and $$M$$ a finitely generated graded $$A$$-module. Then there exists an exact sequence with degree 0 maps


 * $$ 0 \to F_n \to F_{n-1} \to \cdots \to F_1 \to F_0 \to M \to 0 $$

where the $$F_i$$ are free modules.

Proof idea
The key tools of the proof are the symmetry of Tor, and the Koszul complex.

The idea is that the Koszul complex of $$k$$ has length $$n$$ and hence $$\mathrm{Tor}^A_i(k,M) = \mathrm{Tor}^A_i(M,k) = 0$$ for $$i>n$$. Using this symmetry, one can also compute the Tor groups by tensoring a free resolution of $$M$$ by $$k$$. In particular, taking a minimal free resolution (it is easy to see that minimal free resolutions exist) $$F_\bullet$$ of $$M$$, the differentials become 0 upon tensoring with $$k$$ by definition of minimal resolution. Hence it follows that $$F_i = \mathrm{Tor}^A_i(k,M) = 0$$ for $$i>0$$.

Remarks
We are assuming that $$A$$ has the canonical grading, i.e., that the degree $$i$$ part is the vector space generated by the monomials of degree $$i$$.