Hilbert syzygy theorem

This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
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Name

This result is termed the Hilbert syzygy theorem.

Statement

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

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 $\mathrm {Tor} _{i}^{A}(k,M)=\mathrm {Tor} _{i}^{A}(M,k)=0$ 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 $F_{i}=\mathrm {Tor} _{i}^{A}(k,M)=0$ 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 $i$.