# 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 $\mathrm{Tor}^A_i(k,M) = \mathrm{Tor}^A_i(M,k) = 0$ 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 $i$.