# Hilbert's syzygy theorem

In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890) in connection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it explains how far one has to go to reach a clarified situation. It is now considered to be an early result of homological algebra, and through the depth concept, to be a measure of the non-singularity of affine space.

## Formal statement

In modern language, the theorem may be stated as follows. Let k be a field and M a finitely generated module over the polynomial ring

${\displaystyle k[x_{1},\ldots ,x_{n}].}$

Hilbert's syzygy theorem then states that there exists a free resolution of M of length at most n.