# 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

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

## See also

## References

- David Eisenbud,
*Commutative algebra. With a view toward algebraic geometry*. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 Template:MathSciNet - {{#invoke:citation/CS1|citation

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