nLab
syzygy

Idea

For $R$ a ring and $N$ an $R$-module which is finitely generated over $R$ on $n$ generators, the syzygies are the relations between these generators.

Higher order syzygies are relations between these relations, and so forth.

Definition

Let $R$ be a ring, $N$ an $R$-module generated on $n$ generators. Write

$$R^n\to N$$R^n \to N

for the canonical projection from the free module over $R$ on $n$ generators to $N$, which takes these generators to their image in $N$.

The module of syzygies is the kernel of this morphism. This being a submodule of a free module it is itself free under suitable conditions on $R$, and hence the resulting exact sequence looks like

$$R^{n_1}\to R^n\to N$$R^{n_1} \to R^{n} \to N

relations/syzygies $\to$ generators $\to$ elements

Continuing in this way yields, under suitable assumptions on $R$, a projective resolution (actually a free resolution) of $N$ by syzygies and higher order syzygies.

$$\cdots \to R^{n_3}\to R^{n_2}\to R^{n_1}\to R^n\to N.$$\cdots \to R^{n_3} \to R^{n_2} \to R^{n_1} \to R^{n} \to N \,.

Properties

Hilbert’s syzygy theorem

For $k$ a field and $R=k[x_1,\cdots ,x_n]$ the polynomial ring over $k$ in $n$ variables, every finitely-generated $R$-module has a free resolution of length at most $n$.

Related concepts

• projective resolution

• Koszul complex

Non-linear variants of the idea of syzygy are

• homotopical syzygy?

• computad

and

• homological syzygy?.

The idea of a homotopical $n$-syzygy is discussed at

• higher dimensional szyzgy?

References

An exposition is in

• Roger Wiegand, What is… a syzygy? (pdf)

Lecture note discussion in a general context of projective resolutions in homological algebra includes

• E. L. Lady, A course in homological algebra – Syzygies, Projective dimension, regular sequences and depth (1997) (pdf)

and section 4.5 of

• Pierre Schapira, Categories and homological algebra, lecture notes (2011) (pdf)

Discussion in the context of the Koszul complex is in

• Yasuhiro Shimoda, On the syzygy part of Koszul homology on certain ideals, J. Math. Kyoto Univ. (1984) (EUCLID)

A useful source that discusses the use of syzygies in algebraic geometry,

• D.Eisenbud, The Geometry of Syzygies, Graduate Texts in Mathematics, vol. 229, Springer, 2005.