# nLab syzygy

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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.

Similar definitions apply in non-additive contexts. In the theory of group presentations, 2-dimensional homotopical syzygies? are specific cellular representatives for identities among relations of a presentation of a group.

## Definition

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

$R^n \to N$

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

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

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.

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

## 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 $n$.

Non-linear variants of the idea of syzygy are

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

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.

Revised on January 2, 2015 13:10:54 by Tim Porter (127.0.0.1)