syzygy

(also nonabelian homological algebra)

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.

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.

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

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?

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.

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