For a ring and an -module which is finitely generated over on 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 be a ring, an -module generated on generators. Write
for the canonical projection from the free module over on generators to , which takes these generators to their image in .
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 , and hence the resulting exact sequence looks like
relations/syzygies generators elements
Continuing in this way yields, under suitable assumptions on , a projective resolution (actually a free resolution) of by syzygies and higher order syzygies.
Hilbert’s syzygy theorem
For a field and the polynomial ring over in variables, every finitely-generated -module has a free resolution of length at most .
Non-linear variants of the idea of syzygy are
The idea of a homotopical -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
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.