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.
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.
Revised on September 24, 2012 21:28:52
by Urs Schreiber