Homological algebra

homological algebra


nonabelian homological algebra


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For RR a ring and NN an RR-module which is finitely generated over RR on nn 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 RR be a ring, NN an RR-module generated on nn generators. Write

R nN R^n \to N

for the canonical projection from the free module over RR on nn generators to NN, which takes these generators to their image in NN.

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 RR, and hence the resulting exact sequence looks like

R n 1R nN R^{n_1} \to R^{n} \to N

relations/syzygies \to generators \to elements

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

R n 3R n 2R n 1R nN. \cdots \to R^{n_3} \to R^{n_2} \to R^{n_1} \to R^{n} \to N \,.


Hilbert’s syzygy theorem

For kk a field and R=k[x 1,,x n]R = k[x_1, \cdots, x_n] the polynomial ring over kk in nn variables, every finitely-generated RR-module has a free resolution of length at most nn.

Non-linear variants of the idea of syzygy are


  • homological syzygy?.

The idea of a homotopical nn-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 January 2, 2015 13:10:54 by Tim Porter (