Syzygy (mathematics)

In mathematics, a syzygy (from Greek συζυγία 'pair') is a relation between the generators of a module M. The set of all such relations is called the "first syzygy module of M". A relation between generators of the first syzygy module is called a "second syzygy" of M, and the set of all such relations is called the "second syzygy module of M". Continuing in this way, we derive the nth syzygy module of M by taking the set of all relations between generators of the (n − 1)^th syzygy module of M. If M is finitely generated over a polynomial ring over a field, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert's syzygy theorem). The syzygy modules of M are not unique, for they depend on the choice of generators at each step.

The sequence of the successive syzygy modules of a module M is the sequence of the successive images (or kernels) in a free resolution of this module.

Buchberger's algorithm for computing Gröbner bases allows the computation of the first syzygy module: The reduction to zero of the S-polynomial of a pair of polynomials in a Gröbner basis provides a syzygy, and these syzygies generate the first module of syzygies.

Further reading

• Hazewinkel, Michiel, ed. (2001), "Syzygy", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Wiegand, Roger (April 2006), "WHAT IS...a Syzygy?" (PDF), Notices of the American Mathematical Society 53 (4): 456–457, retrieved 23 May 2011