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.