Syzygy module (${\Omega }_R^i(M)$)

Given a free resolution $\cdots \to F_1\to F_0\to M\to 0$, its first syzygy module is $\ker (F_0\to M)$, the module of relations among the chosen generators of $M$. Higher syzygy modules are successive kernels. They depend on the choice of resolution; a minimal graded free resolution gives canonical ranks and degrees of generators over a standard graded ring structure on a polynomial ring.