4.1 Definition

Definition 4..1   Let $I=\{g_1, \ldots, g_q\}\subseteq K[\underline{x}]^r$.
The module of syzygies syz(I) is ker $(K[\underline{x}]^q \to K[\underline{x}]^r, \sum w_i e_i \mapsto \sum w_i g_i)$.

Lemma 4..2   The module of syzygies of I is

$$(g_1(\underline{x} )-e_{r+1},\ldots , g_q(\underline{y} )-e_{r+q}) \cap \{0\}^r \times K[\underline{x}]^q$$

in $(K[x_1,\ldots , x_m]/J)^q$ .

Remark 4..3   Use a module ordering with $e_i>e_j \forall i \leq r<j$ and the elimination property of lemma 2.7.

SINGULAR example:

ring R=0,(x,y,z),(c,dp);
ideal I=maxideal(1);
// the syzygies of the (x,y,z)
syz(I);
// syz yields a generating set for the module of syzygies
// but may not be a standard basis !