4.3 Kernel of a module homomorphism

Definition 4..4   Let $\;R=K[x_1,\ldots , x_n]/(h_1,\ldots , h_p)\;$, $\;A \in Mat(m\times r,R)\;$ and $B\in Mat(m\times s,R)\;$ then define

$${\bf modulo(A,B)}:= ker( R^r\stackrel{A}{\longrightarrow} R^m/Im(B))$$

( modulo(A,B) is the preimage of B under the homomorphism given by A.)

Lemma 4..5   Let $\;\{\, (\underline{\alpha}_i ,\underline{\beta}_i ,\underline{\gamma}_i )\,\vert\;i=1,\ldots , k\,\}\subset R^{r+s+p}=:R^N\;$ be a generating set of syz(D) where

$$C=\left( \begin{array}{ccccccccc} h_1 & \cdots & h_p & 0 & \c... ...1 & \cdots & h_p \\ \end{array}\right) \in Mat(m\times pm,R)$$

and

$$D= \left( \begin{array}{ccc\vert ccc\vert ccc} a_{11} & \cdot... ...s & c_{m,pm} \\ \end{array}\right) \in Mat(m\times r+s+pm,R)$$

Then

$$modulo\,(A,B):=(\,\alpha_1\ldots \alpha_k\,)\in Mat(r\times k,R)$$

(see lemma 4.2.)

Remark 4..6   In practice, one need not compute the entire syzygy module of D: it is better to find modulo(A,B) as:

$$\left( \begin{array}{ccc\vert ccc\vert ccc} a_{11} & \cdots &... ...c} 0\\ \vdots \\ 0\\ R\\ \vdots\\ R\\ \end{array}\right)$$

(see sections 4.2, 2.7.)