5. Criteria for row-minimality

 

(5.1) Fix $J= J_0 =\{ 1,...,k \}.$ Consider all those matrices $U_T= \left( \begin{array}{rr} I & T \\ 0 & I \end{array} \right)$ from $I\!\!B_+$ which do not increase ${\cal M}(Q), \ Q= \left( \begin{array}{r} Q_J \\ Q_{J'} \end{array} \right)$, and set

$${\cal N}^{(J)}_Q := \{ T \in \mbox{Mat} (k,r\!-\!k;R) \mid {\cal M}(U_TQ) \subseteq {\cal M}(Q) \}.$$

Because $U_T Q = \left( \begin{array}{c} Q_J + TQ_{J'} \\ Q_{J'} \end{array} \right)$ and ${\cal M}(U_T Q)= \langle Q_J + TQ_{J'} \rangle \subseteq {\cal M}(Q) = \langle Q_J \rangle$ if and only if $\langle TQ_{J'} \rangle \subseteq \langle Q_J \rangle$, we obtain:

$${\cal N}^{(J)}_Q = \{ T \in \mbox{Mat} (k,r\!-\!k;R)\ \mid \ ... ...T} \in \mbox{Mat} (n,n;R): \ TQ_{J'} - Q_J \tilde {T} = 0 \}.$$

Choose a basis $T_1, \ldots ,T_N$ of the module ${\cal N}_Q^{(J)}$ and a set of associated matrices $\tilde T_1 , \ldots , \tilde T_N \in \mbox{Mat} (n,n;R)$.

Proposition 9 ((sufficient condition for J-row-minimality))  
Q is J-row-minimal if either

(i1) $\tilde T_1, \ldots , \tilde T_N$ do not have a unit entry; i.e., $\tilde T_i \, \mbox{mod}\; \underline{m} = 0$
or
(i2) for any $T \in {\cal N}^{(J)}_Q$, the corresponding matrix $\tilde T$ satisfies the condition: $\tilde T \, \mbox{mod} \; \underline{m}$ is nilpotent.

Condition (i1) implies (i2). Assuming (i2), we obtain for any $T \in {\cal N}_Q^{(J)}$

$$\begin{eqnarray*}{\cal M}(U_TQ) & = & \langle Q_J + TQ_{J'} \rangle = \langle ... ...Q_J (I + \tilde T) \rangle = \langle Q_J \rangle = {\cal M}(Q), \end{eqnarray*}$$

because $I + \tilde T$ is invertible. Therefore Q is J-row-minimal.

 

(5.2) In order to obtain a necessary condition for row-minimality, we have to improve the choice of a good representative $\tilde Q$ in the equivalence class of Q, cf. (3.3). Write $\tilde Q$ in block form

$$\tilde Q = \left( \begin{array}{rr} A & 0 \\ C & D \end{array} \right), \qquad A \in \mbox{Mat} (k,l).$$

We may assume that the columns of A form a completely reduced, minimal basis of ${\cal M}(Q)$. This means the columns of A form a minimal system of generators, increasingly ordered, and the l-th generator is reduced with respect to the standard basis generated by the first l-1 generators for appropriate l.

Lemma 10   Let Q be a submodule of R^k, given by a completely reduced, minimal basis $A=(\underline{a}_1,\ldots,\underline{a}_r)$. A submodule P of Q is proper if and only if lead(P), the initial module, does not contain all leading terms of A.

We want to show that for a proper submodule P the minimal leading term not contained in lead(Q) must be in lead(A). Assume the contrary; i.e. the minimal $lead(\underline{q}_*)=q \underline{e}_l\not\in P$ is not contained in lead(A). We introduce the module $S= \langle m \underline{e}_s\vert m\underline{e}_s> q\underline{e}_l \rangle$, generated by all monomials in R^k which are greater than $q \underline{e}_l$.

Considering the whole situation in the factor module Rⁿ/S, we have the equation

$$q\underline{e}_l=\sum_i g_i \underline{a}_i,$$

and, from the assumption on reducedness, it follows that all g_i belong to the maximal ideal ${\underline m}$. The staircases of both modules Q and P are identical up to $q \underline{e}_l$, and, hence, for every generator $\underline{a}_i$ an equation

$$\underline{a}_i=\sum_k h_{ik} \underline{p}_k + c_i q\underline{e}_l$$

with $c_i\in K$ exists. We obtain

$$q\underline{e}_l=\sum_{i,k} g_i h_{ik} \underline{p}_k + \sum_i g_i c_i q\underline{e}_l.$$

The second summand vanishes in Rⁿ/S. Thus, $q \underline{e}_l$ must be a leading term of P, too.

The other implication is a well-known fact about standard basis.

 

(5.3) Without loss of generality we will assume that $Q= \tilde Q = \left( \begin{array}{rr} A & 0 \\ C & D \end{array} \right)$ fulfills the conditions of (5.2). For $T \in {\cal N}^{(J)}_Q$ we obtain $U_TQ = \left( \begin{array}{cc} A+TC & TD \\ C & D \end{array} \right)$ and $\langle TC \rangle + \langle TD \rangle \subseteq \langle A \rangle .$ Therefore, matrices $\tilde T' \in \mbox{Mat} (l,l;R)$