7. Finiteness

To complete the algorithm we have to show the finiteness of the row-reduction process.

Proposition 14  
 After a finite number of reductions, ${\cal M}(Q)$ becomes J-row-minimal.

Assume there is an infinite sequence of row-reductions

$${\cal M}(Q) \supset {\cal M}(U_{T_1}Q) \supset {\cal M}(U_{T_2}Q) \ldots$$

By our choice of the reduction in Proposition 14, at any step the lowest possible leading term of ${\cal M}(Q)$ is removed. But all leading terms of a standard basis of ${\cal M}(Q)$ are leading terms of a standard basis of $\langle Q\rangle$, too, which determines the Hilbert-function of the module M=coker(Q).

Let r_M be the regularity bound of the Hilbert function of M. Then the homogeneous degree of all leading terms of a standard basis of Q (resp. of ${\cal M}(Q)$) is not greater than r_M, contradicting our assumption of an infinite reduction sequence.