3.1 The regularity

In the case of global homogeneous computations a very useful invariant exists for minimal resolutions due to D. Mumford. This invariant is called regularity and denoted by r(I) where $I\subset k[\underline{x}]^r$ is an arbitrary submodule of a free module.

Definition 3..1   Assume that the i-th module of syzygies of I is generated by elements f_jⁱ with $j=1,\ldots,n_j$ then r(I) is the minimum of all integers such that for each i:

$$max_j\{deg(f_j^i)\}\leq r(I)+i.$$

From [BM] we know:

Proposition 3..2   The regularity is a upper semi-continuous function on flat families of modules. Hence,

$$r(I)\leq r(in(I)),$$

where in(I) denotes the ideal of leading terms of I.

Our tests have shown that in almost all cases (exept those which are very close to monomial ideals) the additional computation of the resolution of the module or ideal of leading terms takes less time than one obtains by using this bound for the degrees. Therefore, it is advisable to use the regularity by default for quasihomogeneous, global computations.