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##

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
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*_{j}^{i} with

then

*r*(

*I*) is the minimum of all integers
such that for each i:

From [BM] we know:

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

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.

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