The syzygies of the subset are computed in the -th
subresolution. Thus, we have
to consider only pairs of module elements whose leading term is in the first
components. Let us assume that a reduction of such a pair leads to an
element with leading term not in these components. Then it remains,
of course, a syzygy of . Indeed, because of its leading term it
is a syzygy of
and, hence, it lies in .
As is given as a standard basis the reduction could be
completed
by the computation of the normal form of w.r.t. .

REMARK:Compared with Chapter 3.3 the computation of the standard basis
of the partial module is exactly the computation of the next syzygy module
of the resolution of the extension ideal . The normal form
correspond to the choice of representation of
.

For the original ideal (or, module) the situation is different: The leading term of the new generators may ly within the given leading ideal (or, module). Thus, generators of the standard basis of might be reduced by the extension coming from . In this case, the generator of the standard basis of is replaced by its reductum if it is not contained in the set (which may happen only in the non-homogeneous case).

Moreover, there is a new criterion concerning the dependence from sets of generators for the computation of the standard basis of an ideal :